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Researcher profile

Karl Friston

Karl Friston contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Neurons and CognitionArtificial IntelligenceMachine Learningmath.OCBiological PhysicsInformation Theorymath.DGmath.ITmath.STMathematical SoftwareMethodologyquant-phQuantitative MethodsRoboticsStatistics Theory

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Published work

16 published item(s)

preprint2026arXiv

Online Generalised Predictive Coding

This paper introduces an extension of generalised filtering for online applications. Generalised filtering refers to data assimilation schemes that jointly infer latent states, learn unknown model parameters, and estimate uncertainty in an integrated framework -- e.g., estimate state and observation noise -- at the same time (i.e., triple estimation). This framework appears across disciplines under different names, including variational Kalman-Bucy filtering in engineering, generalised predictive coding in neuroscience, and Dynamic Expectation Maximisation (DEM) in time-series analysis. Here, we specialise DEM for ``online'' data assimilation, through a separation of temporal scales. We describe the variational principles and procedures that allow one to assimilate data in a way that allows for a slow updating of parameters and precisions, which contextualise fast Bayesian belief updating about the dynamic hidden states. Using numerical studies, we demonstrate the validity of online DEM (ODEM) using a non-linear -- and potentially chaotic -- generative model, to show that the ODEM scheme can track the latent states of the generative process, even when its functional form differs fundamentally from the dynamics of the generative model. Framed from a neuro-mimetic predictive coding perspective, ODEM offers a biologically inspired solution to online inference, learning, and uncertainty estimation in dynamic environments.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

The Complex Brain Hypothesis: Resolving the Entropy-Content Conundrum in Minimal Phenomenal Experience

Minimal Phenomenal Experiences (MPEs) are states of consciousness in which wakefulness is preserved but phenomenal content is low or absent. The Entropic Brain Hypothesis (EBH) is a model of conscious processes that regards the entropy of spontaneous brain activity as a marker of 'phenomenal richness', exemplified by high-content psychedelic experiences (HCPEs). Yet recent human neuroimaging studies of MPEs induced by meditation -- and possibly 5-MeO-DMT -- suggest that these states, defined by their phenomenological simplicity, also show signs of increased neurophysiological entropy. This presents a conundrum for the EBH: brain entropy is elevated with increased and decreased richness of the phenomenal experience. Here, we put forward the Complex Brain Hypothesis (CBH), which proposes that the richness of experience differentiating MPEs from HCPEs is better indexed by complexity than by entropy. We argue that brain complexity is modulated by the grain of inference through which the brain resolves uncertainty: some HCPEs exemplify a fine-grained regime, in which loosened constraints amplify fluctuations into proliferating content, whereas some MPEs exemplify a coarse-grained regime, in which a simpler model dissolves variety into an experience of 'contentless' awareness. Both regimes can be associated with elevated brain entropy, but they diverge in phenomenology and perturbational signatures. By resolving the entropy-content conundrum, the CBH refines the EBH and highlights MPEs as an important test case for computational theories of consciousness.

0 citations0 reviewsNo code linkedOpen access
preprint2024arXiv

The inner screen model of consciousness: applying the free energy principle directly to the study of conscious experience

This paper presents a model of consciousness that follows directly from the free-energy principle (FEP). We first rehearse the classical and quantum formulations of the FEP. In particular, we consider the inner screen hypothesis that follows from the quantum information theoretic version of the FEP. We then review applications of the FEP to the known sparse (nested and hierarchical) neuro-anatomy of the brain. We focus on the holographic structure of the brain, and how this structure supports (overt and covert) action.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Action and Perception as Divergence Minimization

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.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents

In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric structures to solve these problems efficiently. We show that a wide range of geometric theories emerge naturally in these fields, ranging from measure-preserving processes, information divergences, Poisson geometry, and geometric integration. Specifically, we explain how (i) leveraging the symplectic geometry of Hamiltonian systems enable us to construct (accelerated) sampling and optimisation methods, (ii) the theory of Hilbertian subspaces and Stein operators provides a general methodology to obtain robust estimators, (iii) preserving the information geometry of decision-making yields adaptive agents that perform active inference. Throughout, we emphasise the rich connections between these fields; e.g., inference draws on sampling and optimisation, and adaptive decision-making assesses decisions by inferring their counterfactual consequences. Our exposition provides a conceptual overview of underlying ideas, rather than a technical discussion, which can be found in the references herein.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

GeoSPM: Geostatistical parametric mapping for medicine

The characteristics and determinants of health and disease are often organised in space, reflecting our spatially extended nature. Understanding the influence of such factors requires models capable of capturing spatial relations. Though a mature discipline, spatial analysis is comparatively rare in medicine, arguably a consequence of the complexity of the domain and the inclemency of the data regimes that govern it. Drawing on statistical parametric mapping, a framework for topological inference well-established in the realm of neuroimaging, we propose and validate a novel approach to the spatial analysis of diverse clinical data - GeoSPM - based on differential geometry and random field theory. We evaluate GeoSPM across an extensive array of synthetic simulations encompassing diverse spatial relationships, sampling, and corruption by noise, and demonstrate its application on large-scale data from UK Biobank. GeoSPM is transparently interpretable, can be implemented with ease by non-specialists, enables flexible modelling of complex spatial relations, exhibits robustness to noise and under-sampling, offers well-founded criteria of statistical significance, and is through computational efficiency readily scalable to large datasets. We provide a complete, open-source software implementation of GeoSPM, and suggest that its adoption could catalyse the wider use of spatial analysis across the many aspects of medicine that urgently demand it.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Global dynamics of neural mass models

Neural mass models are used to simulate cortical dynamics and to explain the electrical and magnetic fields measured using electro- and magnetoencephalography. Simulations evince a complex phase-space structure for these kinds of models; including stationary points and limit cycles and the possibility for bifurcations and transitions among different modes of activity. This complexity allows neural mass models to describe the itinerant features of brain dynamics. However, expressive, nonlinear neural mass models are often difficult to fit to empirical data without additional simplifying assumptions: e.g., that the system can be modelled as linear perturbations around a fixed point. In this study we offer a mathematical analysis of neural mass models, specifically the canonical microcircuit model, providing analytical solutions describing dynamical itinerancy. We derive a perturbation analysis up to second order of the phase flow, together with adiabatic approximations. This allows us to describe amplitude modulations as gradient flows on a potential function of intrinsic connectivity. These results provide analytic proof-of-principle for the existence of semi-stable states of cortical dynamics at the scale of a cortical column. Crucially, this work allows for model inversion of neural mass models, not only around fixed points, but over regions of phase space that encompass transitions among semi or multi-stable states of oscillatory activity. In principle, this formulation of cortical dynamics may improve our understanding of the itinerancy that underwrites measures of cortical activity (through EEG or MEG). Crucially, these theoretical results speak to model inversion in the context of multiple semi-stable brain states, such as onset of seizure activity in epilepsy or beta bursts in Parkinsons disease.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

pymdp: A Python library for active inference in discrete state spaces

Active inference is an account of cognition and behavior in complex systems which brings together action, perception, and learning under the theoretical mantle of Bayesian inference. Active inference has seen growing applications in academic research, especially in fields that seek to model human or animal behavior. While in recent years, some of the code arising from the active inference literature has been written in open source languages like Python and Julia, to-date, the most popular software for simulating active inference agents is the DEM toolbox of SPM, a MATLAB library originally developed for the statistical analysis and modelling of neuroimaging data. Increasing interest in active inference, manifested both in terms of sheer number as well as diversifying applications across scientific disciplines, has thus created a need for generic, widely-available, and user-friendly code for simulating active inference in open-source scientific computing languages like Python. The Python package we present here, pymdp (see https://github.com/infer-actively/pymdp), represents a significant step in this direction: namely, we provide the first open-source package for simulating active inference with partially-observable Markov Decision Processes or POMDPs. We review the package's structure and explain its advantages like modular design and customizability, while providing in-text code blocks along the way to demonstrate how it can be used to build and run active inference processes with ease. We developed pymdp to increase the accessibility and exposure of the active inference framework to researchers, engineers, and developers with diverse disciplinary backgrounds. In the spirit of open-source software, we also hope that it spurs new innovation, development, and collaboration in the growing active inference community.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Reclaiming saliency: rhythmic precision-modulated action and perception

Computational models of visual attention in artificial intelligence and robotics have been inspired by the concept of a saliency map. These models account for the mutual information between the (current) visual information and its estimated causes. However, they fail to consider the circular causality between perception and action. In other words, they do not consider where to sample next, given current beliefs. Here, we reclaim salience as an active inference process that relies on two basic principles: uncertainty minimisation and rhythmic scheduling. For this, we make a distinction between attention and salience. Briefly, we associate attention with precision control, i.e., the confidence with which beliefs can be updated given sampled sensory data, and salience with uncertainty minimisation that underwrites the selection of future sensory data. Using this, we propose a new account of attention based on rhythmic precision-modulation and discuss its potential in robotics, providing numerical experiments that showcase advantages of precision-modulation for state and noise estimation, system identification and action selection for informative path planning.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Reward Maximisation through Discrete Active Inference

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.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

A free energy principle for generic quantum systems

The Free Energy Principle (FEP) states that under suitable conditions of weak coupling, random dynamical systems with sufficient degrees of freedom will behave so as to minimize an upper bound, formalized as a variational free energy, on surprisal (a.k.a., self-information). This upper bound can be read as a Bayesian prediction error. Equivalently, its negative is a lower bound on Bayesian model evidence (a.k.a., marginal likelihood). In short, certain random dynamical systems evince a kind of self-evidencing. Here, we reformulate the FEP in the formal setting of spacetime-background free, scale-free quantum information theory. We show how generic quantum systems can be regarded as observers, which with the standard freedom of choice assumption become agents capable of assigning semantics to observational outcomes. We show how such agents minimize Bayesian prediction error in environments characterized by uncertainty, insufficient learning, and quantum contextuality. We show that in its quantum-theoretic formulation, the FEP is asymptotically equivalent to the Principle of Unitarity. Based on these results, we suggest that biological systems employ quantum coherence as a computational resource and - implicitly - as a communication resource. We summarize a number of problems for future research, particularly involving the resources required for classical communication and for detecting and responding to quantum context switches.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A World unto Itself: Human Communication as Active Inference

Work in developmental psychology suggests that humans are predisposed to align their mental states with other individuals. This manifests principally in cooperative communication, that is, intentional communication geared towards aligning mental states. This viewpoint has received ample empirical support. However, this view lacks a formal grounding, and provides no precise neuroscientific hypotheses. To remedy this, we suggest an active inference approach to cooperative communication. We suggest that humans appear to possess an evolved adaptive prior belief that their mental states are aligned with those of conspecifics. Cooperative communication emerges as the principal means to gather evidence for this belief. Our approach has implications for the study of the usage, ontogeny, and cultural evolution of human communication.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Active inference on discrete state-spaces: a synthesis

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Cognition coming about: self-organisation and free-energy

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Markov Blankets in the Brain

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Sophisticated Inference

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.

0 citations0 reviewsNo code linkedOpen access