Predictive computational modelling in the brain (and other animals) Karl Friston, University College London Abstract: This overview of the free energy principle offers an account of embodied exchange with the world that associates conscious operations with actively inferring the causes of our sensations. Its agenda is to link formal (mathematical) descriptions of dynamical systems to a description of perception in terms of beliefs and goals. The argument has two parts: the first calls on the lawful dynamics of any (weakly mixing) ergodic system– from a single cell organism to a human brain. These lawful dynamics suggest that (internal) states can be interpreted as modelling or predicting the (external) causes of sensory fluctuations. In other words, if a system exists, its internal states must encode probabilistic beliefs about external states. Heuristically, this means that if I exist (am) then I must have beliefs (think). The second part of the argument is that the only tenable beliefs I can entertain about myself are that I exist. This may seem rather obvious; however, if we associate existing with ergodicity, then (ergodic) systems that exist by predicting external states can only possess prior beliefs that their environment is predictable. It transpires that this is equivalent to believing that the world – and the way it is sampled – will resolve uncertainty about the causes of sensations. We will conclude by looking at the epistemic behaviour that emerges under these beliefs, using simulations of active inference.   Key words: active inference ∙ autopoiesis ∙ cognitive ∙ dynamics ∙ free energy ∙ epistemic value ∙ self-organization .

Overview Prediction and life Markov blankets and ergodic systems Simulations of a primordial soup The anatomy of inference Graphical models and predictive coding Canonical microcircuits – in the brain Saccadic searches and salience Knowing your place Multi-agent prediction Morphogenesis – in other animals

“How can the events in space and time which take place within the spatial boundary of a living organism be accounted for by physics and chemistry?” (Erwin Schrödinger 1943) The Markov blanket as a statistical boundary (parents, children and parents of children) Sensory states Active states Internal states External states

The Markov blanket in biotic systems Active states External states Internal states Sensory states

The Fokker-Planck equation lemma: any (ergodic random) dynamical system (m) that possesses a Markov blanket will appear to engage in active inference The Fokker-Planck equation And its solution in terms of curl-free and divergence-free components 5

But what about the Markov blanket? Perception Action Value Surprise Entropy Model evidence Reinforcement learning, optimal control and expected utility theory Information theory and minimum redundancy Self-organisation, synergetics and homoeostasis Bayesian brain, active inference and predictive coding Pavlov Barlow Haken Helmholtz

Overview Prediction and life Markov blankets and ergodic systems Simulations of a primordial soup The anatomy of inference Graphical models and predictive coding Canonical microcircuits – in the brain Saccadic searches and salience Knowing your place Multi-agent prediction Morphogenesis – in other animals

Simulations of a (prebiotic) primordial soup Position Short-range forces Strong repulsion Weak electrochemical attraction

A Finding the (principal) Markov blanket Markov blanket matrix: encoding the children, parents and parents of children Markov Blanket = [B · [eig(B) > τ]] Adjacency matrix Markov Blanket 20 Hidden states 40 A 60 80 Sensory states 100 Active states Internal states 120 20 40 60 80 100 120 Element Does action maintain the structural and functional integrity of the Markov blanket (autopoiesis) ? Do internal states appear to infer the hidden causes of sensory states (active inference) ?

Autopoiesis, oscillator death and simulated brain lesions

Decoding through the Markov blanket and simulated brain activation 100 200 300 400 500 -0.4 -0.3 -0.2 -0.1 Time Motion of external state True and predicted motion -5 5 -8 -6 -4 -2 4 6 8 Position Predictability 2 Modes Internal states 10 15 20 25 30 Christiaan Huygens

Interim summary The existence of a Markov blanket necessarily implies a partition of states into internal states, their Markov blanket (sensory and active states) and external or hidden states. Because active states change – but are not changed by – external states they minimize the entropy of internal states and their Markov blanket. This means action will appear to maintain the structural and functional integrity of the Markov blanket (autopoiesis). Internal states appear to infer the hidden causes of sensory states (by maximizing Bayesian evidence) and influence those causes though action (active inference)

Overview Prediction and life Markov blankets and ergodic systems Simulations of a primordial soup The anatomy of inference Graphical models and predictive coding Canonical microcircuits – in the brain Saccadic searches and salience Knowing your place Multi-agent prediction Morphogenesis – in other animals

The Helmholtz machine and the Bayesian brain “Objects are always imagined as being present in the field of vision as would have to be there in order to produce the same impression on the nervous mechanism” - von Helmholtz Hermann von Helmholtz Richard Gregory Geoffrey Hinton The Helmholtz machine and the Bayesian brain Thomas Bayes Richard Feynman

Impressions on the Markov blanket… “Objects are always imagined as being present in the field of vision as would have to be there in order to produce the same impression on the nervous mechanism” - von Helmholtz Hermann von Helmholtz Richard Gregory Impressions on the Markov blanket…

Bayesian filtering and predictive coding prediction update prediction error

sensations – predictions Making our own sensations sensations – predictions Prediction error Action Perception Changing sensations Changing predictions

Hierarchical generative models the Hierarchical generative models what where A simple hierarchy Descending predictions Ascending prediction errors Sensory fluctuations

Predictive coding with reflexes David Mumford Predictive coding with reflexes Action oculomotor signals reflex arc proprioceptive input pons Perception retinal input Prediction error (superficial pyramidal cells) Expectations (deep pyramidal cells) frontal eye fields geniculate Top-down or backward predictions Bottom-up or forward prediction error visual cortex

Interim summary Biological agents minimize their average surprise (entropy) They minimize surprise by suppressing prediction error Prediction error can be reduced by changing predictions (perception) Prediction error can be reduced by changing sensations (action) Perception entails recurrent message passing to optimize predictions Action makes predictions come true (and minimizes surprise)

Overview Prediction and life Markov blankets and ergodic systems Simulations of a primordial soup The anatomy of inference Graphical models and predictive coding Canonical microcircuits – in the brain Saccadic searches and salience Knowing your place Multi-agent prediction Morphogenesis – in other animals

Perception as hypothesis testing – saccades as experiments Sampling the world to minimise expected uncertainty Likelihood Empirical priors Prior beliefs stimulus visual input salience sampling Perception as hypothesis testing – saccades as experiments “I am [ergodic] therefore I think”  “I think therefore I am [ergodic]”

Parietal (where) Pulvinar salience map Fusiform (what) Frontal eye fields Visual cortex Pulvinar salience map Fusiform (what) oculomotor reflex arc Superior colliculus

Saccadic eye movements Saccadic fixation and salience maps 200 400 600 800 1000 1200 1400 -2 2 Action (EOG) time (ms) -5 5 Posterior belief Hidden (oculomotor) states Visual samples vs. Conditional expectations about hidden (visual) states And corresponding percept

Overview Prediction and life Markov blankets and ergodic systems Simulations of a primordial soup The anatomy of inference Graphical models and predictive coding Canonical microcircuits – in the brain Saccadic searches and salience Knowing your place Multi-agent prediction Morphogenesis – in other animals

Knowing your place: multi-agent games and morphogenesis Intercellular signaling Intrinsic signals Exogenous signals Endogenous signals Generative model

(Genetic) encoding of target morphology -4 -2 2 4 -3 -1 1 3 Target form position Position (place code) Signal expression (genetic code) -4 -2 2 4 -3 -1 1 3 Extracellular target signal position

Morphogenesis Expectations Action Morphogenesis Free energy 5 10 15 20 25 30 35 -4 -3 -2 -1 1 2 3 4 Morphogenesis time location 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 5 10 15 20 25 30 5 10 15 20 25 30 time time 5 10 15 20 25 30 -420 -400 -380 -360 -340 Free energy time Softmax expectations cell 1 2 3 4 6 7 8 -4 -2 2 4 -3 -1 1 3 Solution location

Dysmorphogenesis Gradient Intrinsic Extrinsic TARGET

Regeneration Regeneration of head Regeneration of tail morphogenesis 5 10 15 20 25 30 35 -4 -3 -2 -1 1 2 3 4 morphogenesis time location Regeneration of tail

Hermann von Helmholtz “Each movement we make by which we alter the appearance of objects should be thought of as an experiment designed to test whether we have understood correctly the invariant relations of the phenomena before us, that is, their existence in definite spatial relations.” ‘The Facts of Perception’ (1878) in The Selected Writings of Hermann von Helmholtz, Ed. R. Karl, Middletown: Wesleyan University Press, 1971 p. 384

Thank you And thanks to collaborators: And colleagues: Rick Adams Ryszard Auksztulewicz Andre Bastos Sven Bestmann Harriet Brown Jean Daunizeau Mark Edwards Chris Frith Thomas FitzGerald Xiaosi Gu Stefan Kiebel James Kilner Christoph Mathys Jérémie Mattout Rosalyn Moran Dimitri Ognibene Sasha Ondobaka Will Penny Giovanni Pezzulo Lisa Quattrocki Knight Francesco Rigoli Klaas Stephan Philipp Schwartenbeck And colleagues: Micah Allen Felix Blankenburg Andy Clark Peter Dayan Ray Dolan Allan Hobson Paul Fletcher Pascal Fries Geoffrey Hinton James Hopkins Jakob Hohwy Mateus Joffily Henry Kennedy Simon McGregor Read Montague Tobias Nolte Anil Seth Mark Solms Paul Verschure And many others