Biol Cybern
DOI 10.1007/s00422-010-0364-z

ORIGINAL PAPER

Action and behavior: a free-energy formulation
Karl J. Friston · Jean Daunizeau · James Kilner ·Stefan J. Kiebel

Received: 30 October 2009 / Accepted: 19 January 2010
© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract We have previously tried to explain perceptual
inference and learning under a free-energy principle that pursues Helmholtz’s agenda to understand the brain in terms ofenergy minimization. It is fairly easy to show that making
inferences about the causes of sensory data can be cast as
the minimization of a free-energy bound on the likelihoodof sensory inputs, given an internal model of how they werecaused. In this article, we consider what would happen if
the data themselves were sampled to minimize this bound.
It transpires that the ensuing active sampling or inference ismandated by ergodic arguments based on the very existenceof adaptive agents. Furthermore, it accounts for many aspectsof motor behavior; from retinal stabilization to goal-seeking.
In particular, it suggests that motor control can be understood as fulfilling prior expectations about proprioceptive

The free-energy principle is an attempt to explain the structure andfunction of the brain, starting from the fact that we exist: This fact
places constraints on our interactions with the world, which have beenstudied for years in evolutionary biology and systems theory. However,recent advances in statistical physics and machine learning point to asimple scheme that enables biological systems to comply with these
constraints. If one looks at the brain as implementing this scheme(minimizing a free-energy bound on disorder), then many aspects of
its anatomy and physiology start to make sense. In this article, weshow that free-energy can be reduced by selectively sampling sensory
inputs. This leads to adaptive responses and provides a new view of
how movement control might work in the brain. The main conclusion
is that we only need to have expectations about the sensory
consequences of moving in order to elicit movement. This means wethat can replace the notion of desired movements with expectedmovements and understand action in terms of perceptual expectations.
K. J. Friston (B) · J. Daunizeau · J. Kilner · S. J. Kiebel
The Wellcome Trust Centre for Neuroimaging, Institute of Neurology,
University College London, 12 Queen Square, London,
WC1N 3BG, UKe-mail: k.friston@fil.ion.ucl.ac.uk

sensations. This formulation can explain why adaptive
behavior emerges in biological agents and suggests a simple alternative to optimal control theory. We illustrate thesepoints using simulations of oculomotor control and thenapply to same principles to cued and goal-directed movements. In short, the free-energy formulation may provide analternative perspective on the motor control that places it in
an intimate relationship with perception.
Keywords Computational · Motor · Control · Bayesian ·Hierarchical · Priors
List of symbols
 ⊇ {x̃, ṽ, θ , γ }, ⊇ {x̃, ṽ, θ, γ }x̃(t) = [x, x  , x  , . . .]T ,˙ = f (x̃, ṽ, θ ) + w̃x̃(t)ṽ(t) = [v, v  , v  , . . .]Ts̃(t) = g(x̃, ṽ, θ ) + z̃θ ⊇ {θ1 , θ2 , . . .}Unknown causes of sensory
input; variables in bold denotetrue values and those in italics
denote variables assumed by
the agent or model
Generalised hidden-states thatact on an agent. These aretime-varying quantities that
include all high-order temporal
derivatives; they represent a
point in generalised coordinates of motion that encodesa path or trajectory
Generalised forces or causal
states that act on hidden
statesGeneralised sensory statescaused by hidden states
Parameters of the equations of motion and sensory mapping123Biol Cybernγ ⊇ {γ s , γ x , γ v }w̃(t) = [w, w , w  , . . .]Tz̃(t) = [z, z , z , . . .]Tñ(t) = [n, n , n , . . .]T
i := (γ i ) = (γ i )−1g(x̃, ṽ, θ ), f(x̃, ṽ, ã, θ )g(x̃, ṽ, θ ), f (x̃, ṽ, θ )a(t)p(x̃ |m ), p(s̃ |m ) 
D(q||p) = ln(q p) qqm
H (X) = ln p(x̃|m)p
H (S) = ln p(s̃|m)p− ln p(s̃|m)F (s̃, µ) ≥ − ln p(s̃|m)q(|µ)µ = {µ̃(t), µθ , µγ }µ̃ = {µ̃x , µ̃v }η̃(t) = [η, η , η , . . .]T

123

Parameters of the precision of random fluctuations
(γ i ) : i ∈ s, x, v
Generalised random fluctuations of the motion of hidden states
Generalised random fluctuations of sensory states
Generalised random fluctuations of causal states
Precisions or inverse covariances of generalised random fluctuations
Sensory mapping and equations of motion generating
sensory states
Sensory mapping and equations of motion modelingsensory states
Policy: a scalar functionof generalised sensory andinternal states
Ensemble densities; the density of the hidden and sensory states of agents at equilibrium with their environment.
Kullback-Leibler divergenceor cross-entropy between
two densities
Expectation or mean of under
the density q
Model or agent; entailing theform of a generative model
Entropy of generalised hidden and sensory states
Surprise or self-informationof generalised sensory
statesFree-energy bound on surprise
Recognition density on
causes  with sufficient
statistics µ
Conditional or posterior
expectation of the causes
; these are the sufficient
statistics of the Gaussianrecognition density
Prior expectation of generalised causal states

ξi = i ε̃i : i ∈ s, x, v⎡⎤
ε̃s = s̃ − g(µ)ε̃ = ⎣ ε̃x = D µ̃x − f (µ) ⎦ε̃v = µ̃v − η̃Precision-weighted generalised prediction errors
Generalised prediction error
on sensory states, themotion of hidden states andforces or causal states.

1 Introduction
This article looks at motor control from the point of view ofperception; namely, the fitting or inversion of internal models of sensory data by the brain. Critically, the nature of thisinversion lends itself to a relatively simple neural network
implementation that shares many formal similarities with realcortical hierarchies in the brain. The idea that the brain uses
hierarchical inference has been established for years (Mumford 1992; Rao and Ballard 1998; Friston 2005; Friston etal. 2006) and provides a nice explanation for the hierarchical organization of cortical systems. Critically, hierarchicalinference can be formulated as a minimization of free-energy;
where free-energy bounds the surprise inherent in sensory
data, under a model of how those data were caused. This
leads to the free-energy principle, which says that everything
in the brain should change to minimize free-energy. We willsee below that free-energy can be minimized by changing
perceptual representations so that they approximate a posterior or conditional density on the causes of sensations. In short,the free-energy principle entails the Bayesian brain hypothesis (Knill and Pouget 2004; Ballard et al. 1983; Dayan et al.
1995; Lee and Mumford 2003; Rao and Ballard 1998; Friston
2005; Friston and Stephan 2007). However, the free-energy
principle goes further than this. It suggests that our actions
should also minimize free-energy (Friston et al. 2006): We
are open systems in exchange with the environment; the environment acts on us to produce sensory impressions, and weact on the environment to change its states. This exchange
rests upon sensory and effector organs (like photoreceptors
and oculomotor muscles). If we change the environment or
our relationship to it, then sensory input changes. Therefore,
action can reduce free-energy by changing the sensory input
predicted, while perception reduces free-energy by changing
predictions. In this article, we focus in the implications ofsuppressing free-energy through action or behavior.
Traditionally, the optimization of behavior is formulated
as maximizing value or expected reward (Rescorla and Wagner 1972; Sutton and Barto 1981). This theme is seenin cognitive psychology, in reinforcement learning models(Rescorla and Wagner 1972); in computational neuroscienceand machine-learning as variants of dynamic programming,such as temporal difference learning (Sutton and Barto 1981;

Biol Cybern

Watkins and Dayan 1992; Friston et al. 1994; Daw and Doya2006), and in behavioral economics as expected utility theory(Camerer 2003). In computational motor control (Wolpert
and Miall 1996; Todorov and Jordan 2002; Todorov 2006;
Shadmehr and Krakauer 2008), it appears in the form of
optimal control theory. In all these treatments, the problem of
optimizing behavior is reduced to optimizing value (or, conversely, minimizing expected loss or cost). Effectively, thisprescribes an optimal control policy in terms of the value thatwould be expected by pursuing that policy.
Our studies suggest that maximizing value may representa slight misdirection in explaining adaptive behavior, because
the same behaviors emerge in agents that minimize freeenergy (Friston et al. 2009). In brief, the minimization of freeenergy provides a principled basis for understanding both
action and perception, which replaces the optimal polices ofcontrol theory with prior expectations about the trajectory ofan agent’s states. In Sect. 2, we review the free-energy principle and active inference. In Sect. 3, we show how active inference can be used to model reflexive and intentional behavior.
This section deals with visual and proprioceptive models todemonstrate the key role of prior expectations in prescribing
movement. In Sect. 4, we consider the integration of visualand proprioceptive signals in finessing the control of cuedreaching movements. Section 5 addresses how these priorexpectations could be learned and illustrates the acquisitionof goal-directed movements using the mountain-car problem(Sutton 1996; see also Friston et al. 2009). Section 6 revisits
the learning of priors to prescribe autonomous behavior. We
conclude by discussing the relationship between active inference and conventional treatments of computational motor
control.
2 The free-energy principle
In this section, we try to establish the basic motivation forminimizing free-energy. This section rehearses material that
we have used previously to understand perception. It is presented here with a special focus how action maintains a stationary relationship with the environment; and is developed
more formally than in previous descriptions (e.g., Friston andStephan 2007). The arguments for how perception decreases
free-energy can be found in the neurobiological (Friston et al.
2006; Friston 2008) and technical (Friston et al. 2008) literature. These arguments are reviewed briefly but only to a depththat is sufficient to understand the simulations in subsequent
sections.
What is free-energy? In statistics and machine learning,
free-energy is an information theory quantity that bounds theevidence for a model of data (Hinton and von Camp 1993;
MacKay 1995; Neal and Hinton 1998). Here, the data aresensory inputs, and the model is encoded by the brain. Moreprecisely, free-energy is greater than the surprise (negative

log-probability) of some data, given a model of how thosedata were generated. In fact, under simplifying assumptions(see below), it is just the amount of prediction error. It iscalled free-energy because of formal similarities with thermodynamic free-energy in statistical physics; where energies
are just negative log-probabilities (surprise) and free-energy
is a bound on surprise. In what follows, we describe the natureof free-energy, and show why it is minimized by adaptive
agents.
We start with the premise that adaptive agents or phenotypes must occupy a limited repertoire of physical states. Fora phenotype to exist, it must possess defining characteristics
or traits; both in terms of its morphology and exchange withthe environment. These traits essentially limit the agent toa bounded region in the space of all states it could be in.
Once outside these bounds, it ceases to possess that trait (cf.,a fish out of water). This speaks to self-organized autopoietic interactions with the world that ensure these bounds arerespected (cf., Maturana and Varela 1972). Later, we formalize this notion in terms of the entropy or average surprise associated with a probability distribution on the states
an agent experiences. The basic idea is that adaptive agents
must occupy a compact and bounded part of state–space and,
therefore, avoid surprising states (cf., a fish out of water—sic). In terms of dynamical system theory, this set of states
is a random attractor (Crauel and Flandoli 1994). Given thisdefining attribute of adaptive agents, we will look at how
agents might minimize surprise and then consider what thismeans, in terms of their action and perception.
The free-energy principle rests on an ensemble densityp(x̃|m) on generalized states, x̃(t) = [x, x , x , . . .]T , which
affect an agent, m. Generalized states cover position, velocity, acceleration, jerk, and so on (Friston 2008). This meansthat states include the position or configuration of the agent,
its motion, and all influences acting on the agent: i.e., physical forces like gravity; thermodynamic states like ambient
temperature, or physiological states such as hypoglycemia.
Strictly speaking; the dimensionality of generalized states isinfinite because the generalized motion of each state exists
to infinite order. However, in practice one can ignore highorder temporal derivatives because their precision vanishes
and they contain no useful information (i.e., their dispersion gets very large; see Friston 2008; Friston et al. 2008 fordetails). In the simulations below, we only used generalizedstates up to sixth order. In what follows, x(t) refers to a statevector and x̃(t) denotes the corresponding generalized state(i.e., the state and its generalized motion). Note that the density p(x̃|m) is conditioned on the agent or model. We will see
later that the model entails formal constraints on the motionof an agent’s states (i.e., its state-transitions or policy). Thismeans the ensemble density is specific to each class of agent.
The ensemble density can be regarded as the probability of finding an agent in a particular state, when observed123Biol Cybern

on multiple occasions or, equivalently, the density of a large
ensemble of agents at equilibrium with their environment.
Critically, for an agent to exist, the ensemble density shouldhave low entropy. This ensures that agents occupy a limited repertoire of states because a density with low entropy
concentrates its mass in a small subset of state–space (i.e.,
its attractor). This places an important constraint on thestates sampled by an agent; it means agents must somehow counter the dispersive effects of random forces, which
increase entropy. This increase is a consequence of the fluctuation theorem (Evans 2003), which generalizes the second law of thermodynamics and says that the probability ofentropy decreasing vanishes exponentially the longer a system is observed. Adaptive agents resist the implicit dispersionof their states through adaptive behaviors (like eating) thatmaintain their milieu within physiological bounds. In summary, biological agents must resist the natural tendency todisorder or increasing entropy; but how do they do this?

tions of motion and sensory mapping are parameterized by(generally non-overlapping) subsets of some time-invariant
parameters θ.
Under a sensory mapping, the entropy of the hidden states
is bounded by the sensory entropy and a sensory mapping
term, with equality in the absence of sensory noise (seeAppendix 1).

H (X|m) ≤ H (S|m) − p(x̃|m) ln |∂x̃ g|dx̃

2.1 Active agentsThe second term depends on the collective sensitivity|∂x̃ g| := det(∂x̃ g) of sensory inputs to changes in hidden
states, where ∂x̃ g is the derivative of the sensory mappingwith respect to the hidden states. For simplicity, we willassume that this sensitivity is uniformly high over the rangesof states considered. This appeals to the principle of maximum information transfer (Linsker 1990), which has beenvery useful in understanding the emergence of things likereceptive fields (e.g., Olshausen and Field 1996). Under thissimplifying assumption, the second term is constant, and it
is sufficient to minimize sensory entropy to minimize theentropy of hidden states. This is important because the agent
has access to sensory states but not hidden states. Note thatagents cannot minimize sensory entropy by switching off
sensory channels; they can only act by changing hidden
states, not the sensory mapping per se.
Crucially, because the ensemble density on sensory statesis at equilibrium, it can be interpreted as the proportion oftime a single agent entertains these states (the sojourn time).
This ergodic argument (Anosov 2001) means that the sensory
entropy is the long-term average of the surprise − ln p(s̃|m)experienced by a particular agent:

At this point, we introduce the notion of active agents(Schweitzer 2003) that sense some states (with sensory
organs) and can change others (with effector organs). We
can describe this exchange with the environment in terms of
sensory states s̃(t) like the activity of photoreceptors, internal
states µ̃(t) such as neuronal activity, and action a(t) encoding peripheral motor control signals. Because agents only
sense external states of the world through a probabilistic mapping, they are hidden from the agent. We will, therefore, callx̃ ∈ X hidden states and assume that they evolve according tosome complicated equations of motion, x̃˙ = f(x̃, a, θ ) + w̃,where w are random fluctuations. We will see examples ofthis equation later, where x̃(t) represents the angular position and speed of a joint, and θ encodes the elasticity andviscosity of muscles. We will use an unusual convention inwhich true hidden quantities are presented in bold and others
in italics.
Control (e.g., of saccadic eye-movements) is represented
by making the equations of motion a function of action. We
will see below that action depends vicariously on sensory
states and, therefore, constitutes a policy (from the point ofview of optimum control theory). It is this policy that resists
the increasing entropy of hidden states, because the only way
that the agent can change hidden states is through action. We
assume that agents do this by minimizing entropy until the
movement of states toward the agent’s attractor is balanced
by dispersion due to random fluctuations. Sensation (e.g.,
proprioception) can be described with a probabilistic mapping s̃ = g(x̃, θ ) + z̃ from hidden to sensory states s̃ ∈ S,
where z represents sensory noise. Later, we will see examples were g(x̃, θ ) maps from the trajectory of an object inthe visual field to the activity of photoreceptors. The equa-

123

(1)

s̃ = g(x̃, θ ) + z̃x̃˙ = f(x̃, a, θ ) + w̃where the respective entropies are
H (S|m) = − p(s̃|m) ln p(s̃|m)ds̃
H (X|m) = − p(x̃|m) ln p(x̃|m)dx̃

1
H (S|m) = lim −T →∞ T(2)

T
dt ln p(s̃(t)|m)(3)0This means that in order to minimize sensory entropy, one
needs to minimize surprise at all times (i.e., for all successive
sensory samples).
At this point, most (astute) people say “but that means
I should retire to a dark room and cover my ears.” However, is not quite that simple: Recall that entropy is average surprise and surprise depends on the agent. This meansthat it might be surprising to find an agent in a dark room.

Biol Cybern

It helps here to imagine that each class of agent (e.g., species) has equations of motion that induce a random attractor
in state–space (more formally, a compact random invariant
set that attracts all trajectories; Crauel and Flandoli 1994).
Entropy decreases with the volume or measure of this set.
However, it is important not to confuse volume with shapeor topological complexity. The attractor could be very complicated and space-filling but occupy a small volume (e.g.,
the states encountered while driving a car with a small margin of error). In this view, minimizing surprise ensures thatstates evolve according to the expected flow, f(x̃, ã, θ ) andremain in the attractor (e.g., within physiological bounds). In
short, the ensemble and sensory entropy are lower bounded
by a random attractor entailed by an agent’s class. Agents try
to attain this bound by minimizing surprise, while learningand natural selection may reduce the bound by optimizing theattractor per se. This means the notion of a “dark room” agent
is a category error; in the sense that “dark room” or any fixedpoint attractors are not invariant under real world dynamics
and, therefore, cannot exist (e.g., at some point, you will
get thirsty). Perhaps the most potent examples of surprisereducing dynamics are found in physiology, which can beunderstood largely in terms of homeostatic mechanisms thatavoid surprising trajectories of the internal milieu:
The fixity of the milieu supposes a perfection of theorganism such that the external variations are at each
instant compensated for and equilibrated … All of thevital mechanisms, however varied they may be, havealways one goal, to maintain the uniformity of the conditions of life in the internal environment … The stability of the internal environment is the condition for
the free and independent life (Bernard 1974).
This is closely related to perspectives on motor control andsequence learning that “minimize deviations from the desired
state, that is, to minimize disturbances of the homeostasis of
the feedback loop.” See Wörgötter and Porr (2005) for a fuller
discussion. In summary, avoiding surprise is fundamental forsurvival and speaks to the basic need of organisms to maintain equilibrium within their environment. However, there isa problem:
2.2 Free-energy, action and perception
The problem faced by real agents is that they cannot quantify
surprise, because this entails marginalizing over the unknownor hidden causes  ⊃ {x̃, θ } of sensory input that include
time-varying hidden states and time-invariant parameters− ln p(s̃|m) = − ln p(s̃, |m)d(4)
However, there is an alternative and elegant solution tominimizing surprise, which comes from theoretical phys-

ics (Feynman 1972) and machine learning (Hinton and vonCamp 1993; MacKay 1995). This involves minimizing a freeenergy bound on surprise that can be evaluated. Minimizing
this bound implicitly minimizes surprise because the boundis always greater than surprise. The bound is induced by a recognition density q(|µ), which is parameterized by its sufficient statistics µ ⊃ µ̃(t) (e.g., mean or covariance). Thesesufficient statistics are encoded by the internal states of theagent (e.g., by neuronal activity and connection strengths).
The recognition density is a slightly mysterious constructbecause it is an arbitrary probability density specified by theinternal states of the agent. Its role is to induce free-energy,
which is a function of the internal states and sensory inputs.
We will see below that when this density is optimized to minimize free-energy, it becomes the conditional density on thecauses of sensory data. In Bayesian inference, this is known
as the recognition density. In what follows, we summarize
the key ideas behind a large body of study in statistics andmachine learning referred to as ensemble learning or variational Bayes.
The free-energy bound is constructed by simply augmenting surprise with a non-negative divergence between the recognition density and conditional density p(|s̃, m). This is
described Appendix 2. The resulting expression for freeenergy can be expressed in three ways (with the use of Bayes
rule and simple rearrangements):• Energy minus entropy• Divergence plus surprise• Complexity minus accuracy
Mathematically, these correspond to:
F = − ln p(s̃, |m)q + ln q(|µ)q= D(q(|µ)||p(|s̃, m)) − ln p (s̃|m)(5)= D(q(|µ)||p(|m)) − ln p(s̃|, m)q
Here, ·q means the expectation or mean under the density
q and D(·||·) is the cross-entropy or Kullback–Leibler divergence between two densities. The alternative formulations
in Eq. 5 have some important implications: The first showsthat free-energy is a scalar function of sensory data and therecognition density at any particular time. This can be evaluated by an agent; provided it has a probabilistic model of theenvironment. This model is usually expressed as the productof a likelihood and prior, p(s̃, |m) = p(s̃|, m)p(|m).
We will see examples of this model later (e.g., the likelihood
model in Eq. 8).
The second formulation shows that minimizing the freeenergy, by changing internal states (i.e., the sufficient statistics of q(|µ)), reduces the divergence between therecognition and conditional densities. This makes therecognition density an approximate conditional density. This
corresponds to Bayesian inference on the causes of sensory

123Biol Cybern

signals and provides a principled account of perception; i.e.,the Bayesian brain (Helmholtz 1860/1962; Barlow 1969;
Ballard et al. 1983; Mumford 1992; Dayan et al. 1995; Raoand Ballard 1998; Lee and Mumford 2003; Knill and Pouget
2004; Kersten et al. 2004; Friston and Stephan 2007; Deneve2008). Finally, it shows that free-energy is an upper boundon surprise because the divergence cannot be less than zero:
Optimizing the recognition density makes the free-energy
a tight bound on surprise; when the recognition and conditional densities coincide, free-energy is exactly surprise andperception is veridical.
The third equality shows that free-energy can be suppressed by action, through its effects on hidden states andthe ensuing sensory signals. The key term here is the accuracy term, which is the expected surprise of sensory data
under the recognition density (under Gaussian assumptions,this is just the amount of sensory prediction error). Thismeans action will change the motion of sensory states sothat they conform to conditional expectations. This minimizes surprise, provided perception makes free-energy atight bound on surprise. In short, the free-energy principle prescribes an optimum perceptual representation andpolicy∂µ F (s̃, µ) = 0 ⇒ δµµa

In summary, the free-energy principle requires the internal
states of an agent and its action to suppress free-energy.
This corresponds to optimizing a probabilistic model of
how sensations are caused, so that the resulting predictions can guide active sampling of sensory data. The requisite interplay between action and perception (i.e., active
inference) engenders a policy that ensures the hidden statesthat act on the agent have low entropy. Put simply, if
you search out things you expect, you will avoid surprises. This recapitulates the notion that “perception andbehavior can interact synergistically, via the environment”

dtF (s̃, µ) = 0
0

⇒

(6)

T
∂a F (s̃, µ) = 0 ⇒ δadtF (s̃, µ) = 00This policy reduces to sampling input that is expectedunder the recognition density (i.e., sampling selectively what
one expects to experience). In other words, agents must necessarily (if implicitly) make inferences about the causes of
their sensory signals and sample signals that are consistent
with those inferences. Technically, minimizing free-energy
ensures the variation of its path integral, with respect to small
variations in action or internal representations, is zero (seeEq. 6). This time or path-integral is called Action (not to be
confused with action). Under ergodic assumptions, Action isan upper bound on sensory entropy (see Eq. 3), which means
optimal action and perception minimize a bound on sensory
surprise and, implicitly, entropy.
The free-energy principle suggests that we should sample
the world to ensure our predictions become a self-fulfilling
prophecy. In this view, perception is enslaved by action toprovide veridical predictions (more formally, to make thefree-energy a tight bound on surprise) that guide active sampling of the sensorium. This active sampling should not
be confused with active learning (Mumford 1992). Active

123

2.3 Summary

Tµ(t)∗ = arg min F (s̃(t), µ(t))a(t)∗ = arg min F (s̃(t), µ(t))

learning or sequential design tries to harvest data to makerecognition more precise using various criteria, based on
how data inform model parameter estimates; “these criteria depend on the assumption that the hypothesis space iscorrect, which may prove to be their main weakness” (Mumford 1992). Under the free-energy formulation, this weakness
becomes the strength; if the environment delivers data thatare consistent with the agent’s model or hypothesis space,then the agent must be at equilibrium with its environment.

µ=µ∗

to optimize behavior (Verschure et al. 2003). Indeed, Distributed Adaptive Control (Verschure and Voegtlin 1998) relates
closely to the free-energy formulation, because it provides anintegrated solution to the acquisition of probabilistic models
and policies. Active inference is an example of self-referenced learning (Maturana and Varela 1972; Porr and Wörgötter 2003) in which “the actions of the learner influence its own
learning without any valuation process” (Porr and Wörgötter
2003).
In machine learning and statistics, the free-energy is used
as a bound on the log-evidence for a model. This means, given
some data, one can optimize the model to maximize its evidence; i.e., find the model that minimizes the surprise of somefixed data. The free-energy principle turns this optimization
around and considers that the model (agent) is fixed and theaim is to minimize surprise by re-sampling the data. This
minimization is mandated by the fact that average surprise isentropy and agents must have a low entropy ensemble density. This argument rests on noting that the ensemble density
is formally identical to the marginal likelihood in statistics.
In the analysis of time-series data, one optimizes the modelBiol Cybernwith respect to the path integral of free-energy over time (seeFriston 2008): Minimizing free-energy at each point in time
provides a stationary solution for Action (by the fundamental
lemma of variational calculus): this is basically Hamilton’s
principle of stationary Action. All that we are doing here isnoting that Action is a bound on the path-integral of surpriseand, under ergodic assumptions, entropy. This means finding a minimum for Action implicitly finds a minimum forsensory entropy. It is interesting to note that a tendency todisorder or the second law of thermodynamics (which applies
only to closed systems) can be resisted by appealing to themore general tendency of (open) systems to reduce their freeenergy (Gontar 2000; Evans 2003). A related treatment ofself-organization in non-equilibrium systems can be found insynergetics; where “patterns become functional because they
consume in a most efficient manner the gradients which cause
their evolution” (Tschacher and Haken 2007). Here, these
gradients might be regarded as surprise. This section has tried
to establish the theoretical motivation for free-energy minimization. We now turn to how this might be implemented in
the brain.

3 Active inference
In this section, we unpack the theory above to establish thenature of processes underlying action and perception. In theprevious section, we saw that optimal action and perception
could be cast as minimizing free-energy and that free-energy
is defined in relation to a probabilistic generative model of
the world. In what follows, we will examine particular formsof this model, which enable free-energy to be expressed interms of prediction errors (the mismatch between observed
sensory signals and those predicted by the model). This leads
to a formulation of action and perception as the suppression ofprediction errors (or surprise). By assuming the minimizationin Eq. 6 is implemented using gradient decent, we can thenwrite down differential equations that prescribe the dynamics
of action and recognition as functions of prediction error.
In order to see how active inference works, we must first
define an environment and the agent’s model of it. We willassume that both can be cast as dynamical systems with additive random effects. For the environment, we have equationsof motion and a sensory mapping that describe the processes
generating sensory datas̃ = g(x̃, ṽ, θ ) + z̃x̃˙ = f(x̃, ṽ, a, θ ) + w̃(7)This describes how sensory data are actually generated, while
the agent assumes that these data are generated by the following system:s̃ = g(x̃, ṽ, θ ) + z̃x̃˙ = f (x̃, ṽ, θ ) + w̃(8)ṽ = η̃ + ñ
Equation 8 is just a way of expressing a generative model interms of equations that have a formal homology with the true
generative process (Eq. 7). In other words, Eq. 8 represents
a parameterized model of sensory input, which becomes aprobabilistic model, if the agent makes some assumptions
about the probability distribution of the random fluctuations(see below). Note that we have partitioned the hidden states
into those that are autonomous ṽ(t) and those that are notx̃(t) (cf. Eq. 1). The autonomous hidden states (e.g., exogenous forces) allow us to model exogenous perturbations of
the agent’s state. In previous treatments (e.g., Friston et al.
2006; Friston 2005), we have called these casual states. Furthermore, these forces are modeled as mixture of some deterministic prior η̃(t) and random fluctuations ñ(t).
Equations 7 and 8 pertain to the same sensory states andplay a similar role. However, the first is the true stochastic
process generating states, and the second is a probabilistic
model of this process. The equations of motion and sensory
mapping may or may not have the same form, which means
the agent’s model of the world may be different from the realprocess generating sensory data. Furthermore, we allow fora difference between the true unknowns  ⊃ {x̃, ṽ, θ } andthose assumed by the model  ⊃ {x̃, ṽ, θ }.
Gaussian assumptions about the random fluctuations{z̃, w̃, ñ} in Eq. 8 furnish three things; a likelihood model,p(s̃|) = N (g, (γ s )), empirical priors on the generalized motion p(x̃  |ṽ, ) = N (f, (γ x )), and priors on theforces, p(ṽ) = N (η̃, (γ v )). Here, the inverse variances or
precisions  ⊃ γ i : i ∈ s, x, v determine the covariance(γ i ) of the generalized fluctuations. Note that the equations of motion for the true states depend on action, whereas
the generative model has no notion of action; it just produces
predictions that action tries to fulfill. However, the generativemodel contains a prior on exogenous forces, which, as we will
see later, can be used to model the effects of action. Equation 8 is a simple hierarchical dynamic model in generalized
coordinates of motion. This sort of model is considered indetail in Friston (2008). Recall that the equations generating
data (Eq. 7) and those of the generative model (Eq. 8) do nothave to be the same; it is this discrepancy that action tries tocancel.
Given a probabilistic generative model, the free-energy
can now be minimized: This minimization obliges the agent
to infer the states of the world and learn the unknown
parameters responsible for its motion by optimizing thesufficient statistics of its recognition density. This is perceptual inference and learning and can be implemented
in a biologically plausible fashion as described in Friston(2008). In brief, this scheme assumes that a mean-field123

Biol Cybern

approximation: q(|µ) = q(x̃, ṽ|µ̃)q(θ |µθ )q(γ |µγ ) with
Gaussian marginals, whose sufficient statistics are expectations and covariances. Under this Gaussian or Laplaceapproximation, it is sufficient to optimize the expectations
because the covariances can be derived as a function of the
expectations. From now on, we assume that the sufficient
statistics are the conditional expectations of states, parameters, and precisions, µ = {µ̃, µθ , µγ }, where the conditional
expectations of states µ̃(t) = {µ̃x , µ̃v } are in generalized
coordinates of motion. Using these (mean-field and Laplace)approximations, we can formulate Eq. 6 as a gradient descentthat describes perceptual inference, learning, and action (seeFriston et al. 2008; Friston 2008 for details):∂F∂ ε̃T≈ D µ̃x −ξµ̃˙ x = D µ̃x −
∂ µ̃x
∂ µ̃x
∂F
∂ ε̃T
µ̃˙ v = D µ̃v −≈ D µ̃v −
ξ
∂ µ̃v
∂ µ̃v
∂F
∂ ε̃T
≈−
ξ
∂µθ
∂µθ
∂ ε̃T
∂F
= − s ξsȧ = −∂a∂aµ̈θ = −(9)1 T
1ε̃ ε̃ − ln ||22
ξ = ε̃F ≈This equation is used in all the simulations of active inference described later. It has a simple form because the freeenergy in Eq. 5 reduces (effectively) to a sum of squared
errors because of our Gaussian assumptions about the random fluctuations. The approximate equalities in Eq. 9 are dueto the fact that we are neglecting mean-field terms (and retain
only terms that depend on conditional expectations). The
derivative operator D is a block matrix with identity matrices along the first diagonal such that x̃  := [x  , x  , . . .]T =D[x, x  , . . .]T = D x̃. The first pair of equations prescribes
recognition dynamics, in terms of how expected states changeover time. The second terms in these equations are freeenergy gradients. The first terms reflect the fact that weare working in generalized coordinates; they ensure thatµ̃˙ = D µ̃ ⇒ ∂µ̃ F = 0 when free-energy is minimized andits gradient is zero (i.e., they ensure the motion of the expectation is the expected motion). This scheme can be implemented in a biologically plausible manner by assuming thatthe activity of one population of neurons encodes the conditional expectations µ̃(t) and another population encodes
precision-weighed prediction errors⎤⎡ε̃s = s̃ − g(µ)(10)ε̃ = ⎣ ε̃x = D µ̃x − f (µ) ⎦ε̃v = µ̃v − η̃123

The prediction errors ε̃(t) are just the difference between
sensory samples (resp. expected motion of hidden states)
and the predictions afforded by the sensory mapping (resp.
equations of motion) evaluated with the expected states andparameters. The matrix (µγ ) is the expected precision(inverse covariance) of the random effects and effectively
boosts prediction errors in proportion to their precision (seeFriston 2008 for more details). Note that under the Laplaceassumption, minimizing free-energy corresponds to minimizing prediction error. This sort of scheme is also called
predictive coding (Rao and Ballard 1998). Figure 2a showsa schematic of the neuronal circuits that might implement
predictive coding of this sort. These circuits comprise neuronal state-units or populations (white) that encode sensory
input and the conditional expectations of hidden states causing that input. Error-units are shown in red and encode theprediction error for their corresponding state-unit. The dotted lines are connections that are intrinsic to the model, andsolid lines represent coupling between action-units (black)and effectors. The red lines convey prediction errors to thestate (and action) units, while the black lines pass predictions
to the error-units (and effectors). See, Friston (2008), for a
fuller description of the implicit message-passing scheme.
The solutions to the third line in Eq. 9 are the optimum
parameters that may be encoded in the brain with synaptic
efficiency (Friston 2008). These are second-order differential
equations because these expectations optimize Action (thepath-integral of free-energy), due to the fact we know a priori, that they do not change with time. The resulting changes
in synaptic efficacy are formally identical to Hebbian or associative plasticity (under some simplifying assumptions; see
Friston 2008 for details). The final equation describes action
as a gradient descent on free-energy. The only way action canaffect free-energy is through changing the motion of sensory
signals. This means action must suppress sensory prediction
errors, ε̃s = s̃(a)−g(µ). Equation 9 embodies a nice convergence of action and perception: perception tries to suppress
prediction error by adjusting expectations to furnish better
predictions of signals, while action tries to fulfil these predictions by changing those signals.
3.1 Motor control and prediction errors
Optimal behavior is a powerful guide for understanding
motor control. The notion of forward or generative models
has been established in this field for decades (Wolpert et al.
1995). Some have even used forward motor models to motivate generative models in perception (Kawato et al. 1993).
Usually, forward models in the motor control literature areused to finesse control, given motor commands prescribing
a desired movement trajectory (see Sect. 7). The free-energy
formulation offers a simpler view; prediction errors are notthere to finesse motor signals; they are the motor signals.

Biol Cybern

In other words, there are no top-down motor commands; justprior expectations about proprioceptive input that are fulfilled
at the spinal level, given a compliant environment and motor
plant (see Fig. 1). In this view, the central nervous system is
not divided into motor and sensory systems but is one perceptual inference machine that provides predictions of optimal
action, in terms of its expected consequences. This formulation also specifies what movement kinetics should optimize;the only thing that action can affect is the prediction error at
the sensory level. This means action can only suppress theweighted sensory prediction error variance, ε̃sT ξs = ε̃sT s ε̃s .
In generalized coordinates, these include prediction errors on
displacement, velocity, acceleration, jerk, etc. This ensuressmooth trajectories, because the variance of higher temporal derivatives measures the smoothness of trajectories. The
relative weighting of these error variances is controlled by
the optimized precision. This mixture of weighted error variance is related formally to phenomenological cost functions
proposed to account for movement trajectories (cf. Todorov
and Jordan 1998; Nakano et al. 1999; Berret et al. 2008).
3.2 Summary
In this section, we have seen that minimizing free-energy
with respect to neuronal activity encoding the recognition
density can be formulated as a gradient decent. This leads
to recognition dynamics that suppress prediction error, making them formally equivalent to predictive coding. Policies
are prescribed in exactly the same way; however, action can
only affect prediction error at the sensory level. This suggeststhat sensory prediction errors generate motor command signals and predicts an intimate anatomical relationship between
primary motor and sensory systems. This is because motor
commands must emanate from sensory systems responsible for processing the proprioceptive consequences of movement. Empirically, this is seen at cortical, subcortical, andspinal levels. For example, the primary motor and sensory
cortex lie next to each other in the central sulcus and arestrongly interconnected (Huffman and Krubitzer 2001). At
a subcortical level, the superior colliculus provides a site ofconvergence for sensory information (through direct projections from the retina) and predictions (from visual, parietal, and frontal cortex to the intermediate and deep layers).
Neuronal discharges in deep layers, that initiate saccades,
define motor-fields that coincide with visual receptive fields
in superficial layers (Andersen 1989). Finally, at the spinallevel, the ventral (motor) and dorsal horn (proprioceptive)
are juxtaposed (see Fig. 1).
In summary, under active inference, perception tries to
explain away prediction errors by changing predictions,while action tries to explain them away by changing the signals being predicted. This formulation suggests that motor
control may not be the selection of an action but an emergent

property of predictive coding, where proprioceptive prediction errors are quenched by movement. In the next section,
we use the recognition dynamics above and see what sorts of
behavior emerge under active inference.
4 Simulations
4.1 Oculomotor control
In this section, we illustrate active inference using a series ofsimple simulations of oculomotor control. These simulations
assume that the agent has already learned the parameters that
optimize its inference about sensory states. We use these simulations to focus on the nature of active inference and show
how it can lead to phenomena like retinal stabilization, orientating responses, and pursuit movements. We then use thesame model to illustrate how active inference can lead tointentional behavior that is prescribed by prior expectations
about sensory input.
4.1.1 Retinal stabilization and oculomotor reflexes
Here, we use a linear state–space model and a nonlinear sensory mapping to generate high-dimensional sensory data. We
use the same model as a generative model for active inference
and explore the consequences of action under perturbation.
In this model, a single force perturbs the dynamics of two
hidden states to produce a damped transient. These states
control the location of object (a Gabor patch) in retinotopic space. For simplicity, these states cause the eye to move
(as opposed to the object). Sensory signals are sampled byphotoreceptors on a sparse grid rectangular. The equationsof motion are linear and parameterized in such a way that the
perturbing force causes displacement in one direction. This
displacement is then communicated to the other direction,
to induce elliptical motion of the visual target. The sensory
mapping, on the other hand, is a highly nonlinear functionof the hidden states (direction of gaze or stimulus location)
because the visual stimulus has a nonlinear contrast profile.
We will use this system to ask what would happen under
different prior expectations about stimulus movement. This
generative process and model are given by Eqs. 7 and 8 whereg(x)ij = υ i − 34 x1 − N2 υ j − 43 x2 − N2υ(x) = exp − 41 x 2 cos 23 π xf(x) =

1 −4
v+a
x
− 18
2 1
0

(11)

g(x)ij = g(x)ij
1 −4vxf (x) =
− 18
2 1
0

123

Biol Cybern
Fig. 1 Schematic detailing thecomponents of the motor reflex
arc and how it might minimize
sensory prediction error(free-energy) through action.
The basic idea here is that thecentral nervous system provides
predictions about the expected(proprioceptive) consequencesof movements, which action isenslaved to fulfill. In the context
of sensorimotor reflexes, motor
signals emitted by ventral
efferents of the spinal cord try to
suppress prediction errors
induced by an unexpected firingof stretch-receptors. However,from the point of view of the
motor neurons, this prediction
error could also have been
mediated by a top-down
suppression of predicted
stretch-receptor discharge rates.
In short, sensory prediction
error controls movements andtop-down predictions control
sensory prediction error

From reflexes to action

predictionµg (µ )
dorsal horn

dorsal root

s(a)

ξs

actionventral root

ventral hornξs = Πs (s(a) − g (µ))

These equations, which specify the nature of the environment
and model, are used to simulate environmental dynamics
and the agent’s responses. This entails integrating the stochastic differential equation 7 (to evaluate the environment’s
states) and the ordinary differential equation 9 (to evaluate
the agent’s states). See Appendix 3 for details of the integration scheme. Note that these equations are specified in termsof states. The corresponding equations for generalized states
are derived easily using the chain rule (see Friston 2008; Eqs.
1 and 2).
In this example, a force or cause v(t) excites changes in
the hidden states through the equations of motion, f(x). The
hidden states (stimulus location in two dimensions) are thenmapped to sensory data and delivered to the agent, using stateand observation noise with log-precisions (inverse variances)of four and eight respectively. A log-precision of four meansthat the standard deviation is exp(−4/2) = 0.135. In all thesimulations below, we used relatively smooth noise for both
the generative process and model. This smoothness is controlled by the precision of its higher derivatives, which werechosen to give a Gaussian autocorrelation function with astandard deviation of one half of a simulated time bin (usually 1/32 s). In this example, the sensory data are a nonlinear

123

a

∂ε sT
ξs
a=−
∂a

function of stimulus location, generating 6×6 = 36 channelsof data. These functions can be thought of as receptive fields
that map the retinotopic location of the stimulus to the patternof excitation, g(x)ij in an N × N array of sensory receptors(see Fig. 2), where i, j ∈ {1, 2, . . . N} : N = 6.
The agent then infers the hidden states (i.e., stimulus location) by optimizing the sufficient statistics of its recognition
density, as described in the previous section (Eq. 9). The only
extra thing that we need to consider here is action, a(t), whichis part of the generative process but not part of the generative model. The only way that action can change free-energy
is by re-sampling sensory data to reduce sensory prediction
error. From Eq. 9, we can see that this just depends on sensory prediction error and the changes in sensory signals withaction. Appendix 3 provides more details about the ensuingdynamics for the interested reader.
Here, we have deliberately chosen to put action in opposition to the force, so that the agent can oppose perturbations
to its hidden states. By manipulating its prior expectations onthese perturbations, v(t), we can elicit very different behaviors from the agent. This is because the action is used to
explain away prediction error, and prediction error depends
upon the agent’s prior expectations. For example, if weBiol Cybernbasensory prediction and rerro

hidden states (location)21µx (t )

0.8

Visual stimulus

g (µ (t ))

0.6

1

0.4

Sensory
channels

0
0.2
0

-1

-0.2
-0.4

10

20

30

40

50

60

-2

10

20

30

40

50

60

time

time

v
Active inference under flat priors(movement with percept)

a

1.2

1.2

µv (t )

1

µx

µx

1

µv

0.8

ξx

2

perturbation and action

cause (perturbing force)

ξs

s

ξv

1

0.6

0.6

0.4

0.4

0.2

0.2

0

0

-0.2

10

20

30

time

40

v(t)

0.8

50

60

-0.2

a(t)
10

20

30

40

50

60

time

Fig. 2 a Schematic showing the relationship of the generative model(lower panel) to the generative process modeling oculomotor samplingof the world. The schematic of the generative model used by the brain
comprises state-units or populations (white) that encode sensory input
and the conditional expectations of hidden states causing that input.
Error-units are shown in red and encode the prediction error for theircorresponding state-unit. The dotted lines are connections that are
intrinsic to the model, and solid lines represent coupling between actionunits (black) and effectors. The red lines convey prediction errors to
the state (and action) units, while the black lines pass predictions to
the error-units (and effectors). Please see Friston et al. (2008) for a
fuller description of the scheme implied by these circuit diagrams. In
this example, action causes eye movements which change the visual
information sampled by an array of photoreceptors (top panel). The
locations of these receptors are shown by the white dots, superimposed on the Gabor patch-like stimulus. Here, the states of the world
are the direction of gaze in two dimensions. b Upper left These are
the predicted sensory states and corresponding prediction error (redlines); they represent the predicted sensory input over the 36 photoreceptor channels induced by motion of the Gabor patch on the left.
This rather complicated set of transients is the prediction based on asimple elliptical movement of the stimulus described by the expected
hidden states. Upper right The expected hidden states causing sensory
predictions. These can be thought of as vertical and horizontal displacements in retinotopic space. The colored lines correspond to conditional
expectations, and the dashed gray lines to true values (superimposed inthis example). The gray regions correspond to 90% confidence intervals based on the conditional covariance. Lower left The true (dottedline) and conditional expectation (solid line) of the exogenous cause; aGaussian bump function of peristimulus time. In this example, the prior
was zero with a large prior variance. The conditional expectation conforms roughly to the true perturbation. Lower right Action (solid line)
and true cause (dotted line). In this instance, action does not changevery much because it is not needed to explain away prediction error.
This is because a priori the agent expects perturbations of unspecified
amplitude

induce a damped elliptical motion in the target, the agent will
sample the stimulus differently, depending upon whether it
expects the stimulus to move or not.
We first examined the responses of the agent under flat or
uninformative priors on the force. These priors were implemented by making the agent expect large amplitude random
fluctuations (with a log-precision of ln µvγ = −16) about
the prior mean η(t) = 0. This simply means that, a priori,

forces can be very large or small and the prior distribution isvery wide (i.e. flat). Under these priors, we would expect theforce and consequent motion to be perceived, because there
are no constraints on whether perturbations should be large orsmall. Consequently, we would expect the perceived location
to change as it would in the absence of active sampling. The
results in Fig. 2 show this to be the case. Here, we perturbed
the system with a Gaussian bump function of peristimulus123

Biol Cybernab sensory prediction and error

hidden states (location)
2

1

Visual stimulus1

g (µ (t ))0.5µx (t )

0Sensory
channels

-1

0

10

20

30

40

50

60

-2

10

20

time

30

40

50

60

time

v
Active inference under tight priors(no movement or percept)acause (perturbing force)

ξs

s

1

1

µx

1

ξx

2

v(t)

µv (t)

0.8µx

perturbation and action

1.2

0.5

0.6
0
0.4
-0.5

0.2

µv

ξv

a(t)

0
-0.2

-1
10

20

30

time

40

50

60

10

20

30

40

50

60

time

Fig. 3 As per the previous figure but now using informative priors on
the cause. These priors are induced by increasing the conditional logprecision of random fluctuations in the cause; indicated by thick red linein the lower panel of a. The ensuing priors drive action to explain away
prediction error. In this instance, action mirrors the real cause (circled
region in the lower right panel of b, so that their respective influences
cancel and there is an attenuation of perceived movement. Note that theprediction error under these informative priors is larger (red lines). It
is this prediction error that drives action. a This shows the relationshipof the generative model (lower panel) to the generative process modeling oculomotor sampling of the world. b Upper left These are thepredicted sensory states and corresponding prediction error (red lines);
upper right The expected hidden states causing sensory predictions:
The colored lines correspond to conditional expectations, and the grayregions correspond to 90% confidence intervals. Lower left The true
(dotted line) and conditional expectation (solid line) of the exogenous
cause. Lower right Action (solid line) and true cause (dotted line)

time. It can be seen that the conditional expectation andtrue location are roughly the same and that very little action
has been emitted. The results of this simulation should becompared with those in Fig. 3. Here, we placed tight priors(ln µvγ = 16) on the force, so that the agent did not expect
any movement of the stimulus that, essentially, it is foveating.
When the stimulus actually moves, the consequent prediction error is explained away by action, and the excursion of
the stimulus is countered almost immediately. The dynamics
of action can be seen in the lower right panel and, as anticipated, mirror the exogenous force so that it is opposed almost
exactly. The retinotopic trajectories with flat and informative
priors on exogenous forces are plotted in Fig. 4 for comparison. These show that the excursion of the target has been
completely suppressed by precise expectations (of no perturbations). Note that the only difference between the simulations in Figs. 2 and 3 was that we increased the precision of

the priors on the force, as indicated by the thicker red line
at the bottom of the schematic in Fig. 3. We will return to
the key role of precision in controlling behavior in the next
section.
There are several interesting interpretations of this simple
demonstration: If we consider it in the light of vision, thenaction corresponds to saccadic or reflexive eye movements.
Note that the agent can only move its eye in one direction,although the motor plant entailed by the state equations of
the generative process couple the resulting movements inboth directions. Despite this, it is able to stabilize the imageof the stimulus under active sampling. In other words, theimage is stabilized in accordance with the prior expectationof no movement. This may provide a nice metaphor for retinotopic stabilization through visual and vestibular oculomotor reflexes (Tatler and Wade 2003). Alternatively, we could
consider the action as causing a shift in the sensory frame123Biol Cybern

Retinal stabilisation
or tracking induced

Visualby priors

stimulus

under flat priors

under tight priors

2

2

1

1

0

0

-1

-1

-2
-2

x(t )

µx (t )

-2
-1

a

0

1

2

-2

-1

displacement
1

0.5

0.5

0

0

-0.5

-0.5

10

20 30

40 50

0

1

2

displacement

1

-1

real
perceived

60

time

-1

a(t )
µv (t)

v(t )
10

20 30

40

50

actionperceived andtrue perturbation

60

time

Fig. 4 The stimuli and trajectories with uninformative (Fig. 2) and tight(Fig. 3) priors: The upper panel details the stimulus or target in sensory space. Middle panels: These are the displacements of the stimulus
perceived under flat and informative priors. The displacement undertight priors is re-plotted in retinotopic space in the top panel. Note theshrinkage of the perceived excursion toward the prior expectation ofzero under tight priors (middle right). Lower panels: Action and causes
with (right) and without (left) tight priors. Here, action (solid blue line)
is engaged by informative priors to attenuate the perceived excursionof the stimulus (dotted green line)of reference that enables the agent to track a moving target.
This alternative perspective speaks to orientating or tracking
behavior that enables the perception of a moving target as if
it were stationary. Although the target may be moving, thefull extent of this movement is not registered by perception.
This is because the prediction error, under tight priors on
the perturbation, is explained away by action, and action is
not part of the perceptual model. In other words, from thepoint of view of the agent, the target is not moving. Thissuggests that orientating responses might be understood as
active sampling of the sensorium, which conforms to theprior expectation that targets do not move. This is a simple
but generic example of the sort of behavior that emerges withaction under the free-energy principle.

stationary. What would happen if we switched the cause andpriors around? In other words, what would happen if we took
the real cause and used it as a prior expectation but did notactually move the stimulus? (i.e., if we swapped η(t) andv(t) in Eq. 11). In this instance, there would be the same
prediction error (with opposite sign) and an active response.
In this case, the prior expectation is that the target is moving and active sampling will ensure that expectation is met.
This means the agent should saccade away from the target
and then return, once the prior η(t) falls back to zero. The
results in Fig. 5 show an example of this, using a model inwhich we replaced the nonlinear sensory mapping to simulated photoreceptors with a linear mapping to four oculomotor proprioceptors (see the schematic on the left). In thiscase, action reproduces the prior expectation on the cause to
suppress prediction error (because they play the same rolein the generative process; see Eq. 11). The result is apparent
motion of the stimulus, shown in the upper panels of Fig. 6.
This should be contrasted with the corresponding action in

4.1.2 Action and intended movements
In the previous simulations, we used an exogenous force withflat priors to elicit behavior that rendered a moving target

123Biol Cybernabhidden states (location)

sensory prediction and error
2

0.2
1
0.1

Proprioceptive input

0

0

-1

-0.1

v
-0.2
10

20

30

40

50

60

-2

10

20

time

a

s1

s2

s4

s3

30

40

50

60

time

ξs
Active inference under tight priors(movement and percept)

µx

µx

1

ξx

2

cause (prior)perturbation and action

1.2

1

1

0.8

0.8

µv

ξv

0.6

0.6
0.4
0.4
0.2

0.2

0

0

η

-0.2

10

20

30

time

40

50

60

-0.2

10

20

30

40

50

60

time

Fig. 5 As per Fig. 3 but here the true cause and prior expectation havebeen switched: i.e., the prior expectation is a Gaussian bump function,and the true cause is zero. In this instance, action is in the reverse direction to fulfill the prior expectation, causing a saccade away from thetarget and then back again. The resulting trajectory in retinotopic space
is shown in the next figure. In this example, sensory data were generated by proprioceptors in the oculomotor system, as opposed to simulated photoreceptors: This is shown schematically in panel (a), whichshows the relationship of the generative model (lower panel) with thegenerative process modeling oculomotor sampling of the world.
b Upper left: These are the predicted sensory states and corresponding prediction error (red lines); upper right: The expected hidden statescausing sensory predictions. The colored lines correspond to conditional expectations and the gray regions correspond to 90% confidence
intervals. Lower left: The true (dotted line) and conditional expectation(solid line) of the exogenous cause. Lower right: Action (solid line) andtrue cause (dotted line)

the Fig. 4 where action and forces mirror each other, thereby
canceling to mimic the flat prior.
This example suggests that movements can be induced byprior expectations of their sensory consequences. In otherwords, we only have to expect the sensory consequences of amovement to make it happen. Prior expectations induce predictions about what we should be sensing and action automatically moves sensory epithelia to fulfill these predictions. We
are not aware of the movement per se, but register its sensory consequences that are encoded by prior expectations.
In this example, the priors were prescribed by us. In more
realistic hierarchical models, one would imagine that movement-inducing expectations would arise as empirical priors
during inference on a more expansive set of multimodal sensory inputs.

This form of movement specification is a little counterintuitive, but has some appealing and biologically plausible
properties. For example, if we only have to specify the consequences of an intended or expected movement, then theactual action will be robust to variations in the true generative
process or unexpected perturbations to that process. In otherwords, specifying a movement in terms of what we want tosee, as opposed to what we want to do, automatically makes
behavior much more robust. In order to illustrate this, we
added an exogenous but late perturbation to the target stimulus during the intended movement. This is a simple metaphor
for empirical studies of movement trajectories under perturbation (e.g., Paulignan et al. 1991; Fourneret and Jeannerod
1998; Liu and Todorov 2007). The resulting inference andaction in Fig. 6 (middle panels) show that action (blue line)

123Biol Cybern
Fig. 6 These plots show the
trajectory of a perceived
stimulus location (left panels),
under movement induced bypriors and consequent action(right panels). Upper row: This
corresponds to a movement
elicited by priors, where theprior expectation is formally
identical to the true cause of theprevious simulations (seeFig. 3). Middle row: These are
the results of a simulation wherea small exogenous perturbing
force was applied late in the
movement trajectory (arrow inright panel). The impact on the
trajectory is almost
imperceptible (arrow in left
panel). In this instance, action
has compensated for theunanticipated perturbation.
Lower row: Here, we doubled
the sensitivity of the motor plant
to command signals or action.
Note that action has
automatically compensated forthis change and that the ensuing
trajectory, as perceived, has notdeviated from its priorexpectation

trajectories

action and causes

2

Self-generated

1

1
0.5

movements

0

induced by priors

-1

0

-2
-2

-1

0

1

2

2

-0.5

10 20 30 40 50 60

1

real

actionperceived1

perceived cause (prior)
0.5

robust toexogenous cause0

perturbation

0
-1
-2
-2

and change inmotor gain

-1

0

1

2

2

-0.5

10 20 30 40 50 60

1

1
0.5
0
0
-1
-2
-2

-1

0

1

2

-0.5

10 20 30 40 50 60

displacement

explains away the unexpected perturbation with relatively
little impact on the percept (green line). This means that the
perturbation was automatically discounted in terms of the
movement trajectory. As a final example of the robustness of
movement specification under active sampling, we changed
the motor plant by changing the sensitivity of movement toaction in the generative process; by replacing a → 2a in
Eq. 11. Again, because action is simply minimizing prediction error, this had no effect on the perceived excursion of
the target. As seen in Fig. 6 (lower panels), the only effectwas evident in the amplitude of the action elicited. The agent
does not need to know about this change in the dynamics
of the motor plant because action is effectively exploiting
closed-loop feedback (cf. the equilibrium-point hypothesis
discussed later).
4.1.3 Summary
In these simulations, we have seen how active sampling can
mimic retinal stabilization or visual tracking behavior by
implementing a prior expectation that a visual target is stationary. Conversely, we can induce movement through action
by invoking prior expectations about movements. Critically,
specifying an action in terms of its sensory consequences

timerenders action robust to variations in the motor plant or
unexpected exogenous perturbations. This is because action
explains away prediction error, which is prescribed by theprediction and prior expectations. The simulations in thissection used unimodal (visual or proprioceptive) models. In
the next section, we turn to sensorimotor integration in thecontext of multimodal (visual and proprioceptive) inference.
4.2 Sensorimotor integration
In this section, we illustrate the Bayesian integration of proprioceptive and visual signals to optimize computational
motor control of cued reaching movements (Körding and
Wolpert 2004; Disney et al. 2007; Kreisel et al. 2007; Bruyn
and Mason 2009; Diedrichsen and Dowling 2009). Thismethod highlights the ability of active inference to explain
multimodal integration in perception: through the convergence of bottom-up sensory prediction errors that optimize
perceptual representations and multimodal integration inaction; through the convergence of top-down sensory prediction error onto motor commands that optimize action. Thisbilateral role of prediction errors joins sensory and motor
processing and provides a nice perspective on sensorimotor
integration. Our particular focus here will be the weighting of123

Biol Cybern

visual and proprioceptive signals to optimize action, which
emerges from the optimization of precision parameters during recognition.
The model used in this section involves reaching to a target
using a two joint arm. The proprioceptive input corresponds
to the angle of both joints, (x1 , x2 ), which are the hidden
states, while the visual input is modeled as the position andbrightness of a target and the location of the arm’s extremity.
We will treat the location (v1 , v2 ) and brightness v3 of thetarget as autonomous causal states and the location of the arm
as a function of hidden states, J (x). We ignore the complexities of extracting positional information from retinotopically
mapped visual input and assume that the agent has directaccess to locations of the target and arm in visual space:⎡ ⎤xg(x, v) = g(x, v) = ⎣ v ⎦(12)JThe dynamics of the arm conform to Newtonian laws, underwhich action forces the angular position of each joint, withangular inertia, mi . Both joints have an equilibrium position at 90 degrees; with elasticity k = [8, 4] and viscosityκ = [4, 2] giving the following equations of motion
⎤ ⎡ 
⎤
⎡
ẋ1
x1
⎥
⎢ ẋ2 ⎥ ⎢ x 2⎥=⎢ ⎥f(x̃, v) = ⎢π⎦⎣ ẋ 1 ⎦ ⎣ a1 − k1 x1 −2 − κx 1 m1
a2 − k2 x1 − π2 − κx 2 m2
ẋ 2
(13)
However, the agent’s empirical priors had a very different
form: The agents’ generative model ignores elasticity andassumes that the arm’s distal extremity accelerates toward
the target when, and only, when the target is illuminatedbrightly. This prior expectation is encoded as an expected
force ϕ = v3 ([v1 , v2 ]T − J (x)) that acts on both joints
⎤ ⎡ 
⎤
⎡
ẋ1
x1
⎥
⎢ ẋ2 ⎥ ⎢ x ⎥=⎢ 2
 ⎥ (14)f (x̃, v) = ⎢TT⎣ x˙ 1 ⎦ ⎣ (ϕ J2 J OJ1 − κ1 x ) m1 ⎦2
 1
(φ T OJ2 − κ2 x2 ) m2
x˙ 2
Here, Ji (x) corresponds to the relative position of the i-th
joint such that J = J1 + J2 andJ1 =cos(x1 )sin(x1 )J2 =− cos(−x2 − x1 )sin(−x2 − x1 )

O=

0 −1
1 0

(15)
These priors on the dynamics of the hidden states cause thearm to reach for the target in a reasonably graceful and plausible way, when the target is brightened. Figure 8 shows atypical simulation under fairly noisy proprioception. Here,
we modulated the brightness of the target with a Gaussianbump function. When the target becomes bright the agent
believes its hand is drawn to the target. This belief induces

123

prediction errors on the generalized angular position (i.e.,
angular force), which action quenches by moving the joints
to comply with prior expectations. Note that sensory predictions are visual and proprioceptive because the generativemodel renders both modalities a function of hidden states,
like forces and torques. In other words, not only does theagent expect to feel a torque but it also expects to see acceleration of its arm. Once the brightness cue has disappeared, thearm remains on target because the agent’s generative model
does not include elastic forces that would return to the arm
to its resting position. This necessitates persistent action toresist elastic forces (see lower middle panel of Fig. 8). This
type of control is quite similar to the Passive Motion Paradigm (Mussa Ivaldi et al. 1988) and the ensuing Kinematic
Network model. This model expresses kinematic transformations implied by elasticity to regularize the ill-posed problem
that arises from redundancy in motor control.
In this simulation, we used a relatively large amountof sensory noise on the proprioceptive input (with a log
precision of four), while visual information was relatively
high precision (a log precision of eight). Despite this, themovement trajectory was remarkably smooth and accurate.
This is because action is driven by both proprioceptive andvisual prediction errors (descending black arrows in Fig. 7).
Although the proprioceptive errors are noisy, the visual errors
are precise and can elaborate accurate predictions of hidden
states (see top left panel of Fig. 8) and precise motor commands. This robustness to proprioceptive noise rests on optimizing the expected precisions in both sensory modalities asin Eq. 9. The ensuing optimization boosts visual prediction
errors relative to proprioception and ensures that the relative
contribution of both modalities is balanced in a Bayes optimal fashion (for both action and perception). This aspect ofmultimodal integration can be illustrated by examining the
expected precisions under factorial manipulations of visualand proprioceptive noise.
Figure 9 shows the expected precisions for proprioceptiveproµγ and visual µvis
γ modalities and the associated movements under all combinations of low (log precision of eight)
and high (log precision of four) levels of sensory noise. It
can be seen that the estimated precisions are almost exactly
correct for all four combinations. In terms of movement, it
is clear that vision can substitute for proprioception and viceversa. Only when both modalities are noisy (i.e., imperfectproprioception and obscuring the arm from sight), then reaching is compromised. This is seen on the lower left, when thearm flails around erratically, as might be seen in an excited
neonate.
4.2.1 Summary
In this section, we have reiterated the idea that apparently purposeful or intentional movements can be prescribed purely inBiol Cybern

ξ s = Π s ( s − g ( µ ))
ξ x = Π x ( D µ x − f ( µ ))
ξ v = Π v ( µv − η )ξ x ,v
µx

ξvis

µv

∂ε T
µ x ,v = D µ x ,v −ξ∂µ x ,v⎡v⎤
svis = ⎢ ⎥ + wvis
⎣J ⎦
(0,0)

x1
from reflexes to action
J1
Jointed arm
ξ pro

s pro

x + w pro

x1

x2

J2

(v1 , v2 )J (x) ϕaFig. 7 Schematic of the simulated cued-reaching models. Right: Motor
plant, comprising a two-jointed arm with two degrees of freedom (hidden states) corresponding to the angular position of each joint. The
position of the arm’s extremity (red circle) is the sum of the vectors
describing the location of each joint. Here, exogenous causes are theposition and brightness of the target (green circle). The masses (angular inertia) of the first (shoulder) and second (elbow) joints were fourand two respectably, with viscosity coefficients of eight and four. The
elasticity of both was one quarter. Left: The agent senses the hidden
states directly and indirectly through the position of the finger in visual
space. In addition, visual input includes the causes representing target

location and brightness. Sensory prediction errors are passed to higher
brain levels to optimize the conditional expectations of the hidden states(angular position of the joints). The ensuing predictions are sent back tosuppress sensory prediction errors. At the same time, sensory prediction
errors are also trying to suppress themselves by changing sensory input
through action. The red lines denote reciprocal message passing amongneuronal populations encoding prediction error and conditional expectations, however, the black lines represent descending motor control
signals from sensory prediction error units. The equations implement
free-energy minimization or recognition, as described in the main text

terms of prior expectations. Here, these priors were encodedby cue-dependent priors on the generalized motion of hidden
states (angular position) of an agent’s motor plant (a two-jointarm). These priors are combined with multimodal sensory
evidence to provide predictions, which drive action. Critically, the evidence from different modalities is integrated in
an optimal way by weighting the ensuing prediction errors by
their optimized precisions. Both recognition and action reston prediction errors and their relative weighting. Sensory prediction errors can derive from may modalities and sensory
epithelia. In general, these will converge through hierarchicalmessage passing to optimize perception and action in a bilateral fashion. In active inference, perception speaks to action
through sensory prediction errors which optimize perception
through their bottom-up effects and optimize action by projecting down to motor control units in the oculomotor systemand spinal cord.
This section has shown how simple cued movements canbe elaborated by priors under the free-energy principle, witha special focus on the role precision in optimizing multisensory integration in motor control. In the next section, we show
how autonomous goal-directed movements can be learned by123

Biol Cybern

prediction and error

hidden states

2.5

2

2

1.5

µx

1

µx

1.5
1

Trajectory

2

1

0

µ x′

0.5
0.5

1

-0.5

-0.5

10

30

20

40

50

60

30

40

50

60

causal states

perturbation and action

1

J ( x, t )

1.5

µv

1

3

µv

0.5

1,2

1.5

v3

1

0.2

v1,2

2
-0.5

a1

0

0.5

1

1.5

position

0

a2

0
-0.2

20

time

0.8

0.4

10

(v1 , v2 )

0.5

2

time
1.2

0.6

µ x′

position

0

0

10

20

30

40

time

50

60

-0.5

10

20

30

40

50

60

time

Fig. 8 Simulating cued movements: Left conditional expectations of
causal and hidden states using the same format as in Figs. 2, 3, and 5. In
this example, the proprioceptive input was quite noisy (gray lines in the
top left panel); necessarily producing large proprioceptive prediction
errors (red lines in the top left panel). Right: Ensuing movement trajec-

tory caused by action (solid lines in the adjacent lower panel). The redcircles indicate the initial and final position of the arm, which is seen
to reach the target (green circle) reasonably quickly and smoothly. The
black lines show the trajectory of the arm, and the colored lines show
the position vectors at each time step

optimizing the parameters encoding prior expectations about
movement trajectories.

(2009) to illustrate how reinforcement learning can be castas free-energy optimization. Here, we reprise the simulations but interpret them in terms of learning a motor policythat manifests as goal-directed movements. The mountaincar problem can be envisaged as follows: one has to movea heavy weight away from the bottom of the well and keep
it there (Fig. 10). However, the weight is too heavy to simply to push it out of the well. This means that the only way
to move the weight to its desired location is to push it up
one side of the well and use its momentum to get it past thesteepest part of the other side. This represents an interesting
problem, when considered in the state–space of position andvelocity, (x, x  ); the agent has to move away from the target
location (x = 1, x  = 0) to attain its goal and execute avery circuitous movement (cf., avoiding obstacles; Jax andRosenbaum 2007). This problem can be specified with thesensory mapping and equations of motion

5 Goal-directed behavior
This section deals with the optimization of action or policies under perceptual learning (cf. Grafton et al. 2008). In
the previous section, we specified the priors that led to cuedmovements. Here, we consider how these priors are learned.
In brief, we will show that quite complicated goal-directedbehaviors can be instilled through supervised learning byplacing an agent in a controlled environment so that it learns
what to expect. When replaced in an uncontrolled or nativeenvironment, the agent seeks out the sensory states that it
has learned to expect. The result is an optimum policy that isrobust to perturbations and constrained only by the agent’s
prior expectations that have been established during training.
A controlled environment can be thought of as one in which
certain behaviors are enforced. In the context of reaching
movements, this control could be exerted by force channels(e.g., Scheidt et al. 2000; Diedrichsen and Dowling 2009).
The model used in this section is a benchmark example indynamic programming: namely, the mountain-car problem.
These simulations have already been reported in Friston et al.

123

g = x̃
f=

ẋ
ẋ


b(x) ==

xb(x) − 41 x + v + σ (a + c(x̃))−2x − 1 :x≤0−x2 (1 + 5x2 )−3/2 − x4 16 : x > 0c(x̃) = θ1 + θ2 x + θ3 x + θ4 xx + θ5 xx + θ6 x x

(16)

Biol Cybern
Fig. 9 Reaching under
different levels of sensory noise.
All panels show movement
trajectories using the sameformat as in the previous figure.
The inset for each panel shows
the conditional expectations of
the precision (inverse varianceor noise level) of proprioceptive
and visual input. It can be seenthat movement is robust to
noise, unless both modalities
deliver degraded sensory
information. Lower row: Noisy
proprioceptive (angular joint
position) input. Right column:
Noisy visual information about
the location of the arm (e.g.,
partial occlusion)µγpro

Conditional
precisions

10

10

8

8

6

6

4

0

Noisy vision

µγvis

4

0

2

2
0

position

0.5

1

1

1.5

1.5

2
-0.5

0

1

0.5

0

0.5

2
-0.5

1.5

0

0.5

10

10

8

8

6

6

4

0

position

Noisy proprioception

0

1.5

1

0.5

1

2
-2

1.5

-1

position0

position

ForcesMountain car problem
3

0.7

2.5

0.6

b( x) = −∂ xϕ ( x)

2
0.5
0.4

1.5

ϕ ( x)

force

heightFig. 10 Schematic of themountain car problem: Left The
landscape or potential energyfunction that defines forces on
the weight. This has a minimum
at x = −0.5. The weight isshown at its uncontrolled stable
position (transparent) and thetarget position at the top of thewell on the right at x = 1(vertical dotted line). Right:
Forces experienced by theweight at different positions dueto the slope of the well (blue).
Critically, at x = 0, the force isunity and cannot be overcomeby the agent, due to thesquashing function applied toaction

2

0

-1

0

0

1

2

1

2
-0.5

1.5

4

-2

2
0.5

1

0.3

1
0.5
0

0.2
-0.5
0.1
0
-2

-1
-1

0

position

1

2

(x)

-1.5
-2

-1

0

position

1

2

(x)

123

Biol Cybern

As above, substitution of these functions into Eq. 7 provides
the equations used to simulate environmental dynamics andgenerate sensory data. The first equality means that the agent
has a noisy sense of the weight’s position and velocity. The
second means that the forces on the weight, ẋ (t) have four
components: a gravitational force b(x) = −∂x ϕ(x), which isjust the spatial gradient of the potential energy function ϕ(x)in Fig. 10 (left panel). The remaining three forces are frictionx /4, an exogenous force v(t), and a controlling force thatis bounded by a squashing (logistic) function: −1 ≤ σ ≤1. This force comprises action and state-dependent control,c(x̃). Control is approximated here with a second-order polynomial expansion of any nonlinear function of the states,
whose parameters are θ = {θ 1 , . . . , θ 6 }. When θ = 0 theenvironment is uncontrolled; otherwise, the weight experiences state-dependent forces that enable control.
In order to create a controlled environment, we simply
optimize the parameters of its equations of motion to maximize the probability that the goal is attained. This entails
adjusting the parameters to maximize the ensemble density, p(x̃|m) at the target location, as described in Friston
et al. (2009). The upper panels of Fig. 11 show the equilibrium densities without control (θ = 0; top row) and for thecontrolled environment (θ = θQ ; middle row) in which thegoal attracts all trajectories. As anticipated, the trajectories inFig. 11 (middle row) move away from the goal initially andthen converge on it. This controlled environment now plays
host to a naïve agent, who must learn its dynamics throughexperience.

5.1 Learning
The agent’s generative model of its sensory inputs comprised
the functions
g = x̃
f =x

b − 41 x  + v + σ (η + c)

(17)

For simplicity, we assumed that f (x̃, ṽ, θ ) was the same as
Eq. 16 but without action. The unknown causes in this model, ⊇ {x̃, ṽ, θ, γ }, comprise the states (position and velocity), exogenous force, parameters controlling state-dependent acceleration, and precisions of the random fluctuations.
The agent was exposed to 16 trials of 32 s time-bins. Simulated training involved integrating Eqs. 7 and 9 with θ = θQ(see Appendix 3). On each trial, the weight was “pushed”
with an exogenous force, sampled from a Gaussian densitywith a standard deviation of eight. This enforced a limited
exploration of state–space. The agent was aware of these perturbations, which entered as priors on the forcing term; i.e.
η̃ = ṽ. During learning, we precluded active inference, such123

that the agent sensed the trajectory passively, as it left thetarget state and returned to it.
Note that the agent does know the true states because
we added a small amount of observation error (with a logprecision of eight) to form sensory inputs. Furthermore, theagent’s model allows for random fluctuations on both position and velocity. When generating sensory data, we used asmall amount of noise on the motion of the velocity (with alog-precision of eight). After 16 trials, the parameters converged roughly to the values that were utilized to construct thecontrol environment. This means that the agent expected theweight to move along state–space trajectories that converge
on the target. These optimum dynamics have been learnedin terms of empirical priors on the generalized motion ofstates encoded by µθ —the expected parameters of the equations of motion. These expectations are shown in the lower
row of Fig. 11 in terms of the trajectories they encode; i.e.,f (x̃, ṽ, µθ ). It can be seen that the nullclines (lower right)
based on the parameters after training have a similar topology
to the controlled environment (middle right), ensuring thatthe fixed-points that have been learnt are the same as thoserequired to attain the goal. Therefore, what would happen if
the agent was placed in an environment that did not conformto its expectations?
5.2 Active inference
In order to demonstrate that the agent has learnt the optimum policy, we placed it in an uncontrolled environment; i.e.,θ = 0, and allowed action to minimize free-energy. We prevented further perceptual learning by fixing the conditional
expectations of the parameters to µθ and the log-precisions toµxγ = µsγ = 8. An example of active inference after learning
is presented in Fig. 12. Again, this involved integrating environmental and recognition dynamics (Eqs. 7, 9), where thesestochastic differential equations are now coupled throughaction. The colored lines show the conditional expectationsof the states, while the gray areas represent 90% confidence
intervals. These are tight because we used low levels of noise.
The dotted red line on the upper left corresponds to the prediction error: namely, the discrepancy between the observed
and predicted states. The ensuing trajectory is superimposedon the nullclines (middle panels) and shows the agent moving
the weight away from its goal initially, to build up the requisite momentum. Once the goal has been attained, action isstill required because, in the uncontrolled environment, it is
not a fixed-point attractor.
In order to illustrate the robustness of this behavior, we
repeated the simulation while applying exogenous forces tothe weight (modeled with a random normal variate smoothedwith a Gaussian kernel of 8 s). Because the agent did
not expect these perturbations, they were explained away
by action and not perceived. Goal-directed behavior wasBiol Cybern

Uncontrolled

velocityflow and density

nullclines

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2

velocity-2

Controlled

-1

0

1

2

2

2
1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-1

0

1

2

θ = θQ-2-2

velocity

-2

1.5

-2

Expected

θ=0

-1

0

1

-2

2

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2
-2

-1

0

position

1

2

-1

0

x′

1

2

θ = µθ

x
-2

-1

0

1

2

positionFig. 11 Left panels: Flow-fields and associated equilibrium densities
for an uncontrolled environment (top), a controlled or optimized environment (middle), and under prior expectations after learning (bottom).
Notice how the flow of states in the controlled environment enforces
trajectories that start by moving away from the target location at position x = 1 (with a small velocity). The red arrows denote the change instate (position and velocity) prescribed by the parameters: this change
is f (x̃, ṽ) evaluated at x3 = 0.5. The ensemble density is the principal eigenfunction of the Fokker–Plank operator associated with theseparameters. For the controlled and expected environments, these arelow entropy equilibria, concentrated on the target location. Right panels: These panels show the flow fields in terms of their nullclines. Nullclines correspond to lines in state–space where the rate of change of
one variable is zero. Here, the nullcline for position is along the x-axis,

where velocity is zero. The nullcline for velocity is when the change invelocity goes from positive (gray) to negative (white). Fixed points correspond to the intersection of these nullclines. It can be seen that in an
uncontrolled environment (top), there is a stable fixed point, where thevelocity nullcline intersects the position nullcline with negative slope.
Under controlled (middle) and expected ( bottom) dynamics, there are
now three fixed points. The rightmost fixed-point is under the maximumof the ensemble density and is stable. The middle fixed-point is halfway
up the hill, and the final fixed-point is at the bottom. Both of these areunstable and repel trajectories so that they are ultimately attracted tothe target location. The red lines depict an exemplar trajectory, under
deterministic flow, from x = x  = 0. In a controlled environment, thisshows the optimum behavior of moving up the opposite side of the wellto gain momentum so that the target (green dot) can be reachedpreserved (lower panels of Fig. 12), with mirror symmetry
between action and the displacing force it counters. Notethat action is greater because it exerts its effects through asquashing function (see Eq. 16).

5.3 Optimal behavior and conditional precision
Optimal behavior depends on the expected precision of thehidden state motion encoded by µxγ . In this example, the123Biol Cybern

prediction and error

hidden states

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5
20

40

60

80

100

-1.5

120

20

40

time

60

80

100

120

100

120

time

behaviour

action
3

2
1.5

2

1

velocity

Fig. 12 Top row: The left panel
shows the predicted sensory
states (position in blue andvelocity in green). The red lines
correspond to the prediction
error based on conditional
expectations of the states (rightpanel). The associatedconditional covariance isdisplayed as 90% confidence
intervals (thin gray areas).
Middle row: The nullclines andimplicit fixed points associatedwith the parameters learnt by theagent, after exposure to acontrolled environment (left).
The actual trajectory throughstate–space is shown in blue;(the red line is the equivalent
trajectory under deterministicflow). The action causing thistrajectory is shown on the rightand shows a polyphasic
response until the desired
position is reached, after whicha small force is required to stop
the weight sliding back into thewell (see Fig. 10). Bottom row:
As per the middle row but now
in the context of a smoothly
varying perturbation (broken
line in the right panel). Note that
this exogenous force has very
little effect on the navigation ofstate–space because it is
unexpected and countered byaction. These simulations used
expected log-precisions of eight

1

0.5
0

0

-0.5

-1

-1
-2

-1.5
-2
-2

-1

0

1

2

-3

behaviour

20

40

60

80

perturbation and action
3

2
1.5

2

velocity

1
1

0.5
0

0

-0.5

-1

-1
-2

-1.5
-2
-2

-1

0

1

position

agent was fairly confident about its prior expectations, but
did not discount sensory evidence completely (with log-precisions of µvγ = µsγ = 8). These conditional precisions
are important quantities, which control the relative influence of bottom-up sensory information relative to top-down
predictions. In a perceptual setting, they may mediate attentional gain (cf., Abbott et al. 1997; Yu and Dayan 2005; Friston 2008). In active inference, they also control whether an

123

2

-3

20

40

60

80

100

120

time

action is emitted or not (i.e., motor intention): Increasing
the relative precision of the hidden states causes more confident behavior, whereas reducing it subverts action. This isbecause prior expectations are discounted by sensory input
and are, therefore, not expressed at the level of sensory
predictions. In biological formulations of the free-energy
principle, modulatory neurotransmitters might encode theprecision of prior expectations (Friston 2008). In this context,

Biol Cybern

hidden states

hidden states

hidden states

0.5

1.5

0.8
0.6

1

0.4

0.5

0.2

0

0

0

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-0.5

-0.6

-1

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-1

20

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100 120

-1

20

40

time

µγx = 480

60

100 120

time

-1.5

20

40

60

80

100 120timeµγx = 5.5µγx = 6µγs = 8Fig. 13 Inferred states, (top row) and trajectories through state–space(bottom row) under different levels of conditional uncertainty or
expected precision. As in previous figures, the inferred sensory states(position in blue and velocity in green) are shown with their 90% confidence intervals, and the trajectories are superimposed on nullclines. As

the expected precision falls, the inferred dynamics are less accountableto prior expectations, which become less potent in generating prediction
errors and action. It is interesting to see that uncertainty about the states(gray area) increases, as precision falls and confidence is losta deficit in dopaminergic neurotransmission would reduce
the operational potency of priors to elicit action and lead to
motor poverty; as seen in Parkinson’s disease, schizophrenia,
and neuroleptic bradykinesia. Through progressively reducing the expected precision of the empirical priors that havebeen instilled during training, we can simulate this poverty.
Figure 13 shows three phases: first a loss of confident behavior, where the agent rocks backward and forward cautiously,
until it has more than sufficient momentum to reach its goal.
Second, a stereotyped behavior (corresponding to a quasiperiodic attractor), in which the agent prevaricates at the bottomof the well (cf., displacement activity, motor stereotypy, or
perseveration). Finally, we get avolitional behavior, where
the agent succumbs to gravity (cf., bradykinesia or psychomotor poverty).

reward or value to optimize these behaviors. It is sufficient
to induce prior expectations through perceptual learning thatprescribe goal-directed behavior in an unsupervised setting.
In the previous section, we considered the relative precision
among sensory signals in different modalities. Here, we kept
the sensory precision fixed and changed the precision of priorexpectations. Under active inference, the precision of sensory
information and empirical priors determine their relative contribution to action. In this context, it is interesting to note thata “mere expectation to move causes attenuation of sensory
signals” (Voss et al. 2008). This is consistent with a relative
increase in the precision and consequent amplitude of prediction errors that mediate empirical priors. In this section, we
focused on goal-directed movements that had a fixed goal orstate. In the final section, we address the learning of behaviors that are specified not in terms of goals but in terms of
movement trajectories per se.

5.4 Summary
We have seen that agents can learn causal structure in theenvironment and sample it in an adaptive and self-supervised
fashion. This results in behavioral policies that reproduce
those optimized by reinforcement learning and dynamic programming. Critically, we do not need to invoke the notion of

6 Learning autonomous behavior
In the previous section, we specified behavior in terms of atarget location in state–space. This is fine if we want agents123Biol Cybern

to occupy specific states: i.e., if we want to induce action
with fixed-point attractors. However, real-world behaviors
are dynamic, calling for the optimization of state trajectories
of an autonomous and itinerant nature. In ethology, thesemight correspond to a succession of states or sequences of
behaviors that are elicited with high-order operant conditioning paradigms. In optimal control, an exemplar problem is bipedal walking (Prinz 2006; Manoonpong et al.
2007). Here, we set ourselves the task of instilling behavioral trajectories in an agent that conform to one of thesimplest systems with autonomous dynamics, the Lorenz
attractor.
Using the same procedure as above, we used the following generative process and model, whose equations of motion
have the form of a Lorenz system
g =x ⎡⎤ ⎡ ⎤θ 1 x2 − θ 1 x1v
1 ⎣
θ 2 x1 − x3 x1 − x2 ⎦ + ⎣ 0 ⎦ + a
f = 64x1 x2 − θ 3 x30(18)
g =x ⎡⎤
θ1 x2 − θ1 x1
1 ⎣f = 64
θ2 x1 − x3 x1 − x2 ⎦x1 x2 − θ3 x3


We set θ = θQ = 10, 32, 83 and initialized the conditional
expectations of the agent to µθ = 0 (with fixed log-precisionsof µsγ = 8 and µxγ = 16). In these simulations, we took thehidden states to represent the position of an agent (or its arm)in a three-dimensional (3D) Euclidean space. This meansthat the agent moves under control as if it was on a Lorenz
attractor and comes to learn what trajectories to expect. After
16 trials of 512 time-bins (16 × 8 s of real time), the agent
was removed from its controlled environment and examinedfor evidence of learning. The upper panels of Fig. 14 show
the ensemble density and flow of the training environment
and a typical 8-s trajectory (upper panels: θ = θQ ). The
middle panels show the corresponding expectations of theagent before (middle panels: θ = 0) and after (lower panels:
θ = µθ ) learning. It can be seen that before learning, theagent expects to move toward a fixed point attractor; however, after learning, it expects to navigate its environment as
if it was on a Lorenz attractor.
Figure 15 shows that these expectations are fulfilled by
active inference, when placed in an unsupervised environment with θ = 0. Critically, the agent’s behavior is impervious to exogenous perturbations applied to the motion of thefirst state (see Eq. 18); this perturbation was modeled witha random Gaussian time-series smoothed with a Gaussian
kernel with a standard deviation of eight time-bins. Despite
these perturbations, the agent adheres to its concept of a Lorenz attractor. As above, the precision with which the agent

123

recapitulates its learned trajectories rests on a high degree ofconfidence in its dynamical priors, assured by high valuesof the log-precision, µxγ . Although an almost trivial example, this way of prescribing desired trajectories may havepragmatic applications in engineering and robotics (cf., Prinz
2006; Manoonpong et al. 2007). This is because the trajectories prescribed under active inference are remarkably robust
to noise and exogenous perturbations.
6.1 Summary
The picture that emerges from these simulations is that aneffective way to instantiate desired movements is to induce
prior expectations through use-dependent plasticity in theagent’s generative model, so that the expected movements are
reproduced automatically. The resulting priors might be thebasis of hand-path priming effects, in which hand movements
around obstacles induce a significant curvature in the trajectory of the movement trajectories that persists after the obstacle has been removed (Jax and Rosenbaum 2007). Indeed,most neuro-rehabilitation therapies rely on motor training
and use-dependent plasticity, where the patient is guided in
the reproduction or forced to reproduce desired movementswith the paretic limb. (e.g., Kreisel et al. 2007). Interestingly,
this use-dependent plasticity rests on neuromodulatory transmitter function (Bütefisch et al. 2002) of the sort that may
encode precision (Friston 2008).
Optimizing policies under the free-energy principle is not
limited to policies that are defined in terms of states withutility. This is because the only things that are required toteach an agent are exemplar trajectories. Heuristically, thiscorresponds to showing the agent what it is like to be optimaland then letting it fulfill its expectations in an unsupervised
setting. In fact, the only things needed to specify behavior
are the equations of desired motion. In principle, it may bepossible to prescribe desired kinematics, say during walking,and then let the agent reproduce the associated trajectories
in an uncontrolled environment with random perturbations.
It is easy to imagine the attractor in Fig. 15 being replacedby more complex and possibly itinerant attractors that havedesired context-sensitive properties and that could be learned
by adaptive agents.

7 Discussion
This article has suggested that active inference is mandatedby population dynamics and may provide a plausible explanation for some aspects of motor behavior. In particular, itsuggests that motor control can be understood as prior expectations about proprioception, which enslave the peripheral
motor system to fulfill them. Furthermore, active inference
provides a mechanistic account of how adaptive behavior

Biol Cybern

densities

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velocity

controlled

velocity

Fig. 14 The behavior of anagent that learns to be Lorenz
attractor depicted as in Fig. 11,
in terms of equilibrium densities(left) and exemplar trajectories(right). Top panels: Dynamics ofthe supervised environment thatoffers control of the agent sothat it can experience and learn
desired behavior. Middle panels:
Equivalent characterization of
behavior before training, when
the agent expects to be drawn toa point attractor. Lower panels:
The same as in the previous row
but after learning, when priorexpectations about theenvironment have beentranscribed from the controlledenvironment by learning

0

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positioncan emerge and suggests a simple alternative to dynamic
programming or reinforcement learning, when prescribing
desired behaviors. There is a vast literature on motor control and sensorimotor integration, which addresses a host ofissues in detail, relating theoretical and empirical perspectives. We have only addressed some of these issues superficially, using simple simulations to impart the intuitions
behind active inference. In the next section, we try to highlight the implications of the ideas presented above for someestablished areas of research.
7.1 Active inference and optimal control
Typically, dynamic programming and related approaches in
optimal control theory posit a value-function of every point instate–space. This is the utility or reward expected under thecurrent policy and is the solution to the relevant Bellman

10

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-20

θ = µθ
-10

0

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position

equation (Bellman 1952). A policy is then optimized toensure that states of high value are visited with greater probability. In control theory, value acts as a guiding function by
establishing gradients, which the agent can ascend (Bellman
1952; Sutton and Barto 1981; Friston et al. 1994). Similarly,
in discrete models, an optimum policy selects states withthe highest value (Watkins and Dayan 1992; Todorov 2006).
However, under the free-energy principle, there is no valuefunction or Bellman equation to solve. The only states thatagents aspire to are those that they expect to frequent.
In brief, active inference replaces optimal polices withprior expectations about state-transitions (e.g., movement
trajectories). These priors may be acquired (empirical priors)
through learning or may be innate (full priors) and optimized
by natural selection. In the mountain-car example, the priorexpectation (optimal policy) was acquired through learning
the equations of motion of a controlled environment. This

123

Biol Cybern

hidden states

prediction and error
50

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learnt
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5
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velocity

Fig. 15 The behavior of anagent that thinks it is a Lorenz
attractor portrayed using theformat in Fig. 2. However, this is
no ordinary attractor; its
trajectories are supported purely
by action (displayed as afunction of time in the lower
right panel). Action tries tosuppress prediction errors on
motion through its three
dimensional state–space (blueline in lower left panel). These
prediction errors (red line inupper left panel) are thedifference between sensed andexpected motion based on theagent’s generative model andexpected hidden states (upper
right panel). Critically, thisautonomous behavior is very
resistant to random forces on theagent. This can be seen bynoting that the first state has
been perturbed with a smooth
exogenous force (broken blueline on the lower right). Notethat action counters thisperturbation and the ensuing
trajectories are essentially
unaffected

0

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positioncan be regarded as a form of supervised learning, because theenvironment was supplied by us to instill a particular behavior. In Friston (2010), we show how the same behavior can be
elicited in an unsupervised setting, using prior expectations
that enforce exploration of state–space until states with highutility are encountered. Similar adaptive behavior emerges
under isotropic sequence order learning in a closed-loop setting (Porr and Wörgötter 2003). Both approaches rest on
eliminating stationary solutions to sensory-motor dynamics
that involve high-cost states. In sequence learning, this can be
implemented by eliminating withdrawal reflexes using predictive reflexes that are learned in a self-referenced fashion.
Does this mean active inference makes optimal control redundant? Not necessarily: active-inference simply provides alarger context, in which optimal policies become prior expectations. It is possible that the brain solves some Bellman equations to specify optimal expectations (policies) that action
could pursue. Interesting developments in machine learning speak this notion; “Recently there is growing interest in
using probabilistic inference methods for decision making
and planning. Promising about such approaches is that theynaturally extend to distributed state representations and effi-

123

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ciently cope with uncertainty”; see Toussaint (2009) for a fuller discussion of probabilistic inference as a model of planned
behavior. There are also alternative self-referenced schemes(e.g., Verschure and Voegtlin 1998; Wörgötter and Porr 2005;
Tschacher and Haken 2007) that may have greater ethological and neuronal plausibility. This theme will be developed
further in a forthcoming article on value-learning and freeenergy.
The perspective on optimum policies as prior expectations
may call for a reappraisal of reinforcement learning as an
explanation for behavior and, in particular, the role of dopamine in coding unpredicted reward (Montague et al. 1995;
Schultz et al. 1997; Doya 2002): If dopamine encodes precision through its classical neuromodulatory effects (cf., Yu
and Dayan 2005) how can this be reconciled with the viewthat it encodes prediction error on reward? From a neurobiological perspective, it may be that dopamine does not
encode the prediction error of value but the value of prediction error: i.e., the precision of prediction errors that drive
perception and action. If this idea is right, then it speaks to asymmetry between the role of dopamine in optimizing precision in anterior (e.g., mesocortical and mesolimbic) systems

Biol Cybern

Free-energy formulation
Desired and inferredstatesForward-inverse formulation
Desired and inferredstatesµx,v

ξx,v

Forward model(generative model){x , v} → s

Inverse model(control policy){x , v} → a

Sensory prediction
error

Environment{x , v, a} → s

Inverse models →aMotor command(action)

µx,v

ξx,v

Environment{x , v, a} → ss ( x , v, a )aMotor command(action)ξsaForward model{x , v, a} → s

Sensory predictionerrors ( x , v, a )ξsEfference copy
Corollary discharge
Fig. 16 Schematic highlighting the differences in functional architecture implied by the free-energy formulation and conventional motor
control theories. Both schemes rest on forward and inverse models (i.e.,internal models) but their nature is fundamentally different. In the freeenergy formulation, action or motor commands are generated throughan inverse model by sensory prediction errors, whereas in conventional

formulations, motor commands are produced by an inverse model that isdriven by inferred and desired states. The red arrows denote ascending
or bottom-up effects and black arrows mean descending or top-down
message passing. The equations in the text boxes indicate the mapping
implicit in the corresponding model

trying to predict proprioceptive and interoceptive sensations(i.e., value-learning) and the role of acetylcholine in optimizing hierarchical inference on exteroceptive input in posterior(e.g., paralimbic and parietal) systems (i.e., attention; Disney
et al. 2007). Furthermore, this perspective on dopaminergic
function fits comfortably with a gating role for dopamine(O’Reilly et al. 2002) in selecting the percepts that guideaction (Redgrave et al. 1999).
Another interesting aspect of precision is that it covers
generalized motion. This means that action is trying to suppress a mixture of [squared] sensory prediction error on position, velocity, acceleration, and jerk etc., where the relative
contributions, encoded by µsγ control the smoothness of the
trajectory. This mixture is formally similar to cost-functionsin the motor-control literature that try to account for smooth
motor trajectories that are observed empirically (Todorov andJordan 1998; Nakano et al. 1999; Berret et al. 2008).

2008), there are two models, an inverse model and a forward model. The inverse model maps from desired sensory
states to action or motor commands, while the forward modelmaps from action to sensory consequences. In order to learn
the inverse model, one needs to evaluate the consequencesof action, which may be sensed with delays and noise. Forward models are used to finesse this learning, by providing surrogate prediction errors (i.e., the difference between
desired sensory consequences and those predicted by theforward model). However, these forward models are notgenerative models of sensory dynamics that are learned in
active inference. There is a fundamental difference between
the functional architecture implied by active inference andconventional models of motor control (see Fig. 16). In conventional models, action is not driven by sensory prediction error but is specified by a control policy or inverse
model. This inverse model computes the optimum control
for achieving some desired states, which entails invertinga generative model of how action changes hidden states. In
active inference, the corresponding inverse model determines
how sensory prediction errors influence action. This implicit
inverse model corresponds to the partial derivatives ε̃a in

7.2 Inverse models in motor control
In conventional motor control theory (Wolpert and Miall
1996; Todorov and Jordan 2002; Shadmehr and Krakauer

123Biol Cybern

Eq. 9, which describes the change in prediction error withrespect to action; i.e., the change in sensory states with action.
In the examples above, we used the true generative process tocompute this mapping (see Eq. A3.3 in Appendix 3). However, real agents do not have access to the generative process:
Does this mean they have to learn an inverse model as in conventional motor control theories? This is certainly possible;
however, in active inference, the inverse model is much simpler than in conventional formulations, because it maps fromsensations to action, not from hidden states to action (seeFig. 16). This is important because simple inverse mappings
can be hard-wired by evolution. For example, the mappingfrom proprioception to action may be part of classical motorreflex arcs.
In Fig. 7, the inverse mapping from proprioceptive (e.g.,
stretch receptor) signals to action is simple because specificalpha and gamma motor neurons stretch specific muscle spindles. In contrast, the mapping between action and visual input
is highly nonlinear and one-to-many. This means that theinverse mapping from visual prediction error to action isunlikely to be implemented in the spinal cord, as impliedby the descending black connection in Fig. 7. Would removing this connection preclude visual input from contributing
to sensorimotor integration? No; the proprioceptive predictions driving action depend on conditional expectations about
hidden states of the body, and these expectations depend
on vision (they explain away visual prediction error). Thismeans that sensory prediction errors in all modalities canaffect proprioceptive predictions through conditional expectations about the deployment and trajectory of the motor
plant. This key interplay between action and perception willbe the subject of a future article.
In short, the free-energy formulation replaces an inverse
model, mapping from inferred states of the world to action,with the inversion of a much simpler mapping between action
and sensory consequences. This may be one way of understanding the prevalence of retinotopic maps (such as thosein the superficial and deep layers of the superior colliculus), which enable changes in sensory input with small eye
movements to be inverted quickly and simply. The mapping
between action and proprioception is even simpler: Motorneuron discharges in the ventral horn cause muscle spindles
to contract and change sensory prediction errors in dorsal
horn cells through classical motor reflex arcs (see Fig. 1). The
argument here is that action changes proprioceptive prediction errors in a stereotyped and simple way. This may enable
the inverse mapping to be specified epigenetically in peripheral or subcortical systems and conserved over context and phylogeny. The advantages of feedback control at the peripheral(e.g., spinal) level include robustness to noise or incomplete
specification of the inverse model (Wolpert and Miall 1996).
Furthermore, generating command signals in the peripheryfinesses many problems induced by temporal delays.

123

7.3 Forward models in motor control
In the free-energy formulation, recognition proceeds by
inversion of a generative or forward model mapping fromhidden states to sensations. In conventional motor control,
the forward model is used to generate predicted sensory consequences of action. These ensuing sensory prediction errors
are used in state-estimation and as teaching signals to optimize the inverse model or control policy. The mechanismsof this estimation and learning vary but are formally related
to perceptual inference and learning (e.g., Kalman filteringand associative plasticity); however, conventional schemes
only represent states, as opposed to the trajectories implicit
in generalized coordinates, which renders them sensitive totemporal delays.
The key difference is that forward models in conventional
motor control include the effects of action whereas, forward
models in active inference do not. Action is not part of inference (i.e., inversion of a generative model) because it is aknown quantity; action is just there to explain away unexpected sensory prediction errors. This does not mean thatagents cannot learn about their motor plant. Indeed, the motor
plant is probably one of the most important aspects of theenvironment for predicting sensory input (see Grafton and
Hamilton 2007). This may be reflected in the preoccupationof infants with moving their limbs (and the role of rattles inpromoting multimodal learning). However, generative models of the motor plant are not conventional forward models
of motor control, because they map from causes to consequences (not action to consequences). In other words, generative models allow exogenous causes of movement to beperceived but not action per se (imagine a percept of actionin the absence of sensations).
7.4 Efference copy and corollary discharge
In modern treatments, efference copy refers to a copy ofthe signals from the control policy (inverse) model that are
passed to the forward model to create corollary discharge signals (predicted sensory consequences of action; see Fig. 16
and Wolpert and Miall 1996). Corollary discharge is thenused to explain sensory input that can be attributed to action(reafference). This enables the consequences of exogenous
influences (exafference) to be isolated. An example in lower
vertebrates is seen in electric fish (Bell and Grant 1989),
which send top-down corollary discharges to inhibit ascending sensory pathways. This allows the fish to disambiguate
between self-generated electric organ discharges and thosefrom other fish. However, in the free-energy formulation,
there is no need to remove the sensory consequences of action
because action is only there to explain away sensory prediction errors. In the free-energy scheme, efference copy andcorollary discharge are simply the bottom-up and top-down

Biol Cybern

signals that mediate perception. In other words, corollary discharge can be understood as top-down predictions that inhibit
ascending sensory pathways by explaining away sensory prediction error. Conversely, sensory prediction error units send
top-down signals to motor control units and ascending prediction errors (efference copy) to optimize predictions (seeFig. 16). If this interpretation of efference copy and corollary discharge holds, then it highlights again the intimate
relationship between action and perception. A relationshipthat is highlighted by formulations of the equilibrium-point
hypothesis, which suggest “action and perception are accomplished in a common spatial frame of reference” (Feldman2009).
7.5 The equilibrium-point hypothesis
The equilibrium-point hypothesis (Feldman and Levin 1995)
suggests that movement is not controlled directly by the central nervous system but by top-down signals that supply the
final point of a movement (in terms of the lengths of tensor and extensor muscles). In this model, muscles and the
peripheral nervous system accommodate un-modeled forcesand viscosity automatically. This is closely related to thecurrent formulation, in which movements are prescribed by
top-down predictions and peripheral systems respond by
suppressing prediction error. Under the equilibrium-point
hypothesis, these predictions correspond to sensory input at
the equilibrium point. Under active inference, the predictions
are of the sensory trajectories that lead to the equilibrium
point. This difference is crucial, because it might account forsome difficulties with the equilibrium-point hypothesis: forexample, in explaining fast movements and overshoots.
Clearly, we have not touched on the detailed physiology
or mechanistic issues that are addressed by conventional efference copy or equilibrium-point models. However, the formulation of sensory prediction error in terms of trajectories(i.e., in generalized coordinates of motion) may be usefulfor movement specification in terms of fixed-point attractors.
Generalized coordinates mean that movements are specified
as predicted trajectories, which include speed, acceleration,jerk etc. This resolves some of the difficulties encounteredwhen considering neuronal propagation delays or unpredicted changes in force or viscosity. In particular, it addresses
the challenge posed to the equilibrium-point hypothesis: “Atheory of movement should offer insight into why we makemovements the way we do and why we activate muscles
in particular patterns” (Gottlieb 1998). It is interesting tonote that recent equilibrium-point formulations focus on the(coordinate) frames of reference that enable anticipation andprediction: “Experimental data also imply that once a frameof reference is chosen, its attributes are modified in a feedforward way, thus enabling the brain to act in an anticipatory
and predictive manner” (Feldman 2009).

7.6 Dynamic systems and optimal control
Generative models in the free-energy formulation are perceptual models that preclude action as a potential cause of
sensory perturbations. Action per se is used to suppress unexplained prediction errors. This means that, from the pointof view of the agent, there is no difference between navigating in a controlled and uncontrolled environment; theagent does not know that the expected sensory trajectories
are being caused by its own action; it experiences the world
as if it were delivering predicable streams of sensory input.
This perspective could reconcile some differences between
dynamic systems and optimal control treatments of computational motor control: “The dynamic system approach emphasizes motor control as a process of self-organization between
an animal and its environment. … In contrast, optimal control
approaches view motor control as the evolutionary or development result of a nervous system that tries to optimize rather
general organizational principles” (see Schaal et al. 2007). In
active inference, recognition dynamics optimize free-energy:i.e., self-organizing dynamical exchanges with the environment that optimizes neural processes and connectivity undera general free-energy principle. Crucially, free-energy andsurprise rest on prior expectations, which can only be optimized by evolutionary or developmental processes.

8 Conclusion
In summary, we have shown how the free-energy principle
can be motivated from the need for agents to maintain theirexchange with the environment in equilibrium. We have considered behavior that would emerge under this principle andhave shown how it can be harnessed to optimize policies usually addressed with optimum control theory. Underpinning
this study is a unifying approach to action and perception:
perceptual learning and inference is necessary to induce priorexpectations about the sensorium and action is engaged toresample the world to fulfill these expectations. This places
perception and action in intimate relation and accounts forthe both with the same principle. Furthermore, this principle
can be implemented in a simple and biologically plausible
fashion. The same scheme used in this article has been used
to simulate a range of biological processes: ranging fromperceptual categorization of bird-song (Kiebel et al. 2008)
to perceptual learning during the mismatch negativity paradigm (Friston et al. 2006). Furthermore, all the simulationsin those cited articles and in this article use just one Matlab
routine (see Appendix 3). If these ideas are valid, then they
speak to a fundamental role for perception in action (Wolpert
et al. 1995; Shadmehr and Krakauer 2008; Bays and Wolpert
2007; Tseng et al. 2007; Wei and Körding 2009).

123Biol Cybern
Acknowledgements We would like to thank Neil Burgess and Florentin Wörgötter for very helpful reviews of a previous version of thiswork.
Open Access This article is distributed under the terms of the Creative
Commons Attribution Noncommercial License which permits any
noncommercial use, distribution, and reproduction in any medium,provided the original author(s) and source are credited.

Appendices
Appendix 1
Here, we show why the entropy of hidden states is boundedby the entropy of sensory states and a sensory mapping term.
We then discuss why the minimizing sensory entropy is sufficient to reduce the entropy of hidden states. The sensory
states s̃ ∈ S are an additive mixture of a function of the hidden states x̃ ∈ X plus some generalized random fluctuationss̃ = g(x̃, θ ) + z̃(A1.1)

Because x̃ and z̃ ∈ Z are statistically independent, we have
(Eq. 6.4.6 in Jones 1979, p. 149)
I (S, Z) = H (S|m) − H (S|Z, m)= H (S|m) − H (X|m) −p(x̃|m) ln |gx̃ |dx̃
(A1.2)

Here and below a subscript denotes differentiation: i.e., gx :=∂x g. In (A1.1) I (S, Z) = D(p(s̃, z̃|m)||p(s̃|m)p(z̃)) ≥ 0
is the mutual information between the sensory states andnoise. This non-negative cross-entropy or Kullback–Leibler
divergence (see Theorem 6.5; Jones 1979, p. 151) means
the entropy of the sensory states is always greater than theentropy of the sensory map of hidden states(A1.3)
H (S|m) ≥ H (X|m) + p(x̃|m) ln |gx̃ |dx̃The gradient gx̃ is of the sensory mapping with respect tothe hidden states. The integral n (A1.3) reflects the fact thatentropy is not invariant to a change of variables and restson the assumption that the sensory mapping g : X → S isdiffeomorphic. This assumption could be considered as a prediction that sensory mappings must be diffeomorphic (i.e.,bijective and smooth). A diffeomorphism requires the dimensionality of the hidden and sensory state–spaces to be equal.
This is assured by the fact we are dealing with generalizedstates that can be truncated at any arbitrarily high order. For
example, if we had n hidden states in m generalized coordinates motion, then we would consider m sensory states inn generalized coordinates, such that dim(x̃) = dim(s̃) =n × m. Finally, rearranging (A1.3) gives Eq. 1 in the main
text.

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Minimizing sensory entropy
Because entropy is not invariant under a change of variables,
the sensory mapping is a key determinant of sensory entropy.
Note that agents cannot change the mapping per se; they can
only change the hidden states that are mapped to sensory
input. However, the astute reader will note that nonlinearities in the sensory mapping mean that changing some hidden
states will affect the sensory mapping of others (e.g., closing ones eyes). Does this mean that sensory entropy can besuppressed by simply avoiding sensory input? No—because
entropy is conditioned on the agent (it is the average surprise given a particular agent). Agents who are not surprisedby the absence of sensations are unlikely to exist because
they are unable to navigate the environment and maintainan equilibrium density. This argument is at the heart of thefree-energy formulation and appeals to the same tautology
as adaptive fitness in natural selection. In the present context, it suggests that sensory channels that are critical forsurvival should be privileged and cannot be occluded (e.g.,
nociceptive pain signals and interoception signaling hunger,
thirst etc.). Second, it suggests that agents should find theabsence of sensory information surprising. We develop thistheme in more detail in the context of value-learning (Friston
et al., in preparation). In brief, prior expectations (cf. optimal
policies) about the trajectories of hidden states induce exploration and sampling of the environment. These priors are partof the model on which entropy is conditioned, which reduce
the entropy of hidden states. This reduces sensory entropy,
which is a lower bound on free-energy. Put simply, the freeenergy formulation assumes that agents have the right priors.
This assumption cannot be violated, because agents who do
not have the right priors cannot exist.
Appendix 2Here, we derive the various formations of free-energy andshow how they relate to each other. We start with the quantity
we want to bound: namely, surprise or log-evidence associated with some sensory states s̃ that have been caused by some
unknown quantities  (dropping conditional dependency onmodel m for clarity)− ln p(s̃) = − ln p(s̃, )d
(A2.1)
We now simply add a non-negative cross-entropy or divergence between some arbitrary (recognition) density q() :=q(|µ) and the posterior density p(|s̃) to create a freeenergy bound on surprise

q()
d
F = − ln p(s̃) + q() lnp(|s̃)= − ln p(s̃) + D(q()||p(|s̃))(A2.2)

Biol Cybern
Fig. 17 Schematic summary of
the relationships among thevarious energies that participate
in the free-energy formulation.
The right hand panel shows that
free-energy upper bounds
sensory surprise, which in turnbounds implicit surprise about
hidden states of the world (undersome simplifying assumptions).
These bounding relationships
are assured by the divergencesthat separate them, which are(by construction) non-negative.
The associated time orpath-integrals of the energies areshown on the left. These arecalled ‘actions’ and are formallyequivalent to entropy (i.e.,
average surprise). See Appendix
2 for a more detailed discussion

The cross entropy term is non-negative by Gibb’s inequality. Because surprise depends only on sensory states, we canbring it inside the integral and use p(, s̃) = p(|s̃)p(s̃) toshow free-energy is the Gibb’s energy − ln p(, s̃) expectedunder q(), minus its entropy


q()
d
p(|s̃)p(s̃)

q()
d
= q() lnp(, s̃)= − q() ln p(, s̃)d + q() ln q()dF =q() ln= − ln p(, s̃)q + ln q()q(A2.3)

A final rearrangement, using p(, s̃) = p(s̃|)p(), showsthat free-energy is also complexity minus accuracy, wherecomplexity is the divergence between the recognition density q() and the prior density p()q()q() lndp(s̃|)p()


q()
d
= − q() ln p(s̃|)d + q() lnp()= − ln p(s̃|)q + D(q()||p())(A2.4)

F =Equations (A2.2), (A2.3) and (A2.4) are the three formulations used in the main text. Figure 17 provides a graphicalsummary of these relationships.

Appendix 3
The simulations in this article involve integrating time-varying states in both the environment and the agent as a single
system, which can be modeled with the following ordinary
differential equation⎡ ˙ ⎤ ⎡⎤s̃
Dg + Dz̃⎢ x̃˙ ⎥ ⎢ f + w̃ ⎥⎢ ⎥ ⎢⎥⎢ ṽ˙ ⎥ ⎢
⎥
D ṽ
⎢ ⎥ ⎢
⎥
⎢ ˙ ⎥ ⎢⎥Dz̃⎢ z̃ ⎥ ⎢⎥⎢ ˙ ⎥ ⎢⎥D w̃u̇ = ⎢ w̃⎥=⎢⎥⎢˙ ⎥ ⎢x⎢ µ̃x ⎥ ⎢ D µ̃ − Fµ̃x ⎥⎥⎢ ⎥ ⎢⎢ µ̃˙ v ⎥ ⎢ D µ̃v − Fµ̃v ⎥
⎥
⎢ ⎥ ⎣
⎦
⎣ η̃˙ ⎦
D η̃
−∂a F
ȧ

(A3.1)In order to update these states, we use a local linearization:u = (exp(t) − I )(t)−1 u̇ over time steps of t, where
∂ u̇
=
=
∂u
⎤
⎡
0
Dgx̃ Dgṽ D 0
0
···
0
⎢
I
fa ⎥
fx̃ fṽ
⎥
⎢
⎥
⎢ .
.
.
⎥
⎢ .
.
.
⎥
⎢ .
D
.
.
0
⎥
⎢
⎢
. ⎥
⎢
. ⎥
D 0
. ⎥
⎢
⎥
⎢
⎥
⎢ 0
··· 0 D0···⎥⎢
⎢ −Fµ̃x ỹ
···0 D − Fµ̃x µ̃x −Fµ̃x µ̃v −Fµ̃x η̃ −Fµ̃x a ⎥⎥⎢⎢ −Fµ̃v ỹ−Fµ̃v µ̃x D − Fµ̃v µ̃v −Fµ̃v η̃ −Fµ̃v η̃ ⎥
⎥
⎢
⎣
0
0
D
0 ⎦
···
0 −Fa µ̃x
−Fa µ̃v−Fa η̃ −Faa
−Fa ỹ

(A3.2)

123

Biol Cybern

The conditional expectations of the parameters and precisions are updated after every simulated trail as describedin Friston (2008). Equation A3.2 may look complicated but
can be evaluated automatically using numerical derivatives.
All the simulations in this article used just one routine—
spm_ADEM.m. Demonstrations of this scheme are available
as part of the SPM software (http://www.fil.ion.ion.ucl.ac.
uk/spm; DEM_demo.m), and reproduce the examples in themain text.
In order to include action in the free-energy scheme, we
simply add action to the states of the generative process producing sensory data and specify its dynamics as a gradient descent on free-energy. Because action can only affectthe free-energy through the sensory data that are sampled,it can only affect sensory prediction error. Therefore, action
dynamics are prescribed byȧ = −Fa = −ε̃aT ξε̃a = gx̃D −i fx̃i−1 fa(A3.3)i

The partial derivative of the error with respect to actionis the partial derivative of the sensory data with respect toaction and is specified by the generative process. In biologically plausible instances of this scheme, this partial derivative would have to be computed on the basis of a mappingfrom action to sensory consequences (see Sect. 7). One mightassume that this mapping was sufficiently simple to be hardwired or, when dependent upon sensory states, was based onsome interpolation or Gaussian process model. This will bedemonstrated in future simulations.
Note that (A3.1) is formulated in generalized coordinatesof motion. Although there is no need to express action in
generalized coordinates (because it is real-world state), it canchange the high-order motion of other environmental states.
For example, in the mountain-car system, action changes thesecond-order motion of position (i.e., acceleration) and plays
the role of a Newtonian force.

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