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Active inference, a corollary of the free energy principle, is a formal way of describing the behavior of certain kinds of random dynamical systems that have the appearance of sentience. In this chapter, we describe how active inference combines Bayesian decision theory and optimal Bayesian design principles under a single imperative to minimize expected free energy. It is this aspect of active inference that allows for the natural emergence of information-seeking behavior. When removing prior outcomes preferences from expected free energy, active inference reduces to optimal Bayesian design, i.e., information gain maximization. Conversely, active inference reduces to Bayesian decision theory in the absence of ambiguity and relative risk, i.e., expected utility maximization. Using these limiting cases, we illustrate how behaviors differ when agents select actions that optimize expected utility, expected information gain, and expected free energy. Our T-maze simulations show optimizing expected free energy produces goal-directed information-seeking behavior while optimizing expected utility induces purely exploitive behavior and maximizing information gain engenders intrinsically motivated behavior., Comment: 19 pages; 3 figures

Under the Bayesian brain hypothesis, behavioural variations can be attributed to different priors over generative model parameters. This provides a formal explanation for why individuals exhibit inconsistent behavioural preferences when confronted with similar choices. For example, greedy preferences are a consequence of confident (or precise) beliefs over certain outcomes. Here, we offer an alternative account of behavioural variability using R\'enyi divergences and their associated variational bounds. R\'enyi bounds are analogous to the variational free energy (or evidence lower bound) and can be derived under the same assumptions. Importantly, these bounds provide a formal way to establish behavioural differences through an $\alpha$ parameter, given fixed priors. This rests on changes in $\alpha$ that alter the bound (on a continuous scale), inducing different posterior estimates and consequent variations in behaviour. Thus, it looks as if individuals have different priors, and have reached different conclusions. More specifically, $\alpha \to 0^{+}$ optimisation leads to mass-covering variational estimates and increased variability in choice behaviour. Furthermore, $\alpha \to + \infty$ optimisation leads to mass-seeking variational posteriors and greedy preferences. We exemplify this formulation through simulations of the multi-armed bandit task. We note that these $\alpha$ parameterisations may be especially relevant, i.e., shape preferences, when the true posterior is not in the same family of distributions as the assumed (simpler) approximate density, which may be the case in many real-world scenarios. The ensuing departure from vanilla variational inference provides a potentially useful explanation for differences in behavioural preferences of biological (or artificial) agents under the assumption that the brain performs variational Bayesian inference., Comment: 23 pages, 5 figures

This paper develops a Bayesian mechanics for adaptive systems. Firstly, we model the interface between a system and its environment with a Markov blanket. This affords conditions under which states internal to the blanket encode information about external states. Second, we introduce dynamics and represent adaptive systems as Markov blankets at steady-state. This allows us to identify a wide class of systems whose internal states appear to infer external states, consistent with variational inference in Bayesian statistics and theoretical neuroscience. Finally, we partition the blanket into sensory and active states. It follows that active states can be seen as performing active inference and well-known forms of stochastic control (such as PID control), which are prominent formulations of adaptive behaviour in theoretical biology and engineering., Comment: 18 pages, 11 figures

Biological agents have meaningful interactions with their environment despite the absence of immediate reward signals. In such instances, the agent can learn preferred modes of behaviour that lead to predictable states -- necessary for survival. In this paper, we pursue the notion that this learnt behaviour can be a consequence of reward-free preference learning that ensures an appropriate trade-off between exploration and preference satisfaction. For this, we introduce a model-based Bayesian agent equipped with a preference learning mechanism (pepper) using conjugate priors. These conjugate priors are used to augment the expected free energy planner for learning preferences over states (or outcomes) across time. Importantly, our approach enables the agent to learn preferences that encourage adaptive behaviour at test time. We illustrate this in the OpenAI Gym FrozenLake and the 3D mini-world environments -- with and without volatility. Given a constant environment, these agents learn confident (i.e., precise) preferences and act to satisfy them. Conversely, in a volatile setting, perpetual preference uncertainty maintains exploratory behaviour. Our experiments suggest that learnable (reward-free) preferences entail a trade-off between exploration and preference satisfaction. Pepper offers a straightforward framework suitable for designing adaptive agents when reward functions cannot be predefined as in real environments., Comment: 23 pages, 15 figures

To interact seamlessly with robots, users must infer the causes of a robot's behavior and be confident about that inference. Hence, trust is a necessary condition for human-robot collaboration (HRC). Despite its crucial role, it is largely unknown how trust emerges, develops, and supports human interactions with nonhuman artefacts. Here, we review the literature on trust, human-robot interaction, human-robot collaboration, and human interaction at large. Early models of trust suggest that trust entails a trade-off between benevolence and competence, while studies of human-to-human interaction emphasize the role of shared behavior and mutual knowledge in the gradual building of trust. We then introduce a model of trust as an agent's best explanation for reliable sensory exchange with an extended motor plant or partner. This model is based on the cognitive neuroscience of active inference and suggests that, in the context of HRC, trust can be cast in terms of virtual control over an artificial agent. In this setting, interactive feedback becomes a necessary component of the trustor's perception-action cycle. The resulting model has important implications for understanding human-robot interaction and collaboration, as it allows the traditional determinants of human trust to be defined in terms of active inference, information exchange and empowerment. Furthermore, this model suggests that boredom and surprise may be used as markers for under and over-reliance on the system. Finally, we examine the role of shared behavior in the genesis of trust, especially in the context of dyadic collaboration, suggesting important consequences for the acceptability and design of human-robot collaborative systems.

The ability to plan ahead efficiently is key for both living organisms and artificial systems. Model-based planning and prospection are widely studied in cognitive neuroscience and artificial intelligence (AI), but from different perspectives--and with different desiderata in mind (biological realism versus scalability) that are difficult to reconcile. Here, we introduce a novel method to plan in POMDPs--Active Inference Tree Search (AcT)--that combines the normative character and biological realism of a leading planning theory in neuroscience (Active Inference) and the scalability of tree search methods in AI. This unification enhances both approaches. On the one hand, tree searches enable the biologically grounded, first principle method of active inference to be applied to large-scale problems. On the other hand, active inference provides a principled solution to the exploration-exploitation dilemma, which is often addressed heuristically in tree search methods. Our simulations show that AcT successfully navigates binary trees that are challenging for sampling-based methods, problems that require adaptive exploration, and the large POMDP problem 'RockSample'--in which AcT reproduces state-of-the-art POMDP solutions. Furthermore, we illustrate how AcT can be used to simulate neurophysiological responses (e.g., in the hippocampus and prefrontal cortex) of humans and other animals that solve large planning problems. These numerical analyses show that Active Tree Search is a principled realisation of neuroscientific and AI planning theories, which offer both biological realism and scalability., Comment: 50 pages, 9 figures, 1 Appendix of two sections with pseudocodes and one encoding example , submitted preprint

The evolution of epidemiological parameters, such as instantaneous reproduction number Rt, is important for understanding the transmission dynamics of infectious diseases. Current estimates of time-varying epidemiological parameters often face problems such as lagging observations, averaging inference, and improper quantification of uncertainties. To address these problems, we propose a Bayesian data assimilation framework for time-varying parameter estimation. Specifically, this framework is applied to Rt estimation, resulting in the state-of-the-art DARt system. With DARt, time misalignment caused by lagging observations is tackled by incorporating observation delays into the joint inference of infections and Rt; the drawback of averaging is overcome by instantaneously updating upon new observations and developing a model selection mechanism that captures abrupt changes; the uncertainty is quantified and reduced by employing Bayesian smoothing. We validate the performance of DARt and demonstrate its power in revealing the transmission dynamics of COVID-19. The proposed approach provides a promising solution for accurate and timely estimating transmission dynamics from reported data., Comment: Xian Yang, Shuo Wang and Yuting Xing contribute equally

This technical report describes the rationale and technical details for the dynamic causal modelling of mitigated epidemiological outcomes based upon a variety of timeseries data. It details the structure of the underlying convolution or generative model (at the time of writing on 6-Nov-20). This report is intended for use as a reference that accompanies the predictions in following dashboard: https://www.fil.ion.ucl.ac.uk/spm/covid-19/dashboard

The perceptual experience of architecture is enacted by the sensory and motor system. When we act, we change the perceived environment according to a set of expectations that depend on our body and the built environment. The continuous process of collecting sensory information is thus based on bodily affordances. Affordances characterize the fit between the physical structure of the body and capacities for movement in the built environment. Since little has been done regarding the role of architectural design in the emergence of perceptual experience on a neuronal level, this paper offers a first step towards the role of architectural design in perceptual experience. An approach to synthesize concepts from computational neuroscience with architectural phenomenology into a computational neurophenomenology is considered. The outcome is a framework under which studies of architecture and cognitive neuroscience can be cast., Comment: 1 title-page, 23 pages, 5 figures

The principle of stationary action is a cornerstone of modern physics, providing a powerful framework for investigating dynamical systems found in classical mechanics through to quantum field theory. However, computational neuroscience, despite its heavy reliance on concepts in physics, is anomalous in this regard as its main equations of motion are not compatible with a Lagrangian formulation and hence with the principle of stationary action. Taking the Dynamic Causal Modelling neuronal state equation, Hodgkin-Huxley model, and the Leaky Integrate-and-Fire model as examples, we show that it is possible to write complex oscillatory forms of these equations in terms of a single Lagrangian. We therefore bring mathematical descriptions in computational neuroscience under the remit of the principle of stationary action and use this reformulation to express symmetries and associated conservation laws arising in neural systems.

An interesting inference drawn by some Covid-19 epidemiological models is that there exists a proportion of the population who are not susceptible to infection -- even at the start of the current pandemic. This paper introduces a model of the immune response to a virus. This is based upon the same sort of mean-field dynamics as used in epidemiology. However, in place of the location, clinical status, and other attributes of people in an epidemiological model, we consider the state of a virus, B and T-lymphocytes, and the antibodies they generate. Our aim is to formalise some key hypotheses as to the mechanism of resistance. We present a series of simple simulations illustrating changes to the dynamics of the immune response under these hypotheses. These include attenuated viral cell entry, pre-existing cross-reactive humoral (antibody-mediated) immunity, and enhanced T-cell dependent immunity. Finally, we illustrate the potential application of this sort of model by illustrating variational inversion (using simulated data) of this model to illustrate its use in testing hypotheses. In principle, this furnishes a fast and efficient immunological assay--based on sequential serology--that provides a (i) quantitative measure of latent immunological responses and (ii) a Bayes optimal classification of the different kinds of immunological response (c.f., glucose tolerance tests used to test for insulin resistance). This may be especially useful in assessing SARS-CoV-2 vaccines.

Active inference is a probabilistic framework for modelling the behaviour of biological and artificial agents, which derives from the principle of minimising free energy. In recent years, this framework has successfully been applied to a variety of situations where the goal was to maximise reward, offering comparable and sometimes superior performance to alternative approaches. In this paper, we clarify the connection between reward maximisation and active inference by demonstrating how and when active inference agents perform actions that are optimal for maximising reward. Precisely, we show the conditions under which active inference produces the optimal solution to the Bellman equation--a formulation that underlies several approaches to model-based reinforcement learning and control. On partially observed Markov decision processes, the standard active inference scheme can produce Bellman optimal actions for planning horizons of 1, but not beyond. In contrast, a recently developed recursive active inference scheme (sophisticated inference) can produce Bellman optimal actions on any finite temporal horizon. We append the analysis with a discussion of the broader relationship between active inference and reinforcement learning., Comment: 18 pages, 3 figures (main text); 37 pages including references and appendix

To learn directed behaviors in complex environments, intelligent agents need to optimize objective functions. Various objectives are known for designing artificial agents, including task rewards and intrinsic motivation. However, it is unclear how the known objectives relate to each other, which objectives remain yet to be discovered, and which objectives better describe the behavior of humans. We introduce the Action Perception Divergence (APD), an approach for categorizing the space of possible objective functions for embodied agents. We show a spectrum that reaches from narrow to general objectives. While the narrow objectives correspond to domain-specific rewards as typical in reinforcement learning, the general objectives maximize information with the environment through latent variable models of input sequences. Intuitively, these agents use perception to align their beliefs with the world and use actions to align the world with their beliefs. They infer representations that are informative of past inputs, explore future inputs that are informative of their representations, and select actions or skills that maximally influence future inputs. This explains a wide range of unsupervised objectives from a single principle, including representation learning, information gain, empowerment, and skill discovery. Our findings suggest leveraging powerful world models for unsupervised exploration as a path toward highly adaptive agents that seek out large niches in their environments, rendering task rewards optional., Comment: Website: https://danijar.com/apd

This technical note introduces parametric dynamic causal modelling, a method for inferring slow changes in biophysical parameters that control fluctuations of fast neuronal states. The application domain we have in mind is inferring slow changes in variables (e.g., extracellular ion concentrations or synaptic efficacy) that underlie phase transitions in brain activity (e.g., paroxysmal seizure activity). The scheme is efficient and yet retains a biophysical interpretation, in virtue of being based on established neural mass models that are equipped with a slow dynamic on the parameters (such as synaptic rate constants or effective connectivity). In brief, we use an adiabatic approximation to summarise fast fluctuations in hidden neuronal states (and their expression in sensors) in terms of their second order statistics; namely, their complex cross spectra. This allows one to specify and compare models of slowly changing parameters (using Bayesian model reduction) that generate a sequence of empirical cross spectra of electrophysiological recordings. Crucially, we use the slow fluctuations in the spectral power of neuronal activity as empirical priors on changes in synaptic parameters. This introduces a circular causality, in which synaptic parameters underwrite fast neuronal activity that, in turn, induces activity-dependent plasticity in synaptic parameters. In this foundational paper, we describe the underlying model, establish its face validity using simulations and provide an illustrative application to a chemoconvulsant animal model of seizure activity.

The aim of this paper is to leverage the free-energy principle and its corollary process theory, active inference, to develop a generic, generalizable model of the representational capacities of living creatures; that is, a theory of phenotypic representation. Given their ubiquity, we are concerned with distributed forms of representation (e.g., population codes), whereby patterns of ensemble activity in living tissue come to represent the causes of sensory input or data. The active inference framework rests on the Markov blanket formalism, which allows us to partition systems of interest, such as biological systems, into internal states, external states, and the blanket (active and sensory) states that render internal and external states conditionally independent of each other. In this framework, the representational capacity of living creatures emerges as a consequence of their Markovian structure and nonequilibrium dynamics, which together entail a dual-aspect information geometry. This entails a modest representational capacity: internal states have an intrinsic information geometry that describes their trajectory over time in state space, as well as an extrinsic information geometry that allows internal states to encode (the parameters of) probabilistic beliefs about (fictive) external states. Building on this, we describe here how, in an automatic and emergent manner, information about stimuli can come to be encoded by groups of neurons bound by a Markov blanket; what is known as the neuronal packet hypothesis. As a concrete demonstration of this type of emergent representation, we present numerical simulations showing that self-organizing ensembles of active inference agents sharing the right kind of probabilistic generative model are able to encode recoverable information about a stimulus array., Comment: 30 pages; 6 figures

Wright and Bourkes compelling article rightly points out that existing models of embryogenesis fail to explain the mechanisms and functional significance of the dynamic connections among neurons. We pursue their account of Dynamic Logic by appealing to the Markov blanket formalism that underwrites the Free Energy Principle. We submit that this allows one to model embryogenesis as self-organisation in a dynamical system that minimises free-energy. The ensuing formalism may be extended to also explain the autonomous emergence of cognition, specifically in the brain, as a dynamic self-assembling process.

At the inception of human brain mapping, two principles of functional anatomy underwrote most conceptions - and analyses - of distributed brain responses: namely functional segregation and integration. There are currently two main approaches to characterising functional integration. The first is a mechanistic modelling of connectomics in terms of directed effective connectivity that mediates neuronal message passing and dynamics on neuronal circuits. The second phenomenological approach usually characterises undirected functional connectivity (i.e., measurable correlations), in terms of intrinsic brain networks, self-organised criticality, dynamical instability, etc. This paper describes a treatment of effective connectivity that speaks to the emergence of intrinsic brain networks and critical dynamics. It is predicated on the notion of Markov blankets that play a fundamental role in the self-organisation of far from equilibrium systems. Using the apparatus of the renormalisation group, we show that much of the phenomenology found in network neuroscience is an emergent property of a particular partition of neuronal states, over progressively larger scales. As such, it offers a way of linking dynamics on directed graphs to the phenomenology of intrinsic brain networks.

The aim of this paper is twofold: (1) to assess whether the construct of neural representations plays an explanatory role under the variational free-energy principle and its corollary process theory, active inference; and (2) if so, to assess which philosophical stance - in relation to the ontological and epistemological status of representations - is most appropriate. We focus on non-realist (deflationary and fictionalist-instrumentalist) approaches. We consider a deflationary account of mental representation, according to which the explanatorily relevant contents of neural representations are mathematical, rather than cognitive, and a fictionalist or instrumentalist account, according to which representations are scientifically useful fictions that serve explanatory (and other) aims. After reviewing the free-energy principle and active inference, we argue that the model of adaptive phenotypes under the free-energy principle can be used to furnish a formal semantics, enabling us to assign semantic content to specific phenotypic states (the internal states of a Markovian system that exists far from equilibrium). We propose a modified fictionalist account: an organism-centered fictionalism or instrumentalism. We argue that, under the free-energy principle, pursuing even a deflationary account of the content of neural representations licenses the appeal to the kind of semantic content involved in the aboutness or intentionality of cognitive systems; our position is thus coherent with, but rests on distinct assumptions from, the realist position. We argue that the free-energy principle thereby explains the aboutness or intentionality in living systems and hence their capacity to parse their sensory stream using an ontology or set of semantic factors., Comment: 35 pages, 4 figures, 1 table

This technical report addresses a pressing issue in the trajectory of the coronavirus outbreak; namely, the rate at which effective immunity is lost following the first wave of the pandemic. This is a crucial epidemiological parameter that speaks to both the consequences of relaxing lockdown and the propensity for a second wave of infections. Using a dynamic causal model of reported cases and deaths from multiple countries, we evaluated the evidence models of progressively longer periods of immunity. The results speak to an effective population immunity of about three months that, under the model, defers any second wave for approximately six months in most countries. This may have implications for the window of opportunity for tracking and tracing, as well as for developing vaccination programmes, and other therapeutic interventions., Comment: 20 pages, 8 figures, 3 tables (technical report)

Active inference is a Bayesian framework for understanding biological intelligence. The underlying theory brings together perception and action under one single imperative: minimizing free energy. However, despite its theoretical utility in explaining intelligence, computational implementations have been restricted to low-dimensional and idealized situations. In this paper, we present a neural architecture for building deep active inference agents operating in complex, continuous state-spaces using multiple forms of Monte-Carlo (MC) sampling. For this, we introduce a number of techniques, novel to active inference. These include: i) selecting free-energy-optimal policies via MC tree search, ii) approximating this optimal policy distribution via a feed-forward `habitual' network, iii) predicting future parameter belief updates using MC dropouts and, finally, iv) optimizing state transition precision (a high-end form of attention). Our approach enables agents to learn environmental dynamics efficiently, while maintaining task performance, in relation to reward-based counterparts. We illustrate this in a new toy environment, based on the dSprites data-set, and demonstrate that active inference agents automatically create disentangled representations that are apt for modeling state transitions. In a more complex Animal-AI environment, our agents (using the same neural architecture) are able to simulate future state transitions and actions (i.e., plan), to evince reward-directed navigation - despite temporary suspension of visual input. These results show that deep active inference - equipped with MC methods - provides a flexible framework to develop biologically-inspired intelligent agents, with applications in both machine learning and cognitive science., Comment: To appear in NeurIPS 2020

Active inference offers a first principle account of sentient behaviour, from which special and important cases can be derived, e.g., reinforcement learning, active learning, Bayes optimal inference, Bayes optimal design, etc. Active inference resolves the exploitation-exploration dilemma in relation to prior preferences, by placing information gain on the same footing as reward or value. In brief, active inference replaces value functions with functionals of (Bayesian) beliefs, in the form of an expected (variational) free energy. In this paper, we consider a sophisticated kind of active inference, using a recursive form of expected free energy. Sophistication describes the degree to which an agent has beliefs about beliefs. We consider agents with beliefs about the counterfactual consequences of action for states of affairs and beliefs about those latent states. In other words, we move from simply considering beliefs about 'what would happen if I did that' to 'what would I believe about what would happen if I did that'. The recursive form of the free energy functional effectively implements a deep tree search over actions and outcomes in the future. Crucially, this search is over sequences of belief states, as opposed to states per se. We illustrate the competence of this scheme, using numerical simulations of deep decision problems.

Recent characterisations of self-organising systems depend upon the presence of a Markov blanket: a statistical boundary that mediates the interactions between what is inside of and outside of a system. We leverage this idea to provide an analysis of partitions in neuronal systems. This is applicable to brain architectures at multiple scales, enabling partitions into single neurons, brain regions, and brain-wide networks. This treatment is based upon the canonical micro-circuitry used in empirical studies of effective connectivity, so as to speak directly to practical applications. This depends upon the dynamic coupling between functional units, whose form recapitulates that of a Markov blanket at each level. The nuance afforded by partitioning neural systems in this way highlights certain limitations of modular perspectives of brain function that only consider a single level of description., Comment: 25 pages, 5 figures, 1 table, Glossary

By equipping a previously reported dynamic causal model of COVID-19 with an isolation state, we modelled the effects of self-isolation consequent on tracking and tracing. Specifically, we included a quarantine or isolation state occupied by people who believe they might be infected but are asymptomatic, and only leave if they test negative. We recovered maximum posteriori estimates of the model parameters using time series of new cases, daily deaths, and tests for the UK. These parameters were used to simulate the trajectory of the outbreak in the UK over an 18-month period. Several clear-cut conclusions emerged from these simulations. For example, under plausible (graded) relaxations of social distancing, a rebound of infections within weeks is unlikely. The emergence of a later second wave depends almost exclusively on the rate at which we lose immunity, inherited from the first wave. There exists no testing strategy that can attenuate mortality rates, other than by deferring or delaying a second wave. A sufficiently powerful tracking and tracing policy--implemented at the time of writing (10th May 2020)--will defer any second wave beyond a time horizon of 18 months. Crucially, this deferment is within current testing capabilities (requiring an efficacy of tracing and tracking of about 20% of asymptomatic infected cases, with less than 50,000 tests per day). These conclusions are based upon a dynamic causal model for which we provide some construct and face validation, using a comparative analysis of the United Kingdom and Germany, supplemented with recent serological studies., Comment: 31 pages, 12 figures, one table. Technical report

Seizure activity is a ubiquitous and pernicious pathophysiology that, in principle, should yield to mathematical treatments of (neuronal) ensemble dynamics - and therefore interventions on stochastic chaos. A seizure can be characterised as a deviation of neural activity from a stable dynamical regime, i.e. one in which signals fluctuate only within a limited range. In silico treatments of neural activity are an important tool for understanding how the brain can achieve stability, as well as how pathology can lead to seizures and potential strategies for mitigating instabilities, e.g. via external stimulation. Here, we demonstrate that the (neuronal) state equation used in Dynamic Causal Modelling generalises to a Fokker-Planck formalism when propagation of neuronal activity along structural connections is considered. Using the Jacobian of this generalised state equation, we show that an initially unstable system can be rendered stable via a reduction in diffusivity (i.e., connectivity that disperses neuronal fluctuations). We show, for neural systems prone to epileptic seizures, that such a reduction can be achieved via external stimulation. Specifically, we show that this stimulation should be applied in such a way as to temporarily mirror epileptic activity in the areas adjoining an affected brain region - thus 'fighting seizures with seizures'. We offer proof of principle using simulations based on functional neuroimaging data collected from patients with idiopathic generalised epilepsy, in which we successfully suppress pathological activity in a distinct sub-network. Our hope is that this technique can form the basis for real-time monitoring and intervention devices that are capable of suppressing or even preventing seizures in a non-invasive manner.

We recently described a dynamic causal model of a COVID-19 outbreak within a single region. Here, we combine several of these (epidemic) models to create a (pandemic) model of viral spread among regions. Our focus is on a second wave of new cases that may result from loss of immunity--and the exchange of people between regions--and how mortality rates can be ameliorated under different strategic responses. In particular, we consider hard or soft social distancing strategies predicated on national (Federal) or regional (State) estimates of the prevalence of infection in the population. The modelling is demonstrated using timeseries of new cases and deaths from the United States to estimate the parameters of a factorial (compartmental) epidemiological model of each State and, crucially, coupling between States. Using Bayesian model reduction, we identify the effective connectivity between States that best explains the initial phases of the outbreak in the United States. Using the ensuing posterior parameter estimates, we then evaluate the likely outcomes of different policies in terms of mortality, working days lost due to lockdown and demands upon critical care. The provisional results of this modelling suggest that social distancing and loss of immunity are the two key factors that underwrite a return to endemic equilibrium., Comment: Technical report: 35 pages, 14 figures, 1 table

This technical report describes a dynamic causal model of the spread of coronavirus through a population. The model is based upon ensemble or population dynamics that generate outcomes, like new cases and deaths over time. The purpose of this model is to quantify the uncertainty that attends predictions of relevant outcomes. By assuming suitable conditional dependencies, one can model the effects of interventions (e.g., social distancing) and differences among populations (e.g., herd immunity) to predict what might happen in different circumstances. Technically, this model leverages state-of-the-art variational (Bayesian) model inversion and comparison procedures, originally developed to characterise the responses of neuronal ensembles to perturbations. Here, this modelling is applied to epidemiological populations to illustrate the kind of inferences that are supported and how the model per se can be optimised given timeseries data. Although the purpose of this paper is to describe a modelling protocol, the results illustrate some interesting perspectives on the current pandemic; for example, the nonlinear effects of herd immunity that speak to a self-organised mitigation process., Comment: Technical report: 40 pages, 13 figures and 2 tables

We used Bayesian model inversion to estimate epidemic parameters from the reported case and death rates from seven countries using data from late January 2020 to April 5th 2020. Two distinct generative model types were employed: first a continuous time dynamical-systems implementation of a Susceptible-Exposed-Infectious-Recovered (SEIR) model and second: a partially observable Markov Decision Process (MDP) or hidden Markov model (HMM) implementation of an SEIR model. Both models parameterise the size of the initial susceptible population (S0), as well as epidemic parameters. Parameter estimation (data fitting) was performed using a standard Bayesian scheme (variational Laplace) designed to allow for latent unobservable states and uncertainty in model parameters. Both models recapitulated the dynamics of transmissions and disease as given by case and death rates. The peaks of the current waves were predicted to be in the past for four countries (Italy, Spain, Germany and Switzerland) and to emerge in 0.5-2 weeks in Ireland and 1-3 weeks in the UK. For France one model estimated the peak within the past week and the other in the future in two weeks. Crucially, Maximum a posteriori (MAP) estimates of S0 for each country indicated effective population sizes of below 20% (of total population size), under both the continuous time and HMM models. With a Bayesian weighted average across all seven countries and both models, we estimated that 6.4% of the total population would be immune. From the two models the maximum percentage of the effective population was estimated at 19.6% of the total population for the UK, 16.7% for Ireland, 11.4% for Italy, 12.8% for Spain, 18.8% for France, 4.7% for Germany and 12.9% for Switzerland. Our results indicate that after the current wave, a large proportion of the total population will remain without immunity.

Biehl et al (2020) present some interesting observations on an early formulation of the free energy principle in (Friston, 2013). We use these observations to scaffold a discussion of the technical arguments that underwrite the free energy principle. This discussion focuses on solenoidal coupling between various (subsets of) states in sparsely coupled systems that possess a Markov blanket - and the distinction between exact and approximate Bayesian inference, implied by the ensuing Bayesian mechanics., Comment: A response to a technical critique [arXiv:2001.06408] of the free energy principle as presented in "Life as we know it"

Active inference is a normative framework for explaining behaviour under the free energy principle -- a theory of self-organisation originating in neuroscience. It specifies neuronal dynamics for state-estimation in terms of a descent on (variational) free energy -- a measure of the fit between an internal (generative) model and sensory observations. The free energy gradient is a prediction error -- plausibly encoded in the average membrane potentials of neuronal populations. Conversely, the expected probability of a state can be expressed in terms of neuronal firing rates. We show that this is consistent with current models of neuronal dynamics and establish face validity by synthesising plausible electrophysiological responses. We then show that these neuronal dynamics approximate natural gradient descent, a well-known optimisation algorithm from information geometry that follows the steepest descent of the objective in information space. We compare the information length of belief updating in both schemes, a measure of the distance traveled in information space that has a direct interpretation in terms of metabolic cost. We show that neural dynamics under active inference are metabolically efficient and suggest that neural representations in biological agents may evolve by approximating steepest descent in information space towards the point of optimal inference., Comment: 12 pages, 3 figures

Active inference is a normative principle underwriting perception, action, planning, decision-making and learning in biological or artificial agents. From its inception, its associated process theory has grown to incorporate complex generative models, enabling simulation of a wide range of complex behaviours. Due to successive developments in active inference, it is often difficult to see how its underlying principle relates to process theories and practical implementation. In this paper, we try to bridge this gap by providing a complete mathematical synthesis of active inference on discrete state-space models. This technical summary provides an overview of the theory, derives neuronal dynamics from first principles and relates this dynamics to biological processes. Furthermore, this paper provides a fundamental building block needed to understand active inference for mixed generative models; allowing continuous sensations to inform discrete representations. This paper may be used as follows: to guide research towards outstanding challenges, a practical guide on how to implement active inference to simulate experimental behaviour, or a pointer towards various in-silico neurophysiological responses that may be used to make empirical predictions., Comment: 36 pages, 5 figures

In contrast to the symmetries of translation in space, rotation in space, and translation in time, the known laws of physics are not universally invariant under transformation of scale. However, the action can be invariant under change of scale in the special case of a scale free dynamical system that can be described in terms of a Lagrangian, that itself scales inversely with time. Crucially, this means symmetries under change of scale can exist in dynamical systems under certain constraints. Our contribution lies in the derivation of a generalised scale invariant Lagrangian - in the form of a power series expansion - that satisfies these constraints. This generalised Lagrangian furnishes a normal form for dynamic causal models (i.e., state space models based upon differential equations) that can be used to distinguish scale invariance (scale symmetry) from scale freeness in empirical data. We establish face validity with an analysis of simulated data and then show how scale invariance can be identified - and how the associated conserved quantities can be estimated - in neuronal timeseries.

Active inference is a first principle account of how autonomous agents operate in dynamic, non-stationary environments. This problem is also considered in reinforcement learning (RL), but limited work exists on comparing the two approaches on the same discrete-state environments. In this paper, we provide: 1) an accessible overview of the discrete-state formulation of active inference, highlighting natural behaviors in active inference that are generally engineered in RL; 2) an explicit discrete-state comparison between active inference and RL on an OpenAI gym baseline. We begin by providing a condensed overview of the active inference literature, in particular viewing the various natural behaviors of active inference agents through the lens of RL. We show that by operating in a pure belief-based setting, active inference agents can carry out epistemic exploration, and account for uncertainty about their environment in a Bayes-optimal fashion. Furthermore, we show that the reliance on an explicit reward signal in RL is removed in active inference, where reward can simply be treated as another observation; even in the total absence of rewards, agent behaviors are learned through preference learning. We make these properties explicit by showing two scenarios in which active inference agents can infer behaviors in reward-free environments compared to both Q-learning and Bayesian model-based RL agents; by placing zero prior preferences over rewards and by learning the prior preferences over the observations corresponding to reward. We conclude by noting that this formalism can be applied to more complex settings if appropriate generative models can be formulated. In short, we aim to demystify the behavior of active inference agents by presenting an accessible discrete state-space and time formulation, and demonstrate these behaviors in a OpenAI gym environment, alongside RL agents.

Models of coupled oscillators are used to describe a wide variety of phenomena in neuroimaging. These models typically rest on the premise that oscillator dynamics do not evolve beyond their respective limit cycles, and hence that interactions can be described purely in terms of phase differences. Whilst mathematically convenient, the restrictive nature of phase-only models can limit their explanatory power. We therefore propose a generalisation of dynamic causal modelling that incorporates both phase and amplitude. This allows for the separate quantifications of phase and amplitude contributions to the connectivity between neural regions. We establish, using model-generated data and simulations of coupled pendula, that phase-only models perform well only under weak coupling conditions. We also show that, despite their higher complexity, phase-amplitude models can describe strongly coupled systems more effectively than their phase-only counterparts. We relate our findings to four metrics commonly used in neuroimaging: the Kuramoto order parameter, cross-correlation, phase-lag index, and spectral entropy. We find that, with the exception of spectral entropy, the phase-amplitude model is able to capture all metrics more effectively than the phase-only model. We then demonstrate, using local field potential recordings in rodents and functional magnetic resonance imaging in macaque monkeys, that amplitudes in oscillator models play an important role in describing neural dynamics in anaesthetised brain states., Comment: arXiv admin note: text overlap with arXiv:1812.06315

Scale free dynamics are observed in a variety of physical and biological systems. These include neural activity in which evidence for scale freeness has been reported using a range of imaging modalities. Here, we derive the ways in which connections within a network must transform - relative to system size - in order to maintain scale freeness and test these theoretical transformations via simulations. First, we explore the known invariance of planetary motion for orbits varying in size. Using parametric empirical Bayesian modelling and a generic dynamical systems model, we show that we recover Kepler's third law from orbital timeseries, using our proposed transformations; thereby providing construct validation. We then demonstrate that the dynamical critical exponent is inversely proportional to the time rescaling exponent, in the context of coarse graining operations. Using murine calcium imaging data, we then show that the dynamical critical exponent can be estimated in an empirical biological setting. Specifically, we compare dynamical critical exponents - associated with spontaneous and task states in two regions of imaged cortex - that are classified as task-relevant and task-irrelevant. We find, consistently across animals, that the task-irrelevant region exhibits higher dynamical critical exponents during spontaneous activity than during task performance. Conversely, the task-relevant region is associated with higher dynamical critical exponents in task vs. spontaneous states. These data support the idea that higher dynamical critical exponents, within relevant cortical structures, underwrite neuronal processing due to the implicit increase in cross-scale information transmission.