Hybrid predictive coding: Inferring, fast and slow.

Predictive coding is an influential model of cortical neural activity. It proposes that perceptual beliefs are furnished by sequentially minimising "prediction errors"-the differences between predicted and observed data. Implicit in this proposal is the idea that successful perception requires multiple cycles of neural activity. This is at odds with evidence that several aspects of visual perception-including complex forms of object recognition-arise from an initial "feedforward sweep" that occurs on fast timescales which preclude substantial recurrent activity. Here, we propose that the feedforward sweep can be understood as performing amortized inference (applying a learned function that maps directly from data to beliefs) and recurrent processing can be understood as performing iterative inference (sequentially updating neural activity in order to improve the accuracy of beliefs). We propose a hybrid predictive coding network that combines both iterative and amortized inference in a principled manner by describing both in terms of a dual optimization of a single objective function. We show that the resulting scheme can be implemented in a biologically plausible neural architecture that approximates Bayesian inference utilising local Hebbian update rules. We demonstrate that our hybrid predictive coding model combines the benefits of both amortized and iterative inference-obtaining rapid and computationally cheap perceptual inference for familiar data while maintaining the context-sensitivity, precision, and sample efficiency of iterative inference schemes. Moreover, we show how our model is inherently sensitive to its uncertainty and adaptively balances iterative and amortized inference to obtain accurate beliefs using minimum computational expense. Hybrid predictive coding offers a new perspective on the functional relevance of the feedforward and recurrent activity observed during visual perception and offers novel insights into distinct aspects of visual phenomenology.

A behavioral and modeling study of control algorithms underlying the translational optomotor response in larval zebrafish with implications for neural circuit function.

The optomotor response (OMR) is central to the locomotory behavior in diverse animal species including insects, fish and mammals. Furthermore, the study of the OMR in larval zebrafish has become a key model system for investigating the neural basis of sensorimotor control. However, a comprehensive understanding of the underlying control algorithms is still outstanding. In fish it is often assumed that the OMR, by reducing average optic flow across the retina, serves to stabilize position with respect to the ground. Yet the degree to which this is achieved, and how it could emerge from the intermittent burst dynamics of larval zebrafish swimming, are unclear. Here, we combine detailed computational modeling with a new approach to free-swimming experiments in which we control the amount of visual feedback produced by a given motor effort by varying the height of the larva above a moving grid stimulus. We develop an account of underlying feedback control mechanisms that describes both the bout initiation process and the control of swim speed during bouts. We observe that the degree to which fish stabilize their position is only partial and height-dependent, raising questions about its function. We find the relative speed profile during bouts follows a fixed temporal pattern independent of absolute bout speed, suggesting that bout speed and bout termination are not separately controlled. We also find that the reverse optic flow, experienced when the fish is swimming faster than the stimulus, plays a minimal role in control of the OMR despite carrying most of the sensory information about self-movement. These results shed new light on the underlying dynamics of the OMR in larval zebrafish and will be crucial for future work aimed at identifying the neural basis of this behavior.

Recurrent, nonequilibrium systems and the Markov blanket assumption.

Markov blankets - statistical independences between system and environment - have become popular to describe the boundaries of living systems under Bayesian views of cognition. The intuition behind Markov blankets originates from considering acyclic, atemporal networks. In contrast, living systems display recurrent, nonequilibrium interactions that generate pervasive couplings between system and environment, making Markov blankets highly unusual and restricted to particular cases.

Predictive Coding Approximates Backprop Along Arbitrary Computation Graphs.

Backpropagation of error (backprop) is a powerful algorithm for training machine learning architectures through end-to-end differentiation. Recently it has been shown that backprop in multilayer perceptrons (MLPs) can be approximated using predictive coding, a biologically plausible process theory of cortical computation that relies solely on local and Hebbian updates. The power of backprop, however, lies not in its instantiation in MLPs but in the concept of automatic differentiation, which allows for the optimization of any differentiable program expressed as a computation graph. Here, we demonstrate that predictive coding converges asymptotically (and in practice, rapidly) to exact backprop gradients on arbitrary computation graphs using only local learning rules. We apply this result to develop a straightforward strategy to translate core machine learning architectures into their predictive coding equivalents. We construct predictive coding convolutional neural networks, recurrent neural networks, and the more complex long short-term memory, which include a nonlayer-like branching internal graph structure and multiplicative interactions. Our models perform equivalently to backprop on challenging machine learning benchmarks while using only local and (mostly) Hebbian plasticity. Our method raises the potential that standard machine learning algorithms could in principle be directly implemented in neural circuitry and may also contribute to the development of completely distributed neuromorphic architectures.

Simulating homeostatic, allostatic and goal-directed forms of interoceptive control using active inference.

The adaptive regulation of bodily and interoceptive parameters, such as body temperature, thirst and hunger is a central problem for any biological organism. Here, we present a series of simulations using the framework of active inference to formally characterize interoceptive control and some of its dysfunctions. We start from the premise that the goal of interoceptive control is to minimize a discrepancy between expected and actual interoceptive sensations (i.e., a prediction error or free energy). Importantly, living organisms can achieve this goal by using various forms of interoceptive control: homeostatic, allostatic and goal-directed. We provide a computationally-guided analysis of these different forms of interoceptive control, by showing that they correspond to distinct generative models within Active inference. We discuss how these generative models can support empirical research through enabling fine-grained predictions about physiological and brain signals that may accompany both adaptive and maladaptive interoceptive control.

How particular is the physics of the free energy principle?

The free energy principle (FEP) states that any dynamical system can be interpreted as performing Bayesian inference upon its surrounding environment. Although, in theory, the FEP applies to a wide variety of systems, there has been almost no direct exploration or demonstration of the principle in concrete systems. In this work, we examine in depth the assumptions required to derive the FEP in the simplest possible set of systems - weakly-coupled non-equilibrium linear stochastic systems. Specifically, we explore (i) how general the requirements imposed on the statistical structure of a system are and (ii) how informative the FEP is about the behaviour of such systems. We discover that two requirements of the FEP - the Markov blanket condition (i.e. a statistical boundary precluding direct coupling between internal and external states) and stringent restrictions on its solenoidal flows (i.e. tendencies driving a system out of equilibrium) - are only valid for a very narrow space of parameters. Suitable systems require an absence of perception-action asymmetries that is highly unusual for living systems interacting with an environment. More importantly, we observe that a mathematically central step in the argument, connecting the behaviour of a system to variational inference, relies on an implicit equivalence between the dynamics of the average states of a system with the average of the dynamics of those states. This equivalence does not hold in general even for linear stochastic systems, since it requires an effective decoupling from the system's history of interactions. These observations are critical for evaluating the generality and applicability of the FEP and indicate the existence of significant problems of the theory in its current form. These issues make the FEP, as it stands, not straightforwardly applicable to the simple linear systems studied here and suggest that more development is needed before the theory could be applied to the kind of complex systems that describe living and cognitive processes.

Whence the Expected Free Energy?

The expected free energy (EFE) is a central quantity in the theory of active inference. It is the quantity that all active inference agents are mandated to minimize through action, and its decomposition into extrinsic and intrinsic value terms is key to the balance of exploration and exploitation that active inference agents evince. Despite its importance, the mathematical origins of this quantity and its relation to the variational free energy (VFE) remain unclear. In this letter, we investigate the origins of the EFE in detail and show that it is not simply "the free energy in the future." We present a functional that we argue is the natural extension of the VFE but actively discourages exploratory behavior, thus demonstrating that exploration does not directly follow from free energy minimization into the future. We then develop a novel objective, the free energy of the expected future (FEEF), which possesses both the epistemic component of the EFE and an intuitive mathematical grounding as the divergence between predicted and desired futures.

µSPIM Toolset: A software platform for selective plane illumination microscopy.

BACKGROUND: Selective Plane Illumination Microscopy (SPIM) is a fluorescence imaging technique that allows volumetric imaging at high spatio-temporal resolution to monitor neural activity in live organisms such as larval zebrafish. A major challenge in the construction of a custom SPIM microscope using a scanned laser beam is the control and synchronization of the various hardware components. NEW METHOD: We present an open-source software, µSPIM Toolset, built around the widely adopted MicroManager platform, that provides control and acquisition functionality for a SPIM. A key advantage of µSPIM Toolset is a series of calibration procedures that optimize acquisition for a given set-up, making it relatively independent of the optical design of the microscope or the hardware used to build it. RESULTS: µSPIM Toolset allows imaging of calcium activity throughout the brain of larval zebrafish at rates of 100 planes per second with single cell resolution. COMPARISON WITH EXISTING METHODS: Several designs of SPIM have been published but are focused on imaging of developmental processes using a slower setup with a moving stage and therefore have limited use for functional imaging. In comparison, µSPIM Toolset uses a scanned beam to allow imaging at higher acquisition frequencies while minimizing disturbance of the sample. CONCLUSIONS: The µSPIM Toolset provides a flexible solution for the control of SPIM microscopes and demonstrated its utility for brain-wide imaging of neural activity in larval zebrafish.

Learning action-oriented models through active inference.

Converging theories suggest that organisms learn and exploit probabilistic models of their environment. However, it remains unclear how such models can be learned in practice. The open-ended complexity of natural environments means that it is generally infeasible for organisms to model their environment comprehensively. Alternatively, action-oriented models attempt to encode a parsimonious representation of adaptive agent-environment interactions. One approach to learning action-oriented models is to learn online in the presence of goal-directed behaviours. This constrains an agent to behaviourally relevant trajectories, reducing the diversity of the data a model need account for. Unfortunately, this approach can cause models to prematurely converge to sub-optimal solutions, through a process we refer to as a bad-bootstrap. Here, we exploit the normative framework of active inference to show that efficient action-oriented models can be learned by balancing goal-oriented and epistemic (information-seeking) behaviours in a principled manner. We illustrate our approach using a simple agent-based model of bacterial chemotaxis. We first demonstrate that learning via goal-directed behaviour indeed constrains models to behaviorally relevant aspects of the environment, but that this approach is prone to sub-optimal convergence. We then demonstrate that epistemic behaviours facilitate the construction of accurate and comprehensive models, but that these models are not tailored to any specific behavioural niche and are therefore less efficient in their use of data. Finally, we show that active inference agents learn models that are parsimonious, tailored to action, and which avoid bad bootstraps and sub-optimal convergence. Critically, our results indicate that models learned through active inference can support adaptive behaviour in spite of, and indeed because of, their departure from veridical representations of the environment. Our approach provides a principled method for learning adaptive models from limited interactions with an environment, highlighting a route to sample efficient learning algorithms.

Generative models as parsimonious descriptions of sensorimotor loops.

The Bayesian brain hypothesis, predictive processing, and variational free energy minimisation are typically used to describe perceptual processes based on accurate generative models of the world. However, generative models need not be veridical representations of the environment. We suggest that they can (and should) be used to describe sensorimotor relationships relevant for behaviour rather than precise accounts of the world.

PID Control as a Process of Active Inference with Linear Generative Models.

In the past few decades, probabilistic interpretations of brain functions have become widespread in cognitive science and neuroscience. In particular, the free energy principle and active inference are increasingly popular theories of cognitive functions that claim to offer a unified understanding of life and cognition within a general mathematical framework derived from information and control theory, and statistical mechanics. However, we argue that if the active inference proposal is to be taken as a general process theory for biological systems, it is necessary to understand how it relates to existing control theoretical approaches routinely used to study and explain biological systems. For example, recently, PID (Proportional-Integral-Derivative) control has been shown to be implemented in simple molecular systems and is becoming a popular mechanistic explanation of behaviours such as chemotaxis in bacteria and amoebae, and robust adaptation in biochemical networks. In this work, we will show how PID controllers can fit a more general theory of life and cognition under the principle of (variational) free energy minimisation when using approximate linear generative models of the world. This more general interpretation also provides a new perspective on traditional problems of PID controllers such as parameter tuning as well as the need to balance performances and robustness conditions of a controller. Specifically, we then show how these problems can be understood in terms of the optimisation of the precisions (inverse variances) modulating different prediction errors in the free energy functional.

A theory of how active behavior stabilises neural activity: Neural gain modulation by closed-loop environmental feedback.

During active behaviours like running, swimming, whisking or sniffing, motor actions shape sensory input and sensory percepts guide future motor commands. Ongoing cycles of sensory and motor processing constitute a closed-loop feedback system which is central to motor control and, it has been argued, for perceptual processes. This closed-loop feedback is mediated by brainwide neural circuits but how the presence of feedback signals impacts on the dynamics and function of neurons is not well understood. Here we present a simple theory suggesting that closed-loop feedback between the brain/body/environment can modulate neural gain and, consequently, change endogenous neural fluctuations and responses to sensory input. We support this theory with modeling and data analysis in two vertebrate systems. First, in a model of rodent whisking we show that negative feedback mediated by whisking vibrissa can suppress coherent neural fluctuations and neural responses to sensory input in the barrel cortex. We argue this suppression provides an appealing account of a brain state transition (a marked change in global brain activity) coincident with the onset of whisking in rodents. Moreover, this mechanism suggests a novel signal detection mechanism that selectively accentuates active, rather than passive, whisker touch signals. This mechanism is consistent with a predictive coding strategy that is sensitive to the consequences of motor actions rather than the difference between the predicted and actual sensory input. We further support the theory by re-analysing previously published two-photon data recorded in zebrafish larvae performing closed-loop optomotor behaviour in a virtual swim simulator. We show, as predicted by this theory, that the degree to which each cell contributes in linking sensory and motor signals well explains how much its neural fluctuations are suppressed by closed-loop optomotor behaviour. More generally we argue that our results demonstrate the dependence of neural fluctuations, across the brain, on closed-loop brain/body/environment interactions strongly supporting the idea that brain function cannot be fully understood through open-loop approaches alone.

Visual associative learning in wood ants.

Wood ants are a model system for studying visual learning and navigation. They can forage for food and navigate to their nests effectively by forming memories of visual features in their surrounding environment. Previous studies of freely behaving ants have revealed many of the behavioural strategies and environmental features necessary for successful navigation. However, little is known about the exact visual properties of the environment that animals learn or the neural mechanisms that allow them to achieve this. As a first step towards addressing this, we developed a classical conditioning paradigm for visual learning in harnessed wood ants that allows us to control precisely the learned visual cues. In this paradigm, ants are fixed and presented with a visual cue paired with an appetitive sugar reward. Using this paradigm, we found that visual cues learnt by wood ants through Pavlovian conditioning are retained for at least 1 h. Furthermore, we found that memory retention is dependent upon the ants' performance during training. Our study provides the first evidence that wood ants can form visual associative memories when restrained. This classical conditioning paradigm has the potential to permit detailed analysis of the dynamics of memory formation and retention, and the neural basis of learning in wood ants.

Multi-neuronal refractory period adapts centrally generated behaviour to reward.

Oscillating neuronal circuits, known as central pattern generators (CPGs), are responsible for generating rhythmic behaviours such as walking, breathing and chewing. The CPG model alone however does not account for the ability of animals to adapt their future behaviour to changes in the sensory environment that signal reward. Here, using multi-electrode array (MEA) recording in an established experimental model of centrally generated rhythmic behaviour we show that the feeding CPG of Lymnaea stagnalis is itself associated with another, and hitherto unidentified, oscillating neuronal population. This extra-CPG oscillator is characterised by high population-wide activity alternating with population-wide quiescence. During the quiescent periods the CPG is refractory to activation by food-associated stimuli. Furthermore, the duration of the refractory period predicts the timing of the next activation of the CPG, which may be minutes into the future. Rewarding food stimuli and dopamine accelerate the frequency of the extra-CPG oscillator and reduce the duration of its quiescent periods. These findings indicate that dopamine adapts future feeding behaviour to the availability of food by significantly reducing the refractory period of the brain's feeding circuitry.

Transient dynamics between displaced fixed points: an alternate nonlinear dynamical framework for olfaction.

Significant insights into the dynamics of neuronal populations have been gained in the olfactory system where rich spatio-temporal dynamics is observed during, and following, exposure to odours. It is now widely accepted that odour identity is represented in terms of stimulus-specific rate patterning observed in the cells of the antennal lobe (AL). Here we describe a nonlinear dynamical framework inspired by recent experimental findings which provides a compelling account of both the origin and the function of these dynamics. We start by analytically reducing a biologically plausible conductance based model of the AL to a quantitatively equivalent rate model and construct conditions such that the rate dynamics are well described by a single globally stable fixed point (FP). We then describe the AL's response to an odour stimulus as rich transient trajectories between this stable baseline state (the single FP in absence of odour stimulation) and the odour-specific position of the single FP during odour stimulation. We show how this framework can account for three phenomena that are observed experimentally. First, for an inhibitory period often observed immediately after an odour stimulus is removed. Second, for the qualitative differences between the dynamics in the presence and the absence of odour. Lastly, we show how it can account for the invariance of a representation of odour identity to both the duration and intensity of an odour stimulus. We compare and contrast this framework with the currently prevalent nonlinear dynamical framework of 'winnerless competition' which describes AL dynamics in terms of heteroclinic orbits. This article is part of a Special Issue entitled "Neural Coding".

Multiscale model of an inhibitory network shows optimal properties near bifurcation.

We present a systematic multiscale reduction of a biologically plausible model of the inhibitory neuronal network of the pheromone system of the moth. Starting from a Hodgkin-Huxley conductance based model we adiabatically eliminate fast variables and quantitatively reduce the model to mean field equations. We then prove analytically that the network's ability to operate on signal amplitudes across several orders of magnitude is optimal when a disinhibitory mode is close to losing stability and the network dynamics are close to bifurcation. This has the potential to extend the idea that optimal dynamic range in the brain arises as a critical phenomenon of phase transitions in excitable media to brain regions that are dominated by inhibition or have slow dynamics.

Competition-based model of pheromone component ratio detection in the moth.

For some moth species, especially those closely interrelated and sympatric, recognizing a specific pheromone component concentration ratio is essential for males to successfully locate conspecific females. We propose and determine the properties of a minimalist competition-based feed-forward neuronal model capable of detecting a certain ratio of pheromone components independently of overall concentration. This model represents an elementary recognition unit for the ratio of binary mixtures which we propose is entirely contained in the macroglomerular complex (MGC) of the male moth. A set of such units, along with projection neurons (PNs), can provide the input to higher brain centres. We found that (1) accuracy is mainly achieved by maintaining a certain ratio of connection strengths between olfactory receptor neurons (ORN) and local neurons (LN), much less by properties of the interconnections between the competing LNs proper. An exception to this rule is that it is beneficial if connections between generalist LNs (i.e. excited by either pheromone component) and specialist LNs (i.e. excited by one component only) have the same strength as the reciprocal specialist to generalist connections. (2) successful ratio recognition is achieved using latency-to-first-spike in the LN populations which, in contrast to expectations with a population rate code, leads to a broadening of responses for higher overall concentrations consistent with experimental observations. (3) when longer durations of the competition between LNs were observed it did not lead to higher recognition accuracy.

Sensitivity and stability: a signal propagation sweet spot in a sheet of recurrent centre crossing neurons.

In this paper we demonstrate that signal propagation across a laminar sheet of recurrent neurons is maximised when two conditions are met. First, neurons must be in the so-called centre crossing configuration. Second, the network's topology and weights must be such that the network comprises strongly coupled nodes, yet lies within the weakly coupled regime. We develop tools from linear stability analysis with which to describe this regime in terms of the connectivity and weight strengths of a network. We use these results to examine the apparent tension between the sensitivity and instability of centre crossing networks.