Dynamic Causal Modelling for evoked responses J. Daunizeau

Transcription

1 Dynamic Causal Modelling for evoked responses J. Daunizeau Institute for Empirical Research in Economics, Zurich, Switzerland Brain and Spine Institute, Paris, France

2 Overview 1 DCM: introduction 2 Neural ensembles dynamics 3 Bayesian inference 4 Conclusion

3 Overview 1 DCM: introduction 2 Neural ensembles dynamics 3 Bayesian inference 4 Conclusion

4 DCM: 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

5 DCM: introduction connections are recruited in a context-dependent fashion meta-analysis on single-word reading (Turkeltaub, 2002)

6 Introduction DCM for evoked responses: 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

7 Overview 1 DCM: introduction 2 Neural ensembles dynamics 3 Bayesian inference 4 Conclusion

8 Neural ensembles dynamics 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

9 Neural ensembles dynamics from micro- to meso-scale: mean-field treatment mean firing rate (Hz) x j : post-synaptic potential of j th neuron within its ensemble 1 N 1 j' j N H x j ' H x p xdx p x dx S mean firing rate ensemble density p(x) S(x) H(x) membrane depolarization (mv) mean membrane depolarization (mv)

10 Neural ensembles dynamics synaptic dynamics membrane depolarization (mv) post-synaptic potential EPSP IPSP kernel PSP t S u t t i/ e Su 2 i/ e 2 i/ e 1 1 K i 1 K e time (ms)

11 Neural ensembles dynamics 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

12 Neural ensembles dynamics from meso- to macro-scale: neural fields local (homogeneous) density of connexions r 1 r 2 local wave propagation equation: t ( i) ( ) 2 c, t c, t t 2 S i r r ( i) ( i') ii' i'

13 Neural ensembles dynamics 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

14 Neural ensembles dynamics 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 main DCM evolution parameters: - action potential firing threshold + ensemble PSP spread - synaptic time constants + axonal propagation delays - effective coupling strengths + modulatory effects

15 Neural ensembles dynamics the observation mapping main DCM observation parameters: - sources location/orientation (ECD) or spatial profile (distributed responses) - relative contribution of cortical layers to measured signal

16 Neural ensembles dynamics a note on causality u x y time 13 3 u u 13 3 u 32 t 3 t t t t 0 t x f x, u,

17 Overview 1 DCM: introduction 2 Neural ensembles dynamics 3 Bayesian inference 4 Conclusion

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

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

20 Bayesian inference likelihood and priors likelihood p y, m u prior p m generative model m posterior p y, m, p y m p y m p m

21 Bayesian inference zooming in the VB algorithm measured data specify generative forward model (with prior distributions of parameters) Variational Bayesian (VB) algorithm iterative procedure: 1. compute model response using current set of parameters 2. compare model response with data 3. improve parameters, if possible 1. posterior distributions of parameters 2. model evidence (free energy)

22 Frequentist versus Bayesian inference testing point hypotheses define the null and the alternative hypothesis in terms of priors, e.g.: H : p H 0 0 H : p H N 0, if 0 0 otherwise p Y H 0 p Y H 1 apply decision rule, i.e.: if 1 P H0 y 1 P H y y then reject H0 Y space of all datasets Savage-Dickey ratios (nested models, i.i.d. priors): 0 p y H1 p y H p 0 y, H 1 0 H1 p

23 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

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

25 Overview 1 DCM: introduction 2 Neural ensembles dynamics 3 Bayesian inference 4 Conclusion

26 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

27 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

28 Many thanks to: Karl J. Friston (London, UK) Klaas E. Stephan (Zurich, Switzerland) Stefan J. Kiebel (Leipzig, Germany)

29 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 approximate (marginal) posterior distributions: q, q p p, y, m or 2, q 1 or 2 y m

30 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