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• Published: 07 August 2023

Experimental validation of the free-energy principle with in vitro neural networks

• Takuya Isomura   ORCID: orcid.org/0000-0003-2960-4919 1 ,
• Kiyoshi Kotani 2 ,
• Yasuhiko Jimbo 3 &
• Karl J. Friston   ORCID: orcid.org/0000-0001-7984-8909 4 , 5  

Nature Communications volume  14 , Article number:  4547 ( 2023 ) Cite this article

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• Computational models
• Computational neuroscience
• Information theory and computation
• Machine learning
• Synaptic plasticity

Empirical applications of the free-energy principle are not straightforward because they entail a commitment to a particular process theory, especially at the cellular and synaptic levels. Using a recently established reverse engineering technique, we confirm the quantitative predictions of the free-energy principle using in vitro networks of rat cortical neurons that perform causal inference. Upon receiving electrical stimuli—generated by mixing two hidden sources—neurons self-organised to selectively encode the two sources. Pharmacological up- and downregulation of network excitability disrupted the ensuing inference, consistent with changes in prior beliefs about hidden sources. As predicted, changes in effective synaptic connectivity reduced variational free energy, where the connection strengths encoded parameters of the generative model. In short, we show that variational free energy minimisation can quantitatively predict the self-organisation of neuronal networks, in terms of their responses and plasticity. These results demonstrate the applicability of the free-energy principle to in vitro neural networks and establish its predictive validity in this setting.

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Introduction

Elucidating the self-organising principles of biological neural networks is one of the most challenging questions in the natural sciences, and should prove useful for characterising impaired brain function and developing biologically inspired (i.e., biomimetic) artificial intelligence. According to the free-energy principle, perception, learning, and action—of all biological organisms—can be described as minimising variational free energy, as a tractable proxy for minimising the surprise (i.e., improbability) of sensory inputs 1 , 2 . By doing so, neuronal (and neural) networks are considered to perform variational Bayesian inference 3 . (Table  1 provides a glossary of technical terms used commonly in the free-energy principle and active inference literature). This inference follows from treating neuronal dynamics as a gradient flow on variational free energy, which can be read as a form of belief updating about the network’s external milieu. The free energy in question is a function of a generative model that expresses a hypothesis about how sensory data are generated from latent or hidden states. However, to apply the free-energy principle at the cellular and synaptic levels, it is necessary to identify the requisite generative model that explains neuronal dynamics (i.e., inference) and changes in synaptic efficacy (i.e., learning).

The activity of neurons has also been modelled with realistic spiking neuron models 4 , 5 , 6 or reduced rate coding models 7 . Moreover, synaptic plasticity—that depends on the firing of pre- and postsynaptic neurons 8 , 9 , 10 , 11 , 12 —has been modelled as Hebbian-type plasticity rules 13 , 14 , 15 . Although a precise link between the equations that underwrite these models—derived from physiological phenomena—and the corresponding equations from the free-energy principle has not been fully established, we recently identified a formal equivalence between neural network dynamics and variational Bayesian inference 16 , 17 , 18 . Specifically, we reverse-engineered a class of biologically plausible cost functions—for canonical neural networks—and showed that the cost function can be cast as variational free energy, under a class of well-known partially observable Markov decision process (POMDP) models. This suggests that any (canonical) neural network, whose activity and plasticity minimise a common cost function, implicitly performs variational Bayesian inference and learning about external states. This ‘reverse engineering’ approach—guaranteed by formal equivalence—allows us, for the first time, to identify the implicit generative model from empirical neuronal activity. Further, it can precisely link quantities in biological neuronal networks with those in variational Bayesian inference. This enables an experimental validation of the free-energy principle, when applied to these kinds of canonical networks.

Having said this, the free-energy principle is sometimes considered to be experimentally irrefutable in the sense that it can describe any observed biological data 19 . However, when applying the free-energy principle to a particular system, one can examine its predictive validity by asking whether it can predict systemic responses 18 . This offers a formal avenue for validation and application of the free-energy principle. To establish predictive validity, one needs to monitor the long-term self-organisation of neuronal networks and compare their dynamics and architecture with theoretical predictions.

To pursue this kind of validation, we used a previously established microelectrode array (MEA) cell culture system for the long-term monitoring of the self-organisation of in vitro neural networks 20 , 21 . We have used this setup to investigate causal inference in cortical cells obtained from rat embryos 22 , 23 . Causal inference is a simple form of Bayesian inference; namely, inferring and disentangling multiple causes of sensory inputs in the sense of blind source separation 24 , 25 , 26 . Although blind source separation is essential to explain the cocktail party effect—the ability of partygoers to distinguish the speech of one speaker from others in a noisy room 27 , 28 —its precise neuronal mechanisms have yet to be elucidated. We previously demonstrated that, upon receiving sensory stimuli, some populations of neurons in in vitro neural networks self-organised (or learned) to infer hidden sources by responding specifically to distinct causes 22 . Subsequently, we showed that this sensory learning is consistent with variational free energy minimisation under a POMDP generative model 23 . These results—and related in vitro work 29 , 30 , 31 , 32 , 33 , 34 , 35 —speak to the tractability and stability of this neuronal system, making it an ideal tool for examining theoretical predictions in a precise and quantitative fashion.

In the present work, we attempted an experimental validation of the free-energy principle by showing that it predicts the quantitative self-organisation of in vitro neural networks using an established in vitro causal inference paradigm. Henceforth, we will refer to in vitro neural networks as neuronal networks and reserve the term neural network for an in silico model. We reverse-engineered an implicit generative model (including prior beliefs), under which a neuronal network operates. We subsequently demonstrated that the free-energy principle can predict the trajectory of synaptic strengths (i.e., learning curve) as well as neuronal responses after learning, based exclusively on empirical neuronal responses at the beginning of training.

Using pharmacological manipulations, we further examined whether the change in baseline excitability of in vitro networks was consistent with the change in prior beliefs about hidden states (i.e., the state prior), confirming that priors over hidden states are encoded by firing thresholds. These results demonstrate that the self-organisation of neuronal networks can be cast as Bayesian belief updating. This endorses the plausibility of the free-energy principle as an account of self-organisation in neural and neuronal networks. We conclude by discussing possible extensions of our reverse engineering approach to in vivo data.

Equivalence between canonical neural networks and variational Bayes

First, we summarise the mathematical (or natural) equivalence between canonical neural networks and variational Bayesian inference, which enables one to apply the free-energy principle to predict empirical data. In this work, we adopted an experimental setup that could be formulated as a simple POMDP generative process that does not exhibit state transitions (Fig.  1a ). Here, two binary hidden sources \(s={\left({s}_{1},{s}_{2}\right)}^{{{{{{\rm{T}}}}}}}\) were sampled at random from a prior categorical distribution \(D={\left({D}_{1},{D}_{0}\right)}^{{{{{{\rm{T}}}}}}}\) in a mutually independent manner, where \({D}_{1}\) and \({D}_{0}\) are prior expectations that satisfy \({D}_{1}+{D}_{0}=1\) . Then, 32 sensory inputs \(o={\left({o}_{1},\ldots,{o}_{32}\right)}^{{{{{{\rm{T}}}}}}}\) were generated from \(s\) with a categorical distribution characterised by a mixing matrix \(A\) . Each element of \(s\) and \(o\) took either a 1 (ON) or a 0 (OFF) state. The left stimuli group ( \({o}_{1},\ldots,{o}_{16}\) ) in Fig.  1a (left) took the value of source 1 with a 75% probability or the value of source 2 with a 25% probability. In contrast, the right group ( \({o}_{17},\ldots,{o}_{32}\) ) took the value of source 1 or 2 with a 25% or 75% probability, respectively. Analogous to the cocktail party effect 27 , 28 , this setup is formally homologous to the task of distinguishing the voices of speakers 1 ( \({s}_{1}\) ) and 2 ( \({s}_{2}\) ) based exclusively on mixed auditory inputs ( \(o\) ), and in the absence of supervision. Here, the mixing (a.k.a., likelihood) matrix ( \(A\) ) determines the mixing of the two voices, and the prior ( \(D\) ) corresponds to the frequency or probability of each speaker generating speech. Hence, neurons must unmix sensory inputs into hidden sources to perceive the underlying causes. Please refer to the Methods section ‘Generative process’ for the formal expression in terms of probability distributions.

In a – c , panels on the left-hand side depict neural (and neuronal) network formation, while panels on the right-hand side depict variational Bayes formation. a Schematics of the experimental setup (left) and corresponding POMDP generative model (right). Two sequences of independent binary hidden sources generate 32 sensory stimuli through a mixing matrix A , which were applied into cultured neurons on an MEA as electrical pulses. Waveforms at the bottom represent the spiking responses to a sensory stimulus (red line). The diagram on the right-hand side depicts the POMDP scheme expressed as a Forney factor graph 67 , 68 , 69 . The variables in bold (e.g., \({{{{{{\bf{s}}}}}}}_{t}\) ) denote the posterior beliefs about the corresponding variables in non-bold italics (e.g., \({s}_{t}\) ). b Equivalence between canonical neural networks and variational Bayesian inference. See the main text and Methods for details. c Procedure for reverse engineering the implicit generative model and predicting subsequent data. (1) The neuronal responses are recorded, and (2) the canonical neural network (rate coding model) is used to explain the empirical responses. (3) The dynamics of the canonical neural network can be cast as the gradient descent on a cost function. Thus, the original cost function L can be reconstructed by taking the integral of the network’s neural activity equation. Free parameters \(\phi\) are estimated from the mean response to characterise L . (4) Identification of an implicit generative model and the ensuing variational free energy F using the equivalence of functional forms in Table  2 . (5) The synaptic plasticity rule is derived as a gradient descent on variational free energy. (6) The obtained plasticity scheme is used to predict self-organisation of neuronal networks . The details are provided in Methods and have been described previously 16 , 17 , 18 .

In this work, we considered that in vitro neurons can be modelled as a canonical neural network comprising a single feed-forward layer of rate coding models (Fig.  1b , top left) 16 . We considered two distinct ensembles of neurons. Upon receiving sensory inputs \(o\) , these neurons computed the weighted sum of sensory inputs weighted by a synaptic strength matrix W to generate a response (firing intensity) \(x={\left({x}_{1},{x}_{2}\right)}^{{{{{{\rm{T}}}}}}}\) . This canonical neural network has a certain biological plausibility because it derives from realistic neuron models 4 , 5 , 6 through some approximations 17 ; further, its fixed point equips the rate coding model 7 with the widely used sigmoid activation function, also known as a neurometric function 36 . We will show below that this canonical neural network is a plausible computational architecture for neuronal networks that receive sensory stimuli.

Previous work has identified a class of biologically plausible cost functions for canonical neural networks that underlie both neuronal responses and synaptic plasticity 16 , 17 . This cost function can be obtained by simply calculating the integral of the neural activity equation (Fig.  1b , middle left; see the Methods section ‘Canonical neural networks’ for details). The reconstructed neural network cost function L is biologically plausible because both neuronal responses and synaptic plasticity equations can be derived as a gradient descent on L . The ensuing synaptic plasticity rule has a biologically plausible form, comprising Hebbian plasticity 13 , accompanied by an activity-dependent homeostatic plasticity 37 (Fig.  1b , bottom left).

Variational Bayesian inference casts belief updating as revising a prior belief to the corresponding (approximate) posterior belief based on a sequence of observations. The experimental setup considered here is expressed as a POMDP generative model 38 , 39 . The inversion of this model—via a gradient descent on variational free energy—corresponds to inference. In other words, the generative model generates sensory consequences from hidden causes (i.e., two sources), while model inversion (i.e., inference) maps from sensory consequences to hidden causes (Fig.  1a , right). Variational free energy F is specified by the sensory input and probabilistic beliefs about hidden states under a generative model. Minimisation of variational free energy, with respect to these beliefs, yields the posterior over hidden states \({{{{{{\bf{s}}}}}}}_{t}\) (Fig.  1b , top right) and parameters \({{{{{\bf{A}}}}}}\) (Fig.  1b , bottom right), realising Bayesian inference and learning, respectively. The explicit forms of posterior beliefs are described in the Methods section ‘Variational Bayesian inference’.

Crucially, previous work has shown that the neural network cost function L can be read as variational free energy F 16 , 17 . This equivalence allows us to identify the physiological implementation of variational Bayesian inference by establishing a one-to-one mapping between neural network quantities and the quantities in Bayesian inference, as summarised in Table  2 . Namely, neural activity ( \(x\) ) of the canonical neural networks corresponds to the posterior expectation about the hidden states ( \({{{{{\bf{s}}}}}}\) ), synaptic strengths ( \(W\) ) correspond to the posterior expectation about the parameters ( \({{{{{\bf{A}}}}}}\) ), and firing threshold factors ( \(\phi\) ) correspond to the initial state prior ( \(D\) ). These mappings establish a formal relationship between a neural network formulation (Fig.  1b , left) and a variational Bayesian formulation (Fig.  1b , right). In summary, the neural activity and plasticity of canonical networks that minimise a common cost function perform variational Bayesian inference and learning, respectively.

This notion is essential because, by observing neuronal responses, we can reverse engineer the implicit generative model—under which the neuronal network operates—from empirical neuronal responses, to characterise the neuronal network in terms of Bayesian inference (Fig.  1c ) 18 . Perhaps surprisingly, using the reverse engineering technique, if one can derive the neural activity equation from experimental data (Fig.  1c , steps 1,2), it is possible to identify the generative model that the biological system effectively employs (steps 3,4). This allows one to link empirical data to quantities in variational Bayesian inference. Subsequently, by computing the derivative of variational free energy under the generative model, one can derive the synaptic plasticity predicted theoretically (step 5). In short, if one has initial neuronal response data, one can predict how synaptic plasticity will unfold over time. This means that if the free-energy principle applies, it will predict the self-organisation of neuronal networks (step 6).

The virtue of the free-energy principle is that it lends an explainability to neuronal network dynamics and architectures, in terms of variational Bayesian inference. Given this generative model, the free-energy principle provides qualitative predictions of the dynamics and self-organisation of neuronal networks, under the given experimental environment. In other words, because neuronal responses and synaptic plasticity are expected to minimise variational free energy by exploiting the shortest path (i.e., a geodesic or path of least action) on the free energy landscape, this property in turn enables us to theoretically predict a plausible synaptic trajectory (i.e., activity-dependent plasticity).

In the remainder of this paper, we examine the plausibility of variational free energy minimisation as the mechanism underlying the self-organisation of neuronal networks . We will compare the empirical encoding of the sources of sensory inputs with a synthetic simulation of ideal Bayesian encoding, and investigate whether variational free energy minimisation can predict the neuronal responses and plasticity of in vitro networks.

Consistency between in vitro neural networks and variational Bayes

In this section, we verify some qualitative predictions of the free-energy principle when applied to our in vitro neural networks in terms of response selectivity (i.e., inference), plasticity (i.e., learning), and effects of pharmacological manipulations on inference and subsequent learning. Using our in vitro experimental setup 20 , 21 , cortical cells obtained from rat embryos were cultured on an MEA dish with 64 microelectrodes on its floor (Fig.  1a , left). Each electrode was used to deliver electrical stimuli and record the spiking response. After approximately 10 days in culture, the neurons self-organised into a network and exhibited spontaneous activity, with clear evoked responses to electrical stimuli. Neurons were stimulated with the above-constructed patterns of sensory inputs (see the preceding section), comprising 32 binary sensory inputs ( o ) that were generated from two sequences of independent binary hidden sources ( s ) in the manner of the POMDP generative model above (Fig.  1a , right). When a sensory input took the value of 1, an electrical pulse was delivered to the cultured neurons. The 32 stimulation electrodes were randomly distributed over 8 × 8 MEAs in advance and fixed over training. Evoked extracellular activity (i.e., the early neuronal response) was recorded from 64 MEA electrodes. Each session lasted 256 s, in which a 256-time-step sequence of random stimulations was delivered every second, followed by a 244-s resting period. The training comprised 100 sessions, each of which was an identical repetition of the 256 s-long random sequence.

Upon electrical simulation—generated by the mixture of the two hidden sources—our previous work showed the emergence of selective neuronal responses to either of the two sources 22 , 23 . Response intensity was defined as the number of spikes 10–30 ms after a stimulation (Fig.  2a ) following the previous treatment 22 (see Supplementary Fig.  1a for other electrodes). This is because a large number of spikes—induced by synaptic input—were observed during that period, while most directly evoked action potentials (which were not the subject of our analyses) occur within 10 ms after stimulation 40 . The recorded neuronal responses were categorised into source 1- and source 2-preferring and no-preference groups, depending on the average response intensity, conditioned upon the hidden source (Fig.  2b ). Note that each electrode can record spiking responses from one or more neurons. Learning was quantified as the emergence of functional specialisation for recognising particular sources. The response intensity of the source 1-preferring neurons changed during the training period to exhibit a strong response selectivity to source 1 (Fig.  2c , left). These neurons self-organised to fire at a high level when source 1 was ON, but had a low response rate when source 1 was OFF. Similarly, source 2-preferring neurons self-organised to respond selectively to source 2 during training (Fig.  2c , right). These changes were inhibited by N -methyl-D-aspartate (NMDA) receptor antagonist, 2-amino-5-phosphonopentanoic acid (APV) to a certain degree (Fig.  2d ), indicating that the observed self-organisation depends on NMDA-receptor-dependent plasticity. These results indicate the occurrence of blind source separation at a cellular level—through activity-dependent synaptic plasticity—supporting the theoretical notion that neural activity encodes the posterior belief (i.e., expectation) about hidden sources or states 1 , 2 .

a Early evoked responses of in vitro neurons recorded at a single electrode, showing a source 1-preferring neuronal response. Raster plot of spiking responses (left) and peristimulus time histogram (PSTH, right) before and after training are shown. The two sources provide four hidden state patterns, \({s}_{t}=\left({{{{\mathrm{1,1}}}}}\right),\, \left({{{{\mathrm{1,0}}}}}\right),\, \left({{{{\mathrm{0,1}}}}}\right),\, \left({{{{\mathrm{0,0}}}}}\right)\) , and responses in these four conditions are plotted in green, red, blue, and black, respectively. Responses in shaded areas (10–30 ms after a stimulus) were used for analyses. b Recorded neuronal responses were categorised into source 1-preferring (red), source 2-preferring (blue), and no-preference (grey) groups. The Pie-chart indicates numbers (electrodes) in each group, obtained from 30 independent experiments. c Changes in evoked responses of source 1- (left) and source 2- (right) preferring neurons, respectively. Response change from session 1 is shown. Lines and shaded areas represent mean values +/– standard errors. Here and throughout, the two-sided Wilcoxon signed-rank test was used for paired comparisons. d Comparison of response specificities in control ( n  = 965 electrodes) and APV-treated ( n  = 296 electrodes from 9 independent experiments) culture groups. Here and throughout, the two-sided Mann‒Whitney U test was used for unpaired comparisons. Box-and-whisker plots in ( d )( f )( h ) follow standard conventions: the central line indicates the median, the bottom and top box edges indicate the first and third quartiles, respectively, and the whiskers extend to the furthest data point within 1.5 times the interquartile range of the first or third quartile. e Simulations of ideal Bayesian observers. The posterior belief about source 1 with varying hidden state priors is shown. Red and blue lines represent how much the posterior expectation changes, when source 1 is ON or OFF, respectively ( n  = 100 simulations for each condition). In ( e ) ( g ), changes in response from session 1 were computed and then the averaged response (trend) in each session was subtracted to focus on response specificity to the preferred source. Lines and shaded areas in ( e ) ( g ) represent mean values +/– standard deviations. f Difference in responses to s 1  = 1 and s 1  = 0, at session 100 (changes from session 1). g Transitions of selective neuronal responses of source 1-preferring neurons under control (middle), hypo- (left), and hyper-excitability (right) conditions. Red and blue lines represent the averaged evoked response of source 1-preferring neurons, when source 1 is ON or OFF, respectively ( n  = 127, 514, 129 electrodes from 7, 30, 6 independent experiments for diazepam, control, and bicuculline conditions, respectively). h Same as ( f ), but for empirical responses. Source data are provided as a Source Data file.

Given the consistency between source-preferring neuronal responses and state posterior, one can then ask about the neuronal substrates for other quantities in variational Bayesian inference. In light of the above, we modelled neuronal networks using a canonical neural network , comprising a single feed-forward layer (Fig.  1b , top left). As noted above, this neural network acts as an ideal Bayesian observer, exhibiting Bayesian belief updating under a POMDP generative model (Fig.  1b , top right), where the firing threshold encodes a prior over initial states (Table  2 ) 16 , 17 . Thus, this in silico model can learn to detect hidden sources successfully when the implicit state prior matches that of the true generative process (in this case, \({D}_{1}=0.5\) ; Fig.  2e , middle). Conversely, both upregulation ( \({D}_{1}=0.8\) ; Fig.  2e , right) and downregulation ( \({D}_{1}=0.2\) ; Fig.  2e , left) of the state prior significantly disrupted this sensory learning (Fig.  2f ). These simulations used the same empirical stimuli applied to neuronal networks. Hence, if this canonical neural network is an apt model for neuronal networks , the firing threshold (i.e., baseline excitability) of the neuronal network should encode the state prior, and changes in baseline excitability should disrupt the inference and ensuing sensory learning.

To examine this hypothesis, we asked whether pharmacological modulations of the baseline excitability of in vitro networks induce the same disruptions of inference as the alterations in the state prior in the in silico network. Pharmacological downregulation of gamma-aminobutyric acid (GABA)-ergic inputs (using a GABA A -receptor antagonist, bicuculline) or its upregulation (using a benzodiazepine receptor agonist, diazepam) altered the baseline excitability of neuronal networks . These substances were added to the culture medium before the training period and were therefore present over training. Average response levels were higher in bicuculline-treated cultures than in control cultures. Conversely, diazepam-treated cultures exhibited lower response levels, but retained sufficient responsiveness to analyse response specificity. Crucially, alterations in neuronal responses—and subsequent learning—were observed when we pharmacologically modulated the GABAergic input level (Fig.  2g ). We observed that both hyper-excitability (Fig.  2g , right) and hypo-excitability (Fig.  2g , left) significantly suppressed the emergence of response specificity at the cellular level (Fig.  2h ). This disruption of learning was observed both for source 1- and 2-preferring neuronal responses.

Effective synaptic connectivity analysis suggested that a certain amount of plasticity occurred even in the presence of bicuculline or diazepam (Supplementary Fig.  1b ). The difference was observed in the specificity of connectivity emerging during the training period (Supplementary Fig.  1c ). Here, the specificity was characterised with a gap in the contribution of a sensory electrode to sources 1- and 2-preferring units. While the specificity increased in all groups, it was significantly inhibited in the presence of bicuculline or diazepam.

Remarkably, our in silico model—under ideal Bayesian assumptions—could predict the effects of this GABAergic modulation on learning using a simple manipulation of the prior belief about hidden states (please compare Fig.  2e, f with Fig.  2g, h ). This involved setting the prior expectations so that sensory causes were generally present (analogous to the GABAergic antagonist effect) or generally absent (analogous to the agonist effect). Physiologically, this corresponds to increasing and reducing the response intensity, respectively, which is consistent with the effects of these pharmacological manipulations on baseline activity. In terms of inference, this manipulation essentially prepares the network to expect the presence or absence of an object (i.e., a hidden source) prior to receiving sensory evidence. The key notion here is that this simple manipulation was sufficient to account for the failure of inference and subsequent learning, as evidenced by the absence of functional specialisation. Thus, changes in the prior (neuronal) representations of states provide a sufficient explanation for aberrant learning.

In summary, the emergence of response specificity observed under normal network excitability was disrupted by pharmacologically induced hyper- or hypo-excitability of the network. The canonical neural network (i.e., ideal Bayesian observer) predicted these empirical effects—of the agonist and antagonist—by reproducing the hypo-excitability (diazepam) condition, analogous to the prior belief that sources are OFF (‘nothing there’), or by the hyper-excitability (bicuculline) condition, analogous to the prior belief that sources are present (ON). In either case, in vitro and in silico networks failed to perform causal inference, supporting our claim that the failure can be attributed to a biased state prior, under which they operated. These results corroborate the theoretical prediction that the firing threshold is the neuronal substrate of the state prior 16 , 17 , validating the proposed equivalence at the cellular level. This further licences an interpretation of neuronal network dynamics in terms of Bayesian inference and learning.

The free-energy principle predicts learning in neuronal networks

In this section, we examine the predictive validity of the free-energy principle by asking whether its application to neuronal networks can predict their self-organisation. We considered that the neuronal responses of source 1- and source 2-encoding ensembles in each in vitro networks are represented by their averaged response intensity and refer to them as x 1 and x 2 , where the offset was subtracted, and the value was normalised in the range between 0 and 1. We then modelled the neuronal responses of in vitro networks in the form of a canonical neural network and estimated the requisite synaptic strengths W (i.e., effective synaptic connectivity) by fitting empirical neuronal responses to the model (Fig.  3a ; see the Methods section ‘Reverse engineering of generative models’ for details). Using these estimates, we depicted the trajectories (i.e., learning curves) evinced by subsequent neuronal responses.

a Schematic of the system architecture comprising the generative process and the in vitro neuronal network modelled as a canonical in silico neural network . Two neural ensembles receive 32 inputs generated from two sources. b Left: Trajectory of empirically estimated synaptic connectivity ( W ) depicted on the landscape of variational free energy ( F ). Red and blue lines show trajectories of red and blue connectivities in ( a ). The slope indicates a theoretically predicted free energy landscape. Darker green represents lower free energy. Whereas, synaptic strengths (i.e., effective synaptic connectivity or efficacy) are calculated using empirical data in sessions 1–100. Right: Predictions of synaptic plasticity during training. The initial conditions (i.e., parameters of a generative model) were identified using neuronal responses from the first 10 sessions. A brighter colour indicates the predicted synaptic trajectory in the absence of empirical response data. c Empirically estimated posterior belief ( A ) about parameter A . d Error in neuronal networks estimating each column of A matrix, defined as the squared error between empirical and ideal A matrices, divided by the squared amplitude of A ( n  = 28, 120, 24 columns for diazepam, control, and bicuculline conditions, respectively). e Correlation between theoretically predicted strengths and strengths estimated from data, at session 100. f Error in predicting synaptic strengths, defined as the squared error between estimated and predicted \(({\widehat{W}}_{1},{\widehat{W}}_{0})\) , divided by the squared Frobenius norm of \(({\widehat{W}}_{1},{\widehat{W}}_{0})\) (see Methods for the definition). g Trajectory of observed (left) and predicted (right) neuronal responses of source 1-coding ensembles, during training. Red and blue lines indicate the responses when source 1 is ON and OFF, respectively. h Comparison of observed (black) and predicted (red) responses in session 100. i Correlation between observed and predicted responses during session 91–100. j Error in predicting neuronal responses, defined as the mean squared error: \({err}={{{{{\rm{E}}}}}}\left[{\left|{x}_{t}-{x}_{t}^{P}\right|}^{2}\right]/2\) . k Synaptic trajectories on free energy landscape under 0, 25, and 50% mix conditions. l Trajectory of variational free energy. Changes from session 1 are plotted ( n  = 4, 30, 4 independent experiments for 0, 25, and 50% mix conditions, respectively). In ( d , f , i , j ,  l ), data from n  = 30 independent experiments under the control condition were used. Lines and shaded areas (or error bars) in ( d , f , g , i , j , l ) represent mean values +/– standard deviations. Grey areas in ( f , g , j ) indicate the first 10 sessions, from which data were used. See Methods for further details. Source data are provided as a Source Data file.

First, we computed the synaptic strengths W that minimised the neural network cost function L using neuronal responses x . This corresponds to a conventional (model-based) connection strength estimation, where the W of the canonical neural network model was optimised to fit the empirical data (see Methods). We then plotted the trajectory of the estimated synaptic strengths on the landscape of variational free energy F (Fig.  3b , left). This landscape was characterised by the state prior (encoded by the firing threshold) estimated using empirical data from only the initial 10 sessions. According to the free-energy principle, synaptic plasticity occurs in a manner that descends on free energy gradients 1 , 2 . As predicted, we found that the trajectory of the empirically computed synaptic connectivity descended the free energy landscape (Fig.  3b , left; see also Supplementary Movie  1 ). This observation suggests that variational free energy minimisation is a plausible description of self-organisation or learning in neuronal networks.

Interestingly, the reverse engineering enables us to map empirically estimated synaptic strengths ( W ) to the posterior expectation ( A ) about parameter matrix A , to identify the generative model that the neuronal network employs (Table  2 ). The reconstructed posterior A precisely captured the characteristics of the true A in the external milieu, such that source 1 has a greater contribution to \(\left({o}_{1},\ldots,{o}_{16}\right)\) , while source 2 to \(\left({o}_{17},\ldots,{o}_{32}\right)\) (Fig.  3c ). An error between empirical and ideal (Bayesian) posteriors significantly decreased with sessions (Fig.  3d , left). The error was larger in bicuculline- or diazepam-treated condition, owing to biased inference and subsequent learning in these neuronal networks (Fig.  3d , right). These results support the theoretical notion that synaptic strengths encode the posterior expectation of the parameter 1 , 2 . As predicted, synaptic plasticity following the free energy gradient entailed a recapitulation of the generative process within the neuronal network architecture.

The observation that the empirical synaptic trajectory pursues a gradient descent on variational free energy implies that one can predict the subsequent learning in the absence of empirical constraints. Once the initial values of synaptic strengths are identified, the subsequent learning process can in principle be predicted using the free-energy principle, under the canonical neural network (i.e., generative) model.

To test this hypothesis, we predicted the neuronal responses ( x ) and synaptic plasticity ( W ) in sessions 11–100 using the neural network cost function L reconstructed based exclusively on the empirical responses in the initial 10 sessions (see the Methods section ‘Data prediction using the free-energy principle’ for details). As established above, this cost function is formally identical to variational free energy F under a class of POMDP generative models 16 , 17 . Thus, evaluating the responses and plasticity that minimise this cost function \(L\) ( \(\equiv F\) )—in the absence of data—furnishes a prediction of neuronal responses and plasticity under the free-energy principle.

We found that the predicted changes in connectivity matched the changes in empirically estimated effective synaptic connectivity (Fig.  3b , right). Specifically, we observed a strong correlation between the synaptic strengths estimated using neuronal data and the strengths predicted in the absence of data (Fig.  3e ). The prediction error was less than 4%, up to the final session (Fig.  3f ; n  = 30 independent experiments). These results indicate that, based on initial conditions, the free-energy principle can predict the self-organisation of neuronal networks.

In addition to the synaptic trajectory, we confirmed that a minimisation of free energy can predict the underlying changes in neuronal responses (Fig.  3g ). The predictions based only on initial conditions were consistent with observed responses. Specifically, the predicted responses were consistent with the observed responses at each time step (Fig.  3h, i ). Quantitatively, we could predict more than 80% of the neuronal responses in session 100, based only on data from sessions 1–10 (Fig.  3j ). These results suggest that the free-energy principle can predict both changes in synaptic efficacy and the time evolution of neuronal responses based only on initial data. Note that this is a highly nontrivial prediction, because synaptic efficacy shows activity-dependent changes and neuronal responses depend upon synaptic efficacy.

Another interesting observation was that when we varied the free energy landscape by manipulating the mixing matrix A in the stimulus generating process, empirical synaptic plasticity kept pursuing a gradient descent on the new variational free energy (Fig.  3k ). This speaks to a generality of this physiological property. Here, we experimentally varied the mixing balance ( A ) of two sources between 0 and 50%, to train neuronal networks with the generated sensory stimuli ( o ), where 0% indicates an unmixed (i.e., easily separable) condition, while 50% indicates a uniformly mixed (i.e., inseparable) condition. Irrespective of various conditions (i.e., forms of generative process and prior beliefs), the reverse engineering could reconstruct generative models and predict subsequent self-organisations of neuronal networks (Supplementary Fig.  2 ; see also Supplementary Movie  1 ).

Finally, we observed that during the process of assimilating sensory information, neuronal networks significantly reduced their variational free energy (Fig.  3l ). Here, variational free energy F for each session was calculated empirically by substituting the observed neuronal responses into the cost function L . As expected, an easier task (i.e., 0% mix condition) entailed a faster (i.e., greater) reduction of variational free energy. These results provide explicit empirical evidence that neuronal networks self-organise to minimise variational free energy.

In summary, we found that the trajectory of the empirically estimated effective synaptic connectivity is consistent with a slow gradient descent on variational free energy. Furthermore, we demonstrated that the free-energy principle can quantitatively predict sensory learning in neuronal networks in terms of both neuronal responses and plasticity. These results suggest that the self-organisation of the neuronal networks—in response to structured sensory input—is consistent with Bayesian belief updating and the minimisation of variational free energy. This endorses the plausibility of variational free energy minimisation as a rule underlying the dynamics and self-organisation of neuronal networks.

The present work has addressed the predictive validity of the free-energy principle at the circuit level by delineating the functional specialisation and segregation in neuronal networks via free-energy minimisation. Identifying the characteristic functions of arbitrary neural networks is not straightforward. However, according to the complete class theorem 41 , 42 , 43 , any system that minimises a cost function under uncertainty can be viewed as Bayesian inference. In light of this, we showed that any neural network—whose activity and plasticity minimise a common cost function—can be cast as performing (variational) Bayesian inference 16 , 17 . Crucially, the existence of this equivalence enables the identification of a natural map from neuronal activity to a unique generative model (i.e., hypothesis about the external milieu), under which a biological system operates. This step is essential to link empirical data—which report the ‘internal’ circuit dynamics (i.e., physiological phenomena)—to the representation of the ‘external’ dynamics (i.e., functions or computations) that the circuit dynamics imply, in terms of variational Bayesian inference. Using this technique, we fitted stimulus-evoked responses of in vitro networks—comprising the cortical cells of rat embryos—to a canonical neural network and reverse engineered an POMDP generative model, apt to explain the empirical data. In other words, we were able to explain empirical responses as inferring the causes of stimuli, under an implicit generative or world model.

Furthermore, reverse engineering a generative model from observed responses specifies a well-defined synaptic plasticity rule. Using this rule, we showed that the self-organisation of in vitro networks follows a gradient descent on variational free energy under the (POMDP) generative model. In short, the virtues of reverse engineering are that: (i) when provided with empirical responses, it systematically identifies what hypothesis (i.e., generative model) the biological system employs to infer the external milieu. Moreover, (ii) it offers quantitative predictions about the subsequent self-organisation (i.e., learning) of the system that can be tested using data. This provides a useful tool for analysing and predicting electrophysiological and behavioural responses and elucidating the underlying computational and self-organisation principle.

Although numerous neural implementations of Bayesian inference have been proposed 44 , 45 , 46 , these approaches generally derive update rules from Bayesian cost functions without establishing the precise relationship between these update rules and the neural activity and plasticity of canonical neural networks. The reverse engineering approach differs conceptually by asking what Bayesian scheme could account for any given neuronal dynamics or neural network. By identifying the implicit inversion scheme—and requisite generative model—one can then lend any given network an interpretability and explainability. In the current application of this approach, we first consider a biologically plausible cost function for neural networks that explains both neural activity and synaptic plasticity. We then identify a particular generative model under which variational free energy is equivalent to the neural network cost function. In this regard, reverse engineering offers an objective procedure for explaining neural networks in terms of Bayesian inference. Further, the synaptic plasticity rule is derived as the gradient descent on the cost function that is determined by the integral of neural dynamics. Crucially, learning through this plasticity rule can be read, formally, as Bayesian belief updating under an appropriate generative model. Conversely, naive Hebbian plasticity rules—with an ad hoc functional form—correspond to Bayesian belief updating under a suboptimal generative model with biased prior beliefs, which cannot solve simple blind source separation problems 16 . As predicted, in vitro neural networks above failed to perform blind source separation, with changed baseline excitability and implicit priors. In short, the free-energy principle is necessary to determine the optimal balance between Hebbian and homeostatic plasticity that enables blind source separation by in vitro networks.

Previous work has established that ensembles of neurons encode posterior expectations 47 and prediction errors 48 ; however, other quantities in Bayesian inference—such as the state prior and parameter posterior—have yet to be fully investigated. The reverse engineering approach enables us to identify the structures, variables, and parameters of generative models from experimental data, which is essential for empirical applications of the free-energy principle. This is a notion referred to as computational phenotyping 49 ; namely inferring the generative model—and in particular, the priors—that best explain empirical responses under ideal Bayesian assumptions. The reverse engineering naturally maps empirical (biological) quantities to quantities in variational Bayesian inference. Our empirical results suggest that neuronal responses encode the hidden state posterior (Fig.  2c ), baseline excitability encodes the state prior (Fig.  2g ), and synaptic efficacies encode the parameter posterior (Fig.  3c ), as predicted theoretically (Table  2 ).

Having said this, because the free-energy principle can arguably describe any observed biological data by its construction 19 , showing the existence of such a mapping alone is insufficient as an empirical validation. Conversely, one can examine the predictive validity, which is a more delicate problem, by asking whether the free-energy principle can predict subsequent self-organisation without reference to empirical data. Such a generalisability on previously unseen (test) data comprises an essential aspect for empirical applications of the free-energy principle.

We demonstrated that, equipped with the initial conditions (i.e., generative model and implicit prior beliefs of the network) characterised by the experimental data, variational free energy minimisation can predict the subsequent self-organisation of in vitro neural networks, in terms of quantitative neuronal responses and plasticity. It further predicted their performance when spontaneously solving source separation problems, including their speed and accuracy. These results not only validate this application of the free-energy principle; they also speak to the neurophysiological plausibility of related theories of the brain 50 , 51 and spiking neural network models that perform Bayesian inference 44 , 45 , 46 .

In essence, the free-energy principle constrains the relationship between neural activity and plasticity because both activity and plasticity follow a gradient descent on a common variational free energy, under ideal Bayesian assumptions. This property in turn enables precise characterisation of plausible self-organisation rules and quantitative prediction of subsequent neuronal activity and plasticity, under a canonical neural network (generative) model.

Our combined in vitro – in silico system showed that variation of the state prior (in silico model) is sufficient to reproduce the changes in neural excitability and inhibitions of sensory learning and inference observed in vitro. These results suggest that a neuronal networks’ excitability is normally tuned so that the ensemble behaviour is close to that of a Bayes optimal encoder under biological constraints. This is reminiscent of previous experimental observation that suggests that the activity of sensory areas encodes prior beliefs 52 .

These empirical data and complementary modelling results also explain the strong influence of prior beliefs on perception and causal inference—and the disruptive effects of drugs on perception in neuronal networks. Both synaptic plasticity and inference depend on convergent neuronal activity; therefore, aberrant inference will disrupt learning. Conversely, inference is not possible without the knowledge accumulated through experience (i.e., learning). Thus, inference is strongly linked to learning about contingencies that generate false inferences. Our findings demonstrate this association both mechanistically and mathematically, in terms of one simple rule that allows prior beliefs to underwrite inferences about hidden states.

Combining mathematical analyses with empirical observations revealed that baseline excitability is a circuit-level encoding of prior beliefs about hidden states. The notion that manipulating the state prior (encoded by neuronal excitability) disrupts inference and learning may explain the perceptual deficits produced by drugs that alter neuronal excitability, such as anxiolytics and psychedelics 53 . This may have profound implications for our understanding of how anxiolytics and psychedelics mediate their effects; namely, a direct effect on baseline activity can alter subsequent perceptual learning. Additionally, aberrant prior beliefs are a plausible cause of the hallucinations and delusions that constitute the positive symptoms of schizophrenia 54 , 55 . This suggests that, in principle, reverse engineering provides a formal avenue for estimating prior beliefs from empirical data—and for modelling the circuit mechanisms of psychiatric disorders (e.g., synaptopathy). Further, the reproduction of these phenomena in in vitro (and in vivo) networks furnishes the opportunity to elucidate the precise pharmacological, electrophysiological, and statistical mechanisms underlying Bayesian inference in the brain.

Importantly, although this paper focused on a comparison of in vitro data and theoretical prediction, the reverse engineering approach is applicable to characterising in vivo neuronal networks, in terms of their implicit generative model with prior beliefs. It can, in principle, be combined with electrophysiological, functional imaging, and behavioural data—and give predictions, if the learning process is continuously measured. Thus, the proposed approach for validating the free-energy principle can be applied to the neural activity data from any experiment that entails learning or self-organisation; irrespective of the species, brain region, task, or measurement technique. Even in the absence of learning, it can be applied, if one can make some theoretical predictions and compare them with experimental data. For accurate predictions, large-scale and continuous measurements of activity data at the population level from pre-learning to post-learning stages would be a prerequisite. In future work, we hope to test, empirically, whether the free-energy principle can quantitatively predict the perception, learning, and behaviour of various biological systems.

The generic mechanisms for acquiring generative models can be used to construct a neuromorphic hardware for universal applications 56 , 57 . Back-propagation is central in many current deep learning methods, but biologically implausible. This has led to various biologically plausible alternatives; e.g., refs.  58 , 59 , 60 , 61 , some of which appeal to predictive coding formulations of variational free energy minimisation. The equivalence between neural networks and variational Bayes could be useful to establish biologically plausible learning algorithms, because Hebbian learning rules derived under this scheme are local (energy-based) algorithms. This is because the contribution to variational free energy as an extensive quantity can be evaluated locally. Such a biomimetic artificial intelligence—that implements the self-organising mechanisms of neuronal networks—could offer an alternative to conventional learning algorithms such as back-propagation, and to have the high data, computational, and energy efficiency of biological computation 62 , 63 . This makes it promising for the next-generation of artificial intelligence. In addition, the creation of biomimetic artificial intelligence may further our understanding of the brain.

In summary, complementary in vitro neural network recordings and in silico modelling suggest that variational free energy minimisation is an apt explanation for the dynamics and self-organisation of neuronal networks that assimilate sensory data. The reverse engineering approach provides a powerful tool for the mechanistic investigation of inference and learning, enabling the identification of generative models and the application of the free-energy principle. The observed sensory learning was consistent with Bayesian belief updating and the minimisation of variational free energy. Thus, variational free energy minimisation could qualitatively predict neuronal responses and plasticity of in vitro neural networks. These results highlight the validity of the free-energy principle as a rule underlying the self-organisation and learning of neuronal networks.

Generative process

The experimental paradigm established in previous work 22 was employed. The blind source separation addressed in this work is an essential ability for biological organisms to identify hidden causes of sensory information, as considered in the cocktail party effect 27 , 28 . This deals with the separation of mixed sensory inputs into original hidden sources in the absence of supervision, which is a more complex problem than naive pattern separation tasks.

Two sequences of mutually independent hidden sources or states \({s}_{t}={({s}_{t}^{(1)},{s}_{t}^{(2)})}^{{{{{{\rm{T}}}}}}}\) generated 32 sensory stimuli \({o}_{t}={({o}_{t}^{(1)},\ldots,{o}_{t}^{(32)})}^{{{{{{\rm{T}}}}}}}\) through a stochastic mixture characterised by matrix A . Each source and observation took values of 1 (ON) or 0 (OFF) for each trial (or time) t . These stimuli were applied to in vitro neural networks as electrical pulses from 32 electrodes (Fig.  1a , left). In terms of the POMDP scheme 38 , 39 , this corresponds to the likelihood mapping A from two sources \({s}_{t}\) to 32 observations \({o}_{t}\) (Fig.  1a , right). The hidden sources \({s}_{t}\) were sampled from a categorical distribution \(P({s}_{t}^{\left(j\right)})={{{{{\rm{Cat}}}}}}({D}^{\left(j\right)})\) . The state priors varied between 0 and 1, in keeping with \({D}_{1}^{\left(j\right)}+{D}_{0}^{\left(j\right)}=1\) . The likelihood of \({o}_{t}^{\left(i\right)}\) is given in the form of a categorical distribution, \(P({o}_{t}^{\left(i\right)}|{s}_{t},A)={{{{{\rm{Cat}}}}}}({A}^{\left(i\right)})\) , each element of which represents \(P({o}_{t}^{\left(i\right)}={j|}{s}_{t}^{\left(1\right)}=k,{s}_{t}^{\left(2\right)}=l,A)={A}_{{jkl}}^{\left(i\right)}\) . Half of the electrodes ( \(1\le i\le 16\) ) conveyed the source 1 signal with a 75% probability or the source 2 signal with a 25% probability. Because each element of the \(A\) matrix represents the conditional probability that \({o}_{t }\) occurs given \({s}_{t }=({s}_{t }^{\left(1\right)},{s}_{t }^{\left(2\right)})\) , this characteristic is expressed as \({A}_{1\cdot \cdot }^{(i)}=(P({o}_{t }^{(i)}=1|{s}_{t }=({{{{\mathrm{1,1}}}}})),\, P({o}_{t }^{(i)}=1|{s}_{t }=({{{{\mathrm{1,}}}}}\, 0)),\, P({o}_{t }^{(i)}=1|{s}_{t }=({{{{\mathrm{0,}}}}}\, 1)),\, P({o}_{t }^{(i)}=1|{s}_{t }=({{{{\mathrm{0,}}}}}\, 0)))=(1,\, 0.75,\, 0.25,\, 0)\) . The remaining electrodes ( \(17\le i\le 32\) ) conveyed the source 1 or 2 signal with a 25% or 75% probability, respectively, \({A}_{1\cdot \cdot }^{\left(i\right)}=\left({{{{\mathrm{1,0.25,0.75,0}}}}}\right)\) . The remaining elements of A were given by \({A}_{0\cdot \cdot }^{\left(i\right)}=1-{A}_{1\cdot \cdot }^{\left(i\right)}\) . The prior distribution of A was given by the Dirichlet distribution \(P({A}^{\left(i\right)})={{{{{\rm{Dir}}}}}}\left({a}^{\left(i\right)}\right)\) with sufficient statistics a . Hence, the generative model was given as follows:

Here, \({o}_{1:t}:\!\!=\left\{{o}_{1},\ldots,\, {o}_{t}\right\}\) represents a sequence of observations, \(P\left(A\right)=\mathop{\prod }\nolimits_{i=1}^{32}P({A}^{(i)})\) , \(P({s}_{\tau })=P({s}_{\tau }^{(1)})P({s}_{\tau }^{(2)})\) , and \(P\left({o}_{\tau }|{s}_{\tau },A\right)=\mathop{\prod }\nolimits_{i=1}^{32}P({o}_{\tau }^{(i)}|{s}_{\tau },A)\) are prior distributions and likelihood that factorise 16 .

Variational Bayesian inference

We considered a Bayesian observer under the generative model in the form of the above POMPD and implemented variational message passing to derive the Bayes optimal encoding of hidden sources or states 38 , 39 . Under the mean-field approximation, the posterior beliefs about states and parameters were provided as follows:

Here, the posterior distributions of \({s}_{\tau }\) and \(A\) are given by categorical \(Q\left({s}_{\tau }\right)={{{{{\rm{Cat}}}}}}\left({{{{{{\bf{s}}}}}}}_{\tau }\right)\) and Dirichlet \(Q\left(A\right)={{{{{\rm{Dir}}}}}}\left({{{{{\bf{a}}}}}}\right)\) distributions, respectively. The bold case variables (e.g., \({{{{{{\bf{s}}}}}}}_{t}\) ) denote the posterior beliefs about the corresponding italic case variables (e.g., \({s}_{t}\) ), and \({{{{{\bf{a}}}}}}\) indicates the Dirichlet concentration parameter. Due to the factorial nature of the states, \({{{{{{\bf{s}}}}}}}_{t}\) and \({{{{{\bf{a}}}}}}\) are the outer products of submatrices (i.e., tensors): see ref. 16 for details.

Variational free energy—or equivalently, the negative of evidence lower bound (ELBO) 3 —is defined as an upper bound of sensory surprise \(F\left({o}_{1:t},Q\left({s}_{1:t},\, A\right)\right):\!\!={{{{{{\rm{E}}}}}}}_{Q\left({s}_{1:t},\, A\right)}\left[-{{{{{\rm{ln}}}}}}P\left({o}_{1:t},\, {s}_{1:t},\;A\right)+{{{{{\rm{ln}}}}}} \, Q\left({s}_{1:t},\;A\right)\right]\) . Given the above-defined generative model and posterior beliefs, ensuing variational free energy of this system is given by:

up to an \({{{{{\mathscr{O}}}}}}\left({{{{{\rm{ln}}}}}}t\right)\) term. This \({{{{{\mathscr{O}}}}}}\left({{{{{\rm{ln}}}}}}t\right)\) corresponds to the parameter complexity expressed using the Kullback–Leibler divergence \({{{{{{\mathcal{D}}}}}}}_{{{{{{\rm{KL}}}}}}}\left[Q\left(A\right){||P}\left(A\right)\right]=\mathop{\sum }\nolimits_{i=1}^{32}\{\left({{{{{{\bf{a}}}}}}}^{\left(i\right)}-{a}^{\left(i\right)}\right)\cdot {{{{{\rm{ln}}}}}}\,{{{{{{\bf{A}}}}}}}^{\left(i\right)}-{{{{{\rm{ln}}}}}}{{{{{\mathcal{B}}}}}}\left({{{{{{\bf{a}}}}}}}^{\left(i\right)}\right)\}\) and is negligible when t is sufficiently large. Note that \(\cdot\) expresses the inner product operator, \({{{{{\rm{ln}}}}}}\,{{{{{{\bf{A}}}}}}}^{\left(i\right)}\) indicates the posterior expectation of \({{{{{\rm{ln}}}}}}\,{A}^{\left(i\right)}\) , and \({{{{{\mathcal{B}}}}}}\left(\bullet \right)\) is the beta function. Inference and learning entail updating posterior expectations about hidden states and parameters, respectively, to minimise variational free energy. Solving the fixed point \(\partial F/\partial {{{{{{\bf{s}}}}}}}_{t}=0\) and \(\partial F/\partial {{{{{\bf{a}}}}}}=O\) yields the following analytic expression:

where \(\sigma \left(\bullet \right)\) is a softmax function, which corresponds to the sigmoid activation function, and \(\bigotimes\) expresses the outer product operator. From Eq. ( 5 ), the parameter posterior is given as \({{{{{\rm{ln}}}}}}\,{{{{{\bf{A}}}}}}=\psi \left({{{{{\bf{a}}}}}}\right)-\psi \left({{{{{{\bf{a}}}}}}}_{1\bullet }+{{{{{{\bf{a}}}}}}}_{0\bullet }\right)\) using the digamma function \(\psi \left(\bullet \right)={\varGamma }^{{\prime} }\left(\bullet \right)/\varGamma \left(\bullet \right)\) . As Eqs. ( 2 )–( 5 ) adopted a simplified notation, please refer to ref.  16 for the detailed derivation taking into account the factorial nature of the states.

Canonical neural networks

The complete class theorem 41 , 42 , 43 suggests that any neural network whose internal states minimise a common cost function can be read as performing Bayesian inference. However, the implicit Bayesian model that corresponds to any given cost function is a more complicated issue. Thus, we reverse engineered cost functions for canonical neural networks to identify the corresponding generative model 16 , 17 .

The neural response \({x}_{t}={\left({x}_{t1},{x}_{t2}\right)}^{{{{{{\rm{T}}}}}}}\) at time t upon receiving sensory inputs \({o}_{t}\) is modelled as the canonical neural network, which is expressed as the following ordinary differential equation:

where \({{{{{{\rm{sig}}}}}}}^{-1}\left({x}_{t}\right)\) indicates the leak current characterised by the inverse of sigmoid function (or equivalently, logit function), \(W\) denotes a 2 × 32 matrix of synaptic strengths, \(W{o}_{t}\) is the synaptic input, and \(h\) is a vector of the adaptive firing thresholds. We considered that \(W:\!\!={W}_{1}-{W}_{0}\) is the sum of excitatory ( \({W}_{1}\) ) and inhibitory ( \({W}_{0}\) ) synaptic strengths. The firing threshold is expressed as \(h:\!\!={h}_{1}-{h}_{0}\) using \({h}_{1}\) and \({h}_{0}\) that are functions of \({W}_{1}\) and \({W}_{0}\) . This model derives from realistic neuron models 4 , 5 , 6 through approximations 17 .

Without loss of generality, Eq. ( 6 ) can be derived as the gradient descent on a cost function L . Following previous work 16 , 17 , this cost function can be identified by taking the integral of the right-hand side of Eq. ( 6 ) with respect to \({x}_{t}\) (referred to as reverse engineering):

up to a negligible \({{{{{\mathcal{C}}}}}}\) term. The overline variable indicates one minus the variable, \(\overline{{x}_{t}}: \!\!=\vec{1}-{x}_{t}\) , where \(\vec{1}:\!\!={\left(1,\ldots,1\right)}^{{{{{{\rm{T}}}}}}}\) is a vector of ones. Equation ( 7 ) ensures that the gradient descent on L with respect to \({x}_{t}\) , \({\dot{x}}_{t}\propto -\partial L/\partial {x}_{t}\) , provides Eq. ( 6 ). The \({{{{{\mathcal{C}}}}}}\) term is a function of \({W}_{1}\) and \({W}_{0}\) , \({{{{{\mathcal{C}}}}}}{{{{{\mathscr{=}}}}}}{{{{{\mathcal{C}}}}}}\left({W}_{1},{W}_{0}\right)\) , and usually considered to be smaller than order t , \({{{{{\mathcal{C}}}}}}={{{{{\mathcal{o}}}}}}\left(t\right)\) 16 \(.\)

The firing thresholds can be decomposed as \({h}_{1}={{{{{\rm{ln}}}}}}\overline{{\widehat{W}}_{1}}\vec{1}+{\phi }_{1}\) and \({h}_{0}={{{{{\rm{ln}}}}}}\overline{{\widehat{W}}_{0}}\vec{1}+{\phi }_{0}\) , respectively, where \({\phi }_{1}\) and \({\phi }_{0}\) are the threshold factors, \({\widehat{W}}_{1}:\!\!={{{{{\rm{sig}}}}}}\left({W}_{1}\right)\) is the sigmoid function of \({W}_{1}\) in the elementwise sense, and \(\overline{{\widehat{W}}_{1}}\) indicates one minus \({\widehat{W}}_{1}\) in the elementwise sense. Subsequently, Eq. ( 7 ) can be transformed as follows:

We showed that this cost function L can be cast as variational free energy F under a class of POMPD generative models 16 , 17 . Equation ( 8 ) is asymptotically equivalent to variational free energy (Eq. ( 3 )) under the generative model defined in Eq. ( 1 ), up to negligible \({{{{{\mathcal{O}}}}}}\left({{{{{\rm{ln}}}}}}t\right)\) and \({{{{{\mathcal{C}}}}}}\) terms. One-to-one correspondences between components of L and F can be observed. Specifically, the neural response \({x}_{t}\) encodes the state posterior \({{{{{{\bf{s}}}}}}}_{t}\) , \(\left(\begin{array}{c}{x}_{\tau }\\ \overline{{x}_{\tau }}\end{array}\right)={{{{{{\bf{s}}}}}}}_{t}\) ; synaptic strengths W encode the parameter posterior \({{{{{\bf{A}}}}}}\) , \({{{{{\rm{ln}}}}}}\left(\begin{array}{cc}{\widehat{W}}_{1} & \overline{{\widehat{W}}_{1}}\\ {\widehat{W}}_{0} & \overline{{\widehat{W}}_{0}}\end{array}\right)={{{{{\rm{ln}}}}}}{{{{{\bf{A}}}}}}\) ; and the threshold factor \(\phi\) encodes the state prior \(D\) , \(\phi=\left(\begin{array}{c}{\phi }_{1}\\ {\phi }_{0}\end{array}\right)={{{{{\rm{ln}}}}}}D\) , as summarised in Table  2 . Hence, the neural network cost function is asymptotically equivalent to variational free energy for sufficiently large t . Further details, including the correspondence between the \({{{{{\mathcal{C}}}}}}\) term in Eq. ( 8 ) and the parameter complexity ( \({{{{{\mathcal{O}}}}}}\left({{{{{\rm{ln}}}}}}t\right)\) term in Eq. ( 3 )), are described in previous work 16 .

The virtue of this equivalence is that it links quantities in the neural network with those in the variational Bayes formation. Moreover, this suggests that a physiologically plausible synaptic plasticity (derived from L ) enables the network to learn the parameter posterior in a self-organising or unsupervised manner 16 , 17 . Further, reverse-engineering can naturally derive variational Bayesian inference—under a particular mean-field approximation defined in Eq.  2 —from a canonical neural network architecture. This representation of posterior beliefs is essential for the networks to encode rapidly changing hidden states ( \({s}_{\tau }\) ) and slow parameters ( \(A\) ) with neural activity ( \(x\) ) and synaptic strengths ( \(W\) ), respectively. In this setting, a mean field approximation implements a kind of adiabatic approximation, in which the separation of timescales between fast neuronal responses and slow learning is leveraged to increase the efficiency of inference. Please see ref. 16 . for further discussion.

Simulations

In Fig.  2 , simulations continued over \(T=25600\) time steps and used the empirical stimuli applied to in vitro neural networks. Synaptic strengths were initialised as values close to 0. Here, \({D}_{1}=0.5\) (Fig.  2e , centre) matched the true process that generates sensory stimuli. Either the upregulation (right, \({D}_{1}=0.8\) ) or downregulation (left, \({D}_{1}=0.2\) ) of the state prior disrupted inference and ensuing learning.

Cell culture

The dataset used for this work comprised data obtained from newly conducted experiments, and those originally used in the previous work 22 . All animal experiments were performed with the approval of the animal experiment ethics committee at the University of Tokyo (approval number C-12-02, KA-14-2) and according to the University of Tokyo guidelines for the care and use of laboratory animals. The procedure for preparing dissociated cultures of cortical neurons followed the procedures described in previous work 22 . Pregnant Wistar rats (Charles River Laboratories, Yokohama, Japan) were anaesthetised with isoflurane and immediately sacrificed. The cerebral cortex was removed from 19-day-old embryos (E19) and dissociated into single cells by treatment with 2.5% trypsin (Life Technologies, Carlsbad, CA, USA) at 37 °C for 20 min, followed by mechanical pipetting. Half a million dissociated cortical cells (a mixture of neurons and glial cells) were seeded on the centre of MEA dishes, where the surface of MEA was previously coated with polyethyleneimine (Sigma‒Aldrich, St. Louis, MO, USA) overnight. These cells were cultured in the CO 2 incubator. Culture medium comprised Neurobasal Medium (Life Technologies) containing 2% B27 Supplement (Life Technologies), 2 mM GlutaMAX (Life Technologies), and 5–40 U/mL penicillin/streptomycin (Life Technologies). Half of the culture medium was changed once every second or third day. These cultures were recorded during the age of 18–83 days in vitro. During this stage, the spontaneous firing patterns of the neurons had reached a developmentally stable period 64 , 65 .

In this work, 21 independent cell cultures were used for the control condition to conduct 30 independent experiments, 6 were treated with bicuculline, 7 with diazepam, 9 with APV, 4 were trained under the 0% mix condition, and 4 under the 50% mix condition. Out of these samples, 7 in the control condition, 6 treated with bicuculline, and 7 with diazepam were obtained from newly conducted experiments, where their response intensities were 3.0 ± 1.1, 3.7 ± 1.9, and 2.3 ± 0.86 spike/trial, respectively (mean ± standard deviation). Other cultures were originally recorded for previous work 22 . The cell-culturing and experimental conditions in the previous work were essentially the same as those recorded for the present work. Note that the same cultures were used more than once for experiments with other stimulation pattern conditions, after at least one day interval. This is justified because the different stimulation patterns were independent of each other, and thus, learning history with other stimulation patterns did not affect the subsequent experiments 22 .

Pharmacological treatment

The excitability of cultured neurons was pharmacologically controlled. To block GABA A -receptor activity, bicuculline, a GABA A -receptor antagonist (Sigma‒Aldrich, St. Louis, MO, USA) was used. Bicuculline was adjusted to 10 mM using phosphate-buffered saline (PBS), and 10 µL was added to the culture medium to a final concentration of 50 µM. To upregulate GABA A -receptor activity, diazepam, a benzodiazepine receptor agonist (Sigma‒Aldrich) was used. Diazepam was adjusted to 100 µM using N,N-dimethylformamide (DMF), and 20 µL was added to the culture medium to a final concentration of 1 µM. After adding the solution to the medium, cultured neurons were placed in a CO 2 incubator for 30 min, and stable activity of the neurons was confirmed before recording.

Electrophysiological experiments

Electrophysiological experiments were conducted using an MEA system (NF Corporation, Yokohama, Japan). This enabled extracellular recordings of evoked spikes from multiple sites immediately after electrical stimulation 20 , 21 . An MEA dish comprises 8×8 microelectrodes (50-µm square each) embedded on its centre, deployed on a grid with 250-µm microelectrodes separation. Recordings were conducted with a 25-kHz sampling frequency and band-pass filter of 100–2000 Hz. The data acquisition was conducted using LabVIEW version 2011. The spike sorting analysis suggested that an electrode was expected to record the activities from up to four neurons. Three-phasic extracellular potentials, as described in previous work 20 , 21 , were recorded from the majority of the electrodes.

The 32 stimulation electrodes were randomly distributed over 8×8 MEAs in advance and fixed over training. A biphasic pulse with a 1-V amplitude and 0.2-ms duration, which efficiently induces activity-dependent synaptic plasticity 22 , was used as a sensory stimulus. A session of training comprised a 256-time-step sequence of stimuli with 1-s intervals, followed by a 244-s resting period. We repeated this training for 100 sessions (approximately 14 h in total). All recordings and stimulation were conducted in a CO 2 incubator.

Data preprocessing

For spike detection, the acquired signals were passed through a digital band-pass filter of 500‒2000 Hz after the removal of the saturated ranges and noises that were caused by electric potential variations associated with the switch from the stimulation circuit to the recording circuit. Subsequently, waveform valleys that fell below 4 times the standard deviation of the signal sequence of each electrode were detected as spikes. Note that for data obtained in the previous work 22 , waveform valleys that fell below 5 times the standard deviation were detected as spikes because of the difference in the noise level.

Irrespective of the presence or absence of bicuculline or diazepam, the peak of evoked response usually fell at 10–20 ms after each stimulus. Accordingly, we defined the intensity of the evoked response to the stimulus by the number of spikes generated until 10–30 ms after each stimulus. We referred to the evoked response at electrode i as \({r}_{{ti}}\) (spike/trial), using discrete time step (or trial) t . Only electrodes at which the all-session average of \({r}_{{ti}}\) was larger than 1 spike/trial were used for subsequent analyses.

The conditional expectation of evoked response \({r}_{{ti}}\) —when a certain source state \(\left({s}_{1},{s}_{2}\right)=\left({{{{\mathrm{1,1}}}}}\right),\, \left({{{{\mathrm{1,0}}}}}\right),\, \left({{{{\mathrm{0,1}}}}}\right),\, \left({{{{\mathrm{0,0}}}}}\right)\) is provided—is given as \({{{{{\rm{E}}}}}}\left[{r}_{{it}}|{s}_{1},{s}_{2}\right]:\!\!={{{{{\rm{E}}}}}}\left[{r}_{{ti}}|{s}_{t}=\left({s}_{1},{s}_{2}\right),\, 1\le t\le 256\right]\) (spike/trial). This \({{{{{\rm{E}}}}}}\left[{r}_{{it}}|{s}_{1},{s}_{2}\right]\) was computed for each session. Recorded neurons were categorised into three groups based on their preference to sources. We referred to a neuron (or electrode) as source 1-preferring when the all-session average of \({{{{{\rm{E}}}}}}\left[{r}_{{it}}|{{{{\mathrm{1,0}}}}}\right]-{{{{{\rm{E}}}}}}\left[{r}_{{it}}|{{{{\mathrm{0,1}}}}}\right]\) was larger than 0.5 spike/trial, as source 2-preferring when the all-session average of \({{{{{\rm{E}}}}}}\left[{r}_{{it}}|{{{{\mathrm{1,0}}}}}\right]-{{{{{\rm{E}}}}}}\left[{r}_{{it}}|{{{{\mathrm{0,1}}}}}\right]\) was smaller than –0.5 spike/trial, or no preference when otherwise. Note that the number of source 1-preferring, source 2-preferring, and no preference electrodes in each sample are 17.1 ± 7.0, 15.0 ± 7.0, and 11.5 ± 6.7, respectively ( n  = 30 samples under the control condition). Sources 1- and 2-preferring ensembles were quantitatively similar because the total contribution from sources 1 and 2 to stimuli \({o}_{t}\) was designed to be equivalent, owing to the symmetric structure of the \(A\) matrix. Under this setting, this similarity was conserved, irrespective of the details of the \(A\) .

Our hypothesis 23 was that the stimulus ( o ) obligatorily excites a subset of neurons in the network, while repeated exposure makes other neurons with appropriate connectivity learn that the patterns of responses are caused by ON or OFF of hidden sources ( s ). Thus, the recorded neuronal responses comprise the activity of neurons directly receiving the input and that of neurons encoding the sources. To identify functionally specialised neurons, we modelled recorded activity as a mixture of the response directly triggered by the stimulus and functionally specialised response to the sources. Most directly triggered responses occur within 10 ms of stimulation, and their number is largely invariant over time, while their latency varies in the range of a few hundred microseconds 40 . Conversely, functionally specialised responses emerge during training, and the majority occur 10–30 ms after stimulation. Thus, analysing the deviation of the number of spikes in this period enables the decomposition of the responses into stimulus- and source-specific components.

The empirical responses were represented as the averaged responses in each group. For subsequent analyses, we defined \({x}_{t1}\) as the ensemble average over source 1-preferring neurons and \({x}_{t2}\) as that over source 2-preferring neurons in each culture. For analytical tractability, we normalised the recorded neural response to ensure that it was within the range of \(0\le {x}_{t1},{x}_{t2}\le 1\) , after subtracting the offset and trend.

Statistical tests

The two-sided Wilcoxon signed-rank test was used for paired comparisons. The two-sided Mann‒Whitney U test was used for unpaired comparisons.

Reverse engineering of generative models

In this section, we elaborate the procedure for estimating the threshold factor ( \(\phi\) ) and effective synaptic connectivity ( \(W\) ) from empirical data, to characterise the landscape of the neural network cost function \(L\) ( \(\equiv F\) ) and further derive the generative model that the biological system employs.

Assuming that the change in threshold factor was sufficiently slow relative to a short experimental period, the threshold factor \(\phi\) was estimated based on the mean response intensity of empirical data. Following the treatment established in previous work 16 , 17 , the constants are estimated for each culture using the empirical data as follows:

where \(\left\langle \cdot \right\rangle : \!\!=\frac{1}{t}\mathop{\sum }\nolimits_{\tau=1}^{t}\cdot\) indicates the average over time. Equation ( 9 ) was computed using data in the initial 10 sessions. Subsequently, the state prior \(D\) was reconstructed from the relationship \({{{{{\rm{ln}}}}}} \, D=\phi\) (Table  2 ). This \(D\) expresses the implicit perceptual bias of an in vitro network about hidden sources.

Synaptic plasticity rules conjugate to Eq. ( 6 ) are derived as the gradient descent on L 16 , 17 , which are asymptotically given as \({\dot{W}}_{1}\propto -\frac{1}{t}\frac{\partial L}{\partial {W}_{1}}=\left\langle {x}_{t}{o}_{t}^{{{{{{\rm{T}}}}}}}\right\rangle -\left\langle {x}_{t}{\vec{1}}{\,\!}^{{{{{{\rm{T}}}}}}}\right\rangle \odot {\widehat{W}}_{1}\) and \({\dot{W}}_{0}\propto -\frac{1}{t}\frac{\partial L}{\partial {W}_{0}}=\left\langle \overline{{x}_{t}}{o}_{t}^{{{{{\rm{T}}}}}}\right\rangle -\left\langle \overline{{x}_{t}}{\vec{1}}{\,\!}^{{{{{\rm{T}}}}}}\right\rangle \odot {\widehat{W}}_{0}\) in the limit of a large t , where \(\odot\) denotes the elementwise product (a.k.a., the Hadamard product). These rules comprise Hebbian plasticity accompanied with an activity-dependent homeostatic term, endorsing the biological plausibility of this class of cost functions. Solving the fixed point of these equations provides the following synaptic strengths:

where \(\oslash\) denotes the elementwise division operator. In this work, we refer to Eq. ( 10 ) as the empirically estimated effective synaptic connectivity, where \(W={W}_{1}-{W}_{0}\) . This was estimated for each session, using empirical neuronal response data \({x}_{t}\) . These synaptic strengths encode the posterior belief about the mixing matrix A (Table  2 ). Further details are provided in previous works 16 , 17 .

These empirically estimated parameters are sufficient to characterise the generative model that an in vitro neural network employs. Owing to the equivalence, the empirical variational free energy F for the in vitro network was computed by substituting empirical neuronal responses x and empirically estimated parameters \(W\) (Eq. ( 10 )) and \(\phi\) (Eq. ( 9 )) into the neural network cost function L (Eq. ( 8 )): see Fig.  3l for its trajectory.

Data prediction using the free-energy principle

The virtues of the free-energy principle are that it offers the quantitative prediction of transitions (i.e., plasticity) of the neural responses and synaptic strengths in future, in the absence of empirical response data. We denote the predicted neural responses and synaptic strengths as \({x}_{t}^{P}\) and \({W}^{P}\) , respectively, to distinguish them from the observed neural responses \({x}_{t}\) and empirically estimated synaptic strengths \(W\) defined above.

The predicted neural response is given as the fixed-point solution of Eq. ( 6 ):

where \({h}^{P}={{{{{\rm{ln}}}}}}\overline{{\widehat{W}}_{1}^{P}}\vec{1}-{{{{{\rm{ln}}}}}}\overline{{\widehat{W}}_{0}^{P}}\vec{1}+{\phi }_{1}-{\phi }_{0}\) denotes the adaptive firing threshold. Empirical \(\phi\) (Eq. ( 9 )) estimated from data in the initial 10 sessions was used to characterise \({h}^{P}\) . Here, predicted synaptic strength matrix \({W}^{P}\) was used instead of the empirically estimated \(W\) . The predicted synaptic strengths are given as follows:

where \({W}^{P}:\!\!={W}_{1}^{P}-{W}_{0}^{P}\) . Here, the predicted neural responses \({x}_{t}^{P}\) were employed to compute the outer products. The initial value of \({W}^{P}\) was computed using empirical response data in the first 10 sessions. By computing Eqs. ( 11 ) and ( 12 ), one can predict the subsequent self-organisation of neuronal networks in sessions 11–100, without reference to the observed neuronal responses.

We note that the reverse engineering approach provides three novel aspects compared to earlier work 22 , 23 . First, previous work assumed the form of the generative model and did not examine whether all elements of the generative model corresponded to biological entities at the circuit level. In the present work, we objectively reverse-engineered the generative model from empirical data and showed a one-to-one mapping between all the elements of the generative model and neural network entities. Second, previous work did not examine the impact of changing prior beliefs on Bayesian inference performed by in vitro neural networks. The present work analysed how Bayesian inference and free energy reduction changed when the prior belief and external environment were artificially manipulated and showed that the results were consistent with theoretical predictions. This work validated the predicted relationship between baseline excitability and prior beliefs about hidden states. Third, previous work did not investigate whether the free-energy principle can quantitatively predict the learning process of biological neural networks based exclusively on initial empirical data. This was demonstrated in the current work.

Reporting summary

Further information on research design is available in the  Nature Portfolio Reporting Summary linked to this article.

Data availability

The neuronal response data are available at GitHub https://github.com/takuyaisomura/reverse_engineering .  Source data are provided with this paper.

Code availability

The simulations and analyses were conducted using MATLAB version R2020a. The scripts are available at GitHub https://github.com/takuyaisomura/reverse_engineering 66 . The scripts are covered under the GNU General Public License v3.0.

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Acknowledgements

We are grateful to Masafumi Oizumi, Asaki Kataoka, Daiki Sekizawa, and other members of the Oizumi laboratory for discussions. T.I. is supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers JP23H04973 and JP23H03465 and the Japan Science and Technology Agency (JST) CREST Grant Number JPMJCR22P1. K.J.F. is supported by funding for the Wellcome Centre for Human Neuroimaging (Ref: 205103/Z/16/Z), a Canada-UK Artificial Intelligence Initiative (Ref: ES/T01279X/1) and the European Union’s Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement No. 945539 (Human Brain Project SGA3). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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Isomura, T., Kotani, K., Jimbo, Y. et al. Experimental validation of the free-energy principle with in vitro neural networks. Nat Commun 14 , 4547 (2023). https://doi.org/10.1038/s41467-023-40141-z

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The two kinds of free energy and the Bayesian revolution

* E-mail: [email protected]

Affiliation Institute of Neural Information Processing, Ulm University, Ulm, Germany

• Sebastian Gottwald, 
• Daniel A. Braun

Published: December 3, 2020

• https://doi.org/10.1371/journal.pcbi.1008420
• Reader Comments

The concept of free energy has its origins in 19th century thermodynamics, but has recently found its way into the behavioral and neural sciences, where it has been promoted for its wide applicability and has even been suggested as a fundamental principle of understanding intelligent behavior and brain function. We argue that there are essentially two different notions of free energy in current models of intelligent agency, that can both be considered as applications of Bayesian inference to the problem of action selection: one that appears when trading off accuracy and uncertainty based on a general maximum entropy principle, and one that formulates action selection in terms of minimizing an error measure that quantifies deviations of beliefs and policies from given reference models. The first approach provides a normative rule for action selection in the face of model uncertainty or when information processing capabilities are limited. The second approach directly aims to formulate the action selection problem as an inference problem in the context of Bayesian brain theories, also known as Active Inference in the literature. We elucidate the main ideas and discuss critical technical and conceptual issues revolving around these two notions of free energy that both claim to apply at all levels of decision-making, from the high-level deliberation of reasoning down to the low-level information processing of perception.

Citation: Gottwald S, Braun DA (2020) The two kinds of free energy and the Bayesian revolution. PLoS Comput Biol 16(12): e1008420. https://doi.org/10.1371/journal.pcbi.1008420

Editor: Samuel J. Gershman, Harvard University, UNITED STATES

Copyright: © 2020 Gottwald, Braun. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This study was funded by the European Research Council under the Starting Grant ERC-StG-2015-ERC termed "BRISC: Bounded Rationality in Sensorimotor Coordination" with Project ID 678082 (D.A.B.). The first author’s (S.G.) salary was payed from this funding. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

1 Introduction

There is a surprising line of thought connecting some of the greatest scientists of the last centuries, including Immanuel Kant, Hermann von Helmholtz, Ludwig E. Boltzmann, and Claude E. Shannon, whereby model-based processes of action, perception, and communication are explained with concepts borrowed from statistical physics. Inspired by Kant’s Copernican revolution and motivated from his own studies of the physiology of the sensory system, Helmholtz was one of the first proponents of the analysis-by-synthesis approach to perception [ 1 ], whereby a perceiver is not simply conceptualized as some kind of tabula rasa recording raw external stimuli, but rather relies on internal models of the world to match and anticipate sensory inputs. The internal model paradigm is now ubiquitous in the cognitive and neural sciences and has even led some researchers to propose a Bayesian brain hypothesis, whereby the brain would essentially be a prediction and inference engine based on internal models [ 2 – 4 ]. Coincidentally, Helmholtz also invented the notion of the Helmholtz free energy that plays an important role in thermodynamics and statistical mechanics, even though he never made a connection between the two concepts in his lifetime.

This connection was first made by Dayan, Hinton, Neal, and Zemel in their computational model of perceptual processing as a statistical inference engine known as the Helmholtz machine [ 5 ]. In this neural network architecture, there are feed-forward and feedback pathways, where the bottom-up pathway translates inputs from the bottom layer into hidden causes at the upper layer (the recognition model), and top-down activation translates simulated hidden causes into simulated inputs (the generative model). When considering log-likelihood in this setup as energy in analogy to statistical mechanics, learning becomes a relaxation process that can be described by the minimization of variational free energy. While it should be emphasized that variational free energy is not the same as Helmholtz free energy, the two free energy concepts can be formally related. Importantly, variational free energy minimization is not only a hallmark of the Helmholtz machine, but of a more general family of inference algorithms, such as the popular expectation-maximization (EM) algorithm [ 6 , 7 ]. In fact, over the last two decades, variational Bayesian methods have become one of the foremost approximation schemes for tractable inference in the machine learning literature. Moreover, a plethora of machine learning approaches use loss functions that have the shape of a free energy when optimizing performance under entropy regularization in order to boost generalization of learning models [ 8 , 9 ].

In the meanwhile, free energy concepts have also made their way into the behavioral sciences. In the economic literature, for example, trade-offs between utility and entropic uncertainty measures that take the form of free energies have been proposed to describe decision-makers with stochastic choice behavior due to limited resources [ 10 – 14 ] or robust decision-makers with limited precision in their models [ 15 , 16 ]. The free energy trade-off between entropy and reward can also be found in information-theoretic models of biological perception-action systems [ 17 – 19 ], some of which have been subjected to experimental testing [ 20 – 25 ]. Finally, in the neuroscience literature the notion of free energy has risen to recent fame as the central puzzle piece in the Free Energy Principle [ 26 ] that has been used to explain a cornucopia of experimental findings including neural prediction error signals [ 27 ], synaptic plasticity rules [ 28 ], neural effects of biased competition and attention [ 29 , 30 ], visual exploration in humans [ 31 ], and more—see the references in [ 32 ]. Over time, the Free Energy Principle has grown out of an application of the free energy concept used in the Helmholtz machine, to interpret cortical responses in the context of predictive coding [ 33 ], and has gradually developed into a general principle for intelligent agency, also known as Active Inference [ 32 , 34 , 35 ]. Consequences and implications of the Free Energy Principle are discussed in neighbouring fields like psychiatry [ 36 , 37 ] and the philosophy of mind [ 38 , 39 ].

Given that the notion of free energy has become such a pervasive concept that cuts through multiple disciplines, the main rationale for this discussion paper is to trace back and to clarify different notions of free energy, to see how they are related and what role they play in explaining behavior and neural activity. As the notion of free energy mainly appears in the context of statistical models of cognition, the language of probabilistic models constitutes a common framework in the following discussion. Section 2 therefore starts with preliminary remarks on probabilistic modelling. Section 3 introduces two notions of free energy that are subsequently expounded in Section 4 and Section 5, where they are applied to models of intelligent agency. Section 6 concludes the paper.

2 Probabilistic models and perception-action systems

Systems that show stochastic behavior, for example due to randomly behaving components or because the observer ignores certain degrees of freedom, are modelled using probability distributions. This way, any behavioral, environmental, and hidden variables can be related by their statistics, and dynamical changes can be modelled by changes in their distributions.

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The arrows (edges) indicate causal relationships between the random variables (nodes). The full joint distribution p 0 over all random variables is sometimes also referred to as a generative model , because it contains the complete knowledge about the random variables and their dependencies and therefore allows to generate simulated data. Such a model could for example be used by a farmer to infer the soil quality S based on the crop yields X through Bayesian inference, which allows to determine a priori unknown distributions such as p ( S | X ) from the generative model p 0 via marginalization and conditionalization.

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In principle, Bayesian inference requires only two different kinds of operations, namely marginalization , i.e., summing out unobserved variables that have not been queried, such as X ′, S ′ and A above, and conditionalization , i.e., renormalizing the joint distribution over observed and queried variables—that may itself be the result from a previous marginalization such as p ( S , X ) above—to obtain the required conditional distribution over the queried variables. In practice, however, inference is a hard computational problem and many more efficient inference methods are available that may provide approximate solutions to the exact Bayes’ posteriors, including belief propagation [ 40 ], expectation propagation [ 41 ], variational Bayesian inference [ 42 ], and Monte Carlo algorithms [ 43 ]. Also note that inference is trivial if the sought-after conditional distribution of the queried variable is already given by one of the conditional distributions that jointly specify the probabilistic model, e.g., p ( X | S ) = p 0 ( X | S ).

Probabilistic models can be used not only as external (observer) models , but also as internal models that are employed by the agent itself, or by a designer of the agent, in order to determine a desired course of action. In this latter case, actions could either be thought of as deterministic parameters of the probabilistic model that influence the future ( influence diagrams ) or as random variables that are part of the probabilistic model themselves ( prior models ) [ 44 ]. Either way, internal models allow making predictions over future consequences in order to find actions or distributions over actions that lead to desirable outcomes, for example actions that produce high rewards in the future. In mechanistic or process model interpretations, some of the specification procedures to find such actions are themselves meant to represent what the agent is actually doing while reasoning, whereas as if interpretations simply use these methods as tools to arrive at distributions that describe the agent’s behavior. Free energy is one of the concepts that appears in both types of methods.

3 The two notions of free energy

While the two notions of free energy that we discuss in the following are vaguely inspired by the physical original, their motivations are rather distinct and the main reason they share the nomenclature is due to their general form (3) resembling the Helmholtz free energy (4) .

3.1 Free energy from constraints

The first notion of free energy is closely tied to the principle of maximum entropy [ 46 ], which virtually appears in all branches of science. From this vantage point, the physical free energy is merely a special instance of a more general inference problem where we hold probabilistic beliefs about unknown quantities (e.g., the exact energy values of the molecules in a gas) and we can only make coarse measurements or observations (e.g., the temperature of the gas) that we can use to update our beliefs about these hidden variables. The principle of maximum entropy suggests that, among the beliefs that are compatible with the observation, we should choose the most “unbiased” belief, in the sense that it corresponds to a maximum number of possible assignments of the hidden variables.

3.1.1 Wallis’ motivation of the maximum entropy principle.

3.1.2 Free energy from constraints and the Boltzmann distribution.

3.1.3 The trade-off between energy and uncertainty.

An important feature of the minimization of the free energies (7) and (9) consists in the balancing of the two competing terms of energy and entropy (cf. Fig 2 ). This trade-off between maximal uncertainty (uniform distribution, or p 0 ) on the one hand and minimal energy (e.g., a delta distribution) on the other hand is the core of the maximum entropy principle. The inverse temperature β plays the role of a trade-off parameter that controls how these two counteracting forces are weighted.

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The maximum entropy principle goes back to the principle of insufficient reason [ 49 – 51 ], which states that two events should be assigned the same probability if there is no reason to think otherwise. It has been hailed as a principled method to determine prior distributions and to incorporate novel information into existing probabilistic knowledge. In fact, Bayesian inference can be cast in terms of relative entropy minimization with constraints given by the available information [ 52 ]. Applications of this idea can also be found in the machine learning literature, where subtracting (or adding) an entropy term from an expected value of a function that must be optimized is known as entropy regularization and plays an important role in modern reinforcement learning algorithms [ 8 , 9 ] to encourage exploration [ 53 ] as well as to penalize overly deterministic policies resulting in biased reward estimates [ 54 ].

From now on, we refer to a free energy expression that is motivated from a trade-off between an energy and an entropy term, such as (7) and (9) , as free energy from constraints , in order to discriminate it from the notion of free energy introduced in the following section, which—despite of its resemblance—has a different motivation.

3.2 Variational free energy

There is another, distinct appearance of the term “free energy” outside of physics, which is a priori not motivated from a trade-off between an energy and entropy term, but from possible efficiency gains when representing Bayes’ rule in terms of an optimization problem. This technique is mainly used in variational Bayesian inference [ 55 ], originally introduced by Hinton and van Camp [ 42 ]. As before, for simplicity all random variables are discrete, but most expressions can directly be translated to the continuous case by replacing probability distributions by probability densities and sums by the corresponding integrals.

3.2.1 Variational Bayesian inference.

3.2.2 Variational free energy, an extension of relative entropy.

In two dimensions, the normalization of a point ϕ = ( ϕ 1 , ϕ 2 ) corresponds to a (non-orthogonal) projection onto the plane of probability vectors ( A ). For continuous domains, where probability distributions are represented by densities, normalization corresponds to a rescaling of ϕ such that the area below the graph equals 1 ( B ). Instead, when minimizing variational free energy (red colour), the trial distributions q are varied until they fit to the shape of the unnormalized function ϕ (perfectly at q = p ϕ ).

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Variational free energy can be regarded as an extension of relative entropy with the reference distribution being replaced by a non-normalized reference function, since in the case when ϕ is already normalized, that is if ∑ z ϕ ( z ) = 1, then the free energy (13) coincides with the KL divergence D KL ( q ‖ ϕ ). In particular, while relative entropy is a measure for the dissimilarity of two probability distributions, where the minimum is achieved if both distributions are equal, variational free energy is a measure for the dissimilarity between a probability distribution q and a (generally non-normalized) function ϕ , where the minimum with respect to q is achieved at p ϕ . Accordingly, we can think of the variational free energy as a specific error measure between probability distributions and reference functions. In principle, one could design many other error measures that have the same minimum. This means that, a statement in a probabilistic setting that a distribution q * minimizes a variational free energy F ( q ‖ ϕ ) with respect to a given reference ϕ , is analogous to a statement in a non-probabilistic setting that some number x = x * minimizes the value of an error measure ϵ ( x , y ) (e.g., the squared error ϵ ( x , y ) = ( x − y ) 2 ) with respect to a given reference value y .

3.2.3 Approximate and iterative inference.

Representing Bayes’ rule as an optimization problem over auxiliary distributions q has two main applications that both can simplify the inference process (cf. Fig 4 ). First, it allows to approximate exact Bayes’ posteriors by restricting the optimization space, for example using a non-exhaustive parametrization, e.g., an exponential family. Second, it enables iterative inference algorithms consisting of multiple simpler optimization steps, for example by optimizing with respect to each term in a factorized representation of q separately. A popular choice is the mean-field approximation , which combines both of these simplifications, as it assumes independence between hidden states, effectively reducing the search space from joint distributions to factorized ones, and moreover it allows to optimize with respect to each factor alternatingly. Note, however, that mean-field approximations have limited use in sequential environments, where independence of subsequent states cannot be assumed and therefore less restrictive assumptions must be used instead [ 56 ].

In practice, this variational representation is often exploited to simplify a given inference problem, either by reducing the seach space of distributions, for example through a restrictive parametrization resulting in approximate inference, or by splitting up the optimization into multiple partial optimization steps that are potentially easier to solve than the original problem but might still converge to the exact solution. These two simplifications can also be combined, for example in the case of mean-field assumptions where the space of distributions is reduced and an efficient iterative inference algorithm is obtained at the same time.

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Many efficient iterative algorithms for exact and approximate inference can be viewed as examples of variational free energy minimization, for example the EM algorithm [ 6 , 57 ], belief propagation [ 40 , 58 ], and other message passing algorithms [ 41 , 59 – 62 ]. While the (Bayesian) EM algorithm [ 7 ] and Pearl’s belief propagation [ 58 ] both can be seen as minimizing the same variational free energy, just with different assumptions on the approximate posteriors, in [ 61 ], it is shown that also many other message passing algorithms such as [ 41 , 59 , 60 ] can be cast as minimizing some type of free energy, the only difference being the choice of the divergence measure as the entropy term. Simple versions of these algorithms have often existed before their free energy formulations were available, but the variational representations usually allowed for extensions and refinements—see [ 6 , 7 , 63 , 64 ] in case of EM and [ 58 , 62 , 65 , 66 ] in case of message passing.

We are now turning to the question of how the two notions of free energy introduced in this section are related to recent theories of intelligent agency.

4 Free energy from constraints in information processing

4.1 the basic idea.

The concept of free energy from constraints as a trade-off between energy and uncertainty can be used in models of perception-action systems, where entropy quantifies information processing complexity required for decision-making (e.g., planning a path for fleeing a predator) and energy corresponds to performance (e.g., distinguishing better and worse flight directions). The notion of decision in this context is very broad and can be applied to any internal variable in the perception-action pipeline [ 67 ], that is not given directly by the environment. In particular, it also subsumes perception itself, where the decision variables are given by the hidden causes that are being inferred from observations.

The utility values U ( x ) could either be objective, for example a monetary gain, or subjective in which case they represent the decision-maker’s preferences. In general, the utility does not have to be defined directly on Ω, but could be derived from utility values that are attached to certain states, for example to the configurations of the playboard in a board game. In the case of perception, utility values are usually given by (log-)likelihood functions, in which case utility maximization without constraints corresponds to greedy inference such as maximum likelihood estimation. Note that, for simplicity, in this section we consider one-step decision problems. Sequential tasks can either be seen as multiple one-step problems where the utility of a given step might depend on the policy over future steps, or as path planning problems where an action represents a full action path or policy [ 18 , 69 – 71 ].

While ideal rational decision-makers are assumed to perfectly optimize a given utility function U , real behavior is often stochastic, meaning that multiple exposures to the same problem lead to different decisions. Such non-deterministic behavior could be a consequence of model uncertainty, as in Bayesian inference or various stochastic gambling schemes, or a consequence of satisficing [ 72 ], where decision-makers do not choose the single best option, but simply one option that is good enough. Abstractly, this means that, the choice of a single decision is replaced by the choice of a distribution over decisions. More generally, also considering prior information that the decision-maker might have from previous experience, the process of deliberation during decision-making might be expressed as the transformation of a prior p 0 to a posterior distribution p .

When assuming that deliberation has a cost C ( p , p 0 ), then arriving at narrow posterior distributions should intuitively be more costly than choosing distributions that contain more uncertainty (cf. Fig 5A ). In other words, deliberation costs must be increasing with the amount of uncertainty that is reduced by the transformation from p 0 to p . Uncertainty reduction can be understood as making the probabilities of options less equal to each other, rigorously expressed by the mathematical concept of majorization [ 73 ]. This notion of uncertainty can also be generalized to include prior information, so that the degree of uncertainty reduction corresponds to more or less deviations from the prior [ 74 ].

A : Decision-making can be considered as a search process in the space of options Ω, where options are progressively ruled out. Deliberation costs are defined to be monotone functions under such uncertainty reduction. B : Exemplary efficiency curve resulting from the trade-off between utility and costs, that separates non-optimal from non-admissible behavior. The points on the curve correspond to bounded-optimal agents that optimally trade off utility against uncertainty, analogous to the rate-distortion curve in information theory.

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4.2 A simple example

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4.3 Critical points

The main idea of free energy in the context of information processing with limited resources is that any computation can be thought of abstractly as a transformation from a distribution p 0 of prior knowledge to a posterior distribution p that encapsulates an advanced state of knowledge resulting from deliberation. The progress that is made through such a transformation is quantitatively captured by two measures: the expected utility 〈 U 〉 p that quantifies the quality of p and C ( p , p 0 ) that measures the cost of uncertainty reduction from p 0 to p . Clearly, the critical point of this framework is the choice of the cost function C . In particular, we could ask whether there is some kind of universal cost function that is applicable to any perception-action process or whether there are only problem-specific instantiations. Of course, having a universal measure that allows applying the same concepts to extremely diverse systems is both a boon and a bane, because the practical insights it may provide for any concrete instance could be very limited. This is the root of a number of critical issues:

• What is the cost C ? An important restriction of all deliberation costs of the form C ( p , p 0 ) is that they only depend on the initial and final distributions and ignore the process of how to get from p 0 to p . When varying a single resource (e.g., processing time) we can use C ( p , p 0 ) as a process-independent proxy for the resource. However, if there are multiple resources involved (e.g., processing time, memory, and power consumption), a single cost cannot tell us how these resources are weighted optimally without making further process-dependent assumptions. In general, the theory makes no suggestions whatsoever about mechanical processes that could implement resource-optimal strategies, it only serves as a baseline for comparison. Finally, simply requiring the measure to be monotonic in the uncertainty reduction, does not uniquely determine the form of C , as there have been multiple proposals of uncertainty measures in the literature (see e.g., [ 86 ]), where relative entropy is just one possibility. However, relative entropy is distinguished from all other uncertainty measures in its additivity property, that for example allows to express optimal probabilistic updates from p 0 to p in terms of additions or subtractions of utilities, such as log-likelihoods for evidence accumulation in Bayesian inference.
• What is the utility? When systems are engineered, utilities are usually assumed to be given such that desired behavior is specified by utility maximization. However, when we observe perception-action systems, it is often not so clear what the utility should be, or in fact, whether there even exists a utility that captures the observed behavior in terms of utility maximization. This question of the identifiability of a utility function is studied extensively in the economic sciences, where the basic idea is that systems reveal their preferences through their actual choices and that these preferences have to satisfy certain consistency axioms in order to guarantee the existence of a utility function. In practice, to guarantee unique identifiability these axioms are usually rather strong, for example ignoring the effects of history and context when choosing between different items, or ignoring the possibility that there might be multiple objectives. When not making these strong assumptions, utility becomes a rather generic concept, like the concept of probability, and additional assumptions like soft-maximization are necessary to translate from utilities to choice probabilities.
• The problem of infinite regress. One of the main conceptual issues with the interpretation of C as a deliberation cost is that the original utility optimization problem is simply replaced by another optimization problem that may even be more difficult to solve. This novel optimization problem might again require resources to be solved and could therefore be described by a higher-level deliberation cost, thus leading to an infinite regress. In fact, any decision-making model that assumes that decision-makers reason about processing resources are affected by this problem [ 87 , 88 ]. A possible way out is to consider the utility-information trade-off simply an as if description, since perception-action systems that are subject to a utility-information trade-off do not necessarily have to reason or know about their deliberation costs. It is straightforward, for example, to design processes that probabilistically optimize a given utility with no explicit notion of free energy, but for an outside observer the resulting choice distribution looks like an optimal free energy trade-off [ 89 ].

In summary, the free energy trade-off between utility and information primarily serves as a normative model for optimal probability assignments in information-processing nodes or networks. Like other Bayesian approaches, it can also serve as a guide for constructing and interpreting systems, although it is in general not a mechanistic model of behavior. In that respect it shares the fate of its cousins in thermodynamics and coding theory [ 90 ] in that they provide theoretical bounds on optimality but devise no mechanism for processes to achieve these bounds.

5 Variational free energy in Active Inference

5.1 the basic idea.

Variational free energy is the main ingredient used in the Free Energy Principle for biological systems in the neuroscience literature [ 26 , 33 , 35 , 91 ], which has been considered as “arguably the most ambitious theory of the brain available today” [ 92 ]. Since variational free energy in itself is just a mathematical construct to measure the dissimilarity between distributions and functions—see Section 3—, the biological content of the Free Energy Principle must come from somewhere else. The basic biological phenomenon that the Free Energy Principle purports to explain is homeostasis , the ability to actively maintain certain relevant variables (e.g., blood sugar) within a preferred range. Usually, homeostasis is applied as an explanatory principle in physiology whereby the actual value of a variable is compared to a target value and corrections to deviation errors are made through a feedback loop. However, homeostasis has also been proposed as an explanatory principle for complex behavior in the cybernetic literature [ 93 – 96 ]—for example, maintaining blood sugar may entail complex feedback loops of learning to hunt, to trade and to buy food. Crucially, being able to exploit the environment in order to attain favorable sensory states, requires implicit or explicit knowledge of the environment that could either be pre-programmed (e.g., insect locomotion) or learnt (e.g., playing the piano).

In a nutshell, the central tenet of the Free Energy Principle states that organisms maintain homeostasis through minimization of variational free energy between a trial distribution q and a reference function ϕ by acting and perceiving. Sometimes the even stronger statement is made that minimizing variational free energy is mandatory for homeostatic systems [ 97 , 98 ].

5.2 A simple example

Ingredients. Applying the Active Inference recipe (cf. Fig 7 ) to our running example from Fig 1 with current and future states S , S ′, current and future observations X , X ′, and action A , we need a generative model p 0 , a desired distribution p des , and trial distributions q . The generative model p 0 ( X , S , A ) is specified by the factors in the decomposition (1) , the desired distribution p des ( X ′) is a given fixed probability distribution over future sensory states X ′, and the trial distributions q are probability distributions over all unknown variables, S , S ′, X ′, and A .

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Free energy minimization. Once the form of the trial distributions q —e.g., by a partial mean-field assumption (23) or a Bethe approximation (see S1 Appendix )—and the reference ϕ are defined, the variational free energy is simply determined by F ( q ‖ ϕ ). In the case of a mean-field assumption, the resulting free energy minimization problem is solved approximately by performing an alternating optimization scheme, in which the variational free energy is minimized separately with respect to each of the variable factors in a factorization of q , for example by alternating between min q ( S ) F , min q ( S ′| A ) F , and min q ( A ) F in the case of the partial mean-field assumption (23) , where in each step the factors that are not optimized are kept fixed (cf. Fig 7 ). In S1 Appendix we derive the update equations for the cases (24) and (25) under mean-field and Bethe approximations for the one-step example discussed in this section. Mean-field solutions for the general case of arbitrarily many timesteps together with their exact solutions can be found in S1 Notebook , where we also highlight the theoretical differences between various proposed formulations of Active Inference. The effect of some of these differences can be seen in the grid world simulations in S2 Notebook .

5.3 Critical points

The main idea behind Active Inference is to express the problem of action selection in a similar manner to the perceptual problem of Bayesian inference over hidden causes. In Bayesian inference, agents are equipped with likelihood models p 0 ( X | Z ) that determine the desirability of different hypotheses Z under known data X . In Active Inference, agents are equipped with a given desired distribution p des ( X ′) over future outcomes that ultimately determines the desirability of actions A . An important difference that arises is that perceptual inference has to condition on past observations X = x , whereas naive inference over actions would have to condition on desired future outcomes X ′ = x ′.

For a single desired future observation x ′, Bayesian inference could be applied in a straightforward way by simply conditioning the generative model p 0 on X ′ = x ′. Similarly, one could condition on a desired distribution p des ( X ′) using Jeffrey’s conditioning rule [ 104 ], resulting in p ( A | p des ) = ∑ x ′ p ( A | x ′) p des ( x ′), which could be implemented by first sampling a goal x ′ ∼ p des ( X ′) and then inferring p ( A | x ′) given the single desired observation x ′. However, one of the problems with such a naive approach is that the choice of a goal is solely determined by its desirability, whereas its realizability for the decision-maker is not taken into account. This is because by conditioning on p des , the decision-maker effectively seeks to choose actions in order to reproduce or match the desired distribution.

To overcome this problem, Control as Inference or Planning as Inference approaches in the machine learning literature [ 77 , 105 – 108 ] do not directly condition on desired future observations but on future success by introducing an auxiliary binary random variable R such that R = 1 encodes the occurence of desired outcomes. The auxiliary variable R comes with a probability distribution p 0 ( R | X ′, …) that determines how well the outcomes satisfy desirability criteria of the decision-maker, usually defined in terms of the reward or utility attached to certain outcomes—see the discussion in ( iii ) below. The extra variable gives the necessary flexibility to infer successful actions by simply conditioning on R = 1. The advantage of such an approach over direct Jeffrey conditionalization given a desired distribution over future observations can be seen in the grid world simulations in S2 Notebook , especially the ability of choosing a desired outcome that is not only desirable but also achievable—see also Fig 8 .

As can be seen from the displayed equations, conditioning on p des (Jeffrey conditionalization) and conditioning on success (Control as Inference/direct Active Inference) only differ in the order of normalizing and taking the expectation over X ′. While conditioning on p des requires to first sample a target outcome from p des before an action from p ( A | x ′) can be planned, conditioning on success directly weighs the desirability of an outcome p des ( x ′) by its realizability p ( x ′| A ). From this point of view, the expected utility approach is very similar to Control as Inference (which can also be seen in the grid world environment S2 Notebook ), since it also weighs the utility of an outcome with its realizability before soft-maximizing. It only differs in how it treats the desired distribution as an exponentiated utility, moving the utility values closer together so that option A = 1 is slightly preferred. The early version [ 34 ] of Active Inference is similar to Jeffrey conditioning, because decision-makers are also assumed to match the desired distribution, by defining the value function Q as a KL divergence between the predicted and desired distributions. In later versions of Q -value Active Inference [ 35 , 99 , 100 ], the value function Q is modified by an additional entropy term that explicitly punishes observations with high variability. Consequently, even when the effect of the action on future observations is kept the same, i.e., the predictive distribution p ( X ′| A ) = ∑ s ′ p 0 ( X ′| s ′) p 0 ( s ′| A ) remains as depicted in the left-hand column, the preference over actions now changes completely depending on p 0 ( X ′| S ′)—whereas in the other approaches, only the predictive distribution p ( X ′| A ) and p des ( X ′) influence planning. While there might be circumstances where this extra punishment of high outcome variability could be beneficial, it is questionable from a normative point of view why anything else other than the predicted outcome probability p ( X ′| A ) should be considered for planning. See S2 Appendix for details about the choices made in the example.

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Active Inference tries to overcome the same problem of reconciling realizability and desirability, but without explicitly introducing extra random variables and without explicitly conditioning on the future. Instead, the desired distribution is combined with the generative model to form a new reference function ϕ such that the posteriors q * resulting from the minimization of the free energy F ( q ‖ ϕ ) contain a baked-in tendency to reach the desired future encoded by ϕ . This approach is the root of a number of critical issues with current formulations of Active Inference:

• How to incorporate the desired distribution into the reference? Instead of using Bayesian conditioning directly in order to condition the generative model p 0 on the desired future, in Active Inference it is required that the reference ϕ contains the desired distribution in a way such that actions sampled from the resulting posterior model are more likely if they lead to the desired future. As can be seen already for the one-step case in (24) and (25) , the method of how to incorporate the desired distribution into the reference function is not unique and does not follow from first principles. There have been essentially two different proposals in the literature on Active Inference of how to combine the two distributions p des and p 0 into ϕ (cf. Fig 7 ): Either a hand-crafted value function Q is designed that specifically modifies the action probability of the generative model, or the probability over futures X ′ under the generative model p 0 is modified by directly multiplying p des to the likelihood p 0 ( X ′| S ′). We discuss both of these proposals in ( ii ) and ( iii ) below.

6 So what does free energy bring to the table?

6.1 a practical tool.

It is unquestionable that the concept of free energy has seen many fruitful practical applications outside of physics in the statistical and machine learning literature. As has been discussed in Section 3, these applications generally fall into one of two categories, the principle of maximum entropy, and a variational formulation of Bayesian inference. Here, the principle of maximum entropy is interpreted in a wider sense of optimizing a trade-off between uncertainty (entropy) and the expected value of some quantity of interest (energy), which in practice often appears in the form of regularized optimization problems (e.g., to prevent overfitting) or as a general inference method allowing to determine unbiased priors and posteriors (cf. Section 3.1). In the variational formulation of Bayes’ rule, free energy plays the role of an error measure that allows to do approximate inference by constraining the space of distributions over which free energy is optimized, but can also inform the design of efficient iterative inference algorithms that result from an alternating optimization scheme where in each step the full variational free energy is optimized only partially, such as the Bayesian EM algorithm, belief propagation, and other message passing algorithms (cf. Section 3.2).

6.2 Theories of intelligent agency

These practical use-cases of free energy formulations have also influenced models of intelligent behavior. In the cognitive and behavioral sciences, intelligent agency has been modelled in a number of different frameworks, including logic-based symbolic models, connectionist models, statistical decision-making models, and dynamical systems approaches. Even though statistical thinking in a broader sense can in principle be applied to any of the other frameworks as well, statistical models of cognition in a more narrow sense have often focused on Bayesian models, where agents are equipped with probabilistic models of their environment allowing them to infer unknown variables in order to select actions that lead to desirable consequences [ 14 , 76 , 111 ]. Naturally, the inference of unknown variables in such models can be achieved by a plethora of methods including the two types of free energy approaches of maximum entropy and variational Bayes. However, both free energy formulations go one step further in that they attempt to extend both principles from the case of inference to the case of action selection: utility optimization with information constraints based on free energy from constraints and Active Inference based on variational free energy.

While sharing similar mathematical concepts, both approaches differ in syntax and semantics. An apparent apple of discord is the concept of utility [ 112 ]. Utility optimization with information constraints requires the determination of a utility function, whereas Active Inference requires the determination of a reference function. In the economic literature, subjective utility functions that quantify the preferences of decision-makers are typically restrictive in order to ensure identifiability when certain consistency axioms are satisfied. In contrast, in Active Inference the reference function involves determining a desired distribution given by the preferred frequency of outcomes. However, these differences start to vanish when weakening the utility concept to something like log-probabilities, such that the utility framework becomes more similar to the concept of probability that is able to explain arbitrary behavior. Moreover, Active Inference has to solve the additional problem of marrying up the agent’s probabilistic model with its desired distribution into a single reference function (cf. Section 5.3). The solution to this problem is not unique, in particular it lies outside the scope of variational Bayesian inference, but it is critical for the resulting behavior because it determines the exact solutions that are approximated by free energy minimization. In fact, as can be seen in simple simulations such as S2 Notebook , the various proposals for this merging that can be found in the Active Inference literature behave very differently.

Also, both approaches differ fundamentally in their motivation. The motivation of utility optimization with information constraints is to capture the trade-off between precision and uncertainty that underlies information processing. This trade-off takes the form of a free energy once an informational cost function has been chosen (cf. Section 4.3). Note that Bayes’ rule can be seen as the minimum of a free energy from constraints with log-likelihoods as utilities, even though this equivalence is not the primary motivation of this trade-off. In contrast, Active Inference is motivated from casting the problem of action selection itself as an inference process [ 34 ], as this allows to express both action and perception as the result of minimizing the same function, the variational free energy. However, there is no mystery in having such a single optimization function, because the underlying probabilistic model already contains both action and perception variables in a single functional format and the variational free energy is just a function of that model. Moreover, while approximate inference can be formulated on the basis of variational free energy, inference in general does not rely on this concept, in particular inference over actions can easily be done without free energy [ 77 , 105 – 107 , 113 ].

However, there are also plenty of similarities between the two free energy approaches. For example, the assumption of a soft-max action distribution in Active Inference is similar to the posterior solutions resulting from utility optimization with information constraints. Moreover, the assumption of a desired future distribution relates to constrained computational resources, because the uncertainty constraint in a desired distribution over future states may not only be a consequence of environmental uncertainty, but could also originate from stochastic preferences of a satisficing decision-maker that accepts a wide range of outcomes. In fact, as we have seen in the discussion around Fig 8 , various methods for inference over actions differ in how they treat preferences given by a distribution over desired outcomes: Some of them try to match the predictive and desired distributions, while others simply seek to reach states whose outcomes have a high desired probability. In S2 Notebook , we provide a comparison of the discussed methods using grid world simulations, in order to see their resulting behavior also in a sequential decision-making task.

A remarkable resemblance among both approaches is the exclusive appearance of relative entropy to measure dissimilarity. In the Active Inference literature it is often claimed that every homeostatic system must minimize variational free energy [ 97 ], which is simply an extension of relative entropy for non-normalized reference functions (cf. Section 3.2.2). In utility-based approaches, the relative entropy (19) is typically used to measure the amount of information processing, even though theoretically other cost functions would be conceivable [ 74 ]. For a given homeostatic process, the KL divergence measures the dissimilarity between the current distribution and the limiting distribution and therefore is reduced while approximating the equilibrium. Similarly, in utility-based decision-making models, relative entropy measures the dissimilarity between the current posterior and the prior. In the Active Inference literature the stepwise minimization of variational free energy that goes along with KL minimization is often equated with the minimization of sensory surprise (see S3 Appendix for a more detailed explanation), an idea that stems from maximum likelihood algorithms, but that has been challenged as a general principle (see [ 114 ] and the response [ 115 ]). Similarly, one could in principle rewrite free energy from constraints in terms of informational surprise, which would however simply be a rewording of the probabilistic concepts in log-space. The same kind of rewording is well-known between probabilistic inference and the minimum description length principle [ 116 ] that also operates in log-space, and thus reformulates the inference problem as a surprise minimization problem without adding any new features or properties.

6.3 Biological relevance

So far we have seen how free energy is used as a technical instrument to solve inference problems and its corresponding appearance in different models of intelligent agency. Crucially, these kinds of models can be applied to any input-output system, be it a human that reacts to sensory stimuli, a cell that tries to maintain homeostasis, or a particle trapped by a physical potential. Given the existing literature that has widely applied the concept of free energy to biological systems, we may ask whether there are any specific biological implications of these models.

Considering free energy from constraints, the trade-off between utility and information processing costs provides a normative model of decision-making under resource constraints, that extends previous optimality models based on expected utility maximization and Bayesian inference. Analogous to rate-distortion curves in information theory, optimal solutions to decision-making problems are obtained that separate achievable from non-achievable regions in the information-utility plane (cf. Fig 5 ). The behavior of real decision-making systems under varying information constraints can be analyzed experimentally by comparing their performance with respect to the corresponding optimality curve. One can experimentally relate abstract information processing costs measured in bits to task-dependent resource costs like reaction or planning times [ 20 , 22 ]. Moreover, the free energy trade-off can also be used to describe networks of agents, where each agent is limited in its ability, but the system as a whole has a higher information processing capacity—for example, neurons in a brain or humans in a group. In such systems different levels of abstraction arise depending on the different positions of decision-makers in the network [ 23 , 71 , 85 ]. As we have discussed in Section 4.3, just like coding and rate-distortion theory, utility theory with information costs can only provide optimality bounds but does not specify any particular mechanism of how to achieve optimality. However, by including more and more constraints one can make a model more and more mechanistic and thereby gradually move from a normative to a more descriptive model, such as models that consider the communication channel capacity of neurons with a finite energy budget [ 24 ].

Considering variational free energy, there is a vast literature on biological applications mostly focusing on neural processing (e.g., predictive coding and dopamine) [ 102 , 117 , 118 ], but there are also a number of applications aiming to explain behavior (e.g., human decision-making and hallucinations) [ 119 ]. Similarly to utility-based models, Active Inference models can be studied in terms of as if models, so that actual behavior can be compared to predicted behavior as long as suitable prior and likelihood models can be identified from the experiment. When applied to brain dynamics, the as if models are sometimes also given a mechanistic interpretation by relating iterative update equations that appear when minimizing variational free energy with dynamics in neuronal circuits. As discussed in Section 3.2.3, the update equations resulting for example from mean-field or Bethe approximations, can often be written in message passing form in the sense that the update for a given variable only has contributions that requires the current approximate posterior of neighbouring nodes in the probabilistic model. These contributions are interpreted as local messages passed between the nodes and might be related to brain signals [ 102 ]. Other interpretations [ 28 , 91 , 100 ] obtain similar update equations by minimizing variational free energy directly through gradient descent, which can again be related to neural coding schemes like predictive coding. As these coding schemes have existed irrespective of free energy [ 120 , 121 ], especially since minimization of prediction errors is already seen in maximum likelihood estimation [ 120 ], the question remains whether there are any specific predictions of the Active Inference framework that cannot be explained with previous models (see [ 39 , 122 ] for recent discussions of this question).

6.4 Conclusion

Any theory about intelligent behavior has to answer three questions: Where am I? , where do I want to go? , and how do I get there? , corresponding to the three problems of inference and perception, goals and preferences, and planning and execution. All three problems can be addressed either in the language of probabilities or utilities. Perceptual inference can either be considered as finding parameters that maximize probabilities or likelihood utilities. Goals and preferences can either be expressed by utilities over outcomes or by desired distributions. The third question can be answered by the two free energy approaches that either determine future utilities based on model predictions, or infer actions that lead to outcomes predicted to have high desired probability or match the desired distribution. In standard decision-making models actions are usually determined by a utility function that ranks different options, whereas perceptual inference is determined by a likelihood model that quantifies how probable certain observations are. In contrast, both free energy approaches have in common that they treat all types of information processing, from action planning to perception, as the same formal process of minimizing some form of free energy. But the crucial difference is not whether they use utilities or probabilities, but how predictions and goals are interwoven into action.

This article started out by tracing back the seemingly mysterious connection between Helmholtz free energy from thermodynamics and Helmholtz’ view of model-based information processing that led to the analysis-by-synthesis approach of perception, as exemplified in predictive coding schemes, and in particular to discuss the role of free energy in current models of intelligent behavior. The mystery starts to dissolve when we consider the two kinds of free energies discussed in this article, one based on the maximum entropy principle and the other based on variational free energy—a dissimilarity measure between distributions and (generally unnormalized) functions that extends the well-known KL divergence from information theory. The Helmholtz free energy is a particular example of an energy information trade-off that results from the maximum entropy principle [ 46 ]. Analysis-by-synthesis is a particular application of inference to perception, where determining model parameters and hidden states can either be seen as a result of maximum entropy under observational constraints or of fitting parameter distributions to the model through variational free energy minimization. Thus, both notions of free energy can be formally related as entropy-regularized maximization of log-probabilities.

Conceptually, however, utility-based models with information constraints serve primarily as ultimate explanations of behavior, this means they do not focus on mechanism, but on the goals of behavior and their realizability under ideal circumstances. They have the appeal of being relatively straightforward generalization of standard utility theory, but they rely on abstract concepts like utility and relative entropy that may not be so straightforwardly related to experimental settings. While these normative models have no immediate mechanistic interpretation, their relevance for mechanistic models may be analogous to the relevance of optimality bounds in Shannon’s information theory for practical codes [ 90 ]. In contrast, Active Inference models of behavior often mix ultimate and proximate arguments of explaining behavior [ 123 , 124 ], because they combine the normative aspect of optimizing variational free energy with the mechanistic interpretation of the particular form of approximate solutions to this optimization. While mean-field approaches of Active Inference may be particularly amenable to such mechanistic interpretations, they are often too simple to capture complex behavior. In contrast, the solutions of direct Active Inference resulting from a Bethe assumption are equivalent to previous Control as Inference approaches [ 77 , 105 – 110 ] that allow for Bayesian message passing formulations whose biological implementability can be debated irrespective of the existence of a free energy functional.

Finally, both kinds of free energy formulations of intelligent agency are so general and flexible in their ingredients that it might be more appropriate to consider them languages or tools to phrase and describe behavior rather than theories that explain behavior, in a sense similar to how statistics and probability theory are not biological or physical theories but simply provide a language in which we can phrase our biological and physical assumptions.

Supporting information

S1 appendix. derivation of exemplary update equations..

We derive update equations of Q -value and direct Active Inference for the example in Section 5.2 under mean-field and Bethe approximations.

https://doi.org/10.1371/journal.pcbi.1008420.s001

S2 Appendix. Uncertain and deterministic options.

We give additional details on the example shown in Fig 8 that illustrates the effects of assuming a particular desired distribution over three outcomes under Jeffrey conditionalization, Control as Inference, expected utility optimization, and Active Inference.

https://doi.org/10.1371/journal.pcbi.1008420.s002

S3 Appendix. Surprise minimization.

Explanation of the relation between free energy minimization, free energy as a bound on surprise, and surprise minimization.

https://doi.org/10.1371/journal.pcbi.1008420.s003

S4 Appendix. Separation of model and state variables.

Discussion of how model and state variables can be separated in variational Bayesian inference which motivates the optimization scheme chosen by Active Inference.

https://doi.org/10.1371/journal.pcbi.1008420.s004

S1 Notebook. Comparison of different formulations of Active Inference.

A detailed comparison of the different formulations of Active Inference found in the literature (2013-2019), including their mean-field and exact solutions in the general case of arbitrary many time steps.

https://doi.org/10.1371/journal.pcbi.1008420.s005

S2 Notebook. Grid world simulations.

We provide implementations of the models discussed in this article in a grid world environment, both as a rendered html file as well as a jupyter notebook that is available on github .

https://doi.org/10.1371/journal.pcbi.1008420.s006

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• v.46(3); 2022 Jun

Friston's free energy principle: new life for psychoanalysis?

Jeremy holmes.

Department of Psychology, University of Exeter, UK

The free energy principle (FEP) is a new paradigm that has gain widespread interest in the neuroscience community. Although its principal architect, Karl Friston, is a psychiatrist, it has thus far had little impact within psychiatry. This article introduces readers to the FEP, points out its consilience with Freud's neuroscientific ideas and with psychodynamic practice, and suggests ways in which the FEP can help explain the mechanisms of action of the psychotherapies.

Today's psychiatrists are pragmatists, on the look-out for what ‘works’ and sceptical about the grand theories that held sway in the previous century. But ideology cannot be wholly avoided, nor theoretical controversy evaded. Current psychiatry's pantheon incudes evidence-based practice, DSM diagnosis and neuroscience. The search for evidence is theory driven. Diagnostic profusion raises questions about the medicalisation of human suffering. Despite extraordinary recent advances in neuroscience, their impact on everyday psychiatric practice has been modest.

The purpose of this article is twofold: first, to introduce readers to an overarching model of brain function associated with the mathematical psychiatrist Karl Friston, the free energy principle (FEP), which has been influential in neuroscience generally, but thus far has caused relatively little stir within psychiatry or clinical psychology. My hope is to redress that. Second, I make the case that FEP can revitalise the psychoanalytic psychotherapies, marginalised by the inexorable rise of cognitive–behavioural therapy (CBT) as the dominant psychological therapy paradigm.

It should be noted that FEP is deliberately described by Friston as a ‘principle’, akin to the principles of natural selection or gravity. The evidence for its validity is circumstantial rather than direct, and its detailed neuronal mechanisms and clinical implications remain to be fully explored.

Friston's forebears

Friston's project builds on the work of a number of pioneering predecessors and their concepts. These include Erwin Schrödinger, Heinrich Helmholtz, the Claudes – Claude Bernard and Claude Shannon – and Thomas Bayes. We live in an entropic universe. Broken cups don't spontaneously reassemble. Coffee cools once poured. Stars burn out. The exception is life itself. Quantum physicist Schrödinger 1 coined the term ‘negentropy’ to describe how living matter, Canute-like for its lifetime, reverses this cosmic tide towards disorder and homogeneity.

The key to negentropy is homeostasis. As Bernard famously put it, the condition of a free life is the stability of the interior milieu – whether one is a unicellular amoeba or, like Schrödinger, a Nobel-prize winning primate. Homeostasis, and the more general processes of allostasis, 2 resist the forces of entropy, physiologically and behaviourally. Inherent in homeostasis are boundaries: cell membranes, the skin, the brain within its skull. Janus-like, homeostasis faces outwards towards the environment and inwards towards the milieu interieur . Temperature sensors in the skin tell us it's a hot day; the sympathetic nervous system activates sweat glands, the brain tells us to fling off jumpers, move into the shade, etc., all in the service of resisting being entropically fried. Note that homeostats vary in ‘precision’ – some are highly sensitive, whereas others tolerate a great range of variation.

Friston had the insight and mathematical sophistication to see that the negentropic homeostatic principle applies not just to the organism as a whole but to the brain itself. 3 , 4 The brain's job is to counteract entropy and to maintain internal stability on behalf of the organism whose processes and behaviour it controls and directs; this applies, reflexively, to itself.

The FEP goes back to the ideas of 19th-century polymath Hermann von Helmholtz, updated by artificial intelligence (AI) neuroscientists Geoffrey Hinton and Peter Dayan. 5 Naively, we tend to think of vision as a camera-like image passively projected onto the visual cortex, or the auditory system as microphone-like, responding indiscriminatingly to the prevailing phonic universe. In the Helmholtz model the brain makes its own world . Our sense organs, external and internal, are constantly bombarded by a vast range of stimuli from an ever-changing environment. To operate with maximum efficiency, the brain selects out the ‘meaning’ of its sensations, attending only to those that are relevant to its ‘affordances’ 6 – its specific ecological niche – and especially to input that is anomalous or novel.

Working in the 1950s at the Bell telephone company laboratory, Claude Shannon saw that this ‘meaning’ could be quantified – as ‘bits’ of information. Gregory Bateson, anthropologist and family therapy guru, called these ‘differences that make a difference’. White noise is chaotic, entropic and devoid of information. Language, whether spoken, sung or gestured, is structured, ordered, negentropic. The measure of informational energy is ‘surprise’, i.e. how unexpected a signal is. In the board game Scrabble, the letter ‘x’ conveys more information than ‘e’ because it is relatively unusual, applying to a smaller range of words, and so in calculating the score, is ‘worth’ more. The brain's aim is constantly to reduce informational entropy and maximise meaning.

A crucial building block for the FEP is the concept of the Bayesian brain. The Reverend Thomas Bayes, a late 18th-century clergyman and founder of probability theory, grasped, Doris Day-like, that the future's not ours to see. Yet, to survive and adapt we need to know, moment to moment, ‘what is going on’ – in ourselves, in the interpersonal world and in the physical world. On the basis of prior experience, the Bayesian brain 7 continuously estimates the likelihood of future events. Probabilities are computed by comparing current states of affairs with past occurrences, estimating the extent of correspondence between them, factoring in the likelihood of errors in both memory and perception, and ending with a portion that represents that which cannot be predicted. This is ‘prediction error’, which must, in the service of negentropy, be minimised as far as is possible – prediction error minimisation or PEM.

The brain, ‘top-down’, uses Bayesian probabilities to clarify ‘bottom-up’ input, extero- and interocaptive: 8 ‘My stomach is complaining, but it's not surprising – I overdid it on the pudding, so it's probably not cancer’; ‘I know that tune, I've heard it so many times – yes of course, it's the Beatles’ Yellow Submarine’; ‘Is that a stick or a snake? Come on, no adders in city centres, probably safe to pick it up’.

Free energy

Now to the free energy principle itself. ‘Energy’ equates to information, albeit physically embodied in patterns of neuronal impulses, synaptic transmission (‘fire together, wire together’ 9 ) and the neurohormonal environment. Prior models of the world, top-down, ‘bind’ incoming bottom-up information. Energy un bound, or prediction error, reflects novelty in need of binding – and so forestall the dangers of entropic chaos.

Circumstantial evidence for the FEP is the fact that more neuronal fibres reach the eye downwards from the brain than travel upward towards the visual cortex. Whenever possible, the brain ‘tells’ the eye what it is likely to be seeing. The FEP postulates a hierarchical series of neuronal interactions, starting from the least to the most complex, from the periphery to the central nervous system, from specificity to abstraction, most of which operate below conscious awareness. At the level of the eye itself the retinal receptors are activated: ‘round, two dots and a straight line between’. Top-down, even in a 1-month baby, this will elicit an answering smile (‘face equals security’). Once language arrives, verbal concepts shape perceptions: ‘Oh of course, that's a face’. At the highest level is mentalising – thinking about thinking: ‘I wonder why bearded faces always make me feel slightly unsettled? Perhaps it's reminiscent of my scary grandfather’.

The FEP visualises a series of ‘conversations’ in which top-down ‘priors’ ‘bind’ bottom-up input into probabilistically recognisable meanings. Each level can be thought of as a meaning–action boundary. Ascending the hierarchy, the Bayesian process ensures that the most mathematically probable pattern prevails across these statistical boundaries or ‘Markov blankets’. 10 Prediction error is minimised by ‘binding’ bottom-up energy (informational as well as physiological) by top-down generative models based on pre-existing patterns and concepts. Thus is order preserved, entropy eschewed. We know what we like and, mostly, see what we want and expect to see.

But there will always be a discrepancy between our pre-existing models of the world and incoming sensations, an excess of energy that cannot be bound and will have to be passed onto the next level up of the hierarchy. Lockdown excepted, we don't live huddled in ‘dark rooms’. 11 The environment is constantly in flux; we need to explore as much as conserve – to find new sources of food, suitable mates, interest and excitement. Surprise, calibrated by the brain as the discrepancy between expectation and incoming sensation, is a proxy for free energy – and hence entropy. Surprise is both vital to survival but also potentially entropic, disruptive or even life-threatening. This represents the prediction error aforementioned. The brain minimises such surprise/error by whatever means possible.

At this point the role of affect becomes important. Free energy is aversive and can be thought of as representing mental pain. Conversely, ‘binding’ free energy is rewarding and therefore motivating. The role of affect, positive and negative, is to drive the free energy minimising processes. This is another ‘AI’ – active inference.

The idea of active inference captures a number of psychological processes central to psychological health. First, action or agency. Given that incoming stimuli are inherently subject to error and imprecision, the brain increases precision by movement – approaching an ambiguous stimulus source, turning the head to use foveal rather than peripheral vision, switching lights on in order to see better, etc. Second, top-down model revision. Now we know what that vague shape really ‘is’ – a cat, clothes strewn on the floor, etc.: ‘Let's listen more carefully. Oh, that's not the Beatles at all, it's the Beach Boys’. Third, and vitally in the case of social species such as our own, active inference is enhanced by recruiting help or ‘twogetherness’: ‘Did you hear something, or was I just imagining it?’; ‘You know about ’70s music – what was that group's name?’. Friston & Frith call this ‘duets for one’ and have worked out the mathematics of such collaborative Markov blankets. 12 Fourth, if all else fails, by choosing or fashioning environments that conform to the brain's pre-existing models of the word: ‘I can't stand modern music. Let's go over to Classic FM’. This last aspect is captured by the psychoanalytic concept of ‘projective identification’, in which we shape our interpersonal world, often deleteriously, to conform with expectations: ‘You psychiatrists are all the same – never there when I need you’.

Free energy and psychopathology

The FEP has clear implications for those who work in mental ill health, and especially who favour psychological methods of treatment. Consider depression, typically triggered by loss, trauma or multiple setbacks. Adversity is widespread – poverty, inequality, racism – but not all succumb. To understand resilience, we need an illness model that encompasses not just events, but individuals’ responses to them. Attachment research shows that those who are securely attached are able to repair the inevitable ruptures to which all are prone, often through the typical sequence of protest, rage, grief and mourning. 13 As children, securely attached people have had caregivers they could depend on to acknowledge their pain, tolerate protest and help them to move on. Repeated episodes of everyday rupture–repair cycles help build this resilience.

The free energy released by the rupture is bound by the child's knowledge that help is at hand and that their epistemically trusted caregiver will provide a generative model to counteract the free energy associated with ruptures: ‘Don't worry love, I'm just going to the loo, I'll be back in a minute’. In the ‘still face’ paradigm, parents are asked to freeze their facial expression for 1 minute while talking or playing with their child. 14 Securely attached children continue actively to try to re-engage with their caregivers in the confident expectation that they will be ‘back soon’. For insecurely attached children, by contrast, rather than rupture–repair, cycles of rupture–despair or rupture–disappear are the norm. Their caregivers have either themselves been overwhelmed by their child's unhappiness and so despairingly abandon attempts to alleviate it; or repress the impact of the child's mental pain and so ‘disappear’ emotionally. Both leave the child alone to find ways to bind the free energy the rupture evokes. When their caregiver's face freezes they look away, become miserable and regressed, and often resort to self-soothing rituals such as rocking or emotional dissociation.

Such insecurely attached children are primed in later life for depression in response to loss or trauma or, in extreme cases, to developing post-traumatic stress disorder. The ingredients of free energy minimisation needed to maintain psychological equilibrium are for them problematic. Active inference is compromised. They tend to be passive rather than active. They stick with limited and simplistic and inflexible ‘top-down’ models such as ‘It's no use trying to make things better, it never works’ or ‘Feelings are dangerous, best to keep them buried’. They find it hard to trust people and so can't ‘borrow’ an intimate other's brain with which to process feelings and build up alternative ways of viewing the world.

Psychotherapeutic implications

The most commonly used therapy for depression, CBT, attempts to address these deficiencies. Therapists encourage patients actively to test their negative ‘hypotheses’ by looking more closely at their experiences and by exploring alternative top-down models to account for them (‘Maybe my boyfriend didn't answer his phone because he'd run out of battery, not because he doesn't love me’). But CBT has its limitations. ‘Treatment-resistant depression’ is common. 15 People with personality disorders do badly with standard CBT, often refusing to engage or dropping out. 16 The FEP provides explanations for this. From an FEP perspective, one way to minimise free energy is to gravitate towards or engender environments that confirm one's view of the world, however negative. Depression relegates sufferers to emotionally impoverished relationships, stereotyped and simplistic top-down models, and thus becomes a self-fulfilling hypothesis, resistant to psychotherapeutic interventions. In addition, these negative top-down priors are ‘inferentially inert’, i.e. inaccessible for modification.

A degree of chaos/uncertainty/free energy needs to be tolerated before new generative models can evolve. Homeostatic imprecision needs to be tolerated for a while. The holding and ‘negative capability’ of the therapist's ‘borrowed brain’ paves the way for a more complex, nuanced top-down reset. Given that people with personality disorders notoriously find it difficult to trust others, the brevity and defocus on the therapeutic relationship in standard CBT limits the scope for such fundamental change.

Moving from depression to an FEP perspective on trauma, the latter creates an overwhelming influx of free energy for which there are no available top-down models with which to bind it. Thoughts of cruelty, neglect and abuse remain in the realm of the unthinkable and are therefore ‘defended against’ by repression or dissociation. 17 However, when jointly considered – under a shared Markov blanket – these bottom-up unprocessed experiences can be bound with the therapist's encouragement and expertise into manageable narratives. However painful, they become less overwhelming, a source of new ways of thinking and psychic reorganisation. As the patient begins to feel that the therapist is safe, reliable, compassionate and empathic, so everyday ruptures – session-endings, holiday breaks and misunderstandings – are repeatedly repaired via model revision (‘Maybe the weekend break does not inevitably mean I'm forgotten’), and the trust this engenders can be generalised into the patient's everyday life.

We can see here how contemporary psychoanalytic psychotherapy and revitalised Freudian ideas resonate with the FEP. Freud started off his working life as a neurologist. Like Friston, he conceptualised the brain's aim as reducing psychic energy, typically through action and ‘word representations’ – i.e. transmuting free energy into thinkable thoughts. He saw unbound energy (which he later transmuted into ‘libido’) as potentially disruptive and responsible for the symptoms of psychological illness. Psychoanalysis was designed first to evoke and then to quieten this trauma-related unbound energy. To achieve this, three key psychoanalytic procedures are free association, dream analysis and analysis of transference.

The ‘virtual’ nature of the psychoanalytic relationship brings both top-down and bottom-up components of the FEP process into focus, enabling them to be mentalised rather than enacted. Free association taps into the mind's normally unvoiced upward-welling stream of consciousness, counteracting the elusiveness of affect seen in the rupture–despair/disappear attachment pattern. This enables the range of top-down responses to be enhanced and aversive free energy minimised. At the top-down level, in a process comparable to the immune system's lexicon of antigen-activated antibodies, dreaming is the means by which the mind generates a repertoire of narratives with which to bind the free energy which life's vicissitudes engender. Transference analysis turns the spotlight on the limited varieties of top-down narratives that sufferers use in their dealings with intimate others to minimise free energy. The enigmatic ambiguity of therapists’ persona enables patients to experience, reconsider and extend the top-down assumptions with which they approach the world of intimate others.

Psychoanalysis has tended to self-isolation, sequestrated from cross-fertilisation by other disciplines. The Friston–Freud consilience opens up new possibilities. Psychoanalytic and attachment-derived mentalisation-based therapy (MBT) is now established as a highly effective therapy for borderline personality disorder, previously considered untreatable. 18 MBT leads to big reductions in medication use, suicide attempts, hospital admission and unemployment among people with borderline personality disorder, as compared with treatment as usual.

MBT is both practically and conceptually consistent with the FEM. It encourages patients (a) to identify the bottom-up feelings that fuel their self-injurious actions, (b) to pause and think of different ways of handling these, i.e. to tolerate a quantum of free energy with the help of the therapists’ ‘borrowed brain’ and (c) through mutual mentalising (therapist and patient together forming a neurobiological ‘bubble’) to generate more complex and adaptive models of the self and significant others. The result is manageable surprise: confounding sufferers’ negative assumptions about the world, becoming less overwhelmed by unbound affect (fewer ‘melt-downs’) and facilitating greater resilience.

Conclusions

If rehabilitation of the psychoanalytic method in the light of the FEP comes as a pleasant surprise, this is consistent with its principles. As in Mark Twain's trope, rumours of psychoanalysis's death have been greatly exaggerated. In place of despair or disappearance, the FEP suggests that repair is possible. FEP-grounded psychoanalytic approaches such as MBT are now known to help those with profound mental distress. They also suggest a scientifically sound account of the interpersonal and neuronal mechanisms by which psychological change comes about.

About the author

Jeremy Holmes is a retired psychiatrist and psychoanalytic psychotherapist. He is a Visiting Professor at Exeter University, UK, and author of many articles and books in the field of attachment theory and psychoanalysis, including The Brain Has a Mind of Its Own: Attachment, Neurobiology, and the New Science of Psychotherapy , in which the ideas of this article are explored in greater detail.

This research received no specific grant from any funding agency, commercial or not-for-profit sectors.

CONCEPTUAL ANALYSIS article

Resurrecting gaia: harnessing the free energy principle to preserve life as we know it.

• 1 Berlin School of Mind and Brain, Humboldt-Universität zu Berlin, Berlin, Germany
• 2 Department of Philosophy, Macquarie University, Sydney, NSW, Australia

This paper applies the Free Energy Principle (FEP) to propose that the lack of action in response to the global ecological crisis should be considered a maladaptive symptom of human activity that we refer to as biophilia deficiency syndrome . The paper is organised into four parts: the characterisation of the natural world under the Gaia Hypothesis, the employment of the FEP as a description of the behavior of self-organising systems, the application of the FEP to Gaia to understand coupling dynamics between living systems and purportedly non-living planetary processes, and the offering of positive interventions for addressing the current state of ecological crisis under this framework. For the latter, we emphasize the importance of perturbing stuck states for healthy development, and the necessary appreciation of life existing as nested systems at multiple levels in a hierarchy. We propose the development of human biophilia virtue in accordance with the FEP as a practical intervention for treating biophilia deficiency syndrome and helping to safeguard the balance of planetary processes and the integrity of living systems that depend on them, offering some examples of what this might look like in practice. Overall, this paper provides novel insights into how to catalyse meaningful ecological change, proposing a deliberate and disruptive approach to addressing the dysfunctional relationship between humans and the rest of the natural world.

1. Introduction

The relation of humanity to the rest of the natural world is essentially one of dependence. Our continued survival and flourishing are utterly contingent upon the state and functioning of our environment, which supplies us with eatable food, drinkable water, and breathable air. In the most practical sense possible, we are existentially bound to the rest of nature. Moreover, extensive empirical evidence demonstrates that connection to natural environments is a key ingredient of mental health. Specifically, spending time in natural settings has been shown to reduce stress, anxiety and depression, and enhance mood, cognitive function, mental health and wellbeing ( White et al., 2019 ; Bratman et al., 2021 ). The psychological construct “connection to the world” correlates with therapeutic outcomes in treatment for depression ( Watts et al., 2022 ), while Berto (2014) found that even exposure to images of natural scenes can improve mood and cognitive function. These results suggest that nature can be a powerful tool for promoting mental health and wellbeing.

However, to frame the value of nature purely in such utilitarian terms would be to gravely miss the point. Recent research in cognitive science suggests that humans possess an innate cognitive disposition towards nature, known as “biophilia.” This concept, coined by biologist Wilson (1984) , purportedly has deep roots in our evolutionary history as the natural psychological disposition of humans to seek out and connect with other living organisms and natural environments ( Clowney, 2013 ; Olivos-Jara et al., 2020 ; Barbiero and Berto, 2021 ). This evolutionary adaptation is believed to be the biological basis for human values of nature and environmental virtue ( Clowney, 2013 ): the sense of connection and identity with, and fondness for (the continued existence of) all forms of life on Earth entails a sense of care towards the rest of the natural world and a willingness to protect it ( Rowlandson, 2015 ). The concept of biophilia is therefore not merely a psychological theory, but rather a fundamental aspect of human nature that has played a crucial role in our survival and adaptation throughout our evolutionary history. The flip-side to this bond is the increasing prevalence of eco-anxiety among young adults and children upon learning about climate change, and observing human inaction towards it ( Pihkala, 2020 ; Baudon and Jachens, 2021 ; Gunasiri et al., 2022 ). The question remains as to whether there can be an effective therapeutic strategy that addresses eco-anxiety without addressing the fundamental ecological crisis, or whether they are two crises which can only be tackled together.

We therefore arrive at a philosophical dilemma. On the one hand, human beings are imbued with biophilia, the “innate emotional affiliation … to other living organisms” ( Wilson, 1993 ; p. 31). This biopsychological disposition suggests that people would be inclined to strive to protect and preserve the environment, including mitigating and ideally overcoming the threats of ecological breakdown. On the other hand, the lack of action to address the ecological crisis appears contradictory to this biophilic inclination. As our population and global footprint continue to expand, humanity's catastrophic impact on the Earth has become increasingly apparent: it is by now beyond doubt that activities such as deforestation, pollution of the oceans, and greenhouse gas emissions are harming our planet.

This paper seeks to address the contradiction between the biophilic inclination towards environmental protection and the lack of action taken to address the current ecological crisis. The paper proceeds in four parts. First, the natural world is characterised by the Gaia Hypothesis (GH). Second, the Free Energy Principle (FEP) is described, and then, third, employed to further develop and augment the GH. Fourth, it is argued that human failure to act on the ecological crisis does not invalidate the FEP. Rather, it is shown that this failure is indicative of an imbalance between active and sensory states, which reflects an unhealthy condition in the human species. If not addressed, this condition will lead to the dissipation of adaptation, the painful decline of our species, and ultimately the end of (human) “life as we know it” ( Friston, 2013 ). The paper concludes by proposing that organised action motivated by biophilia in line with the FEP is necessary to treat this unhealthy condition.

2. The Gaia Hypothesis

“ We have to use the crude tool of metaphor to translate conscious ideas into unconscious understanding.” ( Lovelock, 2007 , p. 178).

The Gaia Hypothesis—named after the goddess who personified Earth in Ancient Greek mythology ( Lovelock, 1972 )—holds as its central tenet that the conditions required for life on our planet are maintained by and for the biosphere ( Lovelock and Margulis, 1974 ). The “biosphere” (synonymous with “Gaia” hereafter) refers to the thin spherical layer of the planet at which life exists, starting where rock meets magma roughly 100 miles below the surface, and extending another 100 miles towards the thermosphere where air meets space ( Lovelock, 2007 ; p. 19; see Figure 1 ). Consisting of rock, soil, water, air, and all of the ~5.5 x 10 14 kg of organic matter in the known universe ( Bar-On et al., 2018 ), Gaia is a geophysiological entity, since the totality of life and its non-living environment are bound as a single interdependent system ( Lovelock, 1989 ). The “abiotic” parts are considered part of the total living organism, like the shell of a snail ( Lovelock, 1972 ).

Figure 1 . Selected layers of Planet Earth. The biosphere, highlighted here in green, encompasses the Earth's crust, the lands and oceans on its surface, and extends out towards the edge of the atmosphere.

The GH is motivated by the fact that during the 3.8 billion years that life has persisted on Earth, conditions have remained remarkably constant and favourable across a huge range of parameters. Most obvious of these is temperature, which has remained within the tight parameters required for life to survive and thrive, despite an increase of the sun's output by at least 30% since life began ( Lovelock and Margulis, 1974 ). The same applies to the pH and salinity of the ocean, despite the incessant depositing of salts from the land into the ocean by rivers ( Lovelock, 2007 ; p. 34), and to the chemical composition of the soil and atmosphere ( Lovelock and Watson, 1982 ). For a quarter of the time the universe has existed, these and countless other variables have been vastly different from what they “should” be, lying far from equilibrium at the levels that just happen to support life, even withstanding countless catastrophic volcanic eruptions, solar flares and asteroid strikes ( Lovelock, 1979 ; p. 33). “For this to have happened by chance,” originator James Lovelock says, “is as unlikely as to survive unscathed a drive blindfolded through rush-hour traffic.” ( Lovelock, 1989 , p. 10).

The Gaia Hypothesis instead takes this constancy to be a natural product of evolution. However, in a modification of canonical Darwinism, rather than the evolution of individual organisms to an inert environment, “what evolves is the whole Earth system with its living and non-living parts existing as a tight coupled entity.” ( Lovelock, 2007 , p. 178). This can be understood through the “hologenome” concept ( Rosenberg and Zilber-Rosenberg, 2011 ): according to this framework, a holobiont—a living host plus all of the smaller organisms living in dependence to it—is the unit of selection, such that changes in environmental parameters can lead to long-term adaptive changes to the holobiont via iterative changes to the constituent living systems. This is corroborated by Pricean formalisations, which demonstrate that natural selection can be extended to accommodate evolution (of Gaia) without reproduction ( Bourrat, 2023 ). Thus, in a neo-Lamarckian turn within a Darwinian framework, the combined genetic wealth of information of the whole system (the hologenome) acquires characteristics that are conducive to its continued existence and in turn affect changes on the environment.

The biosphere thereby constitutes a complex system which “appears to have the … goal of regulating the climate and the chemistry at a comfortable state of life.” (p. 19). This fact was recognised by the scientific community in the 2001 Amsterdam Declaration, where it was acknowledged that the Earth is indeed a self-regulating system ( Moore et al., 2001 ): just as an animal employs homeostatic feedback cycles to maintain viable conditions for its survival, the biosphere is kept within the limits needed for its own continued existence by thermodynamic imperatives ( Karnani and Annila, 2009 ). This self-regulation tends not towards set points but adapts in a flexible manner to support the particular current life forms given the current environmental conditions ( Lovelock, 2007 , p. 19). 1

Some literature on simulations can elucidate this further. “Daisyworld” was originally a caricature of Gaia wherein the only life forms—black and white daisies—absorb and reflect light, respectively ( Watson and Lovelock, 1983 ). Temperature-driven competition between the two creates a balance in populations and therefore albedo effects, such that a planetary temperature favouring daisy growth is maintained, even as the output of the nearby “sun” varies in the simulation ( Watson and Lovelock, 1983 ). Considered as a complex adaptive system, Daisyworld thereby exhibits “emergent self–regulation as a consequence of feedback coupling between life and its environment” ( Lenton and van Oijen, 2002 ).

More recent models have corroborated and extended these findings. Using standard methodology from quantitative genetics, it has been shown that the self-regulating dynamics are explainable purely in terms of the low-level evolutionary dynamics of competition between the daisies: no higher level principle (such as teleology) need be invoked ( Wood and Coe, 2007 ; Makarieva, 2022 ; Bourrat, 2023 ), at least at certain timescales ( Weaver and Dyke, 2012 ). The implications have also been corroborated by models of different properties of organic systems: for instance, the metabolically abstract microorganism system (METAMIC) model simulates nutrient recycling (rather than temperature regulation) given appropriate thermodynamic constraints, and largely recapitulates the relevant behaviour of Daisyworld ( Downing, 2002 , 2003 ). Further, models based on based on Chemical Organization Theory (COT) and the Zero Deficiency Theorem (ZDT) demonstrate the autopoietic properties of the biosphere, and suggest how these might relate to other intrinsic features of living systems, such as autonomy and anticipation ( Rubin et al., 2021 ). Finally, Daisyworld has been used in spatial systems dynamics simulations which demonstrate the importance of spatio-temporal interactions in such systems, thereby bridging the gap between purely abstract models and the three-dimensional coupling involving the biosphere ( Neuwirth et al., 2015 ).

At a broader level, the perspective that the GH brings is appealing for many reasons. Conceptualised as a single complex system, Gaia changes in a way that is adaptive and non-linear ( Lovelock, 1979 ); it displays dynamical, emergent , and sudden tipping behaviour: when bifurcation points are reached, new attractor states form, and higher order is established ( Lovelock, 1979 , 1989 ; Gleick, 1987 ; Capra, 1997 ; Lenton et al., 2020b ); 2 importantly for our purposes, it is self-organising ( Lenton and van Oijen, 2002 ), and strong emphasis is given to the extreme interdependence between the processes of the system ( Capra, 1997 ). Due to this combination of properties, the GH has both informed and been informed by a wide range of fields including ecology, climate science, cybernetics, complex systems theory, chaos mathematics, and the philosophy of mind.

Furthermore, there is an intuitive—and, indeed, poetic—attraction to Gaian thinking, as it allows us to transcend conventional levels of analysis, bridge philosophy and science, and prompt novel discussions on planetary-scale issues ( Ruse, 2013 ). We generally find it intuitively easy to distinguish life from non-life, but only at the (somewhat arbitrary) level of resolution that the range of our senses allows: we would not recognise an E. coli bacterium without a microscope ( Lovelock, 2007 ; p. 174). Similarly, although we struggle to see it that way ( Lenton et al., 2020a ), the natural way for aliens observing the Earth from space would plausibly be not to see a writhing mass of individual organisms on a dead planet, but rather a rock that has come to life ( Oliver, 2020 ; p. 11). As a mode of thought, this visualisation of Gaia, while not necessary to comprehend it, provides a powerful tool for scientific understanding ( de Regt, 2014 ). Thus, like the “overview effect” experienced by astronauts viewing our planet from orbit ( White, 2014 ), the GH offers a glimpse of a fundamentally different way of thinking that in fact dominated most of human history, whereby the world was considered a single living organism—a “Thou” rather than an “It” ( Frankfort et al., 1960 ). This may amount to a revival of some important elements of animism in its most basic form: a sensed appreciation of the vitality of the more-than-human aspects of Earth ( Abram, 1997 ).

However, the GH has been subject to a wide range of criticisms (see e.g., Ruse, 2013 ; Rubin and Crucifix, 2022 ). Of the two most vociferous objections, however, the first worries unnecessarily that the use of language in the formulation of the idea entails commitments concerning the divine, sentient, or otherwise supernatural status of Gaia. Despite being largely compatible with strong claims such as those made in panpsychism, and somewhat ontologically slippery as an edge case of metaphysical categories, Lovelock (2007 ; p. 20) stresses that his illustrative goddess metaphor is just that.

Secondly, it has variously been argued that the GH is either false or unfalsifiable ( Kirchner, 1989 ). Both cannot be correct, and, while it clearly contains and presupposes certain empirical claims (concerning the age of the solar system, the composition of the atmosphere, and so on), the actual idea of the biosphere as a self-regulating system is not a “hypothesis” at all but a perspective , a way of describing the fact of life's existence on Earth (though we will stick to its conventional name to avoid confusion). As per the assertions of Elgin (2019) , truth is defined within this framework, rather than applied to it, and the mode of thinking that the GH engenders “refracts” with other modes, rather than competing with them. This is the “explanatory pluralism” that Kleinhans et al. (2010) refer to in their conclusion from an analysis of the philosophy of earth science.

Like Newtonian physics or psychoanalysis, the GH is therefore much more a framework or a method of investigation than it is a falsifiable empirical claim, and it is therefore to be judged by its usefulness , not its purported correctness, where usefulness is determined by factors such as parsimony, coherence and consistency, plus its power to explain phenomena, produce new facts, and so on, in a given context and a given time-frame ( de Regt and Dieks, 2005 ). Thus, we effectively adopt a broadly pragmatic epistemological view of scientific understanding with regards to GH, similar to that endorsed by de Regt (2009) and others. By analogy, when meeting someone on the street, we conventionally treat the object of the encounter as a single fellow living being, rather than a composite menagerie of over 40 trillion discrete microorganisms in a state of cohabitation ( Sender et al., 2016 ). This “Human Hypothesis” is not more or less correct than a “Microorganism Hypothesis,” it is simply a different (and usually more useful) scale at which to engage with the phenomena. In principle, then, the GH can at worst be called unpragmatic, but not false.

3. The Free Energy Principle for self-organising interacting systems

In this section, we aim to present the fundamental framework of the FEP and active inference. Subsequently, we intend to apply this framework to the planetary-scale processes which comprise Gaia. By doing so, we can apply the FEP to analyse the interdependent dynamics between living systems and (non-living) planetary processes. Specifically, this framework allows us to gain insights into future states of Gaia, formulate predictions, and devise potential interventions to alleviate future dysfunction.

The second law of thermodynamics, which governs the behaviour of energy and entropy in a system, stipulates that all open systems—which means almost all systems in the natural world—tend to dissipate, i.e., tend towards chaos or increased entropy. For example, a cup of hot coffee will always cool down to room temperature, but it will never spontaneously heat back up to its original temperature. The principles of thermodynamics in physics stipulate that the behaviour of open systems is critically influenced by the interaction with the environment. The flow of these interactions cannot be replicated in the opposite direction due to the irreversible nature of time ( Von Bertalanffy, 1950 ; Ptaszyński and Esposito, 2019 ; Pokrovskii, 2020 ; Rovelli, 2023 ).

However, there are displays of “negentropy” ( Schrödinger, 1944 ), pockets of the universe where order and bodily integrity are maintained in the face of the surge towards chaos. These pockets are self-organising systems, or “ things ” ( Hipólito, 2019 ), including—but not limited to—living organisms. These things resist entropy by interacting with the world such that their integrity is maintained, i.e., the process of homeostasis. They use energy to keep themselves within the restricted set of possible states that allows for their continued existence. Hence the workings of feedback cycles—drinking when one is thirsty, retreating from a hot fire, etc.—represent work that can only be done by the energy that is “bound” within the system and therefore useful, whilst the remainder is called “free” energy ( Friston and Stephan, 2007 ). Simply by existing, then—by virtue of seemingly defying the second law of thermodynamics by acting to minimise entropy—all such self-organising systems necessarily act so that their free energy is minimised. This is the Free Energy Principle.

Moreover, a system's free energy bears upon various other related concepts in addition to entropy, including the number of possible states the system can manifest, the predictability of its behaviour, and the level of “surprise” associated with new data ( Table 1 ). 3 Furthermore, at least in Karl Friston's formulation, 4 free energy is equal to information , i.e., the number of new binary facts about the state of the world gleaned by a system from a given data sample ( Solms, 2021 ). This means that, adhering to the FEP, every thing should act so as to minimise the amount of information about the world it needs to process, which amounts to minimizing its free energy, chaos or entropy and thus maintaining its physical integrity.

Table 1 . The relationship between Friston's free energy and related quantities.

Organisms minimise free energy by action. Active inference is a corollary of the FEP that allows us to model and understand a complex system's behaviour. In its formulation, living systems:

will appear to engage in active Bayesian inference . In other words, they will appear to model—and act on—their world to preserve their functional and structural integrity, leading to homoeostasis and a simple form of autopoiesis ( Friston, 2013 ; p. 1, emphasis added).

“Active inference” is a modelling technique that, because it employs a scale-free formalism known as Markov blankets, allows us to understand and make predictions about the coupling dynamics taking place between interacting systems. A living system is equated with “internal states,” while the system it interacts with is “external states.” Because, mathematically, internal and external states are conditionally independent, they do not directly influence one another. The direct influence occurs via yet another set of states: active and sensory states. A balanced reciprocal influence between active and sensory states is fundamental for a system (internal states) to maintain a healthy interaction with—and thereby adaptation to—its environment (external states).

These interdependencies and dynamics are understood by employing a Markov blanket. A Markov blanket is a statistical tool that can be applied to any system that self-organises. More precisely it furnishes probabilistically defined tools to set a system's boundaries through conditional dependence or independence relationships. Because they are scale-free, they can be applied to any level of analysis of the natural world. The concept of Markov blanket involves defining a set of variables, denoted as b, that surround the internal states while labelling all other external variables as η.

The internal and active states are a function of a system's internal and blanket states. Similarly, external and sensory states are a function of external and blanket states. This sparse dynamical coupling means that the state of a system at a moment in time results from the interactive dynamics between internal, sensory and active states; meanwhile, the state of the environment at a moment in time is a result of the dynamics between external, sensory and active states. It follows from this that internal and external states reciprocally (indirectly) influence each other.

There are therefore two ways for a thing to minimise its free energy, and thereby maintain its own integrity and continued existence: the first is to change the model (or belief) that it instantiates so as to more closely resemble the world, by perpetually generating, testing and updating it based on incoming prediction error information. The second is to change the world by acting upon it to bring it in line with predictions, under the generative model. These processes form a feedback loop, from within the system to without and vice versa (as seen in Figure 2 ).

Figure 2 . The partitioning of a system into states. The result is composed of internal (purple) and external or hidden (orange) states, separated by a Markov blanket consisting of sensory (green) and active (blue) states. The schematic highlights directed influences with dotted connectors. Autonomous states are those not influenced by external states, while sensory and autonomous states constitute an entity, namely, blanket and internal states. A system of scientific interest (e.g., a cell, an organ, an organism, a community, an ecosystem) is composed of sensory, active, and internal states, as described in more detail by Hipólito (2019) .

This loop involves the system being described as if using inference to both update its model of the environment and act upon it, as illustrated by Figure 3 in the case of a human. The loop can be described in terms of the following steps (where we can regard free energy as the total amount of prediction error):

1. The generation of a prediction of its sensory input based on its internal model. This prediction is compared to the actual sensory input to generate a prediction error.

2. The updating of its internal model to reduce the prediction error. This involves adjusting the probabilities assigned to various possible causes of the sensory input.

3. Based on an updated internal model, the system is then supposed to generate an action that it “believes” will lead to a predicted outcome in the environment.

4. The system's action upon the environment is seen as generating new sensory input.

5. The loop then repeats, as a system generating a new prediction of sensory input based on the updated internal model, and so on.

Figure 3 . An organism-environment interaction. Active inference and the Free Energy Principle describe a closed causal loop of action and perception within a system, such as a human. This loop involves the system using inference to both update its model of the environment and act upon it.

The system is then understood by scientifically interpreting and predicting its behaviour as a system's internal model is constantly updated in response to new sensory input to improve the accuracy of predictions. This loop therefore minimises the system's uncertainty about the causes of its sensory input, since it is driven by (or amounts to) the minimisation of free energy.

4. Applying the Free Energy Principle to the Gaia Hypothesis

In this section, we utilise the FEP to elucidate the coupling dynamics between biological systems and geophysical planetary processes, which make up the biosphere. In this picture, organisms and ecosystems are self-organising systems which are each part of a larger multiscaled system. The FEP framework facilitates greater understanding of current states and enables us to make predictions about future states. Additionally, it helps to identify and develop interventions to rectify any maladaptive responses, namely actions which are antithetical to maintaining the integrity of a system: human behaviour that leads to an imbalance within the Markov blanket states is therefore to be considered maladaptive or pathological.

As a self-organising system, Gaia meets the requirements to be formalised as part of an active inference system. It exists in non-equilibrium steady states (NESS), viz. its processes and states must persist over time within a range of states that are far from equilibrium (i.e., resisting entropy); and it is distinguishable from its larger scale cosmic environment, defined by conditional independence between its internal and external states (which influence each other only vicariously through blanket states).

The issue of distinguishability—that is, the ability to differentiate the biosphere from the earth's core on one side and the larger-scale solar system, galaxy, and universe on the other—presents a challenge. Experts from diverse fields, from climate and complexity scientists to Buddhist philosophers, have stressed that the biosphere is an integral part of the larger cosmic environment, demonstrating fundamental continuity with and interdependent upon it ( McMichael, 1993 ; Egri et al., 1999 ; Danvers, 2016 ; Kaçar et al., 2021 ; Boulton, 2022 ; Stahel, 2022 ). However, while the ontological continuity of the Earth and the rest of the universe should be emphasised, it is nevertheless possible to employ epistemic (statistical) tools to help us understand the interactions between multiscaled systems, in the same way that we can usefully distinguish between cells and organisms or countries and continents without supposing that they are truly separable.

We can therefore use Markov blankets to elucidate the interdependencies and dynamics between the nested systems, since, as outlined above, a Markov blanket is scale-free, making it suitable for application across various scales in the natural world, depending on the object of scientific interest. For example, we can employ Markov blankets in a multi-scale system to further understand and predict the interactions between human behaviour, Gaia, and the Cosmos.

To envision what this might look like, we can draw from Rubin et al. (2020) who assigned Markov blankets specifically to the Earth's climate system. The authors begin by defining metabolic rates of the biosphere as internal states and the changes in “space weather” (mostly driven by solar radiation) as external states. Further, they define active states as the changes in greenhouse effects and the reflection of sunlight, and sensory states as ocean-driven changes in global temperatures. Internal and external states are conditionally independent, thereby indirectly influencing each other via the ocean's very slow reaction to thermal fluctuation. Through this formalism, “the Earth's climate system [can] be interpreted as an anticipatory system that minimises variational free energy” ( Rubin et al., 2020 ).

As stressed previously, because Markov blankets are a scale-free formalism, we can broaden our object of scientific interest to cover the entire biosphere. If the geophysiological processes of the biosphere are considered internal states, then external states would refer to the external environment, which includes everything outside of the system, such as the outer atmosphere, and other planets and celestial bodies, as well as the molten core of the planet lying below the Earth's crust. Sensory states are the influences the external environmental states have on the planet Earth such as the impact on the Earth's rotation, orbit, and ocean tides by gravitational effects of the sun, moon, and other planets. Earthquakes and volcanoes caused by the influence of by convection currents within the mantle on plate tectonics; climate and weather are influenced by solar radiation and space weather events; cosmic events like meteor impacts can trigger evolutionary changes and cause mass extinctions; the composition of the Earth's atmosphere is similarly influenced by cosmic events. Active states are simply the capacity of the earth to assimilate external influence and adjust to it (i.e., change its properties) in ways that will feed back to the external states. For example, the Earth's gravitational force causes slight perturbations in the orbits of other objects in the solar system, and its magnetic field interacts with the solar wind, affecting phenomena such as the aurora borealis. The Earth also emits radiation that can be detected by other objects in the solar system, even from billions of kilometers away ( Figure 4 ).

Figure 4 . The Markov blankets of Gaia. When we consider the geophysical processes of the biosphere as internal states , the external states refer to everything outside the organism, including the atmosphere, other planets, and celestial bodies. Sensory states include gravitational effects of other celestial bodies, climate and weather, the evolution of life, the composition of the atmosphere. Active states are the Earth's gravitational effects, magnetic field, radiation etc.

The active states of a system embody its primary mechanism for self-regulation, representing its efforts to remain within the bounds required for ongoing existence, based on predictions about the present and future states of the system. For instance, the current levels of greenhouse and albedo effects, soil pH, and the number of living organisms reflect the predictive attempts of the Gaian system to adapt to the anticipated effects of solar events, volcanic eruptions, ocean salinity changes, and other environmental factors in the future. It is in this context that the role of life itself assumes paramount importance in the self-regulation of the biosphere.

In information theory, the dynamics described above can be explained by considering the active states as realizing the prior predictions of the generative model, while the sensory states provide information resulting from the external world, which is used to update the inferred state of the biosphere into posterior beliefs. The system responds to prediction errors, to accommodate this new information, while at the same time acting on the external states to resolve the prediction error, to the extent possible. Note that only certain kinds of prediction errors can be actively resolved. In neurobiology, the prediction errors resolved by movement are the predictions of signals from the muscles, known as proprioception. If prediction errors cannot be resolved via action, then they will be resolved by changing predictions. This can be cast as Bayesian belief updating and an elemental kind of perception. Updating the priors to accommodate this new information causes a change in the macro properties responsible for enabling such fluctuations, thereby updating the model itself such that the mutual information between the internal and external states is maximised, and free energy is minimised. Describing Gaia as a complex system of interacting, self-organising processes that exist in non-equilibrium steady states, actively adapting to changing environmental conditions, affords new syntheses between different conceptual approaches. For instance, combining the greenhouse effect and the role of life in the biosphere provides a unique opportunity to incorporate energy and information as two aspects of the same underlying physical process ( Hermann-Pilath, 2011 ), namely the minimisation of free energy.

Finally, a purported weakness of the FEP actually speaks in favour of its application to the GH. 5 Raja et al. (2021) rightly point out that Markov blankets do not automatically capture every relevant property of biological systems: relational properties such as affordances—what the environment “offers the animal [sic], what it provides or furnishes, either for good or ill” ( Gibson, 1979 ; p. 127)—fall into this bracket. However, the claim that they are precluded by the FEP is not true, at least with respect to the biosphere: on the contrary, affordances correspond well to Gaia as one rung in a ladder of nested Markov blankets—with the system in question at every level ranging up from, say, cells and organs, through animals and ecosystems, to our solar system ( Dennison, 2020 ) and the whole cosmos. In this framework, it is entirely coherent to say that our solar system is one that affords a living planet, or that the geosphere affords a biosphere, when we consider that lower levels or layers in such hierarchies operate at much faster rates of change than higher levels ( Wu, 2013 ). The affordances available in an environment are not determined solely by the physical properties of the environment, but also by the organism's goals, abilities, and previous experiences. In the case of the sun and its affordances for life on the planet, we can apply this theory to understand how the physical properties of the sun, as well as the goals and abilities of organisms on Earth, shape the affordances that the sun provides.

The same principles of dynamics, interdependence and scale apply to the symbiotic evolution and exchange of genetic information throughout the history of life ( Paracer and Ahmadjian, 2000 ; Watson, 2002 ; Gontier, 2016 ), and, increasingly, to the tight coupling between humans and technology and the possible emergence of autonomous symbiotic AI systems ( Wang et al., 2021 ). Hierarchy theory is therefore not only compatible with the FEP as applied to the Gaia Hypothesis, it is a highly tractable vehicle for understanding it, in terms of the complexity in ecosystems ( Allen and Starr, 2017 ), the evolution of the planetary genome across billions of years ( Margulis and Sagan, 1995 ) and changes to the (“non-living”) aspects of ecological landscapes ( King, 1997 ) at different scales of time and space. Overall, there is good evidence that phenomena in earth science are emergent, but should be considered irreducible to the laws of physics ( Kleinhans et al., 2010 ): we therefore aim to marry “bottom-up” principles intrinsic to the FEP from fundamental physics to the higher-level phenomena captured by the GH. As emphasised above, we are not arguing not that this approach to understanding life on Earth is the best one, merely that it can be beneficial.

5. Human behaviour as an aspect of Gaia

We can now employ Markov blankets to further understand and predict the actions of humans within the biosphere. Specifically, we want to understand the extent to which human behaviour is antithetical to the healthy behaviour of the larger system, why this is the case according to the FEP, and what might be done to improve the situation.

Larger systems provide the smaller nested systems they contain with existential challenges as well as affordances: the sun provides energy for photosynthesis and regulation of the climate, but also threatens humans with bad harvests and skin cancer. These challenges demand adaptation in a dynamic fashion. The mechanism for this—as per the FEP—is to assimilate information (in the form of prediction error) of an upcoming threat from the world, and act accordingly on the world such that the system's integrity will be maintained and free energy minimised. Thus, according to the FEP, living beings have a natural tendency to act upon their environment to prevent the dissipation of their integrity. Therefore, if the principle holds true, the knowledge that (for example) climate change caused by continued greenhouse gas emissions will have catastrophic consequences for continued human existence should prompt drastic action from the human species to rectify it, in the form of behaviour that reflects biophilia. The lack of such action can be seen as an imbalance within (Markov) blanket states, therefore indicating a profoundly unhealthy condition in the human species, whereby it is unable to maintain its integrity by adapting to changing environmental conditions. It is the combination of knowing that disrupting the balance in such a way is self-harming, and yet acting as if this were not the case that constitutes a pathological state.

According to the FEP, a system's interaction with the environment consists of using inference to both update the model of the environment and act upon it to make the environment fit the model. 6 Therefore, learning (as we have done) that the global status quo represents a serious threat to human civilisation, one would expect to see behaviour change accordingly. This would amount to gathering more evidence and then acting upon it to minimise free energy and, by extension, preventing the rise of entropy, chaos, and the ultimate dissipation of life.

Instead, the human species either rejects the information available so as to remove the imperative for action or, possessing the information available, still refrains from taking action, or worse, continues the same harmful behaviour and perpetuates the same systems (such as capitalism) that enable it. If the FEP holds, then the observed contradictory behaviour is a maladaptive state, arising as an imbalance within the Markov blanket states ( Figure 5 ).

Figure 5 . Imbalance in the Markov blanket. The figure shows influences occurring within the Markov blanket, which is marked by a dotted box and includes internal, active, and sensory states. This imbalance results in an insulation or echo-chamber effect in internal states that are impermeable to information from external states that demand action.

Using the lens of the FEP to model the dynamics of the relationship between humans and the environment, it is possible to see the current lack of sustainability and pro-environmental behaviour as a pathology of the human species that must receive treatment. We label this condition biophilia deficiency syndrome . To unpack the notion of biophilia deficiency syndrome—and lend it construct validity—it is useful to consider other applications of the FEP to pathology. In general, these applications rest upon instances of false inference, of the sort found in psychiatry and neurology. For example, inferring things are not there when they are, describes certain dissociative and hysterical (e.g., neglect) syndromes. Conversely, inferring things are there when they are not, describes phenomena like hallucinations and delusions.

These pathologies are commonplace in computational psychiatry and generally reduce to a failure to assign the right weight or precision to prediction errors. Perhaps the most prescient example of this is Parkinson's disease, characterised by a failure to initiate movement or action that manifests as bradykinesia. From the perspective of the FEP, this is simply explained by a failure to attenuate evidence from sensors providing evidence that the one is not moving. Put simply, a failure to realise prior predictions can be due to a failure to ignore evidence that predictions are not coming true. This seems to be an apt description of our communal and cultural response to the ecological crisis. In other words, biophilia deficiency syndrome can be seen as a collective or cultural Parkinsonism that inherits from our inability to attenuate the evidence that we are not acting in a remedial or restorative fashion.

From the perspective of the FEP, this is a pernicious pathology because doing nothing is a Bayes optimal response in the face of a pathological attention to various sources of evidence. In computational psychiatry, one then is led to therapeutic interventions that rest either on restoring neurochemical deficits in the brain or engaging in therapy that allows people to become skilled in deploying their attention—and exploring other models of active engagement with the world. In short, enabling patients to escape from particular patterns of active inference in which they are stuck.

By leveraging the concepts of the FEP and active inference, a compelling analogous perspective on how humans relate to the ecological crisis emerges. The FEP and active inference formalisms shed light on climate change and the wider ecological crisis as a predicament that is both human-generated and often disregarded due to a lack of action. According to the FEP, living systems possess a biologically encoded inclination to interact with their environment in order to survive and adapt.

However, the current state of affairs diverges from this natural inclination. Despite climate change being a pressing issue, proactive measures to reverse its effects are not given the priority they deserve. This discrepancy between the expected and observed actions can be understood in the framework of the FEP as an illness. It implies that systems, including human societies, may form erroneous inferences that prevent them from effectively addressing the problem at hand.

Given this characterisation, it is imperative that we recognise the current ecological imbalance as a “stuck state” that requires active disturbance to bring about systemic change (for detail see Hipolito, 2023 ). What we are calling for amounts to a kind of “homeostatic awakening” ( Wong and Bartlett, 2022 ): a deliberately-induced, disruptive shift in trajectory reflecting a prioritisation of planetary homeostasis over infinite growth. To achieve this, we must focus on addressing the reciprocal unbalanced influences between sensory and active states at multiple levels of nested systems, rather than solely at the level of the individual: the best efforts of individuals to change their own behaviour will be in vain so long as the lack of appreciation of dynamics at higher nested levels persists. On the other hand, if we can come to truly understand our relationship to Gaia at a depth that becomes part of cultural common sense, 7 then we will no longer be able to deny our current knowledge, and more appropriate behaviour will follow as a natural reflection of our (FEP-driven) values. Meaningfully addressing the ecological crisis is therefore synonymous with restoring the felt connection between humans and the rest of life on Earth, such that we collectively come to truly embody the fact that we are a part of, not apart from, the natural world ( Seth, 2022 ). This will entail the transformation of the very concepts of “self” and “nature” to reflect a part-to-whole relation, rather than a subject-object dichotomy, in much the same way that integrating the understanding that the Earth revolves around the sun required a shift in what those words actually mean ( Kuhn, 1962 ). 8

It should be clear that individual responses will not suffice for problems that are essentially collective and relational. Thus, beyond enforcing institutional climate commitments at the national and international levels, future work should focus on identifying strategies for action and intervention at community, societal and whole-species levels to stimulate both the integration of ecological knowledge and the practice of biophilia virtue. In this way, both sides of the Markov blanket are addressed simultaneously in an effort to remedy the insulation of internal states (i.e., to address the loss of connection between humans and the Earth; Figure 5 ). Providing a full manifesto for doing so is far beyond the scope of this paper; however, some intuitive examples of measures that can point towards directions of implementation and be taken at the institutional level to catalyse biophilia virtue as conceptualised under the FEP may include the following:

(1) Education: Ecopedagogy is a holistic approach to education that emphasises the interdependence of social, economic, and ecological systems. One of the main goals of ecopedagogy is to integrate environmental sustainability and social justice into educational practices and to encourage learners to take an active role in addressing environmental and social issues. By promoting experiential and place-based learning, ecopedagogy encourages learners to connect with their local environment and community, fostering a deeper understanding of ecological systems and their interrelationships (as opposed to the abstract intellectual knowledge many of us currently harbour). For example, learning about the fundamental role of fungi in the web of life through mycology can foster an appreciation of Gaia's interconnected nature and prompt a shift away from the anthropocentric insistence on thinking of organisms as individuals ( Sheldrake, 2020 ), as well as providing practical skills in food-growing and restorative practices ( Stamets, 2005 ).

Ecopedagogy also emphasises critical thinking skills and encourages learners to engage in social and environmental activism, by promoting, firstly, a true reciprocity between nature and humans ( Varanasi, 2020 ) that recognises that the fate of the latter is dependent on the state of the former, and secondly, non-linear thinking, so that the Earth and its inhabitants can be encountered as the highly complex adaptive systems they are ( Varela et al., 1991 ; Margulis and Sagan, 1995 ; Capra, 1997 ; Duncan, 2018 ; Fried and Robinaugh, 2020 ; Hayes and Andrews, 2020 ). Thus individuals and collectives can become more aware of the impact of human activities on the natural world at multiple levels of analysis, and develop a corresponding sense of responsibility towards the environment ( Zysltra et al., 2014 ; Norat et al., 2016 ; Misiaszek, 2020 ; Hung, 2021 ). A shift in the value pyramid of the educational priorities towards sustainable behaviour not only acts as a perturbation to the biophilia deficiency syndrome but also addresses the raising eco-anxiety symptoms reported by young generations ( Pihkala, 2020 ; Baudon and Jachens, 2021 ; Gunasiri et al., 2022 ).

(2) Urban design : Green spaces, such as parks, green roofs, or community gardens, provide opportunities for individuals to connect with nature, even in urban areas. These spaces offer a range of benefits, from relaxation and recreation to habitat for wildlife. Community gardens, in particular, can promote community engagement and healthy eating, while also fostering a deeper connection to nature through shared stewardship ( Roe and McCay, 2021 ).

(3) Technological and Artificial Intelligence design : Technological design can be used to promote sustainability and reconnect humans with nature. This can be achieved through environmental monitoring to understand the impact of human activities on ecosystems ( Bodini, 2012 ), nature-based gaming to promote education about biodiversity and conservation ( Schneider and Schaal, 2018 ), and sustainable design of buildings and green infrastructure to create and maintain natural environments in urban areas ( Restall and Conrad, 2015 ; Leavell et al., 2019 ; McKewan et al., 2020 ).

(4) Mindful nature practices: Mindfulness practices like yoga or meditation can help individuals develop a greater sense of embodiment and connection with their physical selves, which can translate into a felt appreciation of the “interwoven nature” of the natural world ( Danvers, 2016 ). By cultivating mindfulness practices, individuals can become more attuned to the natural world and develop a greater sense of respect and responsibility towards the environment ( Amel et al., 2009 ; Barbaro and Pickett, 2016 ). Conscious and responsible use of psychedelic plant medicines like ayahuasca and psilocybin can also be fruitful options. Psychedelic use predicts nature connectedness ( Nour et al., 2017 ; Kettner et al., 2019 ), and solidarity with other species ( Pöllänen et al., 2022 ), translating into pro-environmental behaviour ( Forstmann and Sagioglou, 2017 ), as well as both concern for and objective knowledge about climate change ( Sagioglou and Forstmann, 2022 ), while a recent survey on “psychedelically induced biophilia” found a tendency to elicit a “passionate and protective” connection with nature ( Irvine et al., 2023 ). Engagement with these practices would also encourage interaction with cultures from which they originate, affording opportunities for learning from and cooperation with indigenous peoples who currently steward lands with 80% of the world's biodiversity ( Watene and Yap, 2015 ), and increasingly carry the flame for cultural diversity and lost ecological wisdom ( Toledo, 2013 ; Rowlandson, 2015 ; Etchart, 2017 ; George et al., 2019 ).

This list is neither fully developed nor exhaustive, but offered as an exemplar of how the principles provided in this paper can be applied. Despite efforts made by individuals, organisations, and governments to reduce carbon emissions and mitigate the impact of climate change and the wider ecological crisis, progress has been hindered by the multifaceted and global nature of this problem. To catalyse meaningful change, a deliberate, disruptive and holistic approach is necessary to address the symptomatic ways in which humans induce (self-)harm upon Gaia.

6. Conclusion

This paper employs the Free Energy Principle to argue that the lack of action taken in response to threats to planetary life, such as those posed by the ongoing ecological crisis, should be treated as a maladaptive disruption to Gaia's Markov blankets, referred to as biophilia deficiency syndrome . Adopting a pragmatic, pluralist epistemological approach to understanding planetary life processes, the paper proceeded in four parts. Firstly, it characterised life under the Gaia Hypothesis, wherein the biosphere constitutes a single self-organising system of living and non-living planetary processes, arguing that this is a plausible and useful way of looking at life on Earth. It then described the Free Energy Principle as a means of understanding self-organising (living) systems, before employing the FEP to elucidate the coupling dynamics within the biosphere with a view to developing beneficial interventions. Finally, the paper demonstrated the possibility for positive alternatives afforded by this framework to remedy the ongoing crisis of ecological breakdown, emphasising the importance of perturbing stuck states at multiple levels of nested hierarchies to promote biophilia virtue and, by extension, the healthy development of human behaviour as part of Gaia. Ultimately, this paper considers the Gaia Hypothesis through the lens of the Free Energy Principle to reveal insights into how we can restore the balance of planetary processes and safeguard the integrity of living systems that depend on them.

Author contributions

CM conceived the initial idea for the paper and produced the first manuscript. IH revised the thesis and wrote additional sections of the manuscript. All authors generated figures, contributed to manuscript revision, and read and approved the submitted version.

IH was funded by Humboldt-Universität zu Berlin.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

1. ^ For instance, oxygen, which was extremely poisonous to the organisms of the time, was virtually absent from the atmosphere for more than a billion years of life, until the explosion of aerobic life coincided with a dramatic increase of oxygen levels as it was produced by cyanobacteria ( Schirrmeister et al., 2013 ; Lyons et al., 2014 ).

2. ^ These tipping points are partly what make climate change such a difficult challenge, especially since we won't necessarily know when they have been passed ( Lenton et al., 2020b ).

3. ^ Though a technical term, this formal meaning of ‘surprise' is close enough to its use in common parlance.

4. ^ As opposed to Helmholtz's and Gibbs' free energy, which refer to related thermodynamic and chemical quantities, respectively.

5. ^ Note that applications of the FEP to systems such as animals introduces the notion of planning (as inference) in a psychological sense and the consideration of epistemic and pragmatic affordances. However, here, we are applying the FEP to the systems that do not necessarily plan, such as the weather, market economy, evolution, et cetera.

6. ^ Or, fails to attenuate sensory evidence that nothing is — or can — be done about the situation.

7. ^ It is arguably only in WEIRD (Western, educated, industrialised, rich, democratic) cultures that this is not already the case: “Many indigenous belief systems share a view of people and nature as part of an extended ecological family … These different groups … separated by geography, culture and time, have described this phenomenon. The Māori worldview (te ao Māori) acknowledges the interconnectedness of all living and non-living things. The Raramuri worldview includes ‘Iwigara': the total interconnectedness of all life, physical and spiritual. From the point of view of these belief systems, feeling separate from nature would signify a state of disconnectedness and constitute a significant rupture in wellbeing .” ( Watts et al., 2022 , emphasis added).

8. ^ “Part of what [geocentrists] meant by ‘earth' was fixed position. Their earth, at least, could not be moved. Correspondingly, Copernicus' innovation was not simply to move the earth. Rather, it was a whole new way of regarding the problem of physics and astronomy, one that necessarily changed the meaning of both ‘earth' and ‘motion.”' ( Kuhn, 1962 , pp. 149-50).

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Keywords: Free Energy Principle (FEP), Gaia Hypothesis (GH), active inference, biophilia virtue, ecological crisis, nature connectedness

Citation: Montgomery C and Hipólito I (2023) Resurrecting Gaia: harnessing the Free Energy Principle to preserve life as we know it. Front. Psychol. 14:1206963. doi: 10.3389/fpsyg.2023.1206963

Received: 17 April 2023; Accepted: 30 May 2023; Published: 15 June 2023.

Reviewed by:

Copyright © 2023 Montgomery and Hipólito. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Inês Hipólito, ines.hipolito@hu-berlin.de

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Incorporating (variational) free energy models into mechanisms: the case of predictive processing under the free energy principle

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The issue of the relationship between predictive processing (PP) and the free energy principle (FEP) remains a subject of debate and controversy within the research community. Many researchers have expressed doubts regarding the actual integration of PP with the FEP, questioning whether the FEP can truly contribute significantly to the mechanistic understanding of PP or even undermine such integration altogether. In this paper, I present an alternative perspective. I argue that, from the viewpoint of the constraint-based mechanisms approach, the FEP imposes an important constraint, namely variational free energy, on the mechanistic architecture proposed by PP. According to the constraint-based mechanisms approach, high-level cognitive mechanisms are integral parts of extensive heterarchical networks that govern the physiology and behavior of agents. Consequently, mechanistic explanations of cognitive phenomena should incorporate constraints and flows of free energy as relevant components, given that the implemented constraints operate as long as free energy is available. Within this framework, I contend that the FEP provides a relevant constraint for explaining at least some biological cognitive mechanisms described in terms of Bayesian generative models that minimize prediction errors.

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1 Introduction

It has been established that proponents of the PP framework seek mechanistic explanations and that the various models of cognitive functions developed via PP are aimed at this kind of account (Friston et al., 2018 ; Gładziejewski, 2019 ). In line with this view, it has been argued that PP provides a sketch of a mechanism (Gładziejewski, 2019 ; Gordon et al., 2019 ; Harkness, 2015 ; Harkness & Keshava, 2017 ; Hohwy, 2015 ), i.e., an incomplete representation of a target mechanism in which some structural aspects of a mechanistic explanation are omitted (cf. Piccinini & Craver, 2011 ). Understood in this way, the sketch is defined in terms of functional roles played by the respective components, disregarding to some extent their biological or physical implementation. This raises the important question of how to understand the causal structure responsible for predictive mechanisms. It can be a simple multi-level hierarchy from simple neural levels of, e.g., pattern recognition, edge detection, color perception etc. (implemented in the early sensory system), to high-level neural representations (implemented deep in the cortical hierarchy [Sprevak, 2021 ]), to increasingly abstract and general levels related to Bayesian beliefs and concerning the general properties of the world; or it can be a subtler structure implemented by several different, partially independent mechanisms responsible for various phenomena. Footnote 1

The key to this type of practice is the recognition of cognition in the categories of mechanistic causal relations (cf. Gładziejewski, 2019 , p. 665). Gładziejewski suggests that sketches of mechanisms provided by PP should be understood in the sense that these models “share common core assumptions about relevant mechanisms” but do not describe a single cognitive structure (mechanism). This means that “there are a couple of ways in which a collection of mechanisms that fall under a common predictive template could provide a schema-centered explanatory unification” (Gładziejewski, 2019 , p. 666). This author points to four possible research heuristics which, by providing sketches, may allow the identification of actual mechanisms:

There are separate neural mechanisms that follow the same predictive scheme;

Different levels within one hierarchy can explain different cognitive phenomena;

Various aspects of PP mechanisms are explanatory, which means that for a given mechanism, certain aspects of its functioning may explain specific phenomena;

The ways in which distinct PP mechanisms become integrated may play explanatory roles (Gładziejewski, 2019 , pp. 666–667).

Regardless of which of the indicated heuristics is actually employed by PP researchers (whether it be one or a combination of several), there is no doubt that many supporters of PP seek mechanistic explanations.

As can be seen, the thesis about the mechanistic nature of PP is already reasonably well-founded, but it seems that in light of the view advocated by some mechanists (cf. Bechtel, 2019 ; Winning & Bechtel, 2018 ), (at least) some mechanistic explanations should include constraints and flows of free energy as their constitutive component. This view, which I will refer to in this paper as the constraint-based mechanisms approach, Footnote 2 could be of great importance to many debates about PP and FEP theories as it allows for a rethink of the relationship between PP, FEP, and FEP-based Active Inference.

The possibility of a mechanistic integration of PP and the FEP has already been raised by researchers. Some responses have also been offered. There are authors who share the viewpoint that the FEP carries mechanistic implications for PP, asserting that the FEP can be treated as a heuristic guide or regarded as a regulatory principle. Supporters of the first position include Paweł Gładziejewski who, in his paper Mechanistic unity of the predictive mind , states that the FEP is “a powerful heuristic guide for the development of PP” but “only puts extremely general constraint on the causal organization of organisms, perhaps to the point of lacking any non-trivial commitments about it” (Gładziejewski, 2019 , p. 664). Another supporter, Dominic Harkness, claims that “the upshot of this criticism lies within the free energy principle’s potential to act as a heuristic guide for finding multilevel mechanistic explanations” (Harkness, 2015 , p. 2). Jakob Hohwy supports the second position, claiming that the “FEP can be considered a regulatory principle, ‘guiding’ or ‘informing’ the construction of process theories” (Hohwy, 2020 , p. 39), meaning that the FEP provides “distinct process theories explaining perception, action, attention, and other mental phenomena” (Hohwy, 2020 , p. 47).

However, some researchers are not convinced by the FEP or its explanatory relationship with PP. For example, Daniel Williams in his recent paper Is the brain an organ for free energy minimization? argues that “the claim that the FEP implies a substantive constraint on process theories in cognitive science—namely, that they must describe how the brain’s mechanisms implement free energy minimization—rests on a fallacy of equivocation” (Williams, 2021 , p. 8). Similarly, Mateo Colombo and Patricia Palacios in their paper Non-equilibrium thermodynamics and the free energy principle in biology note that “because of a fundamental mismatch between its physics assumptions and properties of its biological targets, model-building grounded in the free energy principle exacerbates a trade-off between generality and biological plausibility” (Colombo & Palacios, 2021 , p. 2). Colombo defends a slightly different position in a paper co-written with Cory Wright, where they take into account that the analysis carried out by the FEP’s supporters can be treated as sketches of mechanisms in the sense of Piccinini and Craver ( 2011 ). They do, however, only treat them as weak explanatory idealizations: “Some of the confusions in recent debates surrounding the FEP, organicism, and mechanism depend on indulging this sort of metaphysics without carefully considering the epistemic and pragmatic roles that ‘rampant and unchecked’ idealizations, like those involved in FEP, play in science” (Colombo & Wright, 2021 , p. 3486).

In this paper I will take a different starting point. I want to demonstrate by reference to the constraint-based mechanisms approach, that the FEP offers an explanatory relevant (variational) constraint for the causal organization of any and all systems equipped with generative models, explained mechanistically by PP. In other words, I will claim that the FEP provides a constraint which determines PP’s scheme of mechanism.

This paper has the following structure: in Sect.  2 , I present an overview of the PP and FEP frameworks and explain why, when analyzing predictive mechanisms, one should take into account the quantity described in the literature on the FEP as variational free energy (VFE). In Sect.  3 , I sketch the new mechanical philosophy and its characteristic systems tradition, describing explanations in terms of the identification and decomposition of mechanisms. I also present the recent position based on mechanism, which I refer to as the constraint-based mechanisms approach and—characteristic for this approach—the so-called heuristics of constraint-based mechanisms. In the following part, I formulate a mechanistic interpretation of PP and wonder if it can meet the norm defined by the heuristics of constraint-based mechanisms. The context of the question is set by the discussion on the FEP and its explanatory relationship with PP. In Sect.  4 , I discuss two main possible interpretations (realistic and instrumental) of the statement that self-organizing systems minimize VFE. Discussing them is important because it provides an initial answer to whether the FEP determines the energetic (in information-theoretic sense) constraint for mechanistic PP. In Sect.  5 , I articulate the position of mechanistic realism, which asserts the feasibility of employing heuristics based on the constraint-based mechanisms approach. I argue that the interpretation of the FEP, which I called moderate realistic, is compatible with mechanistic realism. In Sect.  6 , I discuss Karl Friston’s argument from Bayesian mechanics that VFE coincides with thermodynamic free energy (TFE). If Friston’s perspective is accurate, the FEP serves a similar explanatory role in elucidating living organisms as thermodynamics does in explaining physical systems. However, in this section, I reject Friston’s argument because of its instrumental character, which precludes mechanistic realism and the application of the heuristics of constraint-based mechanisms. As a result, in Sect.  7 , I present an argument in favor of moderate realism regarding the FEP and FEP-based PP. This argument is supported by empirical evidence from investigations into neural computations and the thermodynamics of information. Next I discuss the ontological commitments of this position, and I also formulate a provisional response to the objections of those authors who deny explanatory value to the FEP. In the Conclusion , I summarize the analyses carried out.

2 Predictive processing and variational free energy

PP is a process theory of the brain that provides a computational model of cognitive mechanisms and core processes that underwrite perception and cognition. Some advocates of PP believe that it can be used to unify the models of perception, cognition, and action theoretically (Clark, 2013 ; Hohwy, 2015 ; Seth, 2015 ). Specific versions of PP are grounded in the same process of precision-weighted, hierarchical, and bidirectional message passing and error minimization (Clark, 2013 ; Hohwy, 2020 ). In this framework, perceptual and cognitive processes are conceived as being the result of a computational trade-off between (hierarchical) top-down processing (predictions based on the model of the world) and bottom-up processing (prediction errors tracking the difference between predicted and actually sensed data). A characteristic feature of this view is the assumption that, in order to perceive the world, the cognitive system must resolve its uncertainty about the ‘hidden’ causes of its sense states. This is because the causes of the sensory signals are not directly recognized or detected, but instead must be inferred by a hierarchical, multi-level probabilistic (generative) model. In PP, the activity of the brain (or cognitive system) is understood as instantiating or leveraging a generative model (cf. Clark, 2016 ), which is, generally speaking, a model of the process that generated the sensory data of interest. In short, PP purports to explain the dynamics of the brain by appealing to hierarchically organized bidirectional brain activity, cast as instantiating a generative model.

The generative model is defined as the joint probability of the “observable” data e —sensory state, and h —a hypothesis about these data (trees, birds, glasses etc.). In other words, a generative model is the product of p(h) (priors over states) and p ( e | h ) (likelihood of evidence probability if the hypothesis is true). This means that the generative model is a statistical model of how observations are generated (strictly speaking, a description of causal dependencies in the environment and their relation to sensory signal). It uses prior distributions p(h) (which determine the probability of hypothesis before evidence) that the system applies to the environment about which it makes inferences.

The model minimizes the so-called prediction errors, i.e., the differences between the expectations of the organism—its “best guess” about what would be the case (what caused its sensory states) and what the organism factually observes. To minimize prediction errors, the generative model continuously creates statistical predictions about what is happening or can happen in the world. This means that updating the likelihoods and priors based on prediction errors is a mechanism that can be described in terms of Bayesian inference, i.e., a statistical inference in which a Bayesian rule is used to update the probability for a hypothesis as more evidence or data becomes available.

Technically speaking, according to the Bayesian rule

the generative model p ( h | e ) calculates the posterior probability p ( e | h ), which in practice allows the system to assume the most probable hypothesis explaining the nature and causes of the sensory signal, taking into account the available sensory data. Footnote 3 This hypothesis enables the minimization of the long-term average prediction error (Hohwy, 2020 ). Moving from p ( h | e ) to p ( e | h ), i.e., inverting the likelihood mapping, allows one to update beliefs from prior to posterior beliefs (Smith et al., 2022 , p. 3). Proponents of the PP framework argue that the model approximates Bayesian inference rather than computing it exactly (cf. Clark, 2013 ). In PP, the model implements an algorithm that computes Bayesian inferences so that the prediction error is gradually minimized, which maximizes the posterior probabilities of the hypotheses.

This way, when the model minimizes the prediction error, it also minimizes a certain quantity that is always greater than or equal to the surprisal—negative log probability of an observation/outcome—the surprisal model itself cannot be minimized directly due to ignorance of the underlying causes of the sensory signals (Friston, 2009 , p. 294). This quantity refers to the objective function that is known as VFE or an evidence lower bound (cf. Winn & Bishop, 2005 ). The introduction of VFE helps to convert exact Bayesian inference into approximate Bayesian inference. Footnote 4

Why is this important? Approximate Bayesian inference uses VFE minimization, which can be described as the difference between the approximate posterior distribution of the model and the target distribution. The introduction of an approximate posterior distribution over states, denoted q ( e ) (such that each q ( e ) ∈  Q is a possible approximation to the exact posterior distribution), makes simplifying assumptions about the nature of the true posterior distribution. By iteratively updating the approximate posterior (initially arbitrary), one can find a distribution that approximates the exact posterior. The next step is to measure the similarity between approximated p ( h | e ) and the true posterior p ( e | h ). Formally, this means minimizing the so-called Kullback–Leibler divergence (KL-divergence). It is important that KL-divergence cannot be directly estimated, and therefore the model must optimize a different function (i.e., VFE) which bounds the model evidence. The smaller the VFE, the smaller the KL-divergence. When KL-divergence is zero, then the distributions match. It gets larger the more dissimilar the distributions become. In variational inference, the model iteratively updates approximate posterior q ( e ) until it finds the value that minimizes VFE at which q ( e ) will approximate the true posterior p ( e | h ) (Smith et al., 2022 ; cf. Buckley et al., 2017 ).

The association of PP with VFE helps explain how the generative model minimizes prediction errors by Bayesian inference approximation, which can be interpreted as the way in which neural information processing mechanisms perform variational inference. This remark is crucial for further analyses.

To sum up: predictive mechanisms can be described in terms of the realization of variational principles (cf. Friston et al., 2017 ). In research practice, this means that in order to be able to concretize any variational inference algorithm, we must define the forms of the variational posterior and the generative model, which in the case of PP means (relying on the Laplace assumption) that posterior probability densities are normal (Gaussian). With this assumption in place, free energy can be viewed as the sum of the long-term average prediction error, which is supposed to be linked to the FEP (cf. Friston, 2010 ). It means that in the context of PP, the process involves the minimization of long-term average prediction error through the model’s optimization of the statistics of an approximate posterior distribution. Modelers postulate and refine this distribution to align with the desired target distribution (Millidge et al., 2021 , p. 7). This is an important observation for the very understanding of PP because it allows us to think about the normative function of the predictive mechanisms, which is the long-term average precision-weighted error in terms of free energy minimization.

At this point, however, difficulties arise regarding the linking of the PP framework with the research framework motivated by the FEP. Before discussing them (cf. §4), it is necessary to at least briefly explain what the FEP is.

The FEP was introduced by Karl Friston and colleagues as a mathematical framework that specifies the objective function that self-organizing systems need to minimize in order to change their relationship with the environment and maintain thermodynamic homeostasis (Friston, 2009 , 2010 , 2012 ; Friston & Stephan, 2007 ; Friston et al., 2006 ; cf. Andrews, 2021 ). Originally, the FEP was a principle explaining how the sensory cortex infers the causes of its inputs and learns causal regularities. What distinguished the FEP from other theories of inference (cf. Gregory, 1966 ; Rock, 1983 ) is the fact that all cognitive processes and functions, not only perceptual, can be explained in terms of one unifying principle, which is the minimization of free energy (Bruineberg et al., 2021 , p. 3; cf. Friston, 2010 ). Later, the validity of the FEP was extended from perception and action to organization of all self-organizing systems: from unicellular cells to social networks (cf. Friston, 2009 , p. 293; 2013 ; Wiese & Friston, 2021 ). Footnote 5

According to the current formulation of this principle Footnote 6 any self-organizing system that is at a nonequilibrium steady-state (NESS) with its environment must minimize its free energy. Footnote 7 In other words, any “thing” that achieves NESS can be construed as performing a Bayesian inference with posterior beliefs that are parameterized by the thing’s (model’s) internal states. In other words, the FEP offers an interpretation of mechanical theories of systems as if they possess (Bayesian) beliefs (Ramstead et al., 2023 , p. 2). This is related to the fact that the state flow of a given self-organizing system can be described as a function of their NESS density. The system, if it exists, can be described in terms of a random dynamic system (in terms of Dynamic System Theory—DST) that evolves, which means that it can be said to change over time, subject to random fluctuations. It must be added that any self-organizing system that is at NESS, i.e., one that has an attracting set, can be described in terms of Markov blankets (Friston, 2013 ; Friston et al., 2020 ; Wiese & Friston, 2021 ). Footnote 8

NESS density means a certain probability of finding it in a particular state when the system is observed at random (Friston et al., 2020 , p. 4). In this sense, everything that exists is characterized by properties that remain unchanged or stable enough to be measured over time. In other words, this means that the states of a given system behave as if they are trying to minimize exactly the same quantity: the surprisal of states that constitute the thing, system, and so on. That is, everything that exists will act as if to minimize the entropy of its particular states over time. Thus, open systems that are far away from equilibrium resist the second law of thermodynamics (Friston & Stephan, 2007 ; cf. Davies, 2019 ; Ueltzhöffer, 2019 ). What exists must be in a sense self-evidencing, meaning that it must maximize a particular model evidence or equivalently minimize surprisal (cf. Hohwy, 2016 ). This way, according to Friston and colleagues, it is possible to interpret the flow of (expected) autonomous states of the model as a gradient flow on something what we know as VFE, Footnote 9 and at the same time allows us to think of systems that have Markov blankets as “agents” that optimize the evidence for their own existence. In this sense, their internal states with the blanket surrounding them are (in some sense) autonomous (Kirchhoff et al., 2018 , p. 2; cf. Friston et al., 2020 ). Autonomy understood in this way allows us to think of “agents” as adaptive systems, where adaptivity refers to an ability to operate differentially in certain circumstances. This means that a system that is not adaptive, suggesting that it does not have a Markov blanket and cannot exist. Footnote 10

On the basis of the conducted analyses, it can be concluded that the FEP, as a formal statement—the existential imperatives for any system that manages to survive in a changing environment—can be treated as a generalization of the second law of thermodynamics to NESS (Parr et al., 2020 ). In that sense, the FEP is true for any bounded stationary system that is far from equilibrium, because the FEP applies to all self-organizing systems at NESS (meaning that the FEP applies to all systems equipped with the generative model because NESS density can be described in the terms of generative model [Friston, 2019 , p. 89; cf. Sakthivadivel, 2022 ]). Footnote 11

3 Systems tradition of mechanistic explanation and the constraint-based mechanisms approach

In §1, I drew attention to the fact that many researchers either have doubts about the actual integration of PP with the FEP—where the FEP would offer an explanatory significant contribution to the mechanistic PP (cf. Gładziejewski, 2019 ; Harkness, 2015 ; Hohwy, 2020 ), or even negate such a possibility (cf. Colombo & Palacios, 2021 ; Colombo & Wright, 2021 ; Williams, 2021 ). In this paper, I propose a different research perspective, according to which the FEP imposes an explanatory relevant informational constraint (i.e., VFE) on the mechanistic architecture postulated by PP. In order to justify this view, I will refer to the position I call the constraint-based mechanisms approach. Before I develop my argument, however, it is necessary to explain, albeit briefly, what this approach is.

Scientific research can be described in terms of discovering and describing mechanisms. In many fields of science, it is assumed that in order to formulate a satisfactory explanation of the phenomenon under study, one needs to provide a decomposition of its mechanism. Mechanistic explanations are used with great success in neuroscience as well as in biological, physical, and social sciences (cf. Glennan & Illari, 2018 ). This new mechanistic explanatory program became the dominant view across many debates in the philosophy of science (Bechtel, 2008 ; Bechtel & Richardson, 1993 /2010; Craver, 2007 ; Craver & Darden, 2013 ; Machamer et al., 2000 ).

The introduction of a new mechanism comes with the assumption that a distinction should be made between explanations which are componential or constitutive and etiological explanations, which explain a phenomenon by describing its antecedent causes. Constitutive explanations detail a phenomenon by describing its underlying mechanism, i.e., the relation between the behavior of a mechanism as a whole and the organized activities of its individual components is constitutive (cf. Salmon, 1984 ). Footnote 12 The latter’s explanations assume a strategy of decomposing high-level cognitive capacities into components that are responsible for various information processing operations, and then using various computational models, showing how these operations together explain a given phenomenon. Decomposition is a characteristic determinant of the ‘systems tradition’ (Craver, 2007 ; cf. Bechtel & Richardson, 1993 /2010; Cummins, 1975 ; Fodor, 1968 ; Simon, 1969 ). In this tradition, explanation is understood as a matter of decomposing systems into their parts to show how those parts are organized in such a way to emphasize the explanandum phenomenon.

Systems tradition is currently the dominant approach to explanations formulated in biology, system research, and cognitive neuroscience, while decomposition is the central heuristic strategy in mechanistic explanations besides the identification of mechanisms (Bechtel & Richardson, 1993 /2010; cf. Bechtel, 2008 ; Craver, 2007 ; Illari & Williamson, 2013 ). However, the mechanistic view of explanation has met with controversy (cf. Koutroufinis, 2017 ; Silberstein & Chemero, 2013 ). Moreover, some authors defend dynamical explanation as an alternative to mechanistic explanation (cf. Stepp et al., 2011 ). Footnote 13

3.1 What about constraints?

Some researchers (cf. Bechtel, 2018 , 2019 , 2021 ; Bechtel & Bollhagen, 2021 ; Winning, 2020 ; Winning & Bechtel, 2018 ) point out that the decomposition strategy, as understood by mechanism, assumes that there is a composition or causation relationship (i.e., causal production) between processes present in mechanisms (where one process, an organized set of causal processes is “responsible for” the implementation of another). Such a view, however, ignores two important features of cognitive mechanisms:

Mechanisms of this kind primarily act to control production mechanisms, i.e., mechanisms which are responsible for bodily movement and physiological processes. This type of relationship can be called control, and it is as important for the understanding of the nature of mechanisms and their explanations as the relationships of causation and composition (Winning & Bechtel, 2018 , p. 2). These are, therefore, mechanisms that help to maintain the internal environment of the given organisms. The analysis of control mechanisms is important because they allow organisms to quickly adapt to their environment. Therefore, they perform an important adaptive function and are responsible for the autonomy of the individual, as they contribute to the maintenance of the existence of a given organism. In this sense, they are normative because they contribute to the self-maintenance that is the norm of autonomous living systems (cf. Bickhard, 2003 ). Self-maintenance is the norm (what is good or bad for the system) in the sense that it “is not externally interpreted or derived from an adaptive history but defined intrinsically by the very organization of the system” (Barandiaran & Moreno, 2006 , p. 174);

High-level cognitive mechanisms are components of a highly developed and complex network of heterarchically organized control systems whose aim is to perform a given cognitive task (Bechtel, 2019 , p. 621, cf. Pattee, 1991 ). By heterarchical organization, I mean a such distributed causal network in which a given (production) mechanism is regulated by multiple (control) mechanisms without these control mechanisms being themselves subsumed under a higher-level controller. This means that their organization is horizontal and not vertical, as is the case with hierarchical organization (cf. Bechtel & Bich, 2021 ). Footnote 14

These features (1) and (2) are extremely important and their omission in explaining cognitive mechanisms makes these explanations incomplete, violating the standard of mechanistic explanations (Craver & Kaplan, 2018 ). This may result in “incorrect accounts of cognition” (Bechtel, 2019 , p. 621). Footnote 15 Taking account of these two aspects of cognitive processes, i.e., their function in the production of control mechanisms and their non-autonomous character, leads to the conclusion that their explanation should also cover other components (some of which are flexible and able to be operated on and altered by other mechanisms) than those previously considered. Footnote 16 This means that the mechanisms are organized not only in terms of production and composition, but also in terms of control. Such a view thus presupposes a revision of the systems tradition in which “processes are controlled by other processes, and mechanisms are controlled by other mechanisms, often hierarchically” (Winning & Bechtel, 2018 , p. 3).

A drift from the classical understanding of systems tradition does not mean a departure from the norms of mechanistic explanations, but rather their extension and the recognition that the concept of constraint is also important from the explanatory perspective. The concept of constraint comes from classical mechanics. It was used to describe the reduction of the degree of freedom available to components organized into macroscale objects. Constraints define some limits on independent behavior but also create possibilities (Hooker, 2013 ). For example, in contexts where there is a source of (thermodynamic) free energy, constraints can be used to direct the flow of this energy. This means that elements of biological mechanisms can be used to limit the flow of available free energy so that work is done (which can be used to generate particular phenomena). Some (control) mechanisms are therefore systems of constraints that restrict the flow of free energy to perform work. Therefore, the operation of control mechanisms leads to such behaviors or physiological processes that would not be possible if not for the changes that constraints make in the mechanisms of production. Controlling the production mechanisms is essential because they are constrained to do work as long as free energy is available. The same is true for artifacts. For example: turning on the on/off switch enables the user of a given machine to control it so that it can use energy and carry out its design activities (Bechtel, 2019 , p. 623). Footnote 17

Constraints understood in this way do not only (or at all) function as the context or background conditions in which a given mechanism is implemented, but most of all they are its constitutive (in the sense of being responsible for producing a given phenomenon, resp . mechanism) component because “mechanical systems inherently contain a ‘thicket’ of constraints” (Winning, 2020 , p. 20). Footnote 18

Bechtel ( 2018 , 2019 , 2021 ), Bechtel and Bollhagen ( 2021 ), Winning and Bechtel ( 2018 ), and Winning ( 2020 ) emphasize the need to refer to constraints, linking them with the necessity to include both constraints and energy flows as those elements which, apart from entities and activities, are relevant for the explanation of mechanisms at higher levels of organization. Footnote 19 It is the constraints and the flows of free energy that make living organisms “dissipative structures”, Footnote 20 which means “that they actually use the second law of thermodynamics to their advantage to maintain their organization” (Winning & Bechtel, 2018 , p. 3; cf. Moreno & Mossio, 2014 ). This way, living organisms—unlike most “things”—develop while maintaining their autonomy, rather than being degraded by the flow of energy and interaction with the environment. Footnote 21

Biological mechanisms derive their causal efficacy from being constrained systems: “An active causal power exists when a system within a larger system is internally constrained in such a way as to externally constrain under certain conditions” (Winning, 2020 , p. 28). In other words, constraints determine the causal powers of mechanisms in such a way that they direct the flows of free energy so that biological systems may remain in a state of energy non-equilibrium with the environment. Such mechanisms are part of a heterarchical network of controllers that guarantees the biological autonomy of a given system. Based on this, mechanisms are systems of constraints that restrict the flow of free energy to perform work (Bich & Bechtel, 2021 , p. 2).

Mechanisms are active and serve to maintain the autonomy of biological systems as a result of the constrained flows of free energy. Including these kinds of constraints in the explanation of activities means breaking with the standard account of mechanistic explanation (systems tradition). Footnote 22 If the energetic dimension is ignored, “at some point, such research typically bottoms out” and “this process leaves the active nature of activities unexplained” (Bechtel & Bollhagen, 2021 , p. 17) because “a completely unconstrained system will have no behaviors; it would simply be a disorganized motion of particles” (Winning & Bechtel, 2018 , p. 7). The approach that takes into account the need to refer to constraints and flows of free energy will be referred to as the ‘constraint-based mechanisms approach’ and its postulate as heuristics of constraint-based mechanisms. It is important to emphasize that this approach is not so much a break with the systems tradition, but its significant modification. Footnote 23

3.2 What about predictive processing?

In §1, I have already discussed the mechanistic commitments of PP. We can now take the next step. From the point of the view of the constraint-based mechanisms approach we should note that, if PP explains its phenomena mechanistically, then it is legitimate to ask whether the mechanistic explanations based on the PP framework include constraints and the energy dimension as their constitutive component. This is not a trivial or secondary question, because, according to the heuristics of constraint-based mechanisms, mechanistic PP should also include energy processes. This case is not obvious. Let us note, however, that there are indications that the above heuristic is used by researchers working in the PP framework.

On the one hand, many of PP’s supporters use the term “constraint” in their considerations to refer to perceptual inference in the brain. For example, “the only constraint on the brain’s causal inference is the immediate sensory input” (Hohwy, 2013 , p. 14), but “immediate sensory input is not the only constraint; there are, in addition, general beliefs about the world, specific hypotheses about the current state of the world, and ongoing sensory input” (Anderson, 2017 , p. 3) and “perceptual experience is determined by the mutual constraint between the incoming sensory signal and ongoing neural and bodily processes, and no aspect of that content can be definitively attributed to either influence” (Anderson, 2017 , p. 17). It is also worth adding that the levels of bidirectional hierarchical structure are constraints for each other (Clark, 2013 , p. 183; cf. Gordon et al., 2019 ). Conversely, some have suggested that “without independent constraints on their content, there is a significant risk of post hoc model-fitting” (Williams, 2020 , p. 1753). However, it is not clear in what sense these authors use this term and whether they use it in the same way. Footnote 24

These various uses of the concept of constraint are difficult to relate directly to the understanding of constraints as control mechanisms, which I defend in this paper. The constraints discussed by these authors, however, reveal the non-trivial commitment of PP. Namely: the functioning of predictive mechanisms depends on the existence of various types of constraints, which on the one hand limit the content of the generative model, and on the other hand, enable its adaptation to the environment, making it an effective adaptive tool to maintain the autonomy of the organism. The perspective I defend allows us to specify the functions of constraints in PP and to study them in a more systematic way. What is important is the question of how certain constraints are constitutive of predictive mechanisms. In other words, the point is to demonstrate how such and such organization of predictive mechanisms constrains free energy so that it is possible to perform the work required to generate particular phenomena, resp . predictions.

On the other hand, broadly speaking, we have to note that the findings within the FEP and NESS mathematics (expressed in the language of DST)—according to which, if something exists then it must exhibit properties as if it is optimizing a VFE— look like they coincide with the heuristics of constraint-based mechanisms whereby mechanisms are active and serve to maintain the autonomy of biological systems as a result of the constrained release of free energy. It seems that mechanistic PP should take into account the energetic dimension of predictive mechanisms. Is it really so? The full answer to this question depends on further empirical solutions, and it is certainly not only an a priori answer. Nevertheless, I argue that if the arguments presented above are correct, then it should be asked if FEP-based PP meets the requirements of the constraint-based mechanisms approach and allows one to think of predictive mechanisms as constitutive control mechanisms for autonomous systems armed with a generative model. I will devote my further analysis to answering this question.

4 What does it mean for the system that it minimizes variational free energy?

The connection between PP and the FEP raises a number of doubts, which can be reduced to two main issues: (1) the very interpretation of the FEP as a principle of modeling self-organizing systems armed with generative models; (2) the question of how the FEP determines the energetic (in the information-theoretical sense) constraint for the mechanistic PP. Let me start by outlining the first difficulty. I will devote another section to the second.

I stated earlier that under the mathematical framework of the FEP, PP looks like it coincides with the heuristics of constraint-based mechanisms. But why do I use the terms “looks like” and “as if”? Footnote 25 I do it because this is how some proponents of the FEP define its application to autonomous systems: “physical systems that look as if they encode probabilistic beliefs about the environment”; “self-organising system that looks as if it is modelling its embedding environment” or “all systems that look as if they engage in inference” (Ramstead et al., 2023 , pp. 1, 2, 18) and so on. What does the phrase “as if” mean? Simon McGregor defines its use as follows: “To say that something behaves ‘as if’ it has property X usually implies that it does not, in fact, have property X . However, there is clearly a sense in which a system possessing property X must also behave as if it had property X ; it is in this, less restrictive, sense that we intend the phrase ‘as if’. In other words, we classify both the regulation of temperature by a thermostat, and also the pursuit of prey by an eagle, as ‘as if’ agency” (McGregor, 2017 , p. 72). McGregor distinguishes between two senses of “as if”. In the first one (“instrumental”), the system can be described as if it had a given property, even though it does not actually have it, and in the second (“realistic”), it can be described as if it had a given property precisely because it has it. Footnote 26

This duality allows us to see that the use of the phrase “as if” in relation to systems that are supposed to minimize VFE can be interpreted in at least several ways: from the realistic interpretation, where VFE is a quantity (or means a quantity) that is minimized by biological systems that maintain their organization – in this approach, VFE cannot be reduced to researches’ construction or explained only in terms of the practice of modeling Footnote 27 ; to various anti-realistic or instrumental interpretations in which the FEP is a construction devised by scientists to describe the dynamics of any self-organizing system that is at NESS with its environment without any implications for their actual causal structure. In this approach, VFE looks like a quantity that relates to the models made by scientists, while the FEP serves to designate a model structure on the basis of which specific models are constructed (cf. Andrews, 2021 ). Footnote 28

The discussion so far concerning the ontological and epistemological commitments of the FEP is rich. It is worth mentioning the papers of Andrews ( 2021 , 2022 ), Bruineberg et al. ( 2021 ), Kirchhoff et al. ( 2022 ), Ramstead et al. ( 2022 ) or Van Es ( 2021 ). I will not discuss them here. However, I would like to draw attention to the fact that the mechanistic perspective adopted in this paper is realistic (see §5) and therefore imposes certain theoretical commitments on the understanding of the FEP and VFE, which bring my positions closer to a realistic interpretation of the FEP, which I will call moderate. Footnote 29 It is moderate in the sense that it assumes that systems can be described as if minimizing VFE, because they implement some causal mechanism that can be described (approximately) in terms of minimizing VFE, resp . long-term average prediction error (see §6). Footnote 30

Therefore, considering the goal I have set for myself in this paper, which is to apply the heuristics of constraint-based mechanisms in relation to PP and determine the energetic constraint for the mechanistic architecture proposed by this framework, it is important to acknowledge that the FEP provides a relevant (variational) constraint for the causal organization of all autonomous systems equipped with generative models, as explained mechanistically by PP. If this is true, than the FEP provides a relevant constraint for PP’s scheme of mechanism. Footnote 31

5 Mechanistic realism and the free energy principle

Many mechanists emphasize that there are objective structures in the world that are in some sense richer than mere aggregations of causes. Entities, their hierarchical-heterarchical organization and the operations binding them, produce mechanisms. The task of scientists is to identify and decompose them (cf. Bechtel, 2008 ; Craver, 2007 ; Craver & Darden, 2013 ; Machamer et al., 2000 ). This view can be called mechanistic realism. It is not a clear-cut position, as recently demonstrated by Dewhurst and Isaac ( 2023 ), because its ontological commitments are unclear. There is no space in this paper to discuss this issue in more detail, but I believe it is reasonable to say that the architecture implied by the heuristics of constraint-based mechanisms assumes a certain mechanistic realism in relation to the causal patterns present in the world (cf. Winning, 2020 ). In other words, the fact that production mechanisms are limited and activated in one way or another by specific constraints and flows of free energy suggests that the causal relationships between specific patterns or, in Bayesian modeling terminology, sensory signal statistics cannot be described merely as an aggregation of causes (cf. Craver & Bechtel, 2007 ). This means that there must be some facts about the structure of mechanisms that explain them and determine what mechanisms should be and what components and operations will appear at a given level of their hierarchical-heterarchical structure (Dewhurst & Isaac, 2023 ; cf. Craver, 2013 ).

Because of this realistic nature of the mechanistic explanations, I argue that if the heuristics of constraint-based mechanisms can be applied to VFE-constrained predictive mechanisms, then realism must be assumed for the FEP. Instrumentalism imposes no commitments on the explanations regarding the architecture of the mechanisms, and treats the mechanisms themselves as useful fictions. The heuristic interpretation of the FEP defended by Gładziejewski ( 2019 ) and Harkness ( 2015 ), while not excluding realism in relation to mechanisms, denies any explanatory power to the FEP. Footnote 32 In this sense, it does not allow VFE flows to be treated as significant for the functioning of predictive mechanisms.

I can now present the realistic interpretation of what it means to say that self-organizing systems minimize VFE. The moderate realistic interpretation, which I defend, does not impose strong commitments on mechanistic architecture that would involve committing the literalist fallacy (cf. Kirchhoff et al., 2022 ). Moderate realism assumes that the concepts implied by Bayesian modeling are not precisely mapped to the target phenomena. Thus, they can be treated as approximations (cf. Laudan, 1981 ; Weisberg, 2007 ). Bayesian formal structures are rather non-arbitrary (in the instrumental sense) interpretations of causal patterns in the world, which, according to mechanistic realism, have specific structures that cannot be reduced to being aggregates of causes.

I argue that the proper interpretation that allows PP to be integrated with the FEP framework in accordance with the heuristics of constraint-based mechanisms, follows a moderate realistic approach to the FEP. Why? In order to answer this question, it is necessary to look at the arguments that concern the possibility of linking VFE with TFE.

It seems that the first step in demonstrating that VFE is a relevant constraint for predictive architecture has been made. To sum up: in accordance with the realistic approach to the FEP, VFE is not only a scientists’ construct, but in a sense models the actual property of the target phenomena, which do not have to be treated as exact representations of formal structures.

We thus come to the second difficulty, which I indicated at the beginning of §4: does the FEP determine the energetic (in the information-theoretical sense) constraint for the mechanistic PP, and to what extent?

6 Is variational free energy the same thing as thermodynamic free energy?

Let us first cite the observation of William Bechtel, who explicitly states that “The notion of free energy invoked in mechanical action is distinct from the free-energy principle articulated by Friston (…). The conception of free energy required in the account of mechanisms is that appealed to in mechanics to explain work of any form” (Bechtel, 2019 , p. 634; cf. Bich & Bechtel, 2021 , p. 52). This claim seems to exclude the idea of using VFE as a constraint for mechanistic PP, at least in the sense that Bechtel and colleagues propose. However, it seems that it is doubtful, however, whether Bechtel rightly excludes Fristonian VFE. In the quoted paper, he refers to a 2010 piece by Friston. In this work, free energy is understood as “an information theory measure that bounds or limits (by being greater than) the surprise on sampling some data, given a generative model” (Friston, 2010 , p. 127) and as such it is distinguished from the thermodynamic free energy referred to by Bechtel (cf. Moreno & Mossio, 2014 ). However, in more recent papers, Friston argues, based on the mathematical relationships between non-equilibrium dynamics, variational inference, and stochastic thermodynamics, that VFE is the same as TFE, because VFE “is consistent with the notion of free energy as the thermodynamic energy available to do work when an ensemble is far from equilibrium” (Friston, 2019 , pp. 66–67; Parr et al., 2020 ). Footnote 33 This statement, as I will soon show, may raise reasonable doubts and ultimately does not justify the belief that VFE is a constraint for mechanistic PP.

What is Friston’s argument for equating VFE with TFE, and why is it important? I will start with the second point. Let us recall: the fact that cognitive mechanisms are active and can serve to maintain the autonomy and self-organization of biological systems is a result of the constrained flows of free energy. It is important to explain “how that free-energy is converted into a specific activity” (Bechtel & Bollhagen, 2021 , p. 3). It seems that Friston goes a step further: mechanisms are constrained and made active not only by the energy in the thermodynamic sense, but also the energy in the information-theoretical sense (i.e., VFE) that the system optimizes to achieve NESS (cf. Friston, 2013 ; Wiese & Friston, 2021 ). If Friston is right, then there are some phenomena that need to be explained by taking into account the energetic constraint of VFE. This means that there are mechanisms that are implemented because they minimize the VFE quantity.

Let us now return to the identification of VFE with TFE. If VFE coincides with TFE, then it looks like the FEP (as a framework for explaining minimization of VFE) is fundamental to explaining many biological and cognitive mechanisms by analogy with the scientific importance of explanations using statistical mechanics and the concept of TFE. In the latest papers, Friston and colleagues introduce the concept of Bayesian mechanics, which “is a probabilistic mechanics, comprising tools that enable us to model systems endowed with a particular partition (i.e., into particles), where the internal states (or the trajectories of internal states) of a particular system encode the parameters of beliefs about external states (or their trajectories)” (Ramstead et al., 2023 , p. 1). In other words, according to these authors, Bayesian mechanics is exactly the same as all these other mechanics but with the added variational energy constraint (i.e., the assumption of Markov blankets) (Friston, 2019 , p. 122). We will therefore take a closer look at Friston’s argument in favor of equating VFE with TFE. I will call it an argument from the Bayesian mechanics.

In this perspective, the assumption of the compatibility of TFE and VFE can entail both ontic commitments characteristic of the realistic interpretation of the FEP and epistemic commitments characteristic of instrumentalism. In the latter case, instrumentalism would imply treating TFE and VFE as constructs of scientists, or useful fictions. As I will argue in the remainder of this paper, however, that from the point of view of the constraint-based mechanisms approach to mechanistic PP, instrumentalism cannot be reconciled with mechanistic realism. This means, therefore, that the application of the heuristics of constraint-based mechanisms to mechanistic PP is possible only in two cases: either when the compatibility of TFE and VFE is justified in realistic terms, or when both of these quantities are treated as independent interpretations of such and such patterns or causal structures present in the world (cf. Weisberg, 2013 ).

6.1 The Bayesian mechanics argument

According to Friston and colleagues, the concept of VFE can only be applied on the basis of Bayesian mechanics: “At the core of Bayesian mechanics is the variational free energy principle (FEP)” (Ramstead et al., 2023 , p. 2). This belief, however, reveals a deeper assumption about the nature of mechanics: every kind of mechanics has its own reified constructs (such as thermodynamic energy, temperature, second law or very VFE). It means, Friston claims, that the existence of this type of construct is justified by a given type of mechanics (classical, statistical, quantum, or Bayesian). For example, from the point of view of quantum mechanics, the temperature construct has no object reference. According to Friston, recognizing the existence of this type of reified constructs presupposes the so-called ensemble assumption (that all particles in your ensemble are exchangeable) which entails a weak coupling between fast and slow modes (Friston, 2019 , p. 47). Footnote 34 In statistical physics or thermodynamics, the ensemble assumption is an idealization according to which there are collections of a very large number of systems in different (quantum) states with common macroscopic attributes. The ensemble is distinguished by which thermodynamic variables are held constant (cf. Gibbs, 1902 ). This means that their properties result from the laws of classical or quantum mechanics. The ensemble assumption, Friston argues, translates into a weak coupling between internal particles and their Markov blanket, which means that the states of the ensemble are partitioned so that the states of each constituent particle can be identified with the homologous states of another. This makes it possible to associate the NESS density with an ensemble density. This means that instead of describing the probability of a given particle appearing in a certain state over time, the NESS density describes a greater number of particles that occupy the same (or adjacent) states (Friston, 2019 , p. 64).

So, how does the use of the ensemble assumption in Bayesian mechanics differ from its use in other mechanics? Friston claims that Bayesian mechanics adds a variational energy constraint (i.e., assumption of Markov blankets). With this additional constraint in place, one can speak of states of something as relative to something else, which is directly applicable to living organisms or neural structures. According to Friston, only Bayesian mechanics can do this (cf. Parr et al., 2020 ). The other types of mechanics assume that a Markov blanket and the states outside the blanket can be ignored, which is related to, for example, talking about a heat bath or a thermal reservoir in terms of statistical mechanics (Friston, 2019 , p. 122).

This is where the important question arises as to why only Bayesian mechanics should allow the separation of internal and external states. Why should this not be possible to achieve through, for instance, a constraint-based mechanistic approach as understood by Bechtel and colleagues or the biological autonomy approach as characterized by Barandiaran, Moreno, Varela and so on? According to Friston and colleagues, it is important to bear in mind that the above-mentioned approaches already assume solutions that are only enabled by the Bayesian mechanics. The new mechanical philosophy of neural mechanisms or an account of biological autonomy based on autopoiesis are only possible on the basis of the ensemble assumption with a variational constraint. To be more precise: the statement that there are some mechanisms, presupposes the mechanistic nature of certain phenomena. Friston claims that without the ensemble assumption, it seems impossible. Nevertheless, it is not difficult to see that the ensemble assumption follows from the assumptions of statistical mechanics, thermodynamics, or even mechanistic realism (cf. Dewhurst & Isaac, 2023 ). However, Friston argues, it is only on the basis of Bayesian mechanics that one can recognize active mechanisms (e.g., information processing neuronal mechanisms) that are characteristic for the organization of living systems such as, for example, bacteria and our brains (Friston, 2019 , p. 1). In other words, only Bayesian mechanics allows us to explain why biological systems “ exist the way they do” (Sakthivadivel, 2022 , p. 2), i.e., indicate what physical mechanisms and constraints enable biological systems to be what they are, rather than being inanimate matter.

According to Friston, VFE can be applied only in the realm of Bayesian mechanics and thus refers to autonomous or active things, while TFE can only be applied in the realm of the statistical ensemble. Thus, both of these mechanics are based on quantum mechanics (cf. Friston et al., 2022 , pp. 5–6). For this reason, it can be said that VFE and TFE are two consequences or expressions of the same thing of a more elemental mechanistic or quantum nature. Footnote 35

6.2 Instrumental interpretation of the Bayesian mechanics argument

Note that, if the Bayesian mechanics argument is valid, then VFE is an explanatory relevant constraint for the PP’s mechanistic architecture. This means that according to the heuristics of constraint-based mechanisms, predictive mechanisms are active because they are a result of the constrained release of free energy (both in terms of TFE that crucial for physical mechanisms and VFE as constitutive for information processing neural mechanisms). From my point of view the main difficulty in accepting this argument lies in its instrumental interpretation defended by Friston and colleagues (cf. Friston, 2019 ; Friston et al., 2022 ).

In instrumental interpretation this argument assumes that TFE and VFE turn out to be two sides or aspects of some more primal dynamics, which, depending on the measurement tools, in one case reveals properties are thermodynamic, in another variational. In this sense, “Bayesian and stochastic mechanics are equivalent formulations of the same thing. One can either regard Bayesian inference is a necessary consequence of thermodynamics (i.e., gradient flows on a thermodynamic potential). Alternatively, Bayesian mechanics is a corollary of thermodynamics” (Friston, 2019 , p. 119). As Friston claims, each type of mechanics posits a different kind of reified constructs, and what they all have in common are random dynamic systems. This means that VFE and TFE can be understood as constructs that are relativized to the description and method of measurement, and each type of mechanics is a complementary description of the behavior of dynamic systems.

Therefore, we should distinguish the map (models developed by science) from the territory (what the models represent) (cf. Friston, 2019 , p. 123; Andrews, 2021 ). In the instrumental interpretation, the FEP allows for the construction of “a map of that part of the territory which behaves as if it were a map” (Ramstead et al., 2022 , p. 8). In this sense, VFE is a tool that is used to explain the dynamics of self-organizing systems (given the state of our knowledge) (Ramstead et al., 2022 , p. 17) without making any ontological commitments regarding the representational or architectural properties of these systems. Therefore, the FEP is only a tool for modeling phenomena. It is arbitrary in the sense in which the choice of measurement tools or labels to name objects is arbitrary.

From this perspective, FEP-based models address the causal structure of the world in the sense that they are epistemically useful. Their use in modeling some empirical data may speak in their favor (cf. Smith et al., 2022 ). However, it is difficult to talk about their mechanistic character in this case (at least if mechanisms are understood ontically). This means that the constructions postulated by the FEP or PP can be treated as useful fictions (cf. Ramstead et al., 2020 ; van Es, 2021 ; van Es & Hipólito, 2020 ).

We have to conclude that instrumental interpretation does not allow for a satisfactory mechanistic integration of the FEP and PP from the perspective of the constraint-based mechanisms approach, because instrumentalism imposes no mechanistic commitments regarding the causal structures under study. Therefore, it is challenging to regard it as compatible with the earlier-discussed mechanistic realism, which I deemed normative for the using of heuristics of constraint-based mechanisms (cf. §5). In such a situation, the only possible position justifying the mechanistic integration of the FEP and PP seems to be moderate realism about the FEP. Is it really so?

7 Moderate realism about the free energy principle and predictive processing

According to the moderate realistic interpretation of the FEP, the system minimizes VFE because it implements some causal mechanism that can be described (approximately) in terms of minimizing VFE. In this sense, VFE can be treated as a constraint of such active mechanisms, which researchers explain in terms of the minimization of long-term average prediction errors. In other words, there are causal structures whose organization cannot be reduced to an aggregation of causes and must be explained in terms of mechanisms constrained by quantity flows described in terms of minimizing VFE or maximizing mutual information between sensory states and internal states (cf. Friston, 2010 ; Friston et al., 2022 ). Footnote 36 In the moderate interpretation that I defend, this means that some mechanisms are systems of constraints that restrict the flow of information to perform work (cf. Bich & Bechtel, 2021 , p. 2) in such a way that they minimize the discrepancy (i.e., prediction error) between estimate-based predictions of the system and the actual sensory stimulation coming from the input to stay at NESS. Why are these systems VFE-users and not just prediction error-users? Because, the minimization of prediction errors by the approximation of Bayesian inference happens through VFE minimization (cf. §2).

The argument from neural computation supports the adoption of a moderate realistic interpretation of the FEP. According to this argument, there is a trade-off between neural information processing and thermodynamic energy consumption, the explanation of which makes it possible to understand how some states of biological systems have characteristically low Shannon entropy, which enables them to adapt and survive in the environment.

7.1 Argument from neural computation

Research on the thermodynamics of information clearly indicates the existence of a trade-off between neural information processing and thermodynamic energy consumption. There is an energetic cost of information processing (cf. Levitin, 1998 ; Niven & Laughlin, 2008 ; Sagava & Ueda, 2011 ). This energy cost can be associated both with Landauer’s principle, according to which information erasure increases the entropy of the environment, i.e., energy dissipation (Landauer, 1961 ; cf. Sartori et al., 2014 ), and with Gregory Bateson’s observation that information (a single bit of information) is a difference which makes a difference, which in the case of living organisms means that the power of a given process by metabolic energy depends precisely on the difference (information) contained in certain states of the organism. For this reason, Bateson claims that the mechanical interaction of muscles can be treated as a computational model (Bateson, 1987 , p. 322).

It has recently been shown that the minimum energy required by a biological sensor to detect a change in an environmental signal is proportional to the amount of information processed during this event (Sartori et al., 2014 ). Sengupta et al. ( 2013 ) proved that minimizing VFE is a significant constraint to the tendency to maximize both metabolic and statistical efficiency in the sense that the motivation for minimizing VFE is to maintain a constant external environment that is encoded by the physical variables measured by TFE. Thus, the reference to the VFE constraint allows for the explanation of the homeostatic nature of neural processes, which mathematically means that states of biological systems have characteristically low Shannon entropy, understood—according to the ergodic theory—as the long-term average of self-information or surprise. Without reference to informational VFE, we would not be able to explain not only the homeostatic nature of neural computational mechanisms, but also their energy consumption, which is related to their ability to transmit information (cf. Laughlin, 2001 ). In other words, from this perspective, it follows that the use of only thermodynamics to explain the work of the brain is not fully justified.

A full explanation of how the brain works, i.e., what makes neural mechanisms active and able to perform their functions, requires taking into account information constraints that can be characterized in terms of VFE minimization. They are responsible for action potentials in the brain’s sensory system, forming a neural code that efficiently represents sensory information by minimizing the number of spikes needed to transmit a given signal according to Barlow’s ( 1961 ) principle of efficient coding (cf. Abbot & Dayan, 2005 , pp. 123–150).

The argument from neural computation can be formally justified by the interpretation of Jarzynski equality (Jarzynski, 1997 ) proposed by Friston ( 2019 ). According to Friston, Jarzynski equality shows that whenever you do any belief updating by changing the information inherent in the configuration of any dynamical system (e.g., belief updating in the Bayesian generative model), there is necessarily a thermodynamic work cost. Any Bayesian belief updating involves a change in biophysical encoding of these beliefs, or any belief updating has to have a concomitant energy expenditure in terms of thermodynamic free energy. Furthermore, it is important to highlight that this thermodynamic cost we actually measure that in brain imagining using brain mapping to detect the thermodynamic activity in terms of activation foci in the brain (cf. Davatzikos et al., 2001 ). Footnote 37

I argue that both empirical and formal findings will most probably determine that there are such phenomena (e.g., the neural computations performed by brains), the explanation of which, according to the constraint-based mechanisms approach, should take into account the energetic constraint of VFE. Otherwise, such an explanation fails to capture the characteristic properties that distinguish the biotic systems that are at NESS from those that can be thermodynamically described as a heat bath.

7.2 Ontological commitments of the moderate realism

If it is true that the free energy flows constitutive of the active mechanisms can be described in terms of minimization of VFE, then it seems that there are no formal obstacles to acknowledging that the mechanistic decomposition of generative models minimizing the average prediction error should refer to the minimization of VFE as a constitutive constraint for these mechanisms. For this reason, I argue that one should adopt moderate realism about the FEP and PP. Its legitimacy is supported by explanatory considerations, integration possibilities regarding PP and perhaps other research frameworks, as well as relatively weak ontological commitments regarding the architecture of target phenomena. Moderate realism allows one to maintain the quantity of VFE without incurring the debts of adopting instrumentalism.

Let’s take a closer look at these ontological commitments that result from adopting moderate realism about PP and FEP, resp . VFE. Firstly, this position assumes that formal structures such as generative models, VFE or TFE, are interpreted as part of explanations in the ontic sense, i.e., the exhibitions “of the ways in which what is to be explained fits into natural patterns or regularities … [and] usually takes the patterns and regularities to be causal” (Salmon, 1984 , p. 293, cf. Craver, 2013 ). In this sense, moderate realism corresponds to mechanistic realism and the constraint-based mechanisms approach. In practice, this means that moderate realism does not map literally the formal structure (generative model or Bayesian network) onto the target phenomena, which would involve committing the literalist fallacy, but assumes that there are structures that cannot be reduced solely to the aggregation of causes and which implement some causal mechanism that can be described (approximately) in terms of generative models minimizing VFE, resp . long-term average prediction error. Therefore, it is important to assert that the formal structures (Bayesian modeling in our case) are such and such , because the world has genuinely causal structures, at least some of which are entities and activities organized to form mechanisms responsible for the phenomena that are described in terms of Bayesian optimization.

This view can be further elucidated through the findings of Kirchhoff, Kiverstein, and Robertson. These authors state that realism in science does not mean that all entities postulated by a given theory or model are literally true (Kirchhoff et al., 2022 , p. 12). A theory may incorporate both “OK-entities” (such as electrons and similar entities) and “supposedly non-OK-entities” (such as numbers or theoretical ideals) (Psillos, 2011 , p. 6). Consequently, it is important to acknowledge that each model includes parts that are fictional entities, which bear resemblance to target systems in various ways. These fictional entities facilitate the understanding of real system dynamics within the model (Kirchhoff et al., 2022 , p. 13), but they do not themselves represent specific causal structures in a literal sense. The expectation of a literal interpretation of fictional entities gives rise to the literalist fallacy, as mentioned earlier. One such fictional entity is VFE. Therefore, process theories like PP should be viewed as approximations of the actual causal structures or patterns in the world. They are approximations due to the inherent complexity of target systems. Hence, I argue that moderate realism posits that a given model fits the data without a literal mapping. Instead, it is approximately true in relation to the data (cf. Kirchhoff et al., 2022 , p. 16; Stanford, 2003 ).

Let us now delve into the relationship between the FEP and PP. Friston argues that Bayesian mechanics provides a “formal description of lifelike particles” (Friston, 2019 , p. 1). This means that the Bayesian mechanics, by establishing a relationship between TFE and VFE, tells researchers something about mathematical models, i.e., formal structures, and only about them. Consequently, process theories such as PP are indispensable for addressing target phenomena. In line with the stance I advocate, the existence of control mechanisms that constrain the flow of free energy (both in terms of TFE and VFE) enables the formulation of theorems regarding the interplay between state theory (the FEP) and process theory (PP). Therefore, it is crucial to distinguish between three distinct elements: the FEP as a formal principle, PP as a computational modeling framework grounded in this formal principle, and the biological systems that PP is employed to model, which are independent of the FEP.

How, then, is the transition from the FEP to target phenomena possible? On one hand, if the view presented in this paper is correct, mechanistic PP, employing the heuristics of constraint-based mechanisms, is utilized to model control mechanisms and systems. One such control system is the brain, modeled by predictive coders as a hierarchical generative model that approximates Bayesian inference. On the other hand, the relationship between VFE and TFE established by Bayesian mechanics informs us about target phenomena because computational models of these systems in PP are constructed using the mathematics of the FEP. Ultimately, this implies that the position of moderate realism concerns not only the FEP and Bayesian mechanics themselves, but rather the application of the FEP in a specific process theory, such as PP, which is a concrete FEP-based model. It is important to note that the FEP, as a formal principle, does not imply any ontological commitments or resolutions (cf. Andrews, 2021 ). Footnote 38 These commitments and resolutions arise at the level of applying the FEP through a particular process theory. The use of the constraint-based mechanisms approach justifies why such an understanding of PP should be interpreted in terms of moderate realism.

There are also further benefits of the FEP and PP interpretation presented here. According to the position defended by Friston, FEP is a (normative) state theory that things may or may not conform to it, and PP is a process theory—a hypothesis on how that principle is realized (Friston et al., 2018 , p. 21). It means that PP as the process theory provides “a possible (mechanistic) story about how the FEP is implemented in real-world, target systems” (Kirchoff et al., 2022 , p. 6). Footnote 39 The proposed mechanistic integration of PP with FEP reveals that the FEP serves as a normative theory for PP, setting a norm that mechanistically non-trivial PP models should strive to meet, assuming the utilization of the constraint-based mechanisms approach and its heuristics. According to this norm, PP models should have an energetic component if they are to be mechanistic. Footnote 40

The view I defend can be treated as a voice in the discussion on the status of PP and its relation to the FEP, because FEP not only constrains the space of possible algorithms for PP (cf. Spratling, 2017 ), but also indicates energetic constraint for the causal organization of all autonomous systems, including those that are armed with generative models and are or should be the subject of (mechanistic) explanations formulated on the basis of PP. In practice, this means that all autonomous systems that can be described in terms of (Bayesian) generative models realizing updating priors and likelihood based on (average) prediction error should be treated as if they approximate Bayesian inference constrained by VFE. In other words: FEP offers a normative framework for the PP process theory, and that the PP explains the (biologically reliable) implementation of the FEP in terms of hierarchical and heterarchical active mechanisms that implement the generative model.

7.3 Why the free energy principle is not a heuristic or a regulatory principle or an idealization

The analyses carried out in this paper allow to refer to various positions concerning the explanatory status of FEP and its relation to PP. If the approach proposed here is valid, it has certain consequences for a number of discussions among PP and FEP researchers (see §1). Due to the limited space, I can only give provisional answers to the questions raised.

Foremost, I think that the presented approach allows for a new way of describing the PP-FEP relationship. If the FEP refers to self-organizing adaptive systems, as described in DST and that are at NESS with their environment, then with the appropriate interpretation of the notion of mechanism, dynamical FEP models may in fact turn out to be descriptions of mechanisms: “dynamical models and dynamical analyses may be involved in both covering law and mechanistic explanations—what matters is not that dynamical models are used, but how they are used” (Zednik, 2008 , p. 1459). Footnote 41 In this view, the FEP provides specific constraint for a PP’s scheme of mechanism.

Therefore, it is a stronger commitment than that suggested by Gładziejewski ( 2019 ) and Harkness ( 2015 ), stating that the FEP offers (only) heuristics. The approach I propose suggests that the FEP is not so much a heuristic that can aid the process of designing experiments or constructing a space of possible mechanisms, but above all points to a constitutive constraint—VFE, which is needed “not just for mechanisms to perform work, but also to maintain the mechanisms themselves” (Winning & Bechtel, 2018 , p. 11). VFE as a constraint determines the causal powers of mechanisms in such a way that the flows of (variational) free energy guarantee that biological systems may remain in a state of energy non-equilibrium with the environment. Such mechanisms are part of a heterarchical network of controllers that guarantees the biological autonomy of a given system. From this point of view, biotic mechanisms are systems of constraints that restrict the flow of free energy to perform work. Footnote 42

For the above reasons, it is also difficult to agree with Hohwy’s thesis that the FEP is a regulatory principle. Surely Hohwy is right when he states that the “FEP itself (does not) implies cognitive architecture” and adds that “notions of architecture will need to build on assumptions about the particular system in question, which will constrain processes for message passing structure” (Hohwy, 2021 , p. 47). However, the constraint relationship is reciprocal: on one hand, a particular system constrains flow of VFE, and on the other hand, those flows constrain the system to perform given work. Therefore, the FEP, as an explication of the dynamics of flows of VFE, possesses a specific explanatory power in the explanation of cognitive phenomena, distinct from its regulatory function. Therefore, it is agreeable to conclude, following Tomasz Korbak, that the FEP can be regarded as a functional principle that offers a general framework for understanding the mechanisms involved in free energy minimization, which can then be further specified through concrete models applied to specific phenomena (Korbak, 2021 , p. 2754).

It seems that these considerations may also shed some light on a number of critical works concerning either the FEP itself or its relationship with the PP. In Introduction , I referred to the papers of Williams, Colombo, Palacios and Wright. Let us recall: Colombo and Palacios ( 2021 ) emphasize that there is an inalienable tension between the “physics assumptions and properties of its biological targets”, which in practice makes it impossible to use the FEP to explain living organisms or, in other words, to integrate it with models developed by mechanists and/or organicists (cf. Colombo & Wright, 2021 ). This objection seems to be thwarted by emphasizing, as I do in my paper, the mechanistic status of explanations of biological phenomena offered in terms of constraints and free energy flows. If, for living organisms, autonomy is a constitutive property (cf. Moreno & Mossio, 2014 ; Ruiz-Mirazo & Moreno, 2004 ; Varela, 1979 ), then the FEP—contrary to what Colombo and Palacios claim—offers specific constraints to mechanistic explanations formulated on the basis of biology and neuroscience, in the sense that it allows one to treat descriptions, using the language of DST, as sketches of mechanisms.

From this perspective, it is also difficult to agree with the belief of Colombo and Wright that the FEP offers a weak explanatory idealization. Even if, as these authors claim, the analyses carried out by FEP supporters can be treated as (weak explanatory) sketches of mechanisms, then in the light of the constraint-based mechanisms approach and arguments presented here, sketches of free energy flow mechanisms can be used in the formulation of schemes of mechanisms with specific explanatory powers.

Finally, let’s note that conducting a detailed discussion that addresses all the aforementioned positions and responds to every objection exceeds the scope of the intended framework for this analysis. Nevertheless, I believe that the general direction of the response has been set.

8 Conclusions

In this paper, I defended the view that the FEP indicates an explanatory relevant constraint (i.e., VFE) for cognitive mechanisms that can be mechanistically explained by PP. The arguments made here were based on the postulate of some mechanists about the need to include in the explanations such constitutive components as constraints for mechanisms and free energy flows. I found that the position defined by me as the constraint-based mechanisms approach has important implications for PP, because the actual research practice in this framework corresponds to the heuristics of constraint-based mechanisms and is related to those approaches that assume the FEP to be a normative framework for the process theory realized by PP. According to the presented approach, non-trivial PP models should include an energetic component, if they are to be mechanistic. The discussion presented here has great importance for considering the relationship between PP, the FEP, and Active Inference.

The advantage of the position I defend—moderate realism about the FEP and PP —is, firstly, that it implies only minimal commitments regarding the architecture of target phenomena; and secondly, it does not reduce the constructions used by scientists to their purely instrumental functions, recognizing them, for example, as useful fictions. I argue that the approach presented here may also contribute to the formulation of a mechanism scheme, which would be defined by a common predictive template combining various mechanisms under one PP flag. Last but not least, this approach (I believe) also enables fruitful discussions with those researchers who regard the FEP as an explanatory weak heuristic, idealization or regulatory idea, as well as with those who deny any explanatory power to the FEP.

Regardless of how to understand the exact causal basis of the implementation of predictive mechanisms, the mechanistic strategy of reconstructing these mechanisms by providing their sketches certainly corresponds to the actual practice of PP researchers (cf. Gordon et al., 2019 ; Keller & Mrsci-Flogel, 2018 ).

This approach is based on the recent papers of William Bechtel and colleagues and, in a sense, unifies their views as presented in various papers. The very concept of constraint-based mechanisms approach has not appeared in the literature so far and, as such, is a novelty. The same is the case with heuristics of constraint-based mechanisms, which can be taken as a distinctive feature of this approach.

In this sense, the model update proceeds in a rational manner.

VFE was introduced by Richard Feynman to solve an intractable inference problem in quantum electrodynamics (Feynman, 1998 , cf. Friston et al., 2006 , p. 221). Minimization of a computable objective function will approximate the minimization of the evidence. This evidence is always upper bounded by VFE. This means that by introducing VFE, an intractable integration problem was converted into a tractable optimization problem; namely minimizing VFE (Dayan et al., 1995 ; Friston, 2011 ). Thus, in variational inference, the model does not directly compute the intractable true posterior. Instead, it optimizes a tractable upper bound on this divergence, called the VFE. VFE is a tractable quantity because it is the discrepancy between two qualities (which we know as modeling subjects) i.e., the variational approximate posterior and the generative model. And because VFE is an upper bound, minimizing it brings us closer to true posterior.

In the light of the analyses carried out, one can invoke Jakob Hohwy’s observation that the FEP as a mathematical principle is a regulatory principle. Hohwy is probably right when he states that the FEP itself does not imply cognitive architecture (Hohwy, 2021 , p. 47). However, it is important to answer whether the FEP is a regulatory principle or has a specific explanatory power in the explanation of neurocognitive mechanisms modelled by the PP framework.

I use the term “current” because the FEP and the Active Inference framework are constantly modified by their proponents. This can of course be explained by the internal dynamics of the theory development, but for this reason, for the opponents of using this research framework “FEP can appear like a moving target, each time introducing new constructs that make the previous criticism inapplicable” (Bruineberg et al., 2021 , p. 2).

The notion of NESS comes from statistical mechanics, in which it denotes the energy dynamics between the system and the surrounding heat bath. NESS is best understood as a breach of this balance.

The full presentation of Markov blankets goes beyond these considerations, so I will only discuss them to the extent necessary for further analysis. The concept of Markov blankets comes from research on Bayesian inference, Bayesian networks, and graphical modeling (Pearl, 1988 ; cf. Bruineberg et al., 2021 ), and basically means a set of random variables which “shield” another set of random variables from other variables in the system. One set of variables (we can call them states) makes states internal to the blanket conditionally independent of external states. For a Bayesian network (described in terms of a directed acyclic graphical model) the Markov blanket comprises the parents, children, and parents of the children of a state. Markov blankets allow for the division of blanket states into internal and external states via their conditional independence. Then the blanket states can be further divided into sensory and active states where sensory states are not influenced by internal states, and active states are not influenced by external states. Internal and external states can only influence each other through a blanket (Friston, 2013 ). Understanding of Markov blankets proposed by Friston differs from that introduced by Pearl. The latter understands blankets in an instrumental way, as a mathematical construct. According to Friston, they gain an “ontic” interpretation that is not “philosophically innocent” (Bruineberg et al., 2021 ; see also: Beni, 2021 ). Without going into detail, I emphasize that in these analyses, I refer to Markov blankets in a Fristonian manner.

Information geometry is also related to the parameterizing states. Information geometry offers a formalism for describing the distance between probability distributions in an abstract space. In this space, each point represents a possible probability distribution. According to Friston ( 2019 ), all systems with NESS distribution and Markov blankets can be described in terms of information geometry (cf. Friston et al., 2020 , pp. 9–11). The analysis of this issue, however, goes beyond the scope of this paper.

Not all existing self-organizing systems are alive. The FEP also applies to such systems—non-biological agents—which have a certain degree of independence from the environment (Wiese & Friston, 2021 , p. 3).

This corresponds in some way to the concept of living organisms defended by mechanists as autonomous dissipative structures, i.e., those “that […] actually use the second law of thermodynamics to their advantage to maintain their organization” (Winning & Bechtel, 2018 , p. 3; cf. Friston & Stephan, 2007 ; Kirchhoff et al., 2018 ; Ueltzhöffer, 2019 ).

In this paper, by “explaining” I mean “constitutive explanations”.

My goal here is not to argue with models of explanations that are alternative to mechanism, or to discuss their validity, especially since there are strong arguments that dynamic models are ultimately mechanistic (cf. Bechtel & Abrahamsen, 2010 ; Kaplan & Bechtel, 2011 ; Zednik, 2008 ). I am rather interested in the discussion that took place within mechanism about the limitations of this view (cf. Bechtel, 2018 , 2019 , 2021 ; Bechtel & Bollhagen, 2021 ; Winning & Bechtel, 2018 ; Winning, 2020 ).

“In both machines and human institutions, control mechanisms are often organized hierarchically. In a hierarchy, individual control mechanisms are themselves controlled by higher-level control mechanism, with a single controller ultimately in charge. The system is organized as a pyramid. In living systems, however, control mechanisms are typically organized heterarchically” (Bich & Bechtel, 2021 , p. 2). The notion of heterarchy first introduced McCulloch ( 1945 ). See also Cumming ( 2016 ).

This is not to say that the systems tradition does not recognize the importance of constraints (cf. Craver, 2007 ; Darden, 2006 ). I do claim, however, that it treats constraints as background conditions or as factors that limit the space of possible mechanisms. In the constraint-based mechanisms approach, the constraints are primarily control mechanisms.

Certain mechanists have engaged in discussions regarding specific control mechanisms, such as circadian mechanisms (Bechtel & Abrahamsen, 2010 ) and feedback mechanisms (Bechtel, 2008 , Ch. 7). Nevertheless, they did not talk much about the effects of these mechanisms on others within certain complexes.

The concept of constraint, as used in this context, was originally proposed by Howard Pattee ( 1972 ) and David Marr ( 1982 ). Marr drew attention to the fact that specific processes can be defined by indicating and separating physical or natural constraints. The importance of Marr’s observation was not duly noted by mechanists at first, but in recent years, several authors have advocated the necessity of referring to various types of constraints, either in explaining neuronal mechanisms (cf. Weiskopf, 2016 ) or in explaining wide cognition (Miłkowski et al., 2018 ).

It is important that such a view to constraints is conditioned by the research perspective. However, an explanatory strategy that favors certain constraints at the expense of others must be distinguished from the fact that these constraints exist and define a given organism or structure (Pattee, 1972 ).

“Higher-level activities, just as those at the bottom-out level, depend upon the release of energy. Higher-level entities also constrain those at the bottom level, determining how energy released in molecular motors, ion pumps, etc. results in activities at higher levels” (2021, p. 21).

Far from the equilibrium state, these are stable stationary states, the formation of which is accompanied by an increase in order.

For the purposes of the analyses, I assume that biological autonomy and the related self-organization and integrity (which enable living organisms (systems) to achieve, maintain, and propagate a high degree of complexity) define the “situatedness” of biological systems in their environment and their “grounding” in thermodynamics. Thanks to this, biological systems do not disintegrate: they construct, maintain and replicate themselves in a changing environment. It means that an organism lives as long as it remains in an energetic non-equilibrium with the environment (cf. Friston & Stephan, 2007 ; Moreno & Mossio, 2014 ). A paradigmatic example of such a system is a living cell that uses metabolic processes to convert energy and materials from the environment into chemical energy and organic molecules, which are essential for the processes that keep the cell alive. All living autonomous organisms “must procure matter and energy from their environment and use these to construct and repair themselves” (Bich & Bechtel, 2021 , p. 1).

Earlier, Darden ( 2006 , p. 272) drew attention to this, claiming that the process of decomposition of selected mechanisms consists in constructing, evaluating and revising them in relation to empirical and experimental limitations. In other words: constraints limit the space of possible mechanisms to a specific area that the model is to reconstruct (cf. Craver, 2007 ).

This modification assumes the need to analyze (at least) some mechanisms in terms of heterarchical organization and network organization of constraints.

One can also point to the “model” understanding of the concept of constraint concerning the very architecture of model building in PP (Millidge et al., 2020 ). It is worth adding that Sprevak has recently drawn attention to the difficulties faced by PP regarding the inclusion of the explanation of constraints: “In general, it is not obvious how predictive coding should reconcile two opposing forces: (i) permitting the implementation to be complex, idiosyncratic, and varied in ways that we do not yet understand; and (ii) imposing some constraints on which physical states do and do not implement the model in order to render the view empirically testable” (Sprevak, 2021 , p. 26).

I use those two terms interchangeably in this context.

Indeed, in the latter sense (as one reviewer pointed out to me), there is no need to describe a system as behaving “as if” it had a given property if this is indeed how the system behaves. However, we can still relate the phrase “as if” to our best models or simulations and assert that the given model or simulation behaves “as if” it had a certain property of the target system. In this context, the term “as if” serves to acknowledge the use of models or simulations as approximations or representations that imitate certain aspects of the target system’s behavior.

This interpretation assumes that systems can be described as if minimizing VFE, because they implement some causal mechanism that can be described (approximately) in terms of minimizing VFE, resp. long-term average prediction error. In other words, there is a definite causal pattern that is the object of scientific interpretation.

In addition to the above-mentioned positions, one should also take account of the views of authors such as Williams ( 2021 ), Colombo and Wright ( 2021 ) or Colombo and Palacios ( 2021 ), who treat the FEP as (at best) a general idealization. Their views can be collectively described as eliminativism about the FEP.

The notion of moderate realism I have proposed can be related to some extent to Hilary Putnam’s ( 1978 , cf. Hacking, 1981 ) distinction between metaphysical realism and internal realism. The first position assumes that the world contains a specific set of objects that exist independently of the human mind and the ways of its conceptual articulation. Our theories are true if they denote what the world is like. The position of internal realism (i.e., the moderate realism I propose) assumes that objects in the world depend on accepted conceptual schemes. Thus, there may be different objects, depending on the conceptual schemes adopted. This means that there is no fixed set of objects that exists independently of conceptual schemes.

This interpretation should be distinguished from the approaches that treat self-organizing systems as literally minimizing VFE, while the use of the phrase “as if” implies that systems behave as if they were minimizing VFE, because in fact they implement the mechanism of VFE minimization, resp . long-term average prediction error (in this view, the phrase “as if” is redundant – see footnote 26). In this sense, my analysis corresponds to the critique of what Kirchhoff, Kiverstein & Robertson ( 2022 ) call the literalist fallacy. The fallacy is that the instrumentalist position is accepted or adopted due to the belief that FEP-based models are not literally mapped onto real target systems.

It is important to bear in mind that based on the difference between realistic and instrumental, resp . antirealistic approaches to the FEP, one can distinguish between free energy minimizing systems that use gradients (VFE-users) and systems that are just minimizers of those gradients (VFE-minimizers) (Kuhn, 2022 , pp. 94–95). Consequently, if there are any VFE-users that exist, they must actually minimize VFE and not just be described as minimizing VFE. This would mean that the FEP indicates an energetic constraint that has significant causal powers necessary for the implementation of specific mechanisms regulating the work of those systems. From this perspective, it is reasonable to claim that the human brain armed with the generative model is actually a VFE-user (cf. Kuhn, 2022 , p. 95).

Let us recall: in line with the classic view of Herbert Simon, heuristics strategies allow researchers to limit their investigations to particular regions within a given space (cf. Simon, 1977 ). However, it is important to emphasize that heuristics as such “cannot itself provide evidence for any particular hypothesis over an empirically equivalent alternative” (Zednik & Jäkel, 2016 , p. 3969). “They are not adequate explanations” and “often provide only the illusion of understanding a mechanism” (Craver, 2006 , pp. 361, 373).

Friston earlier integrated predictive coding with the FEP (Friston, Kilner, & Harrison, 2006 ) by identifying the Rao and Ballard’s energy function (Rao & Ballard, 1999 ) with VFE.

For example: mechanisms that underwrite self-organization rest upon bottom-up causation and top-down causation, which means top and bottom-up causation is necessary in the sense that it defines what variables and relevant variables (in the language of the renormalization group) matter (define the coupling and the shape of the coupling). Top down causation means that these variables also have a very slow dynamic, and crucially contextualize and constrain the dynamic at the lower faster level (cf. Ellis, 2012 ).

“The ensuing Bayesian mechanics is compatible with quantum, statistical, and classical mechanics and may offer a formal description of lifelike particles” (Friston, 2019 , p. 1).

Such mechanisms can be associated with the existence of systems that I previously defined after Peter Kuhn as VFE-users (see footnote 31).

Jarzynski equality can be used in two ways. Either as formal support for the argument from neural computation, or, as suggested by Friston ( 2019 ), as justification for the choice of the Bayesian mechanics as the appropriate explanatory framework for systems armed with generative models which are “shielded” by Markov blankets. The latter solution leads, of course, to the difficulties I pointed out in my discussion of and instrumentalism about the FEP.

For this reason, it can be argued that there should ultimately be no moderate realistic interpretation of the FEP itself. However, if the perspective I am advocating is correct, then the integration of the FEP with the PP based on it can be seen as part of a broader scientific view that could align with a properly developed moderate realism. This perspective largely aligns with what Kirchhoff, Kiverstein & Robertson describe as scientific realism, which asserts that one reasonable goal of our best scientific theories and models is to provide descriptions and explanations of reality that are either literally true, probably true, or approximately true (Kirchhoff et al., 2022 , p. 1).

In the sense, that „The free energy minimizing dynamics at play are implemented by different kinds of mechanisms in different individual organisms and species, as a function of the coupling between their evolved phenotypes and biobehavioural patterns and the niches they inhabit and the scales under scrutiny” (Ramstead et al., 2017 , p. 6). In this view, the FEP can be regarded as a target-directed model in the Weisberg sense ( 2013 ) (cf. Andrews, 2021 ; Kirchoff et al., 2022 ).

It is worth adding that research on systems responding to a stochastic driving signal emphasizes that there is a profound connection between the effective use of information and efficient thermodynamic operation: “any system constructed to keep memory about its environment and to operate with maximal energetic efficiency has to be predictive” (Still et al., 2012 , p. 1).

An example of this type of practice can be found, among others, in Badcock et al., ( 2019 , p. 105): “mechanisms involve a dynamic, bidirectional relationship between specialized functional processing mediated by dense, short-range connections intrinsic to that scale (i.e., its local integration); and their global (functional) integration with other neural subsystems via relatively sparse, long-range (e.g., extrinsic cortico-cortical) connections”.

“Higher-level activities, just as those at the bottom-out level, depend upon the release of energy. Higher-level entities also constrain those at the bottom level, determining how energy released in molecular motors, ion pumps, etc. results in activities at higher levels” (Bich & Bechtel, 2021 , p. 21).

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Acknowledgements

I am grateful to Majid D. Beni, Stephen Fox, Karl J. Friston, Peter Kuhn, Marcin Miłkowski, Maxwell J. D. Ramstead, Noor Sajid and Wanja Wiese for helpful comments and discussions on the draft of this paper. Previous versions of this paper was discussed at the Theoretical Neurobiology meeting, during the East European Network for Philosophy of Science 2022 in University of Tartu and at the Philosophy of Cognitive Science seminar held at the Institute of Philosophy and Sociology at the Polish Academy of Sciences. I would like to thank the organizers and participants of these events for inspiring discussions. I also thank the three anonymous reviewers for this journal for helpful discussion and for their comments on previous versions of this paper.

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Piekarski, M. Incorporating (variational) free energy models into mechanisms: the case of predictive processing under the free energy principle. Synthese 202 , 58 (2023). https://doi.org/10.1007/s11229-023-04292-2

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Biden-Harris Administration Delivers Historic Milestones, New Actions for Clean Energy on Public Lands

Surpassed milestone of permitting 25 gigawatts by 2025 

 Finalized rule to further promote responsible solar and wind energy development on public lands, including through 80% lower fees

Announced full operational status for two California solar projects that will supply 139,000 homes with clean energy

Date: Thursday, April 11, 2024 Contact:  [email protected]

WASHINGTON — The Biden-Harris administration today announced a series of historic milestones and actions to promote responsible clean energy development on public lands and help achieve President Biden’s goal of creating a carbon pollution-free power sector by 2035. 

Secretary of the Interior Deb Haaland today announced that the Department has now permitted more than 25 gigawatts of clean energy projects – surpassing a major milestone ahead of 2025 – enough clean energy to power more than 12 million homes across the country. This includes solar, wind and geothermal projects, as well as gen-tie lines on public lands that are essential for connecting clean electricity projects on both federal and non-federal land to the grid.

“Since Day One, the Biden-Harris administration has worked tirelessly to expand responsible clean energy development to address climate change, enhance America’s energy security and create good-paying union jobs. Surpassing our goal of permitting 25 gigawatts of clean energy by 2025 underscores the significant progress we have made in helping build modern, resilient climate infrastructure that protects our communities from the worsening impacts of climate change,” said Secretary Deb Haaland . “The Interior Department will continue to advance projects that will add enough clean energy to the grid to power millions more homes and help ensure a livable planet for future generations.”

The Department today also announced a final Renewable Energy rule from the Bureau of Land Management (BLM) that will lower consumer energy costs and the cost of developing solar and wind projects, improve renewable energy project application processes, and incentivize developers to continue responsibly developing solar and wind projects on public lands. Consistent with the Biden-Harris administration’s commitment to create high-quality jobs in the clean energy economy and support American manufacturing, the final rule includes additional incentives for use of project labor agreements and American-made materials. 

“Our public lands are playing a critical role in the clean energy transition,” said Acting Deputy Secretary Laura Daniel-Davis . “Finalizing the Renewable Energy rule is a significant milestone that will allow the Interior Department to continue leading the way on renewable energy while furthering President Biden’s commitment to building a clean energy economy, tackling the climate crisis, protecting lands and waters, promoting American energy security, and creating jobs in communities across the country.”

In addition, the BLM announced that two solar projects – the Arica and Victory Pass projects in California – are now fully operational, adding 465 megawatts of clean electricity to the grid. With these two projects coming online, more than 10 gigawatts of clean energy are currently being generated on public lands, powering more than 5 million homes across the West. 

“Renewable energy projects like Arica and Victory Pass on public lands create good-paying jobs and are crucial in achieving the Biden-Harris administration’s goal of a carbon pollution-free power sector by 2035,” said BLM Director Tracy Stone-Manning. “Investing in clean and reliable renewable energy represents the BLM's commitment to addressing climate change. BLM personnel are working tirelessly to efficiently review and approve projects, with significant and thoughtful engagement from states, Tribes and other partners, to ensure we supply families and communities with clean energy that will lower costs and help tackle climate change.” 

Surpassed President Biden’s Goal of 25 Gigawatts by 2025 

The Department and BLM have worked diligently to review and approve dozens of new clean energy projects, including solar, wind, and geothermal projects, as well as interconnected gen-tie lines that are vital to clean energy projects proposed on non-federal land.

Today’s announcement that the Department has surpassed the goal to permit 25 gigawatts of renewable energy includes the approval of more than double the number of projects than were approved during the previous Administration. The Department has now permitted nearly 29 gigawatts of clean energy – enough to power more than 12 million homes across the country. In addition to specific project approvals, the Department has also leased eight new areas in Solar Energy Zones with the capacity to generate nearly 2.5 gigawatts of additional clean energy. 

As the Department continues its momentum to spur a clean energy future, the BLM is currently processing permits for an additional 66 utility-scale clean energy projects proposed on public lands in the western United States. These projects have the combined potential to create thousands of good-paying jobs, add more than 32 additional gigawatts of renewable energy to the western electric grid and power millions of more homes. The BLM is also undertaking a preliminary review of about 200 applications for solar and wind development, as well as more than 100 applications for solar and wind energy site area testing. The BLM continues to track this clean energy permitting progress through an online dashboard .

These investments in a clean energy future help further the President’s Bidenomics strategy and Investing in America agenda, which are growing the American economy from the middle out and bottom up – from rebuilding our nation’s infrastructure, to driving over half a trillion dollars in new private sector manufacturing and clean energy investments in the United States, to creating good-paying jobs and building a clean energy economy that will combat the climate crisis and make our communities more resilient. 

Finalized Renewable Energy Rule to Continue Responsible Development 

The Department today also announced the update of its renewable energy regulations to promote the development of solar and wind energy on public lands. The final Renewable Energy Rule will reduce capacity fees for these projects by 80 percent and facilitate development in priority areas by streamlining application review, delivering greater certainty for the private sector and the opportunity for more clean energy for American households.

The Energy Act of 2020 authorized the BLM to reduce acreage rents and capacity fees to promote the greatest use of wind and solar energy resources. The BLM initially reduced these fees through guidance in 2022. Today’s final rule codifies further reductions, improving financial predictability for developers pursuing long-term projects on public land. 

The final Renewable Energy Rule will facilitate development in identified priority areas for wind and solar energy while maintaining appropriate flexibility to ensure a fair return for the use of public lands. It expands the BLM’s ability to accept leasing applications in these priority areas without first going through a full auction but retains the BLM’s ability to hold competitive auctions where appropriate.  

The final rule continues the Biden-Harris administration’s commitment to creating American manufacturing jobs while helping to build a clean energy economy, including by providing financial incentives for developers to use project labor agreements and domestic materials. The BLM sought comment on these additional incentives in last year’s proposed rule and developed the final provisions following public feedback, including from labor unions and a wide range of clean energy industry stakeholders.  

Today’s rule also complements the BLM’s ongoing efforts to advance responsible clean energy development by updating the Western Solar Plan. The BLM is currently taking comment on a draft analysis of the Utility-Scale Solar Energy Programmatic Environmental Impact Statement, with the goal of streamlining the BLM’s framework for siting solar energy projects across the West in order to support current and future national clean energy goals, long-term energy security, climate resilience, and improved conservation outcomes.  

Announced California Solar Projects are Fully Operational 

In another step towards achieving President Biden’s vision of a fully carbon pollution-free power sector by 2035, the Department today announced the Arica and Victory Pass solar projects are both fully operational. These projects, the first two approved under the Desert Renewable Energy Conservation Plan (DRECP), are located in eastern Riverside County, California. With the completion of these two solar projects, the BLM has also surpassed 10 gigawatts of renewable energy generation from projects on public lands. 

The two projects represent a combined infrastructure investment of about $689 million, will generate $5.9 million in annual operational economic benefit, provide power to nearly 139,000 homes, and add up to 465 megawatts of clean energy generating capacity and 400 megawatts of battery storage. The Department issued final approval for construction of the Arica and Victory Pass solar projects in 2022.  

The DRECP is a landscape-level plan created in collaboration with the State of California for more than 22 million acres, focused on 10.8 million acres of public lands, in the desert regions of seven California counties that balances renewable energy development with the conservation of unique and valuable desert ecosystems and outdoor recreation opportunities. To approve these sites for renewable energy projects, the Department and the BLM worked with Tribal governments, local communities, state regulators, industry and other federal agencies. 

The BLM today also announced the beginning of construction for the Camino Solar project in Kern County, California. The 44-megawatt solar photovoltaic facility is expected to power nearly 13,400 homes. The project will employ around 150 people during peak construction, include a 34.5-kV underground electrical collector line, and connect to the existing Southern California Edison Whirlwind Substation through the Manzana Wind Substation and associated 220 kV generation-tie line. 

Interior Department Finalizes Action to Ensure Fair Return to Taxpayers, Strengthen Accountability…

Interior Department Finalizes Action to Strengthen Endangered Species Act

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