Effective connectivity & Dynamic Causal Modelling (DCM)

1. Effective connectivity& Dynamic Causal Modelling (DCM) Meike J. Grol • Leiden Institute for Brain and Cognition (LIBC), Leiden, The Netherlands • Leiden University - Institute for Psychological Research (LU-IPR), Leiden, The Netherlands • Department of Radiology, Leiden University Medical Center • F. C. Donders Centre for Cognitive NeuroImaging, Nijmegen, The Netherlands Zürich SPM CourseFebruary 29, 2008 Thanks to Klaas Enno Stephan and Miranda van Turennout for slides

2. Functional Segregation Where are the regional responses to an experimental input?

3. ? ? Functional Integration How do regions influence each other?

4. System analyses in functional neuroimaging Functional specialisation Analyses of regionally specific effects: which areas constitute a neuronal system? Functional integration Analyses of inter-regional effects: what are the interactions between the elements of a given neuronal system? Functional connectivity = the temporal correlation between spatially remote neurophysiological events Effective connectivity = the influence that the elements of a neuronal system exert over another MECHANISM-FREE MECHANISTIC MODEL

5. Functional connectivity: methods • Seed-voxel correlation analyses • Eigenimage analysis • Principal Components Analysis (PCA) • Singular Value Decomposition (SVD) • Partial Least Squares (PLS) • Independent Component Analysis (ICA)

6. seed voxel Seed-voxel correlation analysis (SVCA) • hypothesis-driven choice of a ‘seed voxel’ → reference timeseries • correlation with timeseries of all other voxels in the brain

7. SVCA example Finger-tapping task: all voxels (p<0.005, uncorrected) that showed changed functional connectivity with the left ant. cerebellum in schizophrenic patients after medication with olanzapine Stephan et al., Psychol. Med. (2001) p<0.005, uncorrected

8. Functional Connectivity Analyses • Pros: • useful when we have no model of what caused the data (e.g. sleep, hallucinatons, etc.) • Cons: • usually suboptimal for situations where we have a priori knowledge and experimental control about the system of interest • resulting modes/patterns can be difficult to interpret • no mechanistic insight into the neural system of interest models of effective connectivity necessary

9. Models of effective connectivity • Structural Equation Modelling (SEM) • Psycho-physiological interactions (PPI) • Multivariate autoregressive models (MAR)& Granger causality techniques • Kalman filtering • Volterra series • Dynamic Causal Modelling (DCM)

10. Psycho-physiological interaction (PPI) • bilinear model of how the influence of area A on area B changes by the psychological context C: A x C  B • a PPI corresponds to differences in regression slopes for different contexts.

11. SPM{Z} V5 activity time V1 V5 V5 attention V5 activity no attention V1 activity PPI example: attentional modulation of V1→V5 Attention = V1 x Att. Friston et al. 1997, NeuroImage 6:218-229 Büchel & Friston 1997, Cereb. Cortex 7:768-778

12. limited causal interpretability in neural terms, more powerful models needed DCM! Pros & Cons of PPIs • Pros: • given a single source region, we can test for its context-dependent connectivity across the entire brain • Cons: • very simplistic model: only allows to model contributions from a single area • ignores time-series properties of data • operates at the level of BOLD time series

13. Dynamic Causal Modelling • Basic idea • Neural level in DCM • Haemodynamic level in DCM • Priors and Parameter estimation • Interpretation and inference of parameters • Practical steps of a DCM study

14. Contextual inputs Stimulus-free - u2(t) {e.g. cognitive set/time} BA39 Perturbing inputs Stimuli-bound u1(t) {e.g. visual words} STG V4 y y y y y V1 BA37 The aim of DCM Functional integration and the modulation of specific pathways

15. z λ y DCM for fMRI: the basic idea • A cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI). • The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ). The aim of DCM is to estimate parameters at the neuronal level(computed separately for each area)such that the modelled BOLD signals are maximally similar to the experimentally measured BOLD signals.

16. Dynamic Causal Modelling • Basic idea • Neural level in DCM • Haemodynamic level in DCM • Priors and Parameter estimation • Interpretation and inference of parameters • Practical steps of a DCM study

17. What is a system? Input u(t) System = a set of elements which interact in a spatially and temporally specific fashion connectivity parameters  systemstate z(t) State changes of a system are dependent on: • the current state • external inputs • its connectivity • time constants & delays

18. RVF LVF u2 u1 LG left FG right LG right FG left Example: linear dynamic system LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF)visual field. z4 z3 z1 z2

19. Modulation of connectivity Intrinsic connectivity Direct inputs LG left FG right LG right FG left Extension: bilinear dynamic system z4 z3 z1 z2 CONTEXT RVF LVF u2 u3 u1

21. Dynamic Causal Modelling • Basic idea • Neural level in DCM • Hemodynamic level in DCM • Priors and Parameter estimation • Interpretation and inference of parameters • Practical steps of a DCM study

22. The hemodynamic “Balloon” model • 5 hemodynamic parameters: important for model fitting, but of no interest for statistical inference • Empirically determineda priori distributions. • Computed separately for each area (like the neural parameters) region-specific HRFs! Friston et al. 2000,NeuroImage

23. RVF LVF LG left LG right FG right FG left Example: modelled BOLD signal black: observed BOLD signal red: modelled BOLD signal

24. Dynamic Causal Modelling • Basic idea • Neural level in DCM • Haemodynamic level in DCM • Priors & Parameter estimation • Interpretation and inference of parameters • Practical steps of a DCM study

25. Bayesian estimation • Models of • Responses in a single region • Neuronal interactions • Constraints on • Connections • Biophysical parameters Bayesian estimation posterior  likelihood ∙ prior

26. Priors in DCM • needed for Bayesian estimation, embody constraints on parameter estimation • express our prior knowledge or “belief” about parameters of the model • hemodynamic parameters:empirical priors • Coupling parameters of self-connections: principled prior • coupling parameters other connections:shrinkage priors Bayes Theorem posterior  likelihood ∙ prior

27. Shrinkage Priors Small & variable effect Large & variable effect Small but clear effect Large & clear effect

28. Dynamic Causal Modelling • Basic idea • Neural level in DCM • Haemodynamic level in DCM • Priors & Parameter estimation • Interpretation and inference about parameters • Practical steps of a DCM study

29. DCM parameters = rate constants Integration of a first-order linear differential equation gives anexponential function: The coupling parameter a thus describes the speed ofthe exponential change in z(t) Coupling parameter a is inverselyproportional to the half life  of z(t):

30. u 1 u 2 Z 1 Z 2 Example: context-dependent decay u1 stimuli u1 context u2 u2 - + - Z1 z1 + z2 + Z2 - - Penny, Stephan, Mechelli, Friston NeuroImage (2004)

31. Inference about DCM parameters:Bayesian single-subject analysis • Bayesian parameter estimation in DCM: Gaussian assumptions about the posterior distributions of the parameters • Use of the cumulative normal distribution to test the probability by which a certain parameter (or contrast of parameters cT ηθ|y) is above a chosen threshold γ: • γ can be chosen as zero ("does the effect exist?") or as a function of the expected half life τ of the neural process: γ= ln 2 / τ  ηθ|y

32. Inference about DCM parameters:group analyses Bayesian fixed-effects group analysis: Because the likelihood distributions from different subjects are independent, one can combine their posterior densities by using the posterior of one subject as the prior for the next. “Random effects” analyses: 2nd level analyses can be applied to DCM parameters. Separate fitting of identical models for each subject Selection of bilinear parameters of interest RmANOVA:e.g. in case of multiple sessions per subject one-sample t-test:parameter > 0 ? paired t-test:parameter 1 > parameter 2 ?

33. Dynamic Causal Modelling • Basic idea • Neural level in DCM • Haemodynamic level in DCM • Priors & Parameter estimation • Interpretation and inference about parameters • Practical steps of a DCM study

34. Planning a DCM-compatible study • Suitable experimental design: • preferably multi-factorial (e.g. 2 x 2) • e.g. one factor that varies the driving (sensory) input • and one factor that varies the contextual input • Hypothesis and model: • Define specific a priori hypothesis • Which parameters are relevant to test this hypothesis? • Ensure that intended model is suitable to test this hypothesis → simulations • Define criteria for inference • What are the alternative models to test?

35. Task factor Stim1/ Task A Stim2/Task A Task B Task A TA/S1 TB/S1 Stim 1 GLM X1 X2 Stimulus factor Stim 2 TB/S2 TA/S2 Stim 1/ Task B Stim 2/ Task B Stim1 DCM X1 X2 Stim2 Task A Task B Multifactorial design: explaining interactions with DCM Let’s assume that an SPM analysis shows a main effect of stimulus in X1 and a stimulus  task interaction in X2. How do we model this using DCM?

36. Simulated data A1 +++ Stim1 + Stim 1Task B Stim 2Task B Stim 2Task A Stim 1Task A A1 A2 +++ Stim2 +++ + + Task A Task B A2