Dynamic Causal Modelling for EEG/MEG: principles J. Daunizeau

Transcription

1 Dynamic Causal Modelling for EEG/MEG: principles J. Daunizeau Motivation, Brain and Behaviour group, ICM, Paris, France

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 Introduction structural, functional and effective connectivity structural connectivity functional connectivity effective connectivity 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 O. Sporns 2007, Scholarpedia

5 Introduction from functional segregation to functional integration localizing brain activity: functional segregation effective connectivity analysis: functional integration A B? A B u 1 u 2 u 1 u 2 u 1 u 1 X u 2 «Where, in the brain, did my experimental manipulation have an effect?» «How did my experimental manipulation propagate through the network?»

6 Introduction DCM: evolution and observation mappings Hemodynamic observation model: temporal convolution Electromagnetic observation model: spatial convolution fmri neural states dynamics x f ( x, u, ) EEG/MEG simple neuronal model realistic observation model realistic neuronal model simple observation model inputs

7 Introduction DCM: a parametric statistical approach DCM: model structure y g x, x f x, u, likelihood p y,, m u DCM: Bayesian inference parameter estimate: model evidence: ˆ E y, m priors on parameters,, p y m p y m p m p m dd

8 Introduction DCM for EEG-MEG: auditory mismatch negativity sequence of auditory stimuli S S S D S S S S D S standard condition (S) S-D: reorganisation of the connectivity structure rifg deviant condition (D) la1 lstg rifg ra1 rstg lstg rstg la1 ra1 t ~ 200 ms Daunizeau, Kiebel et al., Neuroimage, 2009

9 Introduction DCM for fmri: audio-visual associative learning auditory cue or visual outcome or P(outcome cue) response time (ms) Put 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 theory motivation u x y time 13 3 u u 13 3 u 32 t 3 t t t x t 0 t 0 t t?

12 Dynamical systems theory exponentials

13 Dynamical systems theory initial values and fixed points

14 Dynamical systems theory time constants

15 Dynamical systems theory matrix exponential

16 Dynamical systems theory eigendecomposition of the Jacobian

17 Dynamical systems theory dynamical modes

18 Dynamical systems theory spirals

19 Dynamical systems theory spirals

20 Dynamical systems theory spiral state-space

21 Dynamical systems theory embedding

22 Dynamical systems theory kernels and convolution

23 Dynamical systems theory summary 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: o D>0: fixed points o D>1: spirals o D>1: limit cycles (e.g., action potentials) o 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 dynamics DCM for M/EEG: systems of neural populations macro-scale meso-scale micro-scale Golgi Nissl EI external granular layer external pyramidal layer EP II internal granular layer internal pyramidal layer mean-field firing rate synaptic dynamics

26 Neural ensembles dynamics DCM for M/EEG: from micro- to meso-scale mean firing rate (Hz) xj t : post-synaptic potential of j th neuron within its ensemble 1 N 1 j' j N H x j ' t H xt p xtdx S mean-field firing rate ensemble density p(x) S(x) S(x) H(x) membrane depolarization (mv) mean membrane depolarization (mv)

27 Neural ensembles dynamics DCM for M/EEG: synaptic dynamics membrane depolarization (mv) post-synaptic potential EPSP IPSP i/ e S( ) 2 i/ e 2 i/ e 1 1 K i 1 K e time (ms)

28 Neural ensembles dynamics DCM for M/EEG: intrinsic connections within the cortical column Golgi Nissl 7 8 S( ) e 0 e 8 e 7 external granular layer external pyramidal layer internal granular layer internal pyramidal layer inhibitory interneurons spiny stellate cells pyramidal cells S( ) e 0 e 4 e 1 intrinsic 1 2 connections S( ) e 1 e 5 e 2 x 3 6 S( ) i 7 i 6 i 3 3

29 Neural ensembles dynamics DCM for M/EEG: from meso- to macro-scale lateral (homogeneous) density of connexions i t r, t ( i) ( ) r 1 r 2 local wave propagation equation (neural field): t ( i) ( ) 2 c, t c, t t 2 S i r r ( i) ( i') ii' i' 0 th -order approximation: standing wave

30 Neural ensembles dynamics DCM for M/EEG: extrinsic connections between brain regions 7 8 (( IS ) ( )) e B L 3 0 e 8 e 7 extrinsic lateral connections S( ) L 0 inhibitory interneurons spiny stellate cells extrinsic forward connections S( ) F (( I) S( ) u) e F L 1 0 u e 4 e pyramidal cells (( ) S( ) S( )) e B L e 5 e 2 x 3 6 S( ) i 4 7 i 6 i 3 extrinsic backward connections S( ) B 0

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

32 Bayesian inference forward and inverse problems forward problem p y, m likelihood posterior distribution p y, m inverse problem

33 Bayesian inference the electromagnetic forward problem y ( i) ( i) ( ij) t L w t t i 0 j j

34 Bayesian paradigm deriving the likelihood function - Model of data with unknown parameters: y f e.g., GLM: f X - But data is noisy: y f - Assume noise/residuals is small : f exp p P Distribution of data, given fixed parameters: p y y f exp 2

35 Bayesian paradigm likelihood, priors and the model evidence Likelihood: Prior: generative model m Bayes rule:

36 y=f(x) y = f(x) Bayesian inference model comparison Principle of parsimony : «plurality should not be assumed without necessity» Model evidence:, p y m p y m p m d Occam s razor : x model evidence p(y m) space of all data sets

37 Bayesian inference the variational Bayesian approach KL ln p y m ln p, y m S q D q ; p y, m free energy : functional of q q mean-field: approximate marginal posterior distributions: q, q p p, y, m or 2, q 1 or 2 y m

38 Bayesian inference DCM: key model parameters u u 13 3 u 21, 32, 13 state-state coupling u 3 input-state coupling u 13 input-dependent modulatory effect

39 differences in log- model evidences Bayesian inference model comparison for group studies ln 1 ln p y m2 p y m m 1 m 2 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 Conclusion back to the auditory mismatch negativity sequence of auditory stimuli S S S D S S S S D S standard condition (S) S-D: reorganisation of the connectivity structure rifg deviant condition (D) la1 lstg rifg ra1 rstg lstg rstg la1 ra1 t ~ 200 ms

42 Conclusion DCM for EEG/MEG: variants input depolarization st and 2d order moments second-order mean-field DCM 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) DCM for induced responses DCM for phase coupling

43 Conclusion planning 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)