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Despite the wealth of empirical data in neuroscience,
there are relatively few global theories about how the
brain works. A recently proposed free-energy principle
for adaptive systems tries to provide a unified account
of action, perception and learning. Although this prin-
ciple has been portrayed as a unified brain theory1, its
capacity to unify different perspectives on brain function
has yet to be established. This Review attempts to place
some key theories within the free-energy framework, in
the hope of identifying common themes. I first review
the free-energy principle and then deconstruct several
global brain theories to show how they all speak to the
same underlying idea.
The free-energy principle
The free-energy principle (BOX 1) says that any self-
organizing system that is at equilibrium with its environ-
ment must minimize its free energy2. The principle is
essentially a mathematical formulation of how adaptive
systems (that is, biological agents, like animals or brains)
resist a natural tendency to disorder3–6. What follows is
a non-mathematical treatment of the motivation and
implications of the principle. We will see that although the
motivation is quite straightforward, the implications are
complicated and diverse. This diversity allows the prin-
ciple to account for many aspects of brain structure and
function and lends it the potential to unify different per-
spectives on how the brain works. In subsequent sections,
I discuss how the principle can be applied to neuronal
systems as viewed from these perspectives. This Review
starts in a rather abstract and technical way but then tries
to unpack the basic idea in more familiar terms.
Motivation: resisting a tendency to disorder. The
defining characteristic of biological systems is that
they maintain their states and form in the face of a
constantly changing environment3–6. From the point
of view of the brain, the environment includes both
the external and the internal milieu. This maintenance
of order is seen at many levels and distinguishes bio-
logical from other self-organizing systems; indeed, the
physiology of biological systems can be reduced almost
entirely to their homeostasis7. More precisely, the rep-
ertoire of physiological and sensory states in which an
organism can be is limited, and these states define the
organism’s phenotype. Mathematically, this means that
the probability of these (interoceptive and exterocep-
tive) sensory states must have low entropy; in other
words, there is a high probability that a system will
be in any of a small number of states, and a low prob-
ability that it will be in the remaining states. Entropy
is also the average self information or ‘surprise’8
(more formally, it is the negative log-probability of an
outcome). Here, ‘a fish out of water’ would be in a sur-
prising state (both emotionally and mathematically).
A fish that frequently forsook water would have high
entropy. Note that both surprise and entropy depend
on the agent: what is surprising for one agent (for
example, being out of water) may not be surprising
for another. Biological agents must therefore mini-
mize the long-term average of surprise to ensure that
their sensory entropy remains low. In other words,
biological systems somehow manage to violate the
fluctuation theorem, which generalizes the second law
of thermodynamics9.
The Wellcome Trust Centre
for Neuroimaging,
University College London,
Queen Square, London,
WC1N 3BG, UK.
e‑mail:
k.friston@fil.ion.ucl.ac.uk
doi:10.1038/nrn2787
Published online
13 January 2010
Free energy
An information theory measure
that bounds or limits (by being
greater than) the surprise on
sampling some data, given a
generative model.
Homeostasis
The process whereby an open
or closed system regulates its
internal environment to
maintain its states within
bounds.
Entropy
The average surprise of
outcomes sampled from a
probability distribution or
density. A density with low
entropy means that, on
average, the outcome is
relatively predictable. Entropy
is therefore a measure of
uncertainty.
The free-energy principle:
a unified brain theory?
Karl Friston
Abstract | A free-energy principle has been proposed recently that accounts for action,
perception and learning. This Review looks at some key brain theories in the biological (for
example, neural Darwinism) and physical (for example, information theory and optimal
control theory) sciences from the free-energy perspective. Crucially, one key theme runs
through each of these theories — optimization. Furthermore, if we look closely at what is
optimized, the same quantity keeps emerging, namely value (expected reward, expected
utility) or its complement, surprise (prediction error, expected cost). This is the quantity that
is optimized under the free-energy principle, which suggests that several global brain
theories might be unified within a free-energy framework.
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a
b
Sensations
s
~ = g(x
~, ϑ) + z
~
Action or control signals
a = arg min F(s
~, μ)
Internal states
μ = arg min F(s
~, μ)
External states
˙
x
~ = f(x
~, a, ϑ) + w
~
Environment Agent
Free-energy bound on surprise
F = −<ln p(s
~, ϑ | m)>q + <ln q(ϑ | μ)>q
Action minimizes prediction errors
F = D(q(ϑ | μ) || p(ϑ)) − <ln p(s
~(a) | ϑ, m)>q
a = arg max Accuracy
Perception optimizes predictions
F = D(q(ϑ | μ) || p(ϑ | s
~)) − ln p(s
~ | m)
μ = arg max Divergence
Surprise
(Surprisal or self information.)
The negative log-probability of
an outcome. An improbable
outcome (for example, water
flowing uphill) is therefore
surprising.
Fluctuation theorem
(A term from statistical
mechanics.) Deals with the
probability that the entropy
of a system that is far from the
thermodynamic equilibrium
will increase or decrease over
a given amount of time. It
states that the probability of
the entropy decreasing
becomes exponentially smaller
with time.
Attractor
A set to which a dynamical
system evolves after a long
enough time. Points that
get close to the attractor
remain close, even under
small perturbations.
Kullback-Leibler divergence
(Or information divergence,
information gain or cross
entropy.) A non-commutative
measure of the non-negative
difference between two
probability distributions.
Recognition density
(Or ‘approximating conditional
density’.) An approximate
probability distribution of the
causes of data (for example,
sensory input). It is the product
of inference or inverting a
generative model.
In short, the long-term (distal) imperative — of main-
taining states within physiological bounds — translates
into a short-term (proximal) avoidance of surprise.
Surprise here relates not just to the current state, which
cannot be changed, but also to movement from one state
to another, which can change. This motion can be com-
plicated and itinerant (wandering) provided that it revis-
its a small set of states, called a global random attractor10,
that are compatible with survival (for example, driving a
car within a small margin of error). It is this motion that
the free-energy principle optimizes.
So far, all we have said is that biological agents must
avoid surprises to ensure that their states remain within
physiological bounds (see Supplementary information S1
(box) for a more formal argument). But how do they
do this? A system cannot know whether its sensations
are surprising and could not avoid them even if it did
know. This is where free energy comes in: free energy is
an upper bound on surprise, which means that if agents
minimize free energy, they implicitly minimize surprise.
Crucially, free energy can be evaluated because it is a
function of two things to which the agent has access: its
sensory states and a recognition density that is encoded
by its internal states (for example, neuronal activity
and connection strengths). The recognition density is a
probabilistic representation of what caused a particular
sensation.
This (variational) free-energy construct was
introduced into statistical physics to convert difficult
probability-density integration problems into eas-
ier optimization problems11. It is an information
theoretic quantity (like surprise), as opposed to a
thermo dynamic quantity. Variational free energy has
been exploited in machine learning and statistics to
solve many inference and learning problems12–14. In this
setting, surprise is called the (negative) model evidence.
This means that minimizing surprise is the same as
maximizing the sensory evidence for an agent’s exist-
ence, if we regard the agent as a model of its world. In
the present context, free energy provides the answer to
Box 1 | The free-energy principle
Part a of the figure shows the dependencies among the
quantities that define free energy. These include the
internal states of the brain μ(t) and quantities describing its
exchange with the environment: sensory signals (and their
motion) s˜(t) = [s,s′,s″…]T plus action a(t). The environment
is described by equations of motion, which specify the
trajectory of its hidden states. The causes ϑ ⊃{x˜ , θ, γ} of
sensory input comprise hidden states x˜ (t), parameters θ
and precisions γ controlling the amplitude of the random
fluctuations z˜ (t) and w˜ (t). Internal brain states and action
minimize free energy F(s˜ ,μ), which is a function of sensory
input and a probabilistic representation q(ϑ|μ) of its causes.
This representation is called the recognition density and is
encoded by internal states μ.
The free energy depends on two probability densities:
the recognition density q(ϑ|μ) and one that generates
sensory samples and their causes, p(s˜ ,ϑ|m). The latter
represents a probabilistic generative model (denoted by
m), the form of which is entailed by the agent or brain.
Part b of the figure provides alternative expressions for the
free energy to show what its minimization entails: action
can reduce free energy only by increasing accuracy (that is,
selectively sampling data that are predicted). Conversely,
optimizing brain states makes the representation an
approximate conditional density on the causes of sensory
input. This enables action to avoid surprising sensory
encounters. A more formal description is provided below.
Optimizing the sufficient statistics (representations)
Optimizing the recognition density makes it a posterior or conditional density on the causes of sensory data: this can be
seen by expressing the free energy as surprise –In p(s˜ ,| m) plus a Kullback-Leibler divergence between the recognition and
conditional densities (encoded by the ‘internal states’ in the figure). Because this difference is always positive, minimizing
free energy makes the recognition density an approximate posterior probability. This means the agent implicitly infers or
represents the causes of its sensory samples in a Bayes-optimal fashion. At the same time, the free energy becomes a tight
bound on surprise, which is minimized through action.
Optimizing action
Acting on the environment by minimizing free energy enforces a sampling of sensory data that is consistent with the
current representation. This can be seen with a second rearrangement of the free energy as a mixture of accuracy and
complexity. Crucially, action can only affect accuracy (encoded by the ‘external states’ in the figure). This means that
the brain will reconfigure its sensory epithelia to sample inputs that are predicted by the recognition density — in other
words, to minimize prediction error.
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Generative model
A probabilistic model (joint
density) of the dependencies
between causes and
consequences (data), from
which samples can be
generated. It is usually
specified in terms of the
likelihood of data, given their
causes (parameters of a model)
and priors on the causes.
Conditional density
(Or posterior density.) The
probability distribution of
causes or model parameters,
given some data; that is, a
probabilistic mapping from
observed data to causes.
Prior
The probability distribution or
density of the causes of data
that encodes beliefs about
those causes before observing
the data.
Bayesian surprise
A measure of salience based
on the Kullback-Leibler
divergence between the
recognition density (which
encodes posterior beliefs) and
the prior density. It
measures the information that
can be recognized in the data.
Bayesian brain hypothesis
The idea that the brain uses
internal probabilistic
(generative) models to update
posterior beliefs, using sensory
information, in an
(approximately) Bayes-optimal
fashion.
Analysis by synthesis
Any strategy (in speech coding)
in which the parameters of a
signal coder are evaluated by
decoding (synthesizing) the
signal and comparing it with
the original input signal.
Epistemological automata
Possibly the first theory for why
top-down influences (mediated
by backward connections in
the brain) might be important
in perception and cognition.
Empirical prior
A prior induced by hierarchical
models; empirical priors
provide constraints on the
recognition density in the usual
way but depend on the data.
a fundamental question: how do self-organizing adap-
tive systems avoid surprising states? They can do this by
minimizing their free energy. So what does this involve?
Implications: action and perception. Agents can
suppress free energy by changing the two things it depends
on: they can change sensory input by acting on the world
or they can change their recognition density by chang-
ing their internal states. This distinction maps nicely
onto action and perception (BOX 1). One can see what this
means in more detail by considering three mathematically
equivalent formulations of free energy (see Supplementary
information S2 (box) for a mathematical treatment).
The first formulation expresses free energy as energy
minus entropy. This formulation is important for three
reasons. First, it connects the concept of free energy as
used in information theory with concepts used in sta-
tistical thermodynamics. Second, it shows that the free
energy can be evaluated by an agent because the energy
is the surprise about the joint occurrence of sensations
and their perceived causes, whereas the entropy is sim-
ply that of the agent’s own recognition density. Third, it
shows that free energy rests on a generative model of the
world, which is expressed in terms of the probability of a
sensation and its causes occurring together. This means
that an agent must have an implicit generative model of
how causes conspire to produce sensory data. It is this
model that defines both the nature of the agent and the
quality of the free-energy bound on surprise.
The second formulation expresses free energy as
surprise plus a divergence term. The (perceptual) diver-
gence is just the difference between the recognition den-
sity and the conditional density (or posterior density) of the
causes of a sensation, given the sensory signals. This con-
ditional density represents the best possible guess about
the true causes. The difference between the two densities
is always non-negative and free energy is therefore an
upper bound on surprise. Thus, minimizing free energy
by changing the recognition density (without changing
sensory data) reduces the perceptual divergence, so that
the recognition density becomes the conditional density
and the free energy becomes surprise.
The third formulation expresses free energy as com-
plexity minus accuracy, using terms from the model
comparison literature. Complexity is the difference
between the recognition density and the prior density
on causes; it is also known as Bayesian surprise15 and is the
difference between the prior density — which encodes
beliefs about the state of the world before sensory data are
assimilated — and posterior beliefs, which are encoded
by the recognition density. Accuracy is simply the sur-
prise about sensations that are expected under the recog-
nition density. This formulation shows that minimizing
free energy by changing sensory data (without changing
the recognition density) must increase the accuracy of
an agent’s predictions. In short, the agent will selectively
sample the sensory inputs that it expects. This is known
as active inference16. An intuitive example of this process
(when it is raised into consciousness) would be feeling
our way in darkness: we anticipate what we might touch
next and then try to confirm those expectations.
In summary, the free energy rests on a model of how
sensory data are generated and on a recognition density
on the model’s parameters (that is, sensory causes). Free
energy can be reduced only by changing the recognition
density to change conditional expectations about what is
sampled or by changing sensory samples (that is, sensory
input) so that they conform to expectations. In what fol-
lows, I consider these implications in light of some key
theories about the brain.
The Bayesian brain hypothesis
The Bayesian brain hypothesis17 uses Bayesian probability
theory to formulate perception as a constructive process
based on internal or generative models. The underlying
idea is that the brain has a model of the world18–22 that
it tries to optimize using sensory inputs23–28. This idea is
related to analysis by synthesis20 and epistemological autom-
ata19. In this view, the brain is an inference machine that
actively predicts and explains its sensations18,22,25. Central
to this hypothesis is a probabilistic model that can gener-
ate predictions, against which sensory samples are tested
to update beliefs about their causes. This generative
model is decomposed into a likelihood (the probability of
sensory data, given their causes) and a prior (the a priori
probability of those causes). Perception then becomes the
process of inverting the likelihood model (mapping from
causes to sensations) to access the posterior probability of
the causes, given sensory data (mapping from sensations
to causes). This inversion is the same as minimizing the
difference between the recognition and posterior densi-
ties to suppress free energy. Indeed, the free-energy for-
mulation was developed to finesse the difficult problem
of exact inference by converting it into an easier optimi-
zation problem11–14. This has furnished some powerful
approximation techniques for model identification and
comparison (for example, variational Bayes or ensemble
learning29). There are many interesting issues that attend
the Bayesian brain hypothesis, which can be illuminated
by the free-energy principle; we will focus on two.
The first is the form of the generative model and
how it manifests in the brain. One criticism of Bayesian
treatments is that they ignore the question of how prior
beliefs, which are necessary for inference, are formed27.
However, this criticism dissolves with hierarchical
generative models, in which the priors themselves are
optimized26,28. In hierarchical models, causes in one
level generate subordinate causes in a lower level; sen-
sory data per se are generated at the lowest level (BOX 2).
Minimizing the free energy effectively optimizes empiri-
cal priors (that is, the probability of causes at one level,
given those in the level above). Crucially, because empir-
ical priors are linked hierarchically, they are informed
by sensory data, enabling the brain to optimize its prior
expectations online. This optimization makes every level
in the hierarchy accountable to the others, furnishing an
internally consistent representation of sensory causes at
multiple levels of description. Not only do hierarchical
models have a key role in statistics (for example, ran-
dom effects and parametric empirical Bayes models30,31),
they may also be used by the brain, given the hierarchical
arrangement of cortical sensory areas32–34.
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Sensory
input ξv
(1)
ξx
(1)
˙μv
(i)=Dμv
(i)(i + 1)(i)(i)
− (∂vε)Tξξ
v
−
˙μx
(i)=Dμx
(i)(i)(i)
− (∂xε)Tξ
�μθij = −∂θijεTξ�μγi = ½tr(∂γi
Π(ξξT − Π(μγ)))
ξv
(2)
ξv
(3)
ξx
(2)
Lower cortical areas Higher cortical areas
Synaptic plasticity Synaptic gain
μx
(1)
μv
(1)
μx
(2)
μv
(2)
s
~(t)
Forward:
prediction
error
Backward:
predictions
ξv
(i)=Πv
(i)=Πv
(i)(i)
εv
(i)(μv– g(μ))
(i – 1)
ξx
(i)=Πx
(i)=Πx
(i)(i)
εx
(i)(Dμx– f(μ))
(i )
The second issue is the form of the recognition den-
sity that is encoded by physical attributes of the brain,
such as synaptic activity, efficacy and gain. In general,
any density is encoded by its sufficient statistics (for exam-
ple, the mean and variance of a Gaussian form). The way
the brain encodes these statistics places important con-
straints on the sorts of schemes that underlie recognition:
they range from free-form schemes (for example, particle
filtering26 and probabilistic population codes35–38),
which use a vast number of sufficient statistics, to sim-
pler forms, which make stronger assumptions about
the shape of the recognition density, so that it can be
encoded with a small number of sufficient statistics. The
simplest assumed form is Gaussian, which requires only
the conditional mean or expectation — this is known
as the Laplace assumption39, under which the free energy
is just the difference between the model’s predictions
and the sensations or representations that are predicted.
Minimizing free energy then corresponds to explaining
away prediction errors. This is known as predictive coding
and has become a popular framework for understand-
ing neuronal message passing among different levels of
cortical hierarchies40. In this scheme, prediction error
units compare conditional expectations with top-down
predictions to elaborate a prediction error. This predic-
tion error is passed forward to drive the units in the
level above that encode conditional expectations which
optimize top-down predictions to explain away (reduce)
prediction error in the level below. Here, explaining
away just means countering excitatory bottom-up
inputs to a prediction error neuron with inhibitory syn-
aptic inputs that are driven by top-down predictions
(see BOX 2 and REFS 41,42 for detailed discussion). The
reciprocal exchange of bottom-up prediction errors and
top-down predictions proceeds until prediction error
is minimized at all levels and conditional expectations
are optimized. This scheme has been invoked to explain
many features of early visual responses40,43 and provides
a plausible account of repetition suppression and mis-
match responses in electrophysiology44. FIGURE 1 pro-
vides an example of perceptual categorization that uses
this scheme.
Message passing of this sort is consistent with func-
tional asymmetries in real cortical hierarchies45, where
forward connections (which convey prediction errors)
are driving and backwards connections (which model
the nonlinear generation of sensory input) have both
driving and modulatory characteristics46. This asym-
metrical message passing is also a characteristic feature
of adaptive resonance theory47,48, which has formal simi-
larities to predictive coding.
In summary, the theme underlying the Bayesian brain
and predictive coding is that the brain is an inference
engine that is trying to optimize probabilistic representa-
tions of what caused its sensory input. This optimization
can be finessed using a (variational free-energy) bound
on surprise. In short, the free-energy principle entails
the Bayesian brain hypothesis and can be implemented
by the many schemes considered in this field. Almost
invariably, these involve some form of message passing
or belief propagation among brain areas or units. This
Box 2 | Hierarchical message passing in the brain
The figure details a neuronal architecture that optimizes the conditional expectations of
causes in hierarchical models of sensory input. It shows the putative cells of origin of forward
driving connections that convey prediction error (grey arrows) from a lower area (for
example, the lateral geniculate nucleus) to a higher area (for example, V1), and nonlinear
backward connections (black arrows) that construct predictions41. These predictions try to
explain away prediction error in lower levels. In this scheme, the sources of forward and
backward connections are superficial and deep pyramidal cells (upper and lower triangles),
respectively, where state units are black and error units are grey. The equations represent a
gradient descent on free energy using the generative model below. The two upper equations
describe the formation of prediction error encoded by error units, and the two lower
equations represent recognition dynamics, using a gradient descent on free energy.
Generative models in the brain
To evaluate free energy one needs a generative model of how the sensorium is caused.
Such models p(s˜ ,ϑ) = p(s˜ | ϑ) p(ϑ) combine the likelihood p(s˜ | ϑ) of getting some data given
their causes and the prior beliefs about these causes, p(ϑ). The brain has to explain
complicated dynamics on continuous states with hierarchical or deep causal structure
and may use models with the following form
Nature Reviews | Neuroscience
Z
· HZX θY
U IZX θ\
Z
· K HZKXK θKYK
XKs IZKXK θK\K
……
Here, g(i) and f(i) are continuous nonlinear functions of (hidden and causal) states, with
parameters θ(i). The random fluctuations z(t)(i) and w(t)(i) play the part of observation
noise at the sensory level and state noise at higher levels. Causal states v(t)(i) link
hierarchical levels, where the output of one level provides input to the next. Hidden
states x(t)(i) link dynamics over time and endow the model with memory.
Gaussian assumptions about the random fluctuations specify the likelihood
and Gaussian assumptions about state noise furnish empirical priors in terms of
predicted motion. These assumptions are encoded by their precision (or inverse
variance), П(i)(γ), which are functions of precision parameters γ.
Recognition dynamics and prediction error
If we assume that neuronal activity encodes the conditional expectation of states, then
recognition can be formulated as a gradient descent on free energy. Under Gaussian
assumptions, these recognition dynamics can be expressed compactly in terms
of precision-weighted prediction errors ξ(i) = П(i)(ε)(i) on the causal states and motion of
hidden states. The ensuing equations (see the figure) suggest two neuronal populations
that exchange messages: causal or hidden-state units encoding expected states and
error units encoding prediction error. Under hierarchical models, error units receive
messages from the state units in the same level and the level above, whereas state units
are driven by error units in the same level and the level below. These provide bottom-up
messages that drive conditional expectations μ(i) towards better predictions, which
explain away prediction error. These top-down predictions correspond to g(μ(i)) and f(μ(i)).
This scheme suggests that the only connections that link levels are forward connections
conveying prediction error to state units and reciprocal backward connections that
mediate predictions. See REFS 42,130 for details. Figure is modified from REF. 42.
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a Perceptual inference
b Perceptual categorization
c
0 0.2 0.4 0.6 0.8 1
–20
–10
0
10
20
30
40
50
5,000
4,000
3,000
2,000
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
a
b
c
Time (s)
Time (s)
Estimated causes
Frequency (Hz)
µv1
µv1
10 15 20 25 30 35
1
1.5
2
2.5
3
3.5
a
b
c
v2
v1
Song a Song b Song c
v = v2
v1
Vocal centre Syrinx Sonogram
˙x = f(x, v) = v1x1 − 2x3x1 − x2
18x2 − 18x1
2x1x2 − v2x3
allows us to connect the free-energy principle to another
principled approach to sensory processing, namely
information theory.
The principle of efficient coding
The principle of efficient coding suggests that the brain
optimizes the mutual information (that is, the mutual
predictability) between the sensorium and its internal
representation, under constraints on the efficiency of
those representations. This line of thinking was articu-
lated by Barlow49 in terms of a redundancy reduction
principle (or principle of efficient coding) and formal-
ized later in terms of the infomax principle50. It has been
applied in machine learning51, leading to methods
like independent component analysis52, and in neuro-
biology, contributing to an understanding of the nature
of neuronal responses53–56. This principle is extremely
effective in predicting the empirical characteristics of
classical receptive fields53 and provides a principled
explanation for sparse coding55 and the segregation of
processing streams in visual hierarchies57. It has been
extended to cover dynamics and motion trajectories58,59
and even used to infer the metabolic constraints on neu-
ronal processing60.
At its simplest, the infomax principle says that
neuronal activity should encode sensory information in
an efficient and parsimonious fashion. It considers the
mapping between one set of variables (sensory states)
and another (variables representing those states). At
first glance, this seems to preclude a probabilistic repre-
sentation, because this would involve mapping between
sensory states and a probability density. However, the
infomax principle can be applied to the sufficient sta-
tistics of a recognition density. In this context, the info-
max principle becomes a special case of the free-energy
principle, which arises when we ignore uncertainty
in probabilistic representations (and when there is no
action); see Supplementary information S3 (box) for
mathematical details). This is easy to see by noting that
sensory signals are generated by causes. This means that it
is sufficient to represent the causes to predict these
signals. More formally, the infomax principle can be
understood in terms of the decomposition of free energy
into complexity and accuracy: mutual information is
optimized when conditional expectations maximize
accuracy (or minimize prediction error), and efficiency
is assured by minimizing complexity. This ensures that
no excessive parameters are applied in the generative
model and leads to a parsimonious representation of
sensory data that conforms to prior constraints on their
causes. Interestingly, advanced model-optimization
techniques use free-energy optimization to eliminate
redundant model parameters61, suggesting that free-
energy optimization might provide a nice explanation
for the synaptic pruning and homeostasis that take place
in the brain during neurodevelopment62 and sleep63.
The infomax principle pertains to a forward mapping
from sensory input to representations. How does this
square with optimizing generative models, which map
from causes to sensory inputs? These perspectives can be
reconciled by noting that all recognition schemes based
Figure 1 | Birdsongs and perceptual categorization. a | The generative model of
birdsong used in this simulation comprises a Lorenz attractor with two control parameters
(or causal states) (v1,v2), which, in turn, delivers two control parameters (not shown) to a
synthetic syrinx to produce ‘chirps’ that were modulated in amplitude and frequency (an
example is shown as a sonogram). The chirps were then presented as a stimulus to a
synthetic bird to see whether it could infer the underlying causal states and thereby
categorize the song. This entails minimizing free energy by changing the internal
representation (v1,v2) of the control parameters. Examples of this perceptual inference or
categorization are shown below. b | Three simulated songs are shown in sonogram format.
Each comprises a series of chirps, the frequency and number of which fall progressively
from song a to song c, as a causal state (known as the Raleigh number; v1 in part a) is
decreased. c | The graph on the left depicts the conditional expectations (v1,v2) of the
causal states, shown as a function of peristimulus time for the three songs. It shows that
the causes are identified after around 600 ms with high conditional precision (90%
confidence intervals are shown in grey). The graph on the right shows the conditional
density on the causes shortly before the end of the peristimulus time (that is, the dotted
line in the left panel). The blue dots correspond to conditional expectations and the grey
areas correspond to the 90% conditional confidence regions. Note that these encompass
the true values (red dots) of (v1,v2) that were used to generate the songs. These results
illustrate the nature of perceptual categorization under the inference scheme in BOX 2:
here, recognition corresponds to mapping from a continuously changing and chaotic
sensory input to a fixed point in perceptual space. Figure is reproduced, with permission,
from REF. 130 © (2009) Elsevier.
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Sufficient statistics
Quantities that are sufficient to
parameterize a probability
density (for example, mean and
covariance of a Gaussian
density).
Laplace assumption
(Or Laplace approximation or
method.) A saddle-point
approximation of the integral
of an exponential function, that
uses a second-order Taylor
expansion. When the function
is a probability density, the
implicit assumption is that
the density is approximately
Gaussian.
Predictive coding
A tool used in signal processing
for representing a signal using
a linear predictive (generative)
model. It is a powerful speech
analysis technique and was
first considered in vision to
explain lateral interactions in
the retina.
Infomax
An optimization principle for
neural networks (or functions)
that map inputs to outputs. It
says that the mapping should
maximize the Shannon mutual
information between the inputs
and outputs, subject to
constraints and/or noise
processes.
Stochastic
Governed by random effects.
Biased competition
An attentional effect mediated
by competitive interactions
among neurons representing
visual stimuli; these
interactions can be biased in
favour of behaviourally relevant
stimuli by both spatial and
non-spatial and both
bottom-up and top-down
processes.
on infomax can be cast as optimizing the parameters of a
generative model64. For example, in sparse coding mod-
els55, the implicit priors posit independent causes that
are sampled from a heavy-tailed or sparse distribution42.
The fact that these models predict empirically observed
receptive fields so well suggests that we are endowed
with (or acquire) prior expectations that the causes of
our sensations are largely independent and sparse.
In summary, the principle of efficient coding says
that the brain should optimize the mutual information
between its sensory signals and some parsimonious
neuronal representations. This is the same as optimizing
the parameters of a generative model to maximize the
accuracy of predictions, under complexity constraints.
Both are mandated by the free-energy principle, which
can be regarded as a probabilistic generalization of the
infomax principle. We now turn to more biologically
inspired ideas about brain function that focus on neu-
ronal dynamics and plasticity. This takes us deeper into
neurobiological mechanisms and the implementation of
the theoretical principles outlined above.
The cell assembly and correlation theory
The cell assembly theory was proposed by Hebb65 and
entails Hebbian — or associative — plasticity, which is a
cornerstone of use-dependent or experience-dependent
plasticity66, the correlation theory of von de Malsburg67,68
and other formal refinements to Hebbian plasticity
per se69. The cell assembly theory posits that groups of
interconnected neurons are formed through a strength-
ening of synaptic connections that depends on corre-
lated pre- and postsynaptic activity; that is, ‘cells that fire
together wire together’. This enables the brain to distil
statistical regularities from the sensorium. The correla-
tion theory considers the selective enabling of synaptic
efficacy and its plasticity (also known as metaplastic-
ity70) by fast synchronous activity induced by different
perceptual attributes of the same object (for example, a
red bus in motion). This resolves a putative deficiency
of classical plasticity, which cannot ascribe a presynaptic
input to a particular cause (for example, redness) in the
world67. The correlation theory underpins theoretical
treatments of synchronized brain activity and its role in
associating or binding attributes to specific objects or
causes68,71. Another important field that rests on associa-
tive plasticity is the use of attractor networks as models
of memory formation and retrieval72–74. So how do corre-
lations and associative plasticity figure in the free-energy
formulation?
Hitherto, we have considered only inference on states
of the world that cause sensory signals, whereby condi-
tional expectations about states are encoded by synaptic
activity. However, the causes covered by the recognition
density are not restricted to time-varying states (for
example, the motion of an object in the visual field):
they also include time-invariant regularities that endow
the world with causal structure (for example, objects
fall with constant acceleration). These regularities are
parameters of the generative model and have to be
inferred by the brain — in other words, the conditional
expectations of these parameters that may be encoded
by synaptic efficacy (these are θ in BOX 2) have to be
optimized. This corresponds to optimizing connection
strengths in the brain — that is, plasticity that under-
lines learning. So what form would this learning take? It
transpires that a gradient descent on free energy (that is,
changing connections to reduce free energy) is formally
identical to Hebbian plasticity28,42 (BOX 2). This is because
the parameters of the generative model determine how
expected states (synaptic activity) are mixed to form pre-
dictions. Put simply, when the presynaptic predictions
and postsynaptic prediction errors are highly correlated,
the connection strength increases, so that predictions
can suppress prediction errors more efficiently.
In short, the formation of cell assemblies reflects the
encoding of causal regularities. This is just a restate-
ment of cell assembly theory in the context of a specific
implementation (predictive coding) of the free-energy
principle. It should be acknowledged that the learning
rule in predictive coding is really a delta rule, which
rests on Hebbian mechanisms; however, Hebb’s wider
notions of cell assemblies were formulated from a non-
statistical perspective. Modern reformulations suggest
that both inference on states (that is, perception) and
inference on parameters (that is, learning) minimize
free energy (that is, minimize prediction error) and
serve to bound surprising exchanges with the world. So
what about synchronization and the selective enabling
of synapses?
Biased competition and attention
Causal regularities encoded by synaptic efficacy
control the deterministic evolution of states in the world.
However, stochastic (that is, random) fluctuations in
these states play an important part in generating sen-
sory data. Their amplitude is usually represented as pre-
cision (or inverse variance), which encodes the reliability
of prediction errors. Precision is important, especially
in hierarchical schemes, because it controls the relative
influence of bottom-up prediction errors and top-down
predictions. So how is precision encoded in the brain?
In predictive coding, precision modulates the amplitude
of prediction errors (these are γ in BOX 2), so that pre-
diction errors with high precision have a greater impact
on units that encode conditional expectations. This
means that precision corresponds to the synaptic gain of
prediction error units. The most obvious candidates for
controlling gain (and implicitly encoding precision) are
classical neuromodulators like dopamine and acetylcho-
line, which provides a nice link to theories of attention
and uncertainty75–77. Another candidate is fast synchro-
nized presynaptic input that lowers effective postsynaptic
membrane time constants and increases synchronous
gain78. This fits comfortably with the correlation theory
and speaks to recent ideas about the role of synchronous
activity in mediating attentional gain79,80.
In summary, the optimization of expected precision
in terms of synaptic gain links attention to synaptic gain
and synchronization. This link is central to theories of
attentional gain and biased competition80–85, particularly
in the context of neuromodulation86,87. The theories
considered so far have dealt only with perception.
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Reentrant signalling
Reciprocal message passing
among neuronal groups.
Reinforcement learning
An area of machine learning
concerned with how an agent
maximizes long-term reward.
Reinforcement learning
algorithms attempt to find a
policy that maps states of the
world to actions performed by
the agent.
Optimal control theory
An optimization method
(based on the calculus of
variations) for deriving an
optimal control law in a
dynamical system. A control
problem includes a cost
function that is a function of
state and control variables.
Bellman equation
(Or dynamic programming
equation.) Named after
Richard Bellman, it is a
necessary condition for
optimality associated with
dynamic programming in
optimal control theory.
However, from the point of view of the free-energy
principle, perception just makes free energy a good
proxy for surprise. To actually reduce surprise we need
to act. In the next section, we retain a focus on cell
assemblies but move to the selection and reinforcement
of stimulus–response links.
Neural Darwinism and value learning
In the theory of neuronal group selection88, the emergence
of neuronal assemblies is considered in the light of selec-
tive pressure. The theory has four elements: epigenetic
mechanisms create a primary repertoire of neuronal
connections, which are refined by experience-dependent
plasticity to produce a secondary repertoire of neuro-
nal groups. These are selected and maintained through
reentrant signalling among neuronal groups. As in cell
assembly theory, plasticity rests on correlated pre- and
postsynaptic activity, but here it is modulated by value.
Value is signalled by ascending neuromodulatory trans-
mitter systems and controls which neuronal groups
are selected and which are not. The beauty of neural
Darwinism is that it nests distinct selective processes
within each other. In other words, it eschews a single unit
of selection and exploits the notion of meta-selection
(the selection of selective mechanisms; for example, see
REF. 89). In this context, (neuronal) value confers evolu-
tionary value (that is, adaptive fitness) by selecting neu-
ronal groups that meditate adaptive stimulus–stimulus
associations and stimulus–response links. The capacity
of value to do this is assured by natural selection, in the
sense that neuronal value systems are themselves subject
to selective pressure.
This theory, particularly value-dependent learning90,
has deep connections with reinforcement learning and
related approaches in engineering (see below), such as
dynamic programming and temporal difference mod-
els91,92. This is because neuronal value systems reinforce
connections to themselves, thereby enabling the brain
to label a sensory state as valuable if, and only if, it leads to
another valuable state. This ensures that agents move
through a succession of states that have acquired value to
access states (rewards) with genetically specified innate
value. In short, the brain maximizes value, which may be
reflected in the discharge of value systems (for example,
dopaminergic systems92–96). So how does this relate to
the optimization of free energy?
The answer is simple: value is inversely proportional
to surprise, in the sense that the probability of a pheno-
type being in a particular state increases with the value
of that state. Furthermore, the evolutionary value of
a phenotype is the negative surprise averaged over all
the states it experiences, which is simply its negative
entropy. Indeed, the whole point of minimizing free
energy (and implicitly entropy) is to ensure that agents
spend most of their time in a small number of valuable
states. This means that free energy is the complement of
value, and its long-term average is the complement of
adaptive fitness (also known as free fitness in evolution-
ary biology97). But how do agents know what is valu-
able? In other words, how does one generation tell the
next which states have value (that is, are unsurprising)?
Value or surprise is determined by the form of an agent’s
generative model and its implicit priors — these specify
the value of sensory states and, crucially, are heritable
through genetic and epigenetic mechanisms. This means
that prior expectations (that is, the primary repertoire)
can prescribe a small number of attractive states with
innate value. In turn, this enables natural selection to
optimize prior expectations and ensure they are con-
sistent with the agent’s phenotype. Put simply, valuable
states are just the states that the agent expects to fre-
quent. These expectations are constrained by the form of
its generative model, which is specified genetically and
fulfilled behaviourally, under active inference.
It is important to appreciate that prior expectations
include not just what will be sampled from the world but
also how the world is sampled. This means that natural
selection may equip agents with the prior expectation
that they will explore their environment until states
with innate value are encountered. We will look at this
more closely in the next section, where priors on motion
through state space are cast in terms of policies in
reinforcement learning.
Both neural Darwinism and the free-energy principle
try to understand somatic changes in an individual in
the context of evolution: neural Darwinism appeals to
selective processes, whereas the free energy formulation
considers the optimization of ensemble or population
dynamics in terms of entropy and surprise. The key
theme that emerges here is that (heritable) prior expecta-
tions can label things as innately valuable (unsurprising);
but how can simply labelling states engender adaptive
behaviour? In the next section, we return to reinforce-
ment learning and related formulations of action that try
to explain adaptive behaviour purely in terms of labels
or cost functions.
Optimal control theory and game theory
Value is central to theories of brain function that are
based on reinforcement learning and optimum con-
trol. The basic notion that underpins these treatments
is that the brain optimizes value, which is expected
reward or utility (or its complement — expected loss
or cost). This is seen in behavioural psychology as rein-
forcement learning98, in computational neuroscience
and machine learning as variants of dynamic program-
ming such as temporal difference learning99–101, and in
economics as expected utility theory102. The notion of
an expected reward or cost is crucial here; this is the
cost expected over future states, given a particular policy
that prescribes action or choices. A policy specifies the
states to which an agent will move from any given state
(‘motion through state space in continuous time’). This
policy has to access sparse rewarding states using a cost
function, which only labels states as costly or not. The
problem of how the policy is optimized is formalized
in optimal control theory as the Bellman equation and its
variants99 (see Supplementary information S4 (box)),
which express value as a function of the optimal policy
and a cost function. If one can solve the Bellman equa-
tion, one can associate each sensory state with a value
and optimize the policy by ensuring that the next state
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svisual =+ wvisual
J
V
sprop =+ wprop
x2
x1
V = (v1, v2, v3)
J = J1 + J2 = ( j1, j2)
Movement
trajectory
(0, 0)
x1
J1
J2
x2
Jointed arm
a
ξv
(1)
ξv
(1)
ξx
(1)
ξv
(2)
μx
(1)
μv
(1)
Action
Motor
signals
Predictions
Prediction errors
˙a = −∂aεTξ
Optimal decision theory
(Or game theory.) An area of
applied mathematics
concerned with identifying the
values, uncertainties and other
constraints that determine an
optimal decision.
Gradient ascent
(Or method of steepest
ascent.) A first-order
optimization scheme that finds
a maximum of a function by
changing its arguments in
proportion to the gradient of
the function at the current
value. In short, a hill-climbing
scheme. The opposite scheme
is a gradient descent.
is the most valuable of the available states. In general,
it is impossible to solve the Bellman equation exactly,
but several approximations exist, ranging from simple
Rescorla–Wagner models98 to more comprehensive for-
mulations like Q-learning100. Cost also has a key role in
Bayesian decision theory, in which optimal decisions
minimize expected cost in the context of uncertainty
about outcomes; this is central to optimal decision theory
(game theory) and behavioural economics102–104.
So what does free energy bring to the table? If one
assumes that the optimal policy performs a gradient
ascent on value, then it is easy to show that value is
inversely proportional to surprise (see Supplementary
information S4 (box)). This means that free energy is
(an upper bound on) expected cost, which makes sense
as optimal control theory assumes that action mini-
mizes expected cost, whereas the free-energy principle
states that it minimizes free energy. This is important
because it explains why agents must minimize expected
cost. Furthermore, free energy provides a quantitative
and seamless connection between the cost functions
of reinforcement learning and value in evolutionary
biology. Finally, the dynamical perspective provides a
mechanistic insight into how policies are specified in the
brain: according to the principle of optimality99 cost is the
rate of change of value (see Supplementary information
S4 (box)), which depends on changes in sensory states.
This suggests that optimal policies can be prescribed by
prior expectations about the motion of sensory states.
Put simply, priors induce a fixed-point attractor, and
when the states arrive at the fixed point, value will stop
changing and cost will be minimized. A simple exam-
ple is shown in FIG. 2, in which a cued arm movement
is simulated using only prior expectations that the arm
will be drawn to a fixed point (the target). This figure
illustrates how computational motor control105–109 can
be formulated in terms of priors and the suppression of
sensory prediction errors (K.J.F., J. Daunizeau, J. Kilner
and S.J. Kiebel, unpublished observations). More gener-
ally, it shows how rewards and goals can be considered
as prior expectations that an action is obliged to fulfil16
(see also REF. 110). It also suggests how natural selection
could optimize behaviour through the genetic specifi-
cation of inheritable or innate priors that constrain the
learning of empirical priors (BOX 2) and subsequent goal-
directed action.
It should be noted that just expecting to be attracted
to some states may not be sufficient to attain those states.
This is because one may have to approach attractors vicar-
iously through other states (for example, to avoid obsta-
cles) or conform to physical constraints on action. These
are some of the more difficult problems of accessing
distal rewards that reinforcement learning and opti-
mum control contend with. In these circumstances,
an examination of the density dynamics, on which the
free-energy principle is based, suggests that it is sufficient
to keep moving until an a priori attractor is encountered
(see Supplementary information S5 (box)). This entails
destroying unexpected (costly) fixed points in the envi-
ronment by making them unstable (like shifting to a new
position when sitting uncomfortably). Mathematically,
this means adopting a policy that ensures a positive
divergence in costly states (intuitively, this is like being
pushed through a liquid with negative viscosity or
friction). See FIG. 3 for a solution to the classical
mountain car problem using a simple prior that induces
this sort of policy. This prior is on motion through state
space (that is, changes in states) and enforces exploration
until an attractive state is found. Priors of this sort may
provide a principled way to understand the exploration–
exploitation trade-off111–113 and related issues in evolu-
tionary biology114. The implicit use of priors to induce
dynamical instability also provides a key connection
to dynamical systems theory approaches to the brain
that emphasize the importance of itinerant dynamics,
metastability, self-organized criticality and winner-
less competition115–123. These dynamical phenomena
have a key role in synergetic and autopoietic accounts of
adaptive behaviour5,124,125.
Figure 2 | A demonstration of cued reaching movements. The lower right part of the
figure shows a motor plant, comprising a two-jointed arm with two hidden states, each of
which corresponds to a particular angular position of the two joints; the current position
of the finger (red circle) is the sum of the vectors describing the location of each joint.
Here, causal states in the world are the position and brightness of the target (green
circle). The arm obeys Newtonian mechanics, specified in terms of angular inertia and
friction. The left part of the figure illustrates that the brain senses hidden states directly
in terms of proprioceptive input (Sprop) that signals the angular positions (x1,x2) of the
joints and indirectly through seeing the location of the finger in space (J1,J2). In addition,
through visual input (Svisual) the agent senses the target location (v1,v2) and brightness (v3).
Sensory prediction errors are passed to higher brain levels to optimize the conditional
expectations of hidden states (that is, the angular position of the joints) and causal (that
is, target) states. The ensuing predictions are sent back to suppress sensory prediction
errors. At the same time, sensory prediction errors are also trying to suppress themselves
by changing sensory input through action. The grey and black lines denote reciprocal
message passing among neuronal populations that encode prediction error and
conditional expectations; this architecture is the same as that depicted in BOX 2. The
blue lines represent descending motor control signals from sensory prediction-error
units. The agent’s generative model included priors on the motion of hidden states that
effectively engage an invisible elastic band between the finger and target (when the
target is illuminated). This induces a prior expectation that the finger will be drawn to
the target, when cued appropriately. The insert shows the ensuing movement trajectory
caused by action. The red circles indicate the initial and final positions of the finger,
which reaches the target (green circle) quickly and smoothly; the blue line is the
simulated trajectory.
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f ==
˙x′
˙x
−∇
xϕ − 1⁄8 x′ + σ(a)
x′
Equations of motion
-2 -1 0 12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Position (x)
The mountain car problem
20 40 60 80 100120
–3
–2
–1
0
1
2
3
Time (seconds)
Time (seconds)
20 40 60 80 100120
–5
0
0
0
5
10
15
20
25
30
Conditional expectations
Height
ϕ(x)
μ(t)x
−c(t)
a(t)
Estimated states
Control signal
Position (x)
–2 –1 012
–30
–25
–20
–15
–10
–5
0
5
Loss functions (priors)
Force
c(x)
Position (x)
–2 –1 012
–2
–1
0
1
2
x′
Velocity
Trajectories Action
ab
Principle of optimality
An optimal policy has
the property that whatever the
initial state and initial decision,
the remaining decisions must
constitute an optimal policy
with regard to the state
resulting from the first decision.
Exploration–exploitation
trade-off
Involves a balance between
exploration (of uncharted
territory) and exploitation (of
current knowledge). In
reinforcement learning, it has
been studied mainly through
the multi-armed bandit
problem.
Dynamical systems theory
An area of applied
mathematics that describes
the behaviour of complex
(possibly chaotic) dynamical
systems as described by
differential or difference
equations.
Synergetics
Concerns the self-organization
of patterns and structures in
open systems far from
thermodynamic equilibrium. It
rests on the order parameter
concept, which was generalized
by Haken to the enslaving
principle: that is, the dynamics
of fast-relaxing (stable) modes
are completely determined by
the ‘slow’ dynamics of order
parameters (the amplitudes of
unstable modes).
Autopoietic
Referring to the fundamental
dialectic between structure
and function.
Helmholtzian
Refers to a device or scheme
that uses a generative model to
furnish a recognition density
and learns hidden structures in
data by optimizing the
parameters of generative
models.
In summary, optimal control and decision (game)
theory start with the notion of cost or utility and try to
construct value functions of states, which subsequently
guide action. The free-energy formulation starts with
a free-energy bound on the value of states, which is
specified by priors on the motion of hidden environ-
mental states. These priors can incorporate any cost
function to ensure that costly states are avoided. States
with minimum cost can be set (by learning or evolu-
tion) in terms of prior expectations about motion and
the attractors that ensue. In this view, the problem of
finding sparse rewards in the environment is nature’s
solution to the problem of how to minimize the entropy
(average surprise or free energy) of an agent’s states: by
ensuring they occupy a small set of attracting (that is,
rewarding) states.
Conclusions and future directions
Although contrived to highlight commonalities, this
Review suggests that many global theories of brain
function can be united under a Helmholtzian percep-
tive of the brain as a generative model of the world it
inhabits18,20,21,25 (FIG. 4); notable examples include the
integration of the Bayesian brain and computational
motor control theory, the objective functions shared
by predictive coding and the infomax principle,
hierarchical inference and theories of attention, the
embedding of perception in natural selection and
the link between optimum control and more exotic
phenomena in dynamical systems theory. The constant
theme in all these theories is that the brain optimizes
a (free-energy) bound on surprise or its complement,
value. This manifests as perception (so as to change
Figure 3 | Solving the mountain car problem with prior expectations. a | How paradoxical but adaptive behaviour (for
example, moving away from a target to ensure that it is secured later) emerges from simple priors on the motion of hidden
states in the world. Shown is the landscape or potential energy function (with a minimum at position x = –0.5) that exerts
forces on a mountain car. The car is shown at the target position on the hill at x =1, indicated by the red circle. The equations
of motion of the car are shown below the plot. Crucially, at x = 0 the force on the car cannot be overcome by the agent,
because a squashing function –1≤σ≤1 is applied to action to prevent it being greater than 1. This means that the agent can
access the target only by starting halfway up the left hill to gain enough momentum to carry it up the other side. b | The
results of active inference under priors that destabilize fixed points outside the target domain. The priors are encoded in a
cost function c(x) (top left), which acts like negative friction. When ‘friction’ is negative the car expects to go faster (see
Supplementary information S5 (box) for details). The inferred hidden states (upper right: position in blue, velocity in green
and negative dissipation in red) show that the car explores its landscape until it encounters the target, and that friction then
increases (that is, cost decreases) dramatically to prevent the car from escaping the target (by falling down the hill). The
ensuing trajectory is shown in blue (bottom left). The paler lines provide exemplar trajectories from other trials, with
different starting positions. In the real world, friction is constant. However, the car ‘expects’ friction to change as it changes
position, thus enforcing exploration or exploitation. These expectations are fulfilled by action (lower right).
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˙μv
(i)=Dμv
(i)(i)(i)
−
∂
vεTξ(i + 1)
− ξv
�μθij = −∂θijεTξ
˙a = −∂aεTξ
Infomax and the redundancy
minimization principle
Maximization of the mutual
information between sensations
and representations
Probabilistic neuronal coding
Encoding a recognition density
in terms of conditional
expectations and uncertainty
The Bayesian brain hypothesis
Minimizing the difference between a
recognition density and the conditional
density on sensory causes
Computational motor control
Minimization of sensory
prediction errors
Predictive coding and hierarchical inference
Minimization of prediction error
with recurrent message passing
Perceptual learning and memory
Optimization of synaptic efficacy
to represent causal structure
in the sensorium
Associative plasticity
Optimization of synaptic efficacy
Optimal control and value learning
Optimization of a free-energy
bound on surprise or value
Model selection and evolution
The free-energy principle
Optimizing the agent’s model and
priors through neurodevelopment
and natural selection
Minimization of the free energy of
sensations and the representation
of their causes
Attention and biased competition
Optimization of synaptic gain
representing the precision
(salience) of predictions
m = arg min ∫dtF
a, μ, m = arg min F (s
~, μ | m)
a, μ = arg max V (s
~ | m)
μ = arg max {I (s
~, μ ) − H(μ)}
μθ = arg min ∫dtF
μγ = arg min ∫dtF
μ = arg min DKL(q(ϑ) || (p(ϑ | s
~))
q(ϑ ) = N ( μ, Σ)
predictions) or action (so as to change the sensations
that are predicted). Crucially, these predictions depend
on prior expectations (that furnish policies), which
are optimized at different (somatic and evolutionary)
timescales and define what is valuable.
What does the free-energy principle portend for the
future? If its main contribution is to integrate estab-
lished theories, then the answer is probably ‘not a lot’.
Conversely, it may provide a framework in which cur-
rent debates could be resolved, for example whether
dopamine encodes reward prediction error or sur-
prise126,127 — this is particularly important for under-
standing conditions like addiction, Parkinson’s disease
and schizophrenia. Indeed, the free-energy formulation
has already been used to explain the positive symptoms
of schizophrenia in terms of false inference128. The free-
energy formulation could also provide new approaches
Figure 4 | The free-energy principle and other theories. Some of the theoretical constructs considered in this Review
and how they relate to the free-energy principle (centre). The variables are described in BOXES 1,2 and a full explanation
of the equations can be found in the Supplementary information S1–S4 (boxes).
to old problems that might call for a reappraisal of
conventional notions, particularly in reinforcement
learning and motor control.
If the arguments underlying the free-energy principle
hold, then the real challenge is to understand how it
manifests in the brain. This speaks to a greater appre-
ciation of hierarchical message passing41, the func-
tional role of specific neurons and microcircuits and
the dynamics they support (for example, what is the
relationship between predictive coding, attention
and dynamic co ordination in the brain?129). Beyond
neuroscience, many exciting applications in engineering,
robotics, embodied cognition and evolutionary biology
suggest themselves; although fanciful, it is not difficult to
imagine building little free-energy machines that garner
and model sensory information (like our children) to
maximize the evidence for their own existence.
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Acknowledgments
This work was funded by the Wellcome Trust. I would like to
thank my colleagues at the Wellcome Trust Centre for
Neuroimaging, the Institute of Cognitive Neuroscience and the
Gatsby Computational Neuroscience Unit for collaborations
and discussions.
Competing interests statement
The author declares no competing financial interests.
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