Dynamic Causal Modelling (DCM) for fMRI

1. Dynamic Causal Modelling (DCM) for fMRI Andre Marreiros Wellcome Trust Centre for Neuroimaging University College London

2. Thanks to... Stefan Kiebel Lee Harrison Klaas Stephan Karl Friston

3. Overview Dynamic Causal Modelling of fMRI Definitions & motivation • The neuronal model(bilinear dynamics) • The Haemodynamic model • Estimation: Bayesian framework • DCM latest Extensions

4. Principles of organisation Functional specialization Functional integration

5. Neurodynamics: 2 nodes with input u1 u2 z1 z2 activity in is coupled to via coefficient

6. Neurodynamics: positive modulation u1 u2 z1 z2 modulatory input u2 activity through the coupling

7. Neurodynamics: reciprocal connections u1 u2 z1 z2 reciprocal connection disclosed by u2

8. Haemodynamics: reciprocal connections a11 Simulated response Bold Response a12 Bold Response a22 green: neuronal activity red: bold response

9. Haemodynamics: reciprocal connections a11 Bold with Noise added a12 Bold with Noise added a22 green: neuronal activity red: bold response

10. LG left LG right FG right FG left Example: modelled BOLD signal Underlying model(modulatory inputs not shown) left LG right LG RVF LVF LG = lingual gyrus Visual input in the FG = fusiform gyrus - left (LVF) - right (RVF) visual field. blue: observed BOLD signal red: modelled BOLD signal (DCM)

11. Use differential equations to describe mechanistic model of a system • System dynamics = change of state vector in time • Causal effects in the system: • interactions between elements • external inputs u • System parameters  :specify exact form of system overall system staterepresented by state variables change ofstate vectorin time

12. 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 RVF LVF u2 u1 systemstate input parameters state changes effective connectivity externalinputs

13. LG left FG right LG right FG left Extension: bilinear dynamic system z4 z3 z1 z2 CONTEXT RVF LVF u2 u3 u1

14. Bilinear state equation in DCM/fMRI modulation of connectivity systemstate direct inputs state changes m externalinputs connectivity

15. z λ y Conceptual overview Neuronal state equation The bilinear model effective connectivity modulation of connectivity Input u(t) direct inputs c1 b23 integration neuronal states a12 activity z2(t) activity z3(t) activity z1(t) hemodynamic model y y y BOLD Friston et al. 2003,NeuroImage

16. q = k g t a r h { , , , , } 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

17. Diagram Dynamic Causal Modelling of fMRI Network dynamics Haemodynamic response Priors Model comparison State space Model Model inversion using Expectation-maximization Posterior distribution of parameters fMRI data y

18. Estimation: Bayesian framework • Models of • Hemodynamics in a single region • Neuronal interactions • Constraints on • Connections • Hemodynamic parameters prior likelihood term posterior Bayesian estimation

19. stimulus function u Overview:parameter estimation neuronal state equation • Specify model (neuronal and hemodynamic level) • Make it an observation model by adding measurement errore and confounds X (e.g. drift). • Bayesian parameter estimation using Bayesian version of an expectation-maximization algorithm. • Result:(Normal) posterior parameter distributions, given by mean ηθ|y and Covariance Cθ|y. parameters hidden states state equation ηθ|y observation model modelled BOLD response

20. Haemodynamics: 2 nodes with input Dashed Line: Real BOLD response a11 a22 Activity in z1 is coupled to z2 via coefficient a21

21. Inference about DCM parameters: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:  ηθ|y

22. Pitt & Miyung (2002), TICS Model comparison and selection Given competing hypotheses, which model is the best?

23. SPC SPC V1 V1 V5 V5 Comparison of three simple models Model 1:attentional modulationof V1→V5 Model 2:attentional modulationof SPC→V5 Model 3:attentional modulationof V1→V5 and SPC→V5 Attention Attention Photic Photic Photic SPC 0.55 0.03 0.85 0.86 0.85 0.70 0.75 0.70 0.84 1.36 1.42 1.36 0.89 0.85 V1 -0.02 -0.02 -0.02 0.56 0.57 0.57 V5 Motion Motion Motion 0.23 0.23 Attention Attention Bayesian model selection: Model 1 better than model 2, model 1 and model 3 equal → Decision for model 1: in this experiment, attention primarily modulates V1→V5

24. Extension I: Slice timing model • potential timing problem in DCM: temporal shift between regional time series because of multi-slice acquisition 2 slice acquisition 1 visualinput • Solution: • Modelling of (known) slice timing of each area. • Slice timing extension now allows for any slice timing differences. • Long TRs (> 2 sec) no longer a limitation. • (Kiebel et al., 2007)

25. Single-state DCM Two-state DCM input Extrinsic (between-region) coupling Intrinsic (within-region) coupling Extension II: Two-state model

26. Extension III: Nonlinear DCM for fMRI Here DCM can model activity-dependent changes in connectivity; how connections are enabled or gated by activity in one or more areas. The D matrices encode which of the n neural units gate which connections in the system. attention 0.19 (100%) Can V5 activity during attention to motion be explained by allowing activity in SPC to modulate the V1-to-V5 connection? The posterior density of indicates that this gating existed with 97.4% confidence. SPC 0.03 (100%) 0.01 (97.4%) 1.65 (100%) V1 V5 0.04 (100%) motion

27. Conclusions Dynamic Causal Modelling (DCM) of fMRI is mechanistic model that is informed by anatomical and physiological principles. DCM uses a deterministic differential equation to model neuro-dynamics (represented by matrices A,B and C) DCM uses a Bayesian framework to estimate these DCM combines state-equations for dynamics with observation model (fMRI: BOLD response, M/EEG: lead field). DCM is not model or modality specific (Models will change and the method extended to other modalities e.g. ERPs)