Action and behavior: a free-energy formulation

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 proprioceptive 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

Online Discrimination of Nonlinear Dynamics with Switching Differential Equations

How to recognise whether an observed person walks or runs? We consider a dynamic environment where observations (e.g. the posture of a person) are caused by different dynamic processes (walking or running) which are active one at a time and which may transition from one to another at any time. For this setup, switching dynamic models have been suggested previously, mostly, for linear and nonlinear dynamics in discrete time. Motivated by basic principles of computations in the brain (dynamic, internal models) we suggest a model for switching nonlinear differential equations. The switching process in the model is implemented by a Hopfield network and we use parametric dynamic movement primitives to represent arbitrary rhythmic motions. The model generates observed dynamics by linearly interpolating the primitives weighted by the switching variables and it is constructed such that standard filtering algorithms can be applied. In two experiments with synthetic planar motion and a human motion capture data set we show that inference with the unscented Kalman filter can successfully discriminate several dynamic processes online

Stable representations of dynamic stimuli in perceptual decision making

Models of perceptual decision making, which are based on dynamic stimuli such as random dot motion, are predominantly concerned with how evidence for a stimulus is accumulated over time (e.g., Wang, 2008; Beck, 2008). However, it is unclear how the brain derives this evidence from the sensory dynamics. While it is conceivable that simple feature-detecting neurons can, for example, directly signal evidence for motion in a specific direction, it is less clear how evidence for complex motion, such as human movements, is computed from sensory input. We present a model of the perceptual lower level system which is based on probabilistic inference for dynamical systems (Friston, 2008) and can be used to provide input for higher level decision making systems. We illustrate this mechanism using a random dot motion paradigm, where we (i) consider simple uni-directional motion as typically used in neuroscience experiments and (ii) show that the same system can also infer, i.e. recognize, complex dot motion as generated by humans (cf. point light walkers) in an online fashion. The present model is implemented by a neuronal network and computes stable percepts rapidly, thereby enabling both fast decision (reaction) times and high accuracy. We suggest that the combination of the present model with recent models for evidence accumulation in perceptual decision making may be used to apply neurobiologically plausible decision making strategies to real-world stimuli like movements generated by humans

Free Energy and Dendritic Self-Organization

In this paper, we pursue recent observations that, through selective dendritic filtering, single neurons respond to specific sequences of presynaptic inputs. We try to provide a principled and mechanistic account of this selectivity by applying a recent free-energy principle to a dendrite that is immersed in its neuropil or environment. We assume that neurons self-organize to minimize a variational free-energy bound on the self-information or surprise of presynaptic inputs that are sampled. We model this as a selective pruning of dendritic spines that are expressed on a dendritic branch. This pruning occurs when postsynaptic gain falls below a threshold. Crucially, postsynaptic gain is itself optimized with respect to free energy. Pruning suppresses free energy as the dendrite selects presynaptic signals that conform to its expectations, specified by a generative model implicit in its intracellular kinetics. Not only does this provide a principled account of how neurons organize and selectively sample the myriad of potential presynaptic inputs they are exposed to, but it also connects the optimization of elemental neuronal (dendritic) processing to generic (surprise or evidence-based) schemes in statistics and machine learning, such as Bayesian model selection and automatic relevance determination

Bayesian sparsification for deep neural networks with Bayesian model reduction

Deep learning's immense capabilities are often constrained by the complexity of its models, leading to an increasing demand for effective sparsification techniques. Bayesian sparsification for deep learning emerges as a crucial approach, facilitating the design of models that are both computationally efficient and competitive in terms of performance across various deep learning applications. The state-of-the-art -- in Bayesian sparsification of deep neural networks -- combines structural shrinkage priors on model weights with an approximate inference scheme based on stochastic variational inference. However, model inversion of the full generative model is exceptionally computationally demanding, especially when compared to standard deep learning of point estimates. In this context, we advocate for the use of Bayesian model reduction (BMR) as a more efficient alternative for pruning of model weights. As a generalization of the Savage-Dickey ratio, BMR allows a post-hoc elimination of redundant model weights based on the posterior estimates under a straightforward (non-hierarchical) generative model. Our comparative study highlights the advantages of the BMR method relative to established approaches based on hierarchical horseshoe priors over model weights. We illustrate the potential of BMR across various deep learning architectures, from classical networks like LeNet to modern frameworks such as Vision Transformers and MLP-Mixers

Recognizing recurrent neural networks (rRNN): Bayesian inference for recurrent neural networks

Recurrent neural networks (RNNs) are widely used in computational neuroscience and machine learning applications. In an RNN, each neuron computes its output as a nonlinear function of its integrated input. While the importance of RNNs, especially as models of brain processing, is undisputed, it is also widely acknowledged that the computations in standard RNN models may be an over-simplification of what real neuronal networks compute. Here, we suggest that the RNN approach may be made both neurobiologically more plausible and computationally more powerful by its fusion with Bayesian inference techniques for nonlinear dynamical systems. In this scheme, we use an RNN as a generative model of dynamic input caused by the environment, e.g. of speech or kinematics. Given this generative RNN model, we derive Bayesian update equations that can decode its output. Critically, these updates define a 'recognizing RNN' (rRNN), in which neurons compute and exchange prediction and prediction error messages. The rRNN has several desirable features that a conventional RNN does not have, for example, fast decoding of dynamic stimuli and robustness to initial conditions and noise. Furthermore, it implements a predictive coding scheme for dynamic inputs. We suggest that the Bayesian inversion of recurrent neural networks may be useful both as a model of brain function and as a machine learning tool. We illustrate the use of the rRNN by an application to the online decoding (i.e. recognition) of human kinematics

Perception and Hierarchical Dynamics

In this paper, we suggest that perception could be modeled by assuming that sensory input is generated by a hierarchy of attractors in a dynamic system. We describe a mathematical model which exploits the temporal structure of rapid sensory dynamics to track the slower trajectories of their underlying causes. This model establishes a proof of concept that slowly changing neuronal states can encode the trajectories of faster sensory signals. We link this hierarchical account to recent developments in the perception of human action; in particular artificial speech recognition. We argue that these hierarchical models of dynamical systems are a plausible starting point to develop robust recognition schemes, because they capture critical temporal dependencies induced by deep hierarchical structure. We conclude by suggesting that a fruitful computational neuroscience approach may emerge from modeling perception as non-autonomous recognition dynamics enslaved by autonomous hierarchical dynamics in the sensorium

A Hierarchy of Time-Scales and the Brain

In this paper, we suggest that cortical anatomy recapitulates the temporal hierarchy that is inherent in the dynamics of environmental states. Many aspects of brain function can be understood in terms of a hierarchy of temporal scales at which representations of the environment evolve. The lowest level of this hierarchy corresponds to fast fluctuations associated with sensory processing, whereas the highest levels encode slow contextual changes in the environment, under which faster representations unfold. First, we describe a mathematical model that exploits the temporal structure of fast sensory input to track the slower trajectories of their underlying causes. This model of sensory encoding or perceptual inference establishes a proof of concept that slowly changing neuronal states can encode the paths or trajectories of faster sensory states. We then review empirical evidence that suggests that a temporal hierarchy is recapitulated in the macroscopic organization of the cortex. This anatomic-temporal hierarchy provides a comprehensive framework for understanding cortical function: the specific time-scale that engages a cortical area can be inferred by its location along a rostro-caudal gradient, which reflects the anatomical distance from primary sensory areas. This is most evident in the prefrontal cortex, where complex functions can be explained as operations on representations of the environment that change slowly. The framework provides predictions about, and principled constraints on, cortical structure–function relationships, which can be tested by manipulating the time-scales of sensory input