This paper considers prediction and perceptual categorization as an inference problem that is solved by the brain. We assume that the brain models the world as a hierarchy or cascade of dynamical systems that encode causal structure in the sensorium. Perception is equated with the optimization or inversion of these internal models, to explain sensory data. Given a model of how sensory data are generated, we can invoke a generic approach to model inversion, based on a free energy bound on the model's evidence. The ensuing free-energy formulation furnishes equations that prescribe the process of recognition, i.e. the dynamics of neuronal activity that represent the causes of sensory input. Here, we focus on a very general model, whose hierarchical and dynamical structure enables simulated brains to recognize and predict trajectories or sequences of sensory states. We first review hierarchical dynamical models and their inversion. We then show that the brain has the necessary infrastructure to implement this inversion and illustrate this point using synthetic birds that can recognize and categorize birdsongs.Keywords: generative models; predictive coding; hierarchical; birdsong INTRODUCTIONThis paper reviews generic models of our sensorium and a Bayesian scheme for their inversion. We then show that the brain has the necessary anatomical and physiological equipment to invert these models, given sensory data. Critically, the scheme lends itself to a relatively simple neural network implementation that shares many features with real cortical hierarchies in the brain. The basic idea that the brain tries to infer the causes of sensations dates back to Helmholtz (e.g. Helmholtz 1860/1962Barlow 1961;Neisser 1967;Ballard et al. 1983;Mumford 1992;Kawato et al. 1993;Dayan et al. 1995;Rao & Ballard 1998), with a recent emphasis on hierarchical inference and empirical Bayes (Friston 2003(Friston , 2005Friston et al. 2006). Here, we generalize this idea to cover dynamics in the world and consider how neural networks could be configured to invert hierarchical dynamical models and deconvolve sensory causes from sensory input. This paper comprises four sections. In §1, we introduce hierarchical dynamical models and their inversion. These models cover most of the models encountered in the statistical literature. An important aspect of these models is their formulation in generalized coordinates of motion, which lends them a hierarchal form in both structure and dynamics. These hierarchies induce empirical priors that provide structural and dynamical constraints, which can be exploited during inversion. In §2, we show how inversion can be formulated as a simple gradient ascent using neuronal networks; in §3, we consider how evoked brain responses might be understood in terms of inference under hierarchical dynamical models of sensory input. 1

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 of energy 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 likelihood of sensory inputs, given an internal model of how they were caused. 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 is mandated by ergodic arguments based on the very existence of adaptive agents. Furthermore, it accounts for many aspects of motor behavior; from retinal stabilization to goal-seeking. In particular, it suggests that motor control can be understood as fulfilling prior expectations about proprioceptiveThe free-energy principle is an attempt to explain the structure and function of the brain, starting from the fact that we exist: This fact places constraints on our interactions with the world, which have been studied for years in evolutionary biology and systems theory. However, recent advances in statistical physics and machine learning point to a simple 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, we show 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 we that can replace the notion of desired movements with expected movements and understand action in terms of perceptual expectations.The Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, 12 Queen Square, London, WC1N 3BG, sensations. This formulation can explain why adaptive behavior emerges in biological agents and suggests a simple alternative to optimal control theory. We illustrate these points using simulations of oculomotor control and then apply to same principles to cued and goal-directed movements. In short, the free-energy formulation may provide an alternative perspective on the motor control that places it in an intimate relationship with perception.