The 'free energy principle' (FEP) has been suggested to provide a unified theory of the brain, integrating data and theory relating to action, perception, and learning. The theory and implementation of the FEP combines insights from Helmholtzian 'perception as inference', machine learning theory, and statistical thermodynamics. Here, we provide a detailed mathematical evaluation of a suggested biologically plausible implementation of the FEP that has been widely used to develop the theory. Our objectives are (i) to describe within a single article the mathematical structure of this implementation of the FEP; (ii) provide a simple but complete agent-based model utilising the FEP; (iii) disclose the assumption structure of this implementation of the FEP to help elucidate its significance for the brain sciences. be shown that minimising IFE makes the R-density a good approximation to posterior density of environmental variables given sensory data. Under this interpretation the surprisal term in the IFE becomes more akin to the negative of log model evidence defined in more standard implementations of variational Bayes [30].130 The Action-Perception CycleMinimising IFE by updating the R-density provides an upper-bound on surprisal but cannot minimise it directly. The FEP suggests that organisms also act on their environment to change sensory input, and thus minimise surprisal indirectly [1,2]. The mechanism underlying this process is formally symmet-135 ric to perceptual inference, i.e., rather than inferring the cause of sensory data an organism must infer actions that best make sensory data accord with an internal environmental model [9]. Thus, the mechanism is often referred to as

Tononi et al. ͓Proc. Natl. Acad. Sci. U.S.A. 91, 5033 ͑1994͔͒ proposed a measure of neural complexity based on mutual information between complementary subsystems of a given neural network, which has attracted much interest in the neuroscience community and beyond. We develop an approximation of the measure for a popular Gaussian model which, applied to a continuous-time process, elucidates the relationship between the complexity of a neural system and its structural connectivity. Moreover, the approximation is accurate for weakly coupled systems and computationally cheap, scaling polynomially with system size in contrast to the full complexity measure, which scales exponentially. We also discuss connectivity normalization and resolve some issues stemming from an ambiguity in the original Gaussian model.

When a dynamical system with multiple point attractors is released from an arbitrary initial condition, it will relax into a configuration that locally resolves the constraints or opposing forces between interdependent state variables. However, when there are many conflicting interdependencies between variables, finding a configuration that globally optimizes these constraints by this method is unlikely or may take many attempts. Here, we show that a simple distributed mechanism can incrementally alter a dynamical system such that it finds lower energy configurations, more reliably and more quickly. Specifically, when Hebbian learning is applied to the connections of a simple dynamical system undergoing repeated relaxation, the system will develop an associative memory that amplifies a subset of its own attractor states. This modifies the dynamics of the system such that its ability to find configurations that minimize total system energy, and globally resolve conflicts between interdependent variables, is enhanced. Moreover, we show that the system is not merely ' 'recalling' ' low energy states that have been previously visited but ' 'predicting' ' their location by generalizing over local attractor states that have already been visited. This ' 'self-modeling' ' framework, i.e., a system that augments its behavior with an associative memory of its own attractors, helps us better understand the conditions under which a simple locally mediated mechanism of self-organization can promote significantly enhanced global resolution of conflicts between the components of a complex adaptive system. We illustrate this process in random and modular network constraint problems equivalent to graph coloring and distributed task allocation problems.

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Active inference is emerging as a possible unifying theory of perception and action in cognitive and computational neuroscience. On this theory, perception is a process of inferring the causes of sensory data by minimising the error between actual sensations and those predicted by an inner generative (probabilistic) model. Action on the other hand is drawn as a process that modifies the world such that the consequent sensory input meets expectations encoded in the same internal model. These two processes, inferring properties of the world and inferring actions needed to meet expectations, close the sensory/motor loop and suggest a deep symmetry between action and perception. In this work we present a simple agent-based model inspired by this new theory that offers insights on some of its central ideas. Previous implementations of active inference have typically examined a "perceptionoriented" view of this theory, assuming that agents are endowed with a detailed generative model of their surrounding environment. In contrast, we present an "action-oriented" solution showing how adaptive behaviour can emerge even when agents operate with a simple model which bears little resemblance to their environment. We examine how various parameters of this formulation allow phototaxis and present an example of a different, "pathological" behaviour.