J. Daunizeau Motivation, Brain and Behaviour group, ICM, Paris, France

1. Dynamic Causal Modelling for EEG/MEG: principles J. Daunizeau Motivation, Brain and Behaviour group, ICM, Paris, France Wellcome Trust Centre for Neuroimaging, London, UK

2. Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion

3. Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion

4. Introductionstructural, functional and effective connectivity structural connectivity functional connectivity effective connectivity O. Sporns 2007, Scholarpedia • structural connectivity= presence of axonal connections • functional connectivity = statistical dependencies between regional time series • effective connectivity = causal (directed) influences between neuronal populations • ! connections are recruited in a context-dependent fashion

5. Introductionfrom functional segregation to functional integration localizing brain activity: functional segregation effective connectivity analysis: functional integration u1 u1 ? A B u1 u1 X u2 B A u2 « Where, in the brain, did my experimental manipulation have an effect? » « How did my experimental manipulation propagate through the network? » u2

6. IntroductionDCM: evolution and observation mappings Hemodynamic observation model:temporal convolution Electromagnetic observation model:spatial convolution neural states dynamics fMRI EEG/MEG • simple neuronal model • realistic observation model • realistic neuronal model • simple observation model inputs

7. IntroductionDCM: a parametric statistical approach • DCM: model structure 24 2 likelihood 4 1 3 u • DCM: Bayesian inference parameter estimate: priors on parameters model evidence:

8. standard condition (S) rIFG rIFG rA1 lA1 rSTG lSTG rSTG lSTG deviant condition (D) lA1 rA1 t ~ 200 ms IntroductionDCM for EEG-MEG: auditory mismatch negativity sequence of auditory stimuli … … S D S S S S S S D S S-D: reorganisation of the connectivity structure Daunizeau, Kiebel et al., Neuroimage, 2009

9. time (ms) 0 200 400 600 800 2000 IntroductionDCM for fMRI: audio-visual associative learning auditory cue visual outcome P(outcome|cue) or or Put response PMd PPA FFA FFA PPA Put PMd cue-dependent surprise cue-independent surprise Den Ouden, Daunizeau et al., J. Neurosci., 2010

10. Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion

11. Dynamical systems theorymotivation 1 2 1 2 3 1 2 3 1 2 time 3 3 u

12. Dynamical systems theoryexponentials

13. Dynamical systems theoryinitial values and fixed points

14. Dynamical systems theorytime constants

15. Dynamical systems theorymatrix exponential

16. Dynamical systems theoryeigendecomposition of the Jacobian

17. Dynamical systems theorydynamical modes

18. Dynamical systems theoryspirals

19. Dynamical systems theoryspirals

20. Dynamical systems theoryspiral state-space

21. Dynamical systems theoryembedding

22. Dynamical systems theorykernels and convolution

23. Dynamical systems theorysummary • Motivation: modelling reciprocal influences • Link between the integral (convolution) and differential (ODE) forms • System stability and dynamical modes can be derived from the system’s Jacobian: • D>0: fixed points • D>1: spirals • D>1: limit cycles (e.g., action potentials) • D>2: metastability (e.g., winnerless competition) limit cycle (Vand Der Pol) strange attractor (Lorenz)

24. Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion

25. Neural ensembles dynamicsDCM for M/EEG: systems of neural populations macro-scale meso-scale micro-scale Golgi Nissl external granular layer EI external pyramidal layer EP internal granular layer internal pyramidal layer II mean-field firing rate synaptic dynamics

26. Neural ensembles dynamics DCM for M/EEG: from micro- to meso-scale : post-synaptic potential of jth neuron within itsensemble mean-field firing rate ensemble density p(x) mean firing rate (Hz) S(x) H(x) S(x) membrane depolarization (mV) mean membrane depolarization (mV)

27. Neural ensembles dynamics DCM for M/EEG: synaptic dynamics post-synaptic potential EPSP membrane depolarization (mV) IPSP time (ms)

28. inhibitory interneurons spiny stellate cells pyramidal cells Neural ensembles dynamics DCM for M/EEG: intrinsic connections within the cortical column Golgi Nissl external granular layer external pyramidal layer internal granular layer intrinsic connections internal pyramidal layer

29. Neural ensembles dynamics DCM for M/EEG: from meso- to macro-scale lateral (homogeneous) density of connexions local wave propagation equation (neural field): 0th-order approximation: standing wave

30. inhibitory interneurons spiny stellate cells pyramidal cells Neural ensembles dynamics DCM for M/EEG: extrinsic connections between brain regions extrinsic lateral connections extrinsic forward connections extrinsic backward connections

31. Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion

32. Bayesian inferenceforward and inverse problems forward problem likelihood posterior distribution inverse problem

33. Bayesian inferencethe electromagnetic forward problem

34. Bayesian paradigmderiving the likelihood function - Model of data with unknown parameters: e.g., GLM: - But data is noisy: - Assume noise/residuals is ‘small’: f → Distribution of data, given fixed parameters:

35. Likelihood: Prior: Bayes rule: Bayesian paradigmlikelihood, priors and the model evidence generative modelm

36. Model evidence: y = f(x) x model evidence p(y|m) y=f(x) space of all data sets Bayesian inferencemodel comparison Principle of parsimony : « plurality should not be assumed without necessity » “Occam’s razor”:

37. Bayesian inferencethe variational Bayesian approach free energy : functional of q mean-field: approximate marginal posterior distributions:

38. Bayesian inferenceDCM: key model parameters 1 2 3 u state-state coupling input-state coupling input-dependent modulatory effect

39. Bayesian inferencemodel comparison for group studies m1 differences in log- model evidences m2 subjects fixed effect assume all subjects correspond to the same model random effect assume different subjects might correspond to different models

40. Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion

41. standard condition (S) rIFG rIFG rA1 lA1 rSTG lSTG rSTG lSTG deviant condition (D) lA1 rA1 t ~ 200 ms Conclusionback to the auditory mismatch negativity sequence of auditory stimuli … … S D S S S S S S D S S-D: reorganisation of the connectivity structure

42. ConclusionDCM for EEG/MEG: variants 250 0 input depolarization 1st and 2d order moments 200 -20 150 -40  second-order mean-field DCM 100 -60 50 -80 0 -100 0 100 200 300 0 100 200 300 time (ms) time (ms) time (ms) auto-spectral density LA auto-spectral density CA1 cross-spectral density CA1-LA  DCM for steady-state responses frequency (Hz) frequency (Hz) frequency (Hz) 0 -20 -40  DCM for induced responses -60 -80  DCM for phase coupling -100 0 100 200 300

43. Conclusionplanning a compatible DCM study • Suitable experimental design: • any design that is suitable for a GLM • preferably multi-factorial (e.g. 2 x 2) • e.g. one factor that varies the driving (sensory) input • and one factor that varies the modulatory input • Hypothesis and model: • define specific a priori hypothesis • which models are relevant to test this hypothesis? • check existence of effect on data features of interest • there exists formal methods for optimizing the experimental design • for the ensuing bayesian model comparison • [Daunizeau et al., PLoS Comp. Biol., 2011]

44. Many thanks to: Karl J. Friston (UCL, London, UK) Will D. Penny (UCL, London, UK) Klaas E. Stephan (UZH, Zurich, Switzerland) Stefan Kiebel (MPI, Leipzig, Germany)