Dynamic Causal Modeling of Endogenous Fluctuations Glimpsing at Neuronal Connectivity through the Hemodynamic Veil? What We Can and Cannot Do with Biophysical Dynamic Models of fMRI Connectivity Dynamic Causal Modeling of Endogenous Fluctuations Karl Friston, Wellcome Centre for Neuroimaging, UCL,UK Abstract This talk will present recent advances in dynamic causal modeling (DCM); with a focus on (i) extending DCM to handle state-space models based on random differential equations and (ii) finessing the search over very large model spaces with the Savage-Dickey density ratio. The first advance enables DCM to infer on hidden (endogenous and smooth) fluctuations in neuronal and hemodynamic states. This permits the use of DCM in task-free designs (e.g., resting-state studies). The second advance allows one to discover networks by searching exhaustively over all combinations of connections. These developments resolve the problems faced by causal modeling based on temporal precedence and the theory of Martingales (e.g., Structural Equation Modeling and Granger causality) and circumvents the use of directed acyclic graphs (e.g., Bayes Nets and Structural Causal Modeling).

Outline Dynamic causal modelling for fMRI Modelling endogenous fluctuations Network discovery An empirical example

Functional integration and the enabling of pathways Neuronal network of hidden states Structural perturbations Stimulus-free; e.g., attention, time BA39 Dynamic perturbations Stimuli-bound; e.g., visual words STG V4 V1 BA37 Observation

Forward models and their inversion Observed data Forward model (measurement) Model inversion Forward model (neuronal) input

Model specification and inversion Design experimental inputs Neural dynamics Define likelihood model Observer function Specify priors Invert model Inference on parameters Inference on models Inference

The bilinear (neuronal) model for fMRI Input Dynamic perturbation Structural perturbation average connectivity bilinear and nonlinear connectivity exogenous causes

Output: a mixture of intra- and extravascular signal Hemodynamic models for fMRI basically, a convolution signal The plumbing flow volume dHb 0 8 16 24 sec Output: a mixture of intra- and extravascular signal

A toy example u2 x3 u1 x1 x2 – – Neural population activity 0.4 0.3 0.2 0.1 10 20 30 40 50 60 70 80 90 u2 0.6 0.4 A toy example x3 0.2 10 20 30 40 50 60 70 80 90 0.3 0.2 0.1 BOLD signal change (%) 10 20 30 40 50 60 70 80 90 u1 x1 x2 2 1 – – 10 20 30 40 50 60 70 80 90 3 2 1 -1 10 20 30 40 50 60 70 80 90 2 1 10 20 30 40 50 60 70 80 90

An fMRI study of attention Stimuli 250 radially moving dots at 4.7 degrees/s Pre-Scanning 5 x 30s trials with 5 speed changes (reducing to 1%) Task: detect change in radial velocity Scanning (no speed changes) 4 100 scan sessions; each comprising 10 scans of 4 conditions F A F N F A F N S ................. F - fixation point A - motion stimuli with attention (detect changes) N - motion stimuli without attention S - no motion V5+ PPC Büchel et al 1999

Hierarchical architecture Attentional modulation of prefrontal connections sufficient to explain regionally specific attentional effects Attention .43 .53 Photic SPC .40 .49 .62 V1 .92 .35 IFG .53 Segregation of motion information to V5 Motion V5 .73

Dynamic causal modelling for fMRI Modelling endogenous fluctuations Network discovery An empirical example

Replacing exogenous inputs with endogenous fluctuations Signal and noise Hidden states 1.5 0.2 0.15 1 0.1 0.5 0.05 -0.05 -0.5 -0.1 -1 -0.15 -1.5 -0.2 50 100 150 200 250 50 100 150 200 250 time time Endogenous fluctuations Network or graph generating data Exogenous inputs 0.2 0.15 0.1 0.05 -0.05 -0.1 -0.15 50 100 150 200 250 time

Estimating hidden states with generalised (Bayesian) filtering 1 2 3 4 5 6 7 8 9 -0.5 0.5 True and MAP connections 50 100 150 200 250 -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 Hidden states time (bins) 300 400 500 600 700 800 -0.04 -0.02 0.02 0.04 0.06 True neuronal activity time (sec.) MAP estimate Extrinsic coupling parameter

Dynamic causal modelling for fMRI Modelling endogenous fluctuations 2 3 4 5 6 10 1 Number of models number of nodes Dynamic causal modelling for fMRI Modelling endogenous fluctuations Network discovery An empirical example and the problem of searching large model spaces

c.f., the Savage-Dickey density ratio The concept of reduced models Armani, Calvin Klein and Versace design houses did not refuse this year to offer very brave and reduced models of the “Thong” and “Tango”. The designers consider that a man with the body of Apollo should not obscure the wonderful parts of his body. and their evidence This means that we only have to invert the full model to score all reduced models; c.f., the Savage-Dickey density ratio

Simulating the response of a four-node network 50 100 150 200 250 -1.5 -1 -0.5 0.5 1 1.5 Signal and noise time -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 Hidden states Endogenous fluctuations Network or graph generating data

And recovering (discovering) the true architecture 5 10 15 -0.6 -0.4 -0.2 0.2 0.4 True and MAP connections 20 30 40 50 60 -600 -500 -400 -300 -200 -100 100 Log-evidence model log-probability 1 2 3 4 6 graph size 0.6 0.8 Model posterior probability

Dynamic causal modelling for fMRI Modelling endogenous fluctuations Network discovery An empirical example

An application to real data 200 400 600 800 1000 1200 -5 5 fef: responses 200 400 600 800 1000 1200 -5 5 ag: responses 200 400 600 800 1000 1200 -4 -2 2 ppc: responses 200 400 600 800 1000 1200 -5 5 pfc: responses 200 400 600 800 1000 1200 -2 2 4 sts: responses 200 400 600 800 1000 1200 -4 -2 2 vis: responses With (visual motion) evoked activity time {seconds}

Estimates of hidden neuronal and hemodynamic states 50 100 150 200 250 -2 -1.5 -1 -0.5 0.5 1 1.5 Prediction and error time (bins) -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 Hidden states MAP neuronal activity time (sec) 0.85 0.9 0.95 1.05 1.1 1.15 1.2 Hemodynamic states visually evoked responses showing attentional modulation

And the results of searching model space 0.5 1 1.5 2 2.5 3 3.5 x 10 4 -400 -300 -200 -100 100 Log-posterior model log-probability 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 Model posterior probability 5 10 15 20 25 30 35 -0.6 -0.4 -0.2 MAP connections (full) MAP connections (sparse)

Anatomical space Exploiting estimates of directed coupling in terms of cortical hierarchies 0.00 0.00 -0.57 -0.28 -0.17 -0.31 0.00 0.00 -0.34 0.00 -0.37 -0.42 0.57 0.34 0.00 -0.45 -0.43 -0.51 0.28 0.00 0.45 0.00 0.00 -0.25 0.17 0.37 0.43 0.00 0.00 -0.28 0.31 0.42 0.51 0.25 0.28 0.00 'vis' 'sts' 'pfc' 'ppc' 'ag' 'fef' Functional (embedding) space

Thank you And thanks to CC Chen Jean Daunizeau Olivier David Marta Garrido Lee Harrison Stefan Kiebel Baojuan Li Andre Marreiros Rosalyn Moran Will Penny Klaas Stephan And many others