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Acknowledgements: Special thanks to Thomas Parr for our virtual conversations about Active Inference., Humans display astonishing skill in learning about the environment in which they operate. They assimilate a rich set of affordances and interrelations among different elements in particular contexts, and form flexible abstractions (i.e., concepts) that can be generalised and leveraged with ease. To capture these abilities, we present a deep hierarchical Active Inference model of goal-directed behaviour, and the accompanying belief update schemes implied by maximising model evidence. Using simulations, we elucidate the potential mechanisms that underlie and influence concept learning in a spatial foraging task. We show that the representations formed-as a result of foraging-reflect environmental structure in a way that is enhanced and nuanced by Bayesian model reduction, a special case of structure learning that typifies learning in the absence of new evidence. Synthetic agents learn associations and form concepts about environmental context and configuration as a result of inferential, parametric learning, and structure learning processes-three processes that can produce a diversity of beliefs and belief structures. Furthermore, the ensuing representations reflect symmetries for environments with identical configurations.

A growing body of evidence highlights the intricate linkage of exteroceptive perception to the rhythmic activity of the visceral body. In parallel, interoceptive inference theories of affective perception and self-consciousness are on the rise in cognitive science. However, thus far no formal theory has emerged to integrate these twin domains; instead, most extant work is conceptual in nature. Here, we introduce a formal model of cardiac active inference, which explains how ascending cardiac signals entrain exteroceptive sensory perception and uncertainty. Through simulated psychophysics, we reproduce the defensive startle reflex and commonly reported effects linking the cardiac cycle to affective behaviour. We further show that simulated 'interoceptive lesions' blunt affective expectations, induce psychosomatic hallucinations, and exacerbate biases in perceptual uncertainty. Through synthetic heart-rate variability analyses, we illustrate how the balance of arousal-priors and visceral prediction errors produces idiosyncratic patterns of physiological reactivity. Our model thus offers a roadmap for computationally phenotyping disordered brain-body interaction.

This paper investigates the prospect of developing human-interpretable, explainable artificial intelligence (AI) systems based on active inference and the free energy principle. We first provide a brief overview of active inference, and in particular, of how it applies to the modeling of decision-making, introspection, as well as the generation of overt and covert actions. We then discuss how active inference can be leveraged to design explainable AI systems, namely, by allowing us to model core features of ``introspective'' processes and by generating useful, human-interpretable models of the processes involved in decision-making. We propose an architecture for explainable AI systems using active inference. This architecture foregrounds the role of an explicit hierarchical generative model, the operation of which enables the AI system to track and explain the factors that contribute to its own decisions, and whose structure is designed to be interpretable and auditable by human users. We outline how this architecture can integrate diverse sources of information to make informed decisions in an auditable manner, mimicking or reproducing aspects of human-like consciousness and introspection. Finally, we discuss the implications of our findings for future research in AI, and the potential ethical considerations of developing AI systems with (the appearance of) introspective capabilities.

Predictive coding (PC) accounts of perception now form one of the dominant computational theories of the brain, where they prescribe a general algorithm for inference and learning over hierarchical latent probabilistic models. Despite this, they have enjoyed little export to the broader field of machine learning, where comparative generative modelling techniques have flourished. In part, this has been due to the poor performance of models trained with PC when evaluated by both sample quality and marginal likelihood. By adopting the perspective of PC as a variational Bayes algorithm under the Laplace approximation, we identify the source of these deficits to lie in the exclusion of an associated Hessian term in the PC objective function, which would otherwise regularise the sharpness of the probability landscape and prevent over-certainty in the approximate posterior. To remedy this, we make three primary contributions: we begin by suggesting a simple Monte Carlo estimated evidence lower bound which relies on sampling from the Hessian-parameterised variational posterior. We then derive a novel block diagonal approximation to the full Hessian matrix that has lower memory requirements and favourable mathematical properties. Lastly, we present an algorithm that combines our method with standard PC to reduce memory complexity further. We evaluate models trained with our approach against the standard PC framework on image benchmark datasets. Our approach produces higher log-likelihoods and qualitatively better samples that more closely capture the diversity of the data-generating distribution.

Even though the brain operates in pure darkness, within the skull, it can infer the most likely causes of its sensory input. An approach to modelling this inference is to assume that the brain has a generative model of the world, which it can invert to infer the hidden causes behind its sensory stimuli, that is, perception. This assumption raises key questions: how to formulate the problem of designing brain-inspired generative models, how to invert them for the tasks of inference and learning, what is the appropriate loss function to be optimised, and, most importantly, what are the different choices of mean field approximation (MFA) and their implications for variational inference (VI)., Comment: 7 pages, 2 figures

For this project, we are investigating network-wide effects of induced seizures in a zebrafish model. We apply dynamic causal modelling to light-sheet microscopy data to identify synaptic dynamics that underlie the observed changes in neuronal output at the level of whole neuronal populations.

How is information processed in the brain during perception? Mechanistic insight is achieved only when experiments are employed to test formal or computational models. In analogy to lesion studies, phantom perception may serve as a vehicle to understand the fundamental processing principles underlying auditory perception. With a special focus on tinnitus -- as the prime example of auditory phantom perception -- we review recent work at the intersection of artificial intelligence, psychology, and neuroscience. In particular, we discuss why everyone with tinnitus suffers from hearing loss, but not everyone with hearing loss suffers from tinnitus. We argue that the increase of sensory precision due to Bayesian inference could be caused by intrinsic neural noise and lead to a prediction error in the cerebral cortex. Hence, two fundamental processing principles - being ubiquitous in the brain - provide the most explanatory power for the emergence of tinnitus: predictive coding as a top-down, and stochastic resonance as a complementary bottom-up mechanism. We conclude that both principles play a crucial role in healthy auditory perception., Comment: arXiv admin note: substantial text overlap with arXiv:2010.01914

This paper provides a concise description of the free energy principle, starting from a formulation of random dynamical systems in terms of a Langevin equation and ending with a Bayesian mechanics that can be read as a physics of sentience. It rehearses the key steps using standard results from statistical physics. These steps entail (i) establishing a particular partition of states based upon conditional independencies that inherit from sparsely coupled dynamics, (ii) unpacking the implications of this partition in terms of Bayesian inference and (iii) describing the paths of particular states with a variational principle of least action. Teleologically, the free energy principle offers a normative account of self-organisation in terms of optimal Bayesian design and decision-making, in the sense of maximising marginal likelihood or Bayesian model evidence. In summary, starting from a description of the world in terms of random dynamical systems, we end up with a description of self-organisation as sentient behaviour that can be interpreted as self-evidencing; namely, self-assembly, autopoiesis or active inference., Comment: 44 pages, 9 Figures; 64 pages including graphical abstract, highlights and references

This paper describes a path integral formulation of the free energy principle. The ensuing account expresses the paths or trajectories that a particle takes as it evolves over time. The main results are a method or principle of least action that can be used to emulate the behaviour of particles in open exchange with their external milieu. Particles are defined by a particular partition, in which internal states are individuated from external states by active and sensory blanket states. The variational principle at hand allows one to interpret internal dynamics - of certain kinds of particles - as inferring external states that are hidden behind blanket states. We consider different kinds of particles, and to what extent they can be imbued with an elementary form of inference or sentience. Specifically, we consider the distinction between dissipative and conservative particles, inert and active particles and, finally, ordinary and strange particles. Strange particles can be described as inferring their own actions, endowing them with apparent autonomy or agency. In short - of the kinds of particles afforded by a particular partition - strange kinds may be apt for describing sentient behaviour., Comment: 33 pages (excluding references), 6 figures

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

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 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

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

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��nyi divergences and their associated variational bounds. R��nyi 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 $��$ parameter, given fixed priors. This rests on changes in $��$ 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, $��\to 0^{+}$ optimisation leads to mass-covering variational estimates and increased variability in choice behaviour. Furthermore, $��\to + \infty$ 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 $��$ 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., 23 pages, 5 figures

Twenty years have passed since the dysconnection hypothesis was first proposed (Friston and Frith, 1995; Weinberger, 1993). In that time, neuroscience has witnessed tremendous advances: we now live in a world of non-invasive neuroanatomy, computational neuroimaging and the Bayesian brain. The genomics era has come and gone. Connectomics and large-scale neuroinformatics initiatives are emerging everywhere. So where is the dysconnection hypothesis now? This article considers how the notion of schizophrenia as a dysconnection syndrome has developed – and how it has been enriched by recent advances in clinical neuroscience. In particular, we examine the dysconnection hypothesis in the context of (i) theoretical neurobiology and computational psychiatry; (ii) the empirical insights afforded by neuroimaging and associated connectomics – and (iii) how bottom-up (molecular biology and genetics) and top-down (systems biology) perspectives are converging on the mechanisms and nature of dysconnections in schizophrenia., Schizophrenia Research, 176 (2-3), ISSN:0920-9964, ISSN:1573-2509

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.

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

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 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

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.

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.

© 2019 The Authors This technical note describes a variational or Bayesian implementation of representational similarity analysis (RSA) and pattern component modelling (PCM). It considers RSA and PCM as Bayesian model comparison procedures that assess the evidence for stimulus or condition-specific patterns of responses distributed over voxels or channels. On this view, one can use standard variational inference procedures to quantify the contributions of particular patterns to the data, by evaluating second-order parameters or hyperparameters. Crucially, this allows one to use parametric empirical Bayes (PEB) to infer which patterns are consistent among subjects. At the between-subject level, one can then assess the evidence for different (combinations of) hypotheses about condition-specific effects using Bayesian model comparison. Alternatively, one can select a single hypothesis that best explains the pattern of responses using Bayesian model selection. This note rehearses the technical aspects of within and between-subject RSA using a worked example, as implemented in the Statistical Parametric Mapping (SPM) software. En route, we highlight the connection between univariate and multivariate analyses of neuroimaging data and the sorts of analyses that are possible using component modelling and representational similarity analysis.

Predictive processing (PP) approaches to the mind are increasingly popular in the cognitive sciences. This surge of interest is accompanied by a proliferation of philosophical arguments, which seek to either extend or oppose various aspects of the emerging framework. In particular, the question of how to position predictive processing with respect to enactive and embodied cognition has become a topic of intense debate. While these arguments are certainly of valuable scientific and philosophical merit, they risk underestimating the variety of approaches gathered under the predictive label. Here, we first present a basic review of neuroscientific, cognitive, and philosophical approaches to PP, to illustrate how these range from solidly cognitivist applications-with a firm commitment to modular, internalistic mental representation-to more moderate views emphasizing the importance of 'body-representations', and finally to those which fit comfortably with radically enactive, embodied, and dynamic theories of mind. Any nascent predictive processing theory (e.g., of attention or consciousness) must take into account this continuum of views, and associated theoretical commitments. As a final point, we illustrate how the Free Energy Principle (FEP) attempts to dissolve tension between internalist and externalist accounts of cognition, by providing a formal synthetic account of how internal 'representations' arise from autopoietic self-organization. The FEP thus furnishes empirically productive process theories (e.g., predictive processing) by which to guide discovery through the formal modelling of the embodied mind.

This paper offers a formal account of emotional inference and stress-related behaviour, using the notion of active inference. We formulate responses to stressful scenarios in terms of Bayesian belief-updating and subsequent policy selection; namely, planning as (active) inference. Using a minimal model of how creatures or subjects account for their sensations (and subsequent action), we deconstruct the sequences of belief updating and behaviour that underwrite stress-related responses – and simulate the aberrant responses of the sort seen in post-traumatic stress disorder (PTSD). Crucially, the model used for belief-updating generates predictions in multiple (exteroceptive, proprioceptive and interoceptive) modalities, to provide an integrated account of evidence accumulation and multimodal integration that has consequences for both motor and autonomic responses. The ensuing phenomenology speaks to many constructs in the ecological and clinical literature on stress, which we unpack with reference to simulated inference processes and accompanying neuronal responses. A key insight afforded by this formal approach rests on the trade-off between the epistemic affordance of certain cues (that resolve uncertainty about states of affairs in the environment) and the consequences of epistemic foraging (that may be in conflict with the instrumental or pragmatic value of ‘fleeing’ or ‘freezing’). Starting from first principles, we show how this trade-off is nuanced by prior (subpersonal) beliefs about the outcomes of behaviour – beliefs that, when held with unduly high precision, can lead to (Bayes optimal) responses that closely resemble PTSD.

This technical note presents a framework for investigating the underlying mechanisms of neurovascular coupling in the human brain using multi-modal magnetoencephalography (MEG) and functional magnetic resonance (fMRI) neuroimaging data. This amounts to estimating the evidence for several biologically informed models of neurovascular coupling using variational Bayesian methods and selecting the most plausible explanation using Bayesian model comparison. First, fMRI data is used to localise active neuronal sources. The coordinates of neuronal sources are then used as priors in the specification of a DCM for MEG, in order to estimate the underlying generators of the electrophysiological responses. The ensuing estimates of neuronal parameters are used to generate neuronal drive functions, which model the pre or post synaptic responses to each experimental condition in the fMRI paradigm. These functions form the input to a model of neurovascular coupling, the parameters of which are estimated from the fMRI data. This establishes a Bayesian fusion technique that characterises the BOLD response - asking, for example, whether instantaneous or delayed pre or post synaptic signals mediate haemodynamic responses. Bayesian model comparison is used to identify the most plausible hypotheses about the causes of the multimodal data. We illustrate this procedure by comparing a set of models of a single-subject auditory fMRI and MEG dataset. Our exemplar analysis suggests that the origin of the BOLD signal is mediated instantaneously by intrinsic neuronal dynamics and that neurovascular coupling mechanisms are region-specific. The code and example dataset associated with this technical note are available through the statistical parametric mapping (SPM) software package.

A brief introduction to parameter estimation methods used in DCM

Dynamic Causal Modelling (DCM) is the predominant method for inferring effective connectivity from neuroimaging data. In the 15 years since its introduction, the neural models and statistical routines in DCM have developed in parallel, driven by the needs of researchers in cognitive and clinical neuroscience. In this tutorial, we step through an exemplar fMRI analysis in detail, reviewing the current implementation of DCM and demonstrating recent developments in group-level connectivity analysis. In the first part of the tutorial (current paper), we focus on issues specific to DCM for fMRI, unpacking the relevant theory and highlighting practical considerations. In particular, we clarify the assumptions (i.e., priors) used in DCM for fMRI and how to interpret the model parameters. This tutorial is accompanied by all the necessary data and instructions to reproduce the analyses using the SPM software. In the second part (in a companion paper), we move from subject-level to group-level modelling using the Parametric Empirical Bayes framework, and illustrate how to test for commonalities and differences in effective connectivity across subjects, based on imaging data from any modality.

This tutorial provides a worked example of using Dynamic Causal Modelling (DCM) and Parametric Empirical Bayes (PEB) to characterise inter-subject variability in neural circuitry (effective connectivity). This involves specifying a hierarchical model with two or more levels. At the first level, state space models (DCMs) are used to infer the effective connectivity that best explains a subject's neuroimaging timeseries (e.g. fMRI, MEG, EEG). Subject-specific connectivity parameters are then taken to the group level, where they are modelled using a General Linear Model (GLM) that partitions between-subject variability into designed effects and additive random effects. The ensuing (Bayesian) hierarchical model conveys both the estimated connection strengths and their uncertainty (i.e., posterior covariance) from the subject to the group level; enabling hypotheses to be tested about the commonalities and differences across subjects. This approach can also finesse parameter estimation at the subject level, by using the group-level parameters as empirical priors. We walk through this approach in detail, using data from a published fMRI experiment that characterised individual differences in hemispheric lateralization in a semantic processing task. The preliminary subject specific DCM analysis is covered in detail in a companion paper. This tutorial is accompanied by the example dataset and step-by-step instructions to reproduce the analyses.

Autism is a neurodevelopmental disorder characterized by problems with social-communication, restricted interests and repetitive behavior. A recent and thought-provoking article presented a normative explanation for the perceptual symptoms of autism in terms of a failure of Bayesian inference (Pellicano and Burr, 2012). In response, we suggested that when Bayesian inference is grounded in its neural instantiation—namely, predictive coding—many features of autistic perception can be attributed to aberrant precision (or beliefs about precision) within the context of hierarchical message passing in the brain (Friston et al., 2013). Here, we unpack the aberrant precision account of autism. Specifically, we consider how empirical findings—that speak directly or indirectly to neurobiological mechanisms—are consistent with the aberrant encoding of precision in autism; in particular, an imbalance of the precision ascribed to sensory evidence relative to prior beliefs.

Oscillatory activity in the beta range, in human primary motor cortex (M1), shows interesting dynamics that are tied to behaviour and change systematically in disease. To investigate the pathophysiology underlying these changes, we must first understand how changes in beta activity are caused in healthy subjects. We therefore adapted a canonical (repeatable) microcircuit model used in dynamic causal modelling (DCM) previously used to model induced responses in visual cortex. We adapted this model to accommodate cytoarchitectural differences between visual and motor cortex. Using biologically plausible connections, we used Bayesian model selection to identify the best model of measured MEG data from 11 young healthy participants, performing a simple handgrip task. We found that the canonical M1 model had substantially more model evidence than the generic canonical microcircuit model when explaining measured MEG data. The canonical M1 model reproduced measured dynamics in humans at rest, in a manner consistent with equivalent studies performed in mice. Furthermore, the changes in excitability (self-inhibition) necessary to explain beta suppression during handgrip were consistent with the attenuation of sensory precision implied by predictive coding. These results establish the face validity of a model that can be used to explore the laminar interactions that underlie beta-oscillatory dynamics in humans in vivo. Our canonical M1 model may be useful for characterising the synaptic mechanisms that mediate pathophysiological beta dynamics associated with movement disorders, such as stroke or Parkinson's disease.

In this work, we address the neuronal encoding problem from a Bayesian perspective. Specifically, we ask whether neuronal responses in anin vitroneuronal network are consistent with ideal Bayesian observer responses under the free energy principle. In brief, we stimulated anin vitrocortical cell culture with stimulus trains that had a known statistical structure. We then asked whether recorded neuronal responses were consistent with variational message passing (i.e., belief propagation) based upon free energy minimisation (i.e., evidence maximisation). Effectively, this required us to solve two problems: first, we had to formulate the Bayes-optimal encoding of the causes or sources of sensory stimulation, and then show that these idealised responses could account for observed electrophysiological responses. We describe a simulation of an optimal neural network (i.e., the ideal Bayesian neural code) and then consider the mapping from idealisedin silicoresponses to recordedin vitroresponses. Our objective was to find evidence for functional specialisation and segregation in thein vitroneural network that reproducedin silicolearning via free energy minimisation. Finally, we combined thein vitroandin silicoresults to characterise learning in terms of trajectories in a variational information plane of accuracy and complexity.

To exhibit social intelligence, animals have to recognize who they are communicating with. One way to make this inference is to select among multiple internal generative models of each conspecific. This induces an interesting problem: when receiving sensory input generated by a particular conspecific, how does an animal know which internal model to update? We consider a theoretical and neurobiologically plausible solution that enables inference and learning under multiple generative models by integrating active inference and (online) Bayesian model selection. This scheme fits sensory inputs under each generative model. Model parameters are then updated in proportion to the probability it could have generated the current input (i.e., model evidence). We show that a synthetic bird who employs the proposed scheme successfully learns and distinguishes (real zebra finch) birdsongs generated by several different birds. These results highlight the utility of having multiple internal models to make inferences in complicated social environments.

Energy homeostasis depends on behavior to predictively regulate metabolic states within narrow bounds. Here we review three theories of homeostatic control and ask how they provide insight into the circuitry underlying energy homeostasis. We offer two contributions. First, we detail how control theory and reinforcement learning are applied to homeostatic control. We show how these schemes rest on implausible assumptions; either via circular definitions, unprincipled drive functions, or by ignoring environmental volatility. We argue active inference can elude these shortcomings while retaining important features of each model. Second, we review the neural basis of energetic control. We focus on a subset of arcuate subpopulations that project directly to, and are thus in a privileged position to opponently modulate, dopaminergic cells as a function of energetic predictions over a spectrum of time horizons. We discuss how this can be interpreted under these theories, and how this can resolve paradoxes that have arisen. We propose this circuit constitutes a homeostatic-reward interface that underwrites the conjoint optimisation of physiological and behavioural homeostasis.

Natural selection, Bayesian inference, and Bayesian model selection

Convolutional and Recurrent, deep neural networks have been successful in machine learning systems for computer vision, reinforcement learning, and other allied fields. However, the robustness of such neural networks is seldom apprised, especially after high classification accuracy has been attained. In this paper, we evaluate the robustness of three recurrent neural networks to tiny perturbations, on three widely used datasets, to argue that high accuracy does not always mean a stable and a robust (to bounded perturbations, adversarial attacks, etc.) system. Especially, normalizing the spectrum of the discrete recurrent network to bound the spectrum (using power method, Rayleigh quotient, etc.) on a unit disk produces stable, albeit highly non-robust neural networks. Furthermore, using the $\epsilon$-pseudo-spectrum, we show that training of recurrent networks, say using gradient-based methods, often result in non-normal matrices that may or may not be diagonalizable. Therefore, the open problem lies in constructing methods that optimize not only for accuracy but also for the stability and the robustness of the underlying neural network, a criterion that is distinct from the other.

A poster presented at BrainModes 2017 (Delhi, India)

Prof. Karl J. Friston is a scientist who has made fundamental contributions to areas such as functional brain imaging, statistical techniques for dynamical systems, and the Free Energy Principle (FEP). Here, various topics are formally and informally explored. First there are personal, scientific, and mathematical accounts related to the (origins of the) FEP, and how the FEP diverges from the Predictive coding and Bayesian brain hypotheses. Next, there is a discussion of how the FEP addresses and encompasses the philosophical ideas of David Marr (multiple levels of systems analysis) and Karl Popper (classical statistical inference and hypothesis falsification). The following segment of the interview provides a FEP perspective on human phenomenology, psychopathologies such as schizophrenia and psychosis, and psychology in general. This discussion leads to a clarification of how Active Inference undermines classical ideas related to internal neural representations and clear action-perception dichotomies. The interview closes with an expansion of Prof. Friston’s perspective on self-awareness in non-human systems such as computers and ant colonies., Bayesian brain hypothesis, computational psychiatry, free energy principle, predictive coding

Autobiography, Free Energy Principle, Autism, Education, Childhood

A poster presented at OHBM 2018 (Singapore)

In this technical note, we address an unresolved challenge in neuroimaging statistics: how to determine which of several datasets is the best for inferring neuronal responses. Comparisons of this kind are important for experimenters when choosing an imaging protocol - and for developers of new acquisition methods. However, the hypothesis that one dataset is better than another cannot be tested using conventional statistics (based on likelihood ratios), as these require the data to be the same under each hypothesis. Here we present Bayesian data comparison, a principled framework for evaluating the quality of functional imaging data, in terms of the precision with which neuronal connectivity parameters can be estimated and competing models can be disambiguated. For each of several candidate datasets, neuronal responses are inferred using Dynamic Casual Modelling (DCM) - a commonly used Bayesian procedure for modelling neuroimaging data. Next, the parameters from subject-specific models are summarised at the group level using a Bayesian General Linear Model (GLM). A series of measures, which we introduce here, are then used to evaluate each dataset in terms of the precision of (group-level) parameter estimates and the ability of the data to distinguish similar models. To exemplify the approach, we compared four datasets that were acquired in a study evaluating multiband fMRI acquisition schemes. To enable people to reproduce these analyses using their own data and experimental paradigms, we provide general-purpose Matlab code via the SPM software., Comment: Equal contribution by Peter Zeidman and Samira M Kazan

This paper introduces an active inference formulation of planning and navigation. It illustrates how the exploitation–exploration dilemma is dissolved by acting to minimise uncertainty (i.e., expected surprise or free energy). We use simulations of a maze problem to illustrate how agents can solve quite complicated problems using context sensitive prior preferences to form subgoals. Our focus is on how epistemic behaviour – driven by novelty and the imperative to reduce uncertainty about the world – contextualises pragmatic or goal-directed behaviour. Using simulations, we illustrate the underlying process theory with synthetic behavioural and electrophysiological responses during exploration of a maze and subsequent navigation to a target location. An interesting phenomenon that emerged from the simulations was a putative distinction between ‘place cells’ – that fire when a subgoal is reached – and ‘path cells’ – that fire until a subgoal is reached.

We characterised the pathophysiology of seizure onset in terms of slow fluctuations in synaptic efficacy using EEG in patients with anti-N-methyl-d-aspartate receptor (NMDA-R) encephalitis. EEG recordings were obtained from two female patients with anti-NMDA-R encephalitis with recurrent partial seizures (ages 19 and 31). Focal electrographic seizure activity was localised using an empirical Bayes beamformer. The spectral density of reconstructed source activity was then characterised with dynamic causal modelling (DCM). Eight models were compared for each patient, to evaluate the relative contribution of changes in intrinsic (excitatory and inhibitory) connectivity and endogenous afferent input. Bayesian model comparison established a role for changes in both excitatory and inhibitory connectivity during seizure activity (in addition to changes in the exogenous input). Seizures in both patients were associated with a sequence of changes in inhibitory and excitatory connectivity; a transient increase in inhibitory connectivity followed by a transient increase in excitatory connectivity and a final peak of excitatory–inhibitory balance at seizure offset. These systematic fluctuations in excitatory and inhibitory gain may be characteristic of (anti NMDA-R encephalitis) seizures. We present these results as a case study and replication to motivate analyses of larger patient cohorts, to see whether our findings generalise and further characterise the mechanisms of seizure activity in anti-NMDA-R encephalitis.

In this technical note we compare the performance of four gradient-free MCMC samplers (random walk Metropolis sampling, slice-sampling, adaptive MCMC sampling and population-based MCMC sampling with tempering) in terms of the number of independent samples they can produce per unit computational time. For the Bayesian inversion of a single-node neural mass model, both adaptive and population-based samplers are more efficient compared with random walk Metropolis sampler or slice-sampling; yet adaptive MCMC sampling is more promising in terms of compute time. Slice-sampling yields the highest number of independent samples from the target density — albeit at almost 1000% increase in computational time, in comparison to the most efficient algorithm (i.e., the adaptive MCMC sampler).

This review surveys recent trends in the use of local field potentials—and their non-invasive counterparts—to address the principles of functional brain architectures. In particular, we treat oscillations as the (observable) signature of context-sensitive changes in synaptic efficacy that underlie coordinated dynamics and message-passing in the brain. This rich source of information is now being exploited by various procedures—like dynamic causal modelling—to test hypotheses about neuronal circuits in health and disease. Furthermore, the roles played by neuromodulatory mechanisms can be addressed directly through their effects on oscillatory phenomena. These neuromodulatory or gain control processes are central to many theories of normal brain function (e.g. attention) and the pathophysiology of several neuropsychiatric conditions (e.g. Parkinson's disease).

Recently, there has been a lot of interest in characterising the connectivity of resting state brain networks. Most of the literature uses functional connectivity to examine these intrinsic brain networks. Functional connectivity has well documented limitations because of its inherent inability to identify causal interactions. Dynamic causal modelling (DCM) is a framework that allows for the identification of the causal (directed) connections among neuronal systems — known as effective connectivity. This technical note addresses the validity of a recently proposed DCM for resting state fMRI – as measured in terms of their complex cross spectral density – referred to as spectral DCM. Spectral DCM differs from (the alternative) stochastic DCM by parameterising neuronal fluctuations using scale free (i.e., power law) forms, rendering the stochastic model of neuronal activity deterministic. Spectral DCM not only furnishes an efficient estimation of model parameters but also enables the detection of group differences in effective connectivity, the form and amplitude of the neuronal fluctuations or both. We compare and contrast spectral and stochastic DCM models with endogenous fluctuations or state noise on hidden states. We used simulated data to first establish the face validity of both schemes and show that they can recover the model (and its parameters) that generated the data. We then used Monte Carlo simulations to assess the accuracy of both schemes in terms of their root mean square error. We also simulated group differences and compared the ability of spectral and stochastic DCMs to identify these differences. We show that spectral DCM was not only more accurate but also more sensitive to group differences. Finally, we performed a comparative evaluation using real resting state fMRI data (from an open access resource) to study the functional integration within default mode network using spectral and stochastic DCMs.