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Dynamic Causal Modelling for EEG/MEG: principles. J. Daunizeau Motivation, Brain and Behaviour group, ICM, Paris, France Wellcome Trust Centre for Neuroimaging, London, UK. Overview. 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference
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Dynamic Causal Modelling for EEG/MEG: principles J. Daunizeau Motivation, Brain and Behaviour group, ICM, Paris, France Wellcome Trust Centre for Neuroimaging, London, UK
Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion
Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion
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
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
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
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:
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
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
Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion
Dynamical systems theorymotivation 1 2 1 2 3 1 2 3 1 2 time 3 3 u
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)
Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion
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
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)
Neural ensembles dynamics DCM for M/EEG: synaptic dynamics post-synaptic potential EPSP membrane depolarization (mV) IPSP time (ms)
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
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
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
Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion
Bayesian inferenceforward and inverse problems forward problem likelihood posterior distribution inverse problem
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:
Likelihood: Prior: Bayes rule: Bayesian paradigmlikelihood, priors and the model evidence generative modelm
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”:
Bayesian inferencethe variational Bayesian approach free energy : functional of q mean-field: approximate marginal posterior distributions:
Bayesian inferenceDCM: key model parameters 1 2 3 u state-state coupling input-state coupling input-dependent modulatory effect
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
Overview 1 DCM: introduction 2 Dynamical systems theory 3 Neural states dynamics 4 Bayesian inference 5 Conclusion
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
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
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]
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)