Abstract

1. 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).

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

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

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

5. 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

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

7. 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

8. Neural population activity 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 u2 0.6 0.4 A toy example x3 0.2 0 0 10 20 30 40 50 60 70 80 90 0.3 0.2 0.1 BOLD signal change (%) 0 0 10 20 30 40 50 60 70 80 90 x1 x2 u1 2 1 – – 0 0 10 20 30 40 50 60 70 80 90 3 2 1 0 -1 0 10 20 30 40 50 60 70 80 90 2 1 0 0 10 20 30 40 50 60 70 80 90

9. PPC V5+ 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 Büchel et al 1999

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

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

12. Replacing exogenous inputs with endogenous fluctuations Hidden states Signal and noise 0.2 1.5 0.15 1 0.1 0.5 0.05 0 0 -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 Network or graph generating data Endogenous fluctuations Exogenous inputs 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 50 100 150 200 250 time

13. Hidden states True and MAP connections 0.2 0.5 0.15 0.1 0.05 0 0 -0.05 -0.1 -0.15 -0.5 -0.2 1 2 3 4 5 6 7 8 9 50 100 150 200 250 time (bins) Extrinsic coupling parameter True neuronal activity MAP estimate 0.06 0.06 0.04 0.04 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 time (sec.) time (sec.) Estimating hidden states with generalised (Bayesian) filtering

14. 5 10 4 10 Number of models 3 10 2 10 1 10 0 10 2 3 4 5 6 number of nodes Dynamic causal modelling for fMRI Modelling endogenous fluctuations Network discovery An empirical example and the problem of searching large model spaces

15. 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

16. Hidden states Signal and noise 0.2 1.5 0.15 1 0.1 0.5 0.05 0 0 -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 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 50 100 150 200 250 time Simulating the response of a four-node network

17. Log-evidence True and MAP connections 100 0.4 0 0.2 -100 0 -200 log-probability -300 -0.2 -400 -0.4 -500 -0.6 -600 0 5 10 15 0 10 20 30 40 50 60 model Log-evidence Model posterior 100 1 0 0.8 -100 0.6 -200 log-probability probability -300 0.4 -400 0.2 -500 -600 0 0 1 2 3 4 5 6 0 10 20 30 40 50 60 graph size model And recovering (discovering) the true architecture

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

19. vis: responses 2 0 -2 -4 0 200 400 600 800 1000 1200 ag: responses 5 0 -5 0 200 400 600 800 1000 1200 sts: responses 4 2 0 -2 0 200 400 600 800 1000 1200 ppc: responses 2 0 -2 -4 0 200 400 600 800 1000 1200 fef: responses 5 0 -5 0 200 400 600 800 1000 1200 pfc: responses 5 0 -5 0 200 400 600 800 1000 1200 An application to real data With (visual motion) evoked activity time {seconds}

20. Hidden states Prediction and error 0.2 1.5 0.15 1 0.1 0.5 0.05 0 0 -0.5 -0.05 -1 -0.1 -1.5 -0.15 -2 -0.2 50 100 150 200 250 50 100 150 200 250 time (bins) time (bins) MAP neuronal activity Hemodynamic states 1.2 0.1 1.15 1.1 0.05 1.05 0 1 0.95 -0.05 0.9 -0.1 0.85 50 100 150 200 250 50 100 150 200 250 time (sec) time (sec) Estimates of hidden neuronal and hemodynamic states visually evoked responses showing attentional modulation

21. Log-posterior Model posterior 100 0.9 0.8 0 0.7 0.6 -100 0.5 log-probability probability 0.4 -200 0.3 0.2 -300 0.1 -400 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4 model model x 10 x 10 MAP connections (full) MAP connections (sparse) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 And the results of searching model space

22. 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

23. 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