Dynamic Causal Modelling for event-related responses J. Daunizeau Motivation, Brain and Behaviour group, ICM, Paris, France Wellcome Trust Centre for Neuroimaging, London, UK

Overview DCM: introduction Neural ensembles dynamics Bayesian inference Example Conclusion

Overview DCM: introduction Neural ensembles dynamics Bayesian inference Example Conclusion

Introduction structural, 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

functional segregation functional integration Introduction from functional segregation to functional integration localizing brain activity: functional segregation effective connectivity analysis: functional integration u1 u1 X u2 A B u2 u1 Jusqu’à récemment, on utilisait la neuroimagerie pour localiser les régions impliquées dans telle ou telle tâche cognitive. J’ai travaillé sur le développement de méthodes liées à ce problème pendant ma thèse. Cela a donné lieu à la publication de 8 articles dont 4 en premier auteur. Aujourd’hui, on va plus loin et on analyse la connectivité cérébrale « effective ». En fait, on cherche à savoir quelle est la nature de l’information qui passe d’une région A à une région B. L’approche DCM se base sur des modèles réalistes décrivant la façon dont le cerveau est cablé, et comment il répond à la stimulation expérimentale. En fittant ces modèles aux données, on peut alors montrer qu’une connexion est modulée par une propriété des stimuli qu’on manipule expérimentalement (comme la récompense à un choix). Si tel est le cas, on a montré que cette connexion véhiculait l’information sur la récompense. J’ai contribué aux aspects statistique et neurobiologique de DCM. Cela a mené à la publication de 18 articles dont 8 en premier ou dernier auteur. « Where, in the brain, did my experimental manipulation have an effect? » « How did my experimental manipulation propagate through the network? »

Introduction why do we use dynamical system theory? 1 2 1 2 3 1 2 1 2 3 time 3 3 u

Introduction DCM: evolution and observation mappings Hemodynamic observation model: temporal convolution Electromagnetic observation model: spatial convolution neural states dynamics fMRI EEG/MEG simple neuronal model slow time scale complicated neuronal model fast time scale inputs

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

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) deviant condition (D) t =200 ms S-D: reorganisation of the connectivity structure rIFG rSTG rA1 lSTG lA1 rIFG rSTG rA1 lSTG lA1 Daunizeau, Kiebel et al., Neuroimage, 2009

Overview DCM: introduction Neural ensembles dynamics Bayesian inference Example Conclusion

Neural ensembles dynamics multi-scale perspective 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 from micro- to meso-scale : post-synaptic potential of j th neuron within its ensemble 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 synaptic kinematics post-synaptic potential EPSP depolarization (mV) membrane IPSP time (ms)

inhibitory interneurons Neural ensembles dynamics intrinsic connections within the cortical column Golgi Nissl external granular layer spiny stellate cells inhibitory interneurons pyramidal cells external pyramidal layer internal granular layer intrinsic connections internal pyramidal layer

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

Neural ensembles dynamics extrinsic connections between brain regions lateral connections spiny stellate cells inhibitory interneurons pyramidal cells extrinsic forward connections extrinsic backward connections

Observation mappings the electromagnetic forward model

Overview DCM: introduction Neural ensembles dynamics Bayesian inference Example Conclusion

Bayesian inference forward and inverse problems forward problem likelihood inverse problem posterior distribution

Bayesian inference likelihood and priors generative model m posterior

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

Bayesian inference the variational Bayesian approach free energy : functional of q mean-field: approximate marginal posterior distributions:

Bayesian inference EM in a nutshell Specify generative forward model (with prior distributions of parameters) Evoked responses Expectation-Maximization algorithm Iterative procedure: Compute model response using current set of parameters Compare model response with data Improve parameters, if possible Posterior distributions of parameters Model evidence 23

Bayesian inference DCM: key model parameters 1 2 3 u state-state coupling input-state coupling input-dependent modulatory effect

Bayesian inference model comparison for group studies 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 DCM: introduction Neural ensembles dynamics Bayesian inference Example Conclusion

Example models for deviant response generation Garrido et al., (2007), NeuroImage

Example group-level model comparison Bayesian Model Comparison Group level log-evidence subjects Forward (F) Backward (B) Forward and Backward (FB) Garrido et al., (2007), NeuroImage

Example MMN: temporal hypotheses Models for Deviant Response Generation Do forward and backward connections operate as a function of time? Peristimulus time 1 Peristimulus time 2 Garrido et al., PNAS, 2008 29

Example MMN: (best) model fit time (ms) time (ms) Garrido et al., PNAS, 2008

Example MMN: group-level model comparison across time Bayesian model comparison across subjects. (A) Comparison of the model with backward connections (FB) against the model without (F), across all subjects over the peristimulus interval 180–260 ms. The dots correspond to differences in log-evidence for 11 subjects over time. The solid line shows the average log-evidence differences over subjects [this is proportional to the log-group Bayes factor (Bf) or to the differences in the free energy of the two models (ΔF); see Materials and Methods for details]. The points outside the gray zone imply very strong inference (≥99% confidence that one model is more likely), i.e., model FB supervenes over F for positive points and the converse for negative points. (B) Histogram showing the number of subjects in each of seven levels of inference on models with and without backward connections across the peristimulus interval 180–260 ms. Garrido et al., PNAS, 2008

Overview DCM: introduction Neural ensembles dynamics Bayesian inference Example Conclusion

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

Conclusion DCM for EEG/MEG: variants input depolarization 1st and 2d order moments 100 200 300 50 150 250 100 200 300 -100 -80 -60 -40 -20 100 200 300 -100 -80 -60 -40 -20  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

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