1. Models of effective connectivity & Dynamic Causal Modelling (DCM) Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London Methods & Models for fMRI data analysis 03 December 2008

2. Overview Brain connectivity: types & definitions –anatomical connectivity –functional connectivity –effective connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) –DCM for fMRI: Neural and hemodynamic levels –Parameter estimation & inference Applications of DCM to fMRI data –Design of experiments and models –Some empirical examples and simulations

3. Connectivity A central property of any system Communication systems Social networks (internet) (Canberra, Australia) FIgs. by Stephen Eick and A. Klovdahl; see http://www.nd.edu/~networks/gallery.htm

4. Structural, functional & effective connectivity anatomical/structural connectivity = presence of axonal connections functional connectivity =statistical dependencies between regional time series effective connectivity =causal (directed) influences between neurons or neuronal populations Sporns 2007, Scholarpedia

5. Anatomical connectivity neuronal communication via synaptic contacts visualisation by tracing techniques long-range axons  “association fibres”

6. Diffusion-weighted imaging Parker & Alexander, 2005, Phil. Trans. B

7. Parker, Stephan et al. 2002, NeuroImage Diffusion-weighted imaging of the cortico- spinal tract

8. However, knowing anatomical connectivity is not enough...

9. 1. Connections are recruited in a context-dependent fashion Local functions are context-sensitive: They depend on network activity.

10. 2. Connections show plasticity critical for learning can occur both rapidly and slowly NMDA receptors play a critical role NMDA receptors are regulated by modulatory neurotransmitters like dopamine, serotonine, acetylcholine synaptic plasticity =change in the structure and transmission properties of a chemical synapse Gu 2002, Neuroscience NMDA receptor

11. Different approaches to analysing functional connectivity Seed voxel correlation analysis Eigen-decomposition (PCA, SVD) Independent component analysis (ICA) any other technique describing statistical dependencies amongst regional time series

12. Seed-voxel correlation analyses Very simple idea: –hypothesis-driven choice of a seed voxel → extract reference time series –voxel-wise correlation with time series from all other voxels in the brain seed voxel

13. Drug-induced changes in functional connectivity Finger-tapping task in first-episode schizophrenic patients: voxels that showed changes in functional connectivity (p<0.005) with the left ant. cerebellum after medication with olanzapine Stephan et al. 2001, Psychol. Med.

14. Does functional connectivity not simply correspond to co-activation in SPMs? No, it does not - see the fictitious example on the right: Here both areas A 1 and A 2 are correlated identically to task T, yet they have zero correlation among themselves: r(A 1,T) = r(A 2,T) = 0.71 but r(A 1,A 2 ) = 0 ! task T regional response A 2 regional response A 1 Stephan 2004, J. Anat.

15. Pros & Cons of functional connectivity analyses Pros: –useful when we have no experimental control over the system of interest and no model of what caused the data (e.g. sleep, hallucinatons, etc.) Cons: –interpretation of resulting patterns is difficult / arbitrary –no mechanistic insight into the neural system of interest –usually suboptimal for situations where we have a priori knowledge and experimental control about the system of interest models of effective connectivity necessary

16. For understanding brain function mechanistically, we need models of effective connectivity, i.e. models of causal interactions among neuronal populations.

17. Some models for computing effective connectivity from fMRI data Structural Equation Modelling (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000 regression models (e.g. psycho-physiological interactions, PPIs) Friston et al. 1997 Volterra kernels Friston & Büchel 2000 Time series models (e.g. MAR, Granger causality) Harrison et al. 2003, Goebel et al. 2003 Dynamic Causal Modelling (DCM) bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008

18. Overview Brain connectivity: types & definitions –anatomical connectivity –functional connectivity –effective connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) –DCM for fMRI: Neural and hemodynamic levels –Parameter estimation & inference Applications of DCM to fMRI data –Design of experiments and models –Some empirical examples and simulations

19. Psycho-physiological interaction (PPI) We can replace one main effect in the GLM by the time series of an area that shows this main effect. E.g. let's replace the main effect of stimulus type by the time series of area V1: Task factor Task A Task B Stim 1 Stim 2 Stimulus factor T A /S 1 T B /S 1 T A /S 2 T B /S 2 GLM of a 2x2 factorial design: main effect of task main effect of stim. type interaction main effect of task V1 time series  main effect of stim. type psycho- physiological interaction Friston et al. 1997, NeuroImage

20. V1 attention no attention V1 activity V5 activity SPM{Z} time V5 activity Friston et al. 1997, NeuroImage Büchel & Friston 1997, Cereb. Cortex V1 x Att. = V5 V5 Attention Attentional modulation of V1→V5

21. PPI: interpretation Two possible interpretations of the PPI term: V1 Modulation of V1  V5 by attention Modulation of the impact of attention on V5 by V1 V1V5 V1 V5 attention V1 attention

22. Two PPI variants "Classical" PPI: –Friston et al. 1997, NeuroImage –depends on factorial design –in the GLM, physiological time series replaces one experimental factor –physio-physiological interactions: two experimental factors are replaced by physiological time series Alternative PPI: –Macaluso et al. 2000, Science –interaction term is added to an existing GLM –can be used with any design

23. Is the red letter left or right from the midline of the word? group analysis (random effects), n=16, p<0.05 corrected analysis with SPM2 group analysis (random effects), n=16, p<0.05 corrected analysis with SPM2 Task-driven lateralisation letter decisions > spatial decisions time Does the word contain the letter A or not? spatial decisions > letter decisions Stephan et al. 2003, Science

24. Bilateral ACC activation in both tasks – but asymmetric connectivity ! IPS IFG Left ACC  left inf. frontal gyrus (IFG): increase during letter decisions. Right ACC  right IPS: increase during spatial decisions. left ACC (-6, 16, 42) right ACC (8, 16, 48) spatial vs letter decisions letter vs spatial decisions group analysis random effects (n=15) p<0.05, corrected (SVC) Stephan et al. 2003, Science

25. PPI single-subject example b VS = -0.16 b L =0.63 Signal in left ACC Signal in left IFG b L = -0.19 Signal in right ant. IPS Signal in right ACC b VS =0.50 Left ACC signal plotted against left IFG spatial decisions letter decisions spatial decisions Right ACC signal plotted against right IPS Stephan et al. 2003, Science

26. Pros & Cons of PPIs Pros: –given a single source region, we can test for its context-dependent connectivity across the entire brain –easy to implement Cons: –very simplistic model: only allows to model contributions from a single area –ignores time-series properties of data –application to event-related data relies deconvolution procedure (Gitelman et al. 2003, NeuroImage) –operates at the level of BOLD time series sometimes very useful, but limited causal interpretability; in most cases, we need more powerful models

27. Overview Brain connectivity: types & definitions –anatomical connectivity –functional connectivity –effective connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) –DCM for fMRI: Neural and hemodynamic levels –Parameter estimation & inference Applications of DCM to fMRI data –Design of experiments and models –Some empirical examples and simulations

28. LG left LG right RVFLVF FG right FG left LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. x1x1 x2x2 x4x4 x3x3 u2u2 u1u1 Example: a linear system of dynamics in visual cortex

29. LG left LG right RVFLVF FG right FG left LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. x1x1 x2x2 x4x4 x3x3 u2u2 u1u1 state changes effective connectivity external inputs system state input parameters

30. Extension: bilinear dynamic system LG left LG right RVFLVF FG right FG left x1x1 x2x2 x4x4 x3x3 u2u2 u1u1 CONTEXT u3u3

31. Neural state equation: Electromagnetic forward model: neural activity  EEG MEG LFP Dynamic Causal Modelling (DCM) simple neuronal model complicated forward model complicated neuronal model simple forward model fMRI EEG/MEG inputs Hemodynamic forward model: neural activity  BOLD Stephan & Friston 2007, Handbook of Brain Connectivity

32. Basic idea of DCM for fMRI (Friston et al. 2003, NeuroImage) Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI). The modelled neuronal dynamics (x) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model ( λ ). λ x y The aim of DCM is to estimate parameters at the neuronal level such that the modelled BOLD signals are maximally similar to the experimentally measured BOLD signals.

33. intrinsic connectivity direct inputs modulation of connectivity Neural state equation hemodynamic model λ x y integration BOLD yy y activity x 1 (t) activity x 2 (t) activity x 3 (t) neuronal states t driving input u 1 (t) modulatory input u 2 (t) t Stephan & Friston (2007), Handbook of Brain Connectivity   

34. Bilinear DCM Bilinear state equation: driving input modulation Two-dimensional Taylor series (around x 0 =0, u 0 =0):

35. - x2x2 stimuli u 1 context u 2 x1x1 + + - - - + u 1 Z 1 u 2 Z 2 Example: context-dependent decay u1u1 u2u2 x2x2 x1x1 Penny, Stephan, Mechelli, Friston NeuroImage (2004)

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

37. stimulus functions u t neural state equation hemodynamic state equations Balloon model BOLD signal change equation important for model fitting, but of no interest for statistical inference 6 hemodynamic parameters: Computed separately for each area (like the neural parameters)  region-specific HRFs! The hemodynamic model in DCM Friston et al. 2000, NeuroImage Stephan et al. 2007, NeuroImage

38. stimulus functions u t neural state equation hemodynamic state equations Balloon model BOLD signal change equation The hemodynamic model in DCM Stephan et al. 2007, NeuroImage

39. r ,B r ,A r ,C A B C hh ε How interdependent are our neural and hemodynamic parameter estimates? Stephan et al. 2007, NeuroImage

40. Bayesian statistics posterior  likelihood ∙ prior Bayes theorem allows one to formally incorporate prior knowledge into computing statistical probabilities. In DCM: empirical, principled & shrinkage priors. The “posterior” probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision. new data prior knowledge

41. Shrinkage priors Small & variable effect Large & variable effect Small but clear effect Large & clear effect

42. stimulus function u modelled BOLD response observation model hidden states state equation parameters Combining the neural and hemodynamic states gives the complete forward model. An observation model includes measurement error e and confounds X (e.g. drift). Bayesian parameter estimation by means of a Levenberg-Marquardt gradient ascent, embedded into an EM algorithm. Result: Gaussian a posteriori parameter distributions, characterised by mean η θ|y and covariance C θ|y. Overview: parameter estimation η θ|y neural state equation

43. Inference about DCM parameters: Bayesian single-subject analysis Gaussian assumptions about the posterior distributions of the parameters Use of the cumulative normal distribution to test the probability that a certain parameter (or contrast of parameters c T η θ|y ) is above a chosen threshold γ: By default, γ is chosen as zero ("does the effect exist?").  η θ|y

44. Bayesian single subject inference LG left LG right RVF stim. LVF stim. FG right FG left LD|RVF LD|LVF LD 0.34  0.14 -0.08  0.16 0.13  0.19 0.01  0.17 0.44  0.14 0.29  0.14 Contrast: Modulation LG right  LG links by LD|LVF vs. modulation LG left  LG right by LD|RVF p(c T  >0|y) = 98.7% Stephan et al. 2005, Ann. N.Y. Acad. Sci.

45. Inference about DCM parameters: 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: Under Gaussian assumptions this is easy to compute: group posterior covariance individual posterior covariances group posterior mean individual posterior covariances and means

46. Inference about DCM parameters: group analysis (classical) In analogy to “random effects” analyses in SPM, 2 nd level analyses can be applied to DCM parameters: Separate fitting of identical models for each subject Selection of bilinear parameters of interest one-sample t-test: parameter > 0 ? paired t-test: parameter 1 > parameter 2 ? rmANOVA: e.g. in case of multiple sessions per subject

47. Overview Brain connectivity: types & definitions –anatomical connectivity –functional connectivity –effective connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) –DCM for fMRI: Neural and hemodynamic levels –Parameter estimation & inference Applications of DCM to fMRI data –Design of experiments and models –Some empirical examples and simulations

48. Any design that is good for a GLM of fMRI data. What type of design is good for DCM?

49. GLM vs. DCM DCM tries to model the same phenomena as a GLM, just in a different way: It is a model, based on connectivity and its modulation, for explaining experimentally controlled variance in local responses. No activation detected by a GLM → inclusion of this region in a DCM is useless! Stephan 2004, J. Anat.

50. Multifactorial design: explaining interactions with DCM Task factor Task A Task B Stim 1 Stim 2 Stimulus factor T A /S 1 T B /S 1 T A /S 2 T B /S 2 X1X1 X2X2 Stim2/ Task A Stim1/ Task A Stim 1/ Task B Stim 2/ Task B GLM X1X1 X2X2 Stim2 Stim1 Task ATask B DCM Let’s assume that an SPM analysis shows a main effect of stimulus in X 1 and a stimulus  task interaction in X 2. How do we model this using DCM?

51. Stim 1 Task A Stim 2 Task A Stim 1 Task B Stim 2 Task B Simulated data X1X1 X2X2 +++ X1X1 X2X2 Stimulus 2 Stimulus 1 Task ATask B ++++ + + –– Stephan et al. 2007, J. Biosci.

52. Stim 1 Task A Stim 2 Task A Stim 1 Task B Stim 2 Task B plus added noise (SNR=1) X1X1 X2X2

53. Recent fMRI studies with DCM DCM now an established tool for fMRI analysis >50 studies published, incl. high- profile journals combinations of DCM with computational models

54. Thank you