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The Free-Energy Principle: a Unified Brain Theory?

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The Free-Energy Principle: a Unified Brain Theory? REVIEWS The free-energy principle: a unified brain theory? Karl Friston Abstract | A free-energy principle has been proposed recently that accounts for action, perception and learning. This Review looks at some key brain theories in the biological (for example, neural Darwinism) and physical (for example, information theory and optimal control theory) sciences from the free-energy perspective. Crucially, one key theme runs through each of these theories — optimization. Furthermore, if we look closely at what is optimized, the same quantity keeps emerging, namely value (expected reward, expected utility) or its complement, surprise (prediction error, expected cost). This is the quantity that is optimized under the free-energy principle, which suggests that several global brain theories might be unified within a free-energy framework. Free energy Despite the wealth of empirical data in neuroscience, Motivation: resisting a tendency to disorder. The An information theory measure there are relatively few global theories about how the defining characteristic of biological systems is that that bounds or limits (by being brain works. A recently proposed free-energy principle they maintain their states and form in the face of a greater than) the surprise on for adaptive systems tries to provide a unified account constantly changing environment3–6. From the point sampling some data, given a of action, perception and learning. Although this prin- of view of the brain, the environment includes both generative model. ciple has been portrayed as a unified brain theory1, its the external and the internal milieu. This maintenance Homeostasis capacity to unify different perspectives on brain function of order is seen at many levels and distinguishes bio- The process whereby an open has yet to be established. This Review attempts to place logical from other self-organizing systems; indeed, the or closed system regulates its some key theories within the free-energy framework, in physiology of biological systems can be reduced almost internal environment to homeostasis7 maintain its states within the hope of identifying common themes. I first review entirely to their . More precisely, the rep- bounds. the free-energy principle and then deconstruct several ertoire of physiological and sensory states in which an global brain theories to show how they all speak to the organism can be is limited, and these states define the Entropy same underlying idea. organism’s phenotype. Mathematically, this means that The average surprise of the probability of these (interoceptive and exterocep- outcomes sampled from a entropy probability distribution or The free-energy principle tive) sensory states must have low ; in other density. A density with low The free-energy principle (BOX 1) says that any self- words, there is a high probability that a system will entropy means that, on organizing system that is at equilibrium with its environ- be in any of a small number of states, and a low prob- average, the outcome is ment must minimize its free energy2. The principle is ability that it will be in the remaining states. Entropy relatively predictable. Entropy surprise 8 is therefore a measure of essentially a mathematical formulation of how adaptive is also the average self information or ‘ ’ uncertainty. systems (that is, biological agents, like animals or brains) (more formally, it is the negative log-probability of an resist a natural tendency to disorder3–6. What follows is outcome). Here, ‘a fish out of water’ would be in a sur- a non-mathematical treatment of the motivation and prising state (both emotionally and mathematically). implications of the principle. We will see that although the A fish that frequently forsook water would have high motivation is quite straightforward, the implications are entropy. Note that both surprise and entropy depend The Wellcome Trust Centre for Neuroimaging, complicated and diverse. This diversity allows the prin- on the agent: what is surprising for one agent (for University College London, ciple to account for many aspects of brain structure and example, being out of water) may not be surprising Queen Square, London, function and lends it the potential to unify different per- for another. Biological agents must therefore mini- WC1N 3BG, UK. spectives on how the brain works. In subsequent sections, mize the long-term average of surprise to ensure that e-mail: I discuss how the principle can be applied to neuronal their sensory entropy remains low. In other words, [email protected] doi:10.1038/nrn2787 systems as viewed from these perspectives. This Review biological systems somehow manage to violate the Published online starts in a rather abstract and technical way but then tries fluctuation theorem, which generalizes the second law 13 January 2010 to unpack the basic idea in more familiar terms. of thermodynamics9. NATURE REVIEWS | NEUROSCIENCE VOLUME 11 | FEBRUARY 2010 | 127 © 2010 Macmillan Publishers Limited. All rights reserved REVIEWS Box 1 | The free-energy principle Part a of the figure shows the dependencies among the a quantities that define free energy. These include the Environment Agent internal states of the brain μ(t) and quantities describing its exchange with the environment: sensory signals (and their Sensations T ~ ~ ~ motion) s˜(t) = [s,s′,s″…] plus action a(t). The environment s = g(x, ϑ) + z is described by equations of motion, which specify the trajectory of its hidden states. The causes ⊃ {x˜ , , γ of ϑ θ } External states Internal states sensory input comprise hidden states x˜ (t), parameters θ ~ ~ ~ ~ x˙ = f(x, a, ϑ) + w μ = arg min F(s, μ) and precisions γ controlling the amplitude of the random fluctuations z˜ (t) and w˜ (t). Internal brain states and action minimize free energy F(s˜ ,μ), which is a function of sensory Action or control signals input and a probabilistic representation q(ϑ|μ) of its causes. ~ a = arg min F(s, ) This representation is called the recognition density and is μ encoded by internal states μ. The free energy depends on two probability densities: b the recognition density q(ϑ|μ) and one that generates Free-energy bound on surprise sensory samples and their causes, p(s˜ , m). The latter ~ ϑ| F = −<ln p(s, ϑ | m)>q + <ln q(ϑ | μ)>q represents a probabilistic generative model (denoted by m), the form of which is entailed by the agent or brain. Action minimizes prediction errors Part b of the figure provides alternative expressions for the ~ F = D(q(ϑ | μ) || p(ϑ)) − <ln p(s(a) | ϑ, m)>q free energy to show what its minimization entails: action can reduce free energy only by increasing accuracy (that is, a = arg max Accuracy selectively sampling data that are predicted). Conversely, optimizing brain states makes the representation an Perception optimizes predictions ~ ~ approximate conditional density on the causes of sensory F = D(q(ϑ | μ) || p(ϑ | s)) − ln p(s | m) Surprise input. This enables action to avoid surprising sensory (Surprisal or self information.) μ = arg max Divergence The negative log-probability of encounters. A more formal description is provided below. an outcome. An improbable Optimizing the sufficient statistics (representations) outcome (for example, water Optimizing the recognition density makes it a posterior or conditional density on the causesNa of tursensorye Revie data:ws | Neurthis canoscienc be e flowing uphill) is therefore seen by expressing the free energy as surprise –In p(s˜ ,| m) plus a Kullback-Leibler divergence between the recognition and surprising. conditional densities (encoded by the ‘internal states’ in the figure). Because this difference is always positive, minimizing Fluctuation theorem free energy makes the recognition density an approximate posterior probability. This means the agent implicitly infers or (A term from statistical represents the causes of its sensory samples in a Bayes-optimal fashion. At the same time, the free energy becomes a tight mechanics.) Deals with the bound on surprise, which is minimized through action. probability that the entropy Optimizing action of a system that is far from the Acting on the environment by minimizing free energy enforces a sampling of sensory data that is consistent with the thermodynamic equilibrium current representation. This can be seen with a second rearrangement of the free energy as a mixture of accuracy and will increase or decrease over a given amount of time. It complexity. Crucially, action can only affect accuracy (encoded by the ‘external states’ in the figure). This means that states that the probability of the brain will reconfigure its sensory epithelia to sample inputs that are predicted by the recognition density — in other the entropy decreasing words, to minimize prediction error. becomes exponentially smaller with time. Attractor In short, the long-term (distal) imperative — of main- Crucially, free energy can be evaluated because it is a A set to which a dynamical taining states within physiological bounds — translates function of two things to which the agent has access: its system evolves after a long into a short-term (proximal) avoidance of surprise. sensory states and a recognition density that is encoded enough time. Points that get close to the attractor Surprise here relates not just to the current state, which by its internal states (for example, neuronal activity remain close, even under cannot be changed, but also to movement from one state and connection strengths). The recognition density is a small perturbations. to another, which can change. This motion can be com- probabilistic representation of what caused a particular plicated and itinerant (wandering) provided that it revis- sensation. Kullback-Leibler divergence its a small set of states, called a global random attractor10, This (variational) free-energy construct was (Or information divergence, information gain or cross that are compatible with survival (for example, driving a introduced into statistical physics to convert difficult entropy.) A non-commutative car within a small margin of error).
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