Causality in Time Series - ClopiNet

Causal analysis of fMRIestimate the trajectories of hidden neuronal processes from observed neuroimaging dataif one can formulate an accurate model of the processes leading from neuronal activityto data records. A few years later, this idea was robustly transferred to fMRI data in theform of DCM (Friston et al., 2003). DCM combines three ideas about causal influenceanalysis in fMRI data (or neuroimaging data in general), which can be understood interms of the discussion of the fMRI signal and state space models above (Daunizeauet al., 2009a).First, neuronal interactions are best modeled at the level of unobserved (latent)signals, instead of at the level of observed BOLD signals. This requires a state spacemodel with a dynamic model of neuronal population dynamics and interactions. Theoriginal model that was formulated for the dynamics of neuronal states x = {x 1 ,..., x N }is a bilinear ODE model:∑︁ẋ = Ax + v j B j x + Cv (18)That is, the noiseless neuronal dynamics are characterized by a linear term (with entriesin A representing intrinsic coupling between populations), an exogenous term (with Crepresenting driving influence of experimental variables) and a bilinear term (with B jrepresenting the modulatory influence of experimental variables on coupling betweenpopulations). More recent work has extended this model, e.g. by adding a quadraticterm (Stephan et al., 2008), stochastic dynamics (Daunizeau et al., 2009b) or multiplestate variables per region (Marreiros et al., 2008).Second, the latent neuronal dynamics are related to observed data by a generative(forward) model that accounts for the temporal convolution of neuronal events by slowand variably delayed hemodynamics. This generative forward model in DCM for fMRIis exactly the (simplified) balloon model set out in section 2. Thus, for every selectedregion a single state variable represents the neuronal or synaptic activity of a local populationof neurons and (in DCM for BOLD fMRI) four or five more represent hemodynamicquantities such as capillary blood volume, blood flow and deoxy-hemoglobincontent. All state variables (and the equations governing their dynamics) that servethe mapping of neuronal activity to the fMRI measurements (including the observationequation) can be called the observation model. Most of the physiologically motivatedgenerative model in DCM for fMRI is therefore concerned with an observation modelencapsulating hemodynamics. The parameters in this model are estimated conjointlywith the parameters quantifying neuronal connectivity. Thus, the forward biophysicalmodel of hemodynamics is ‘inverted’ in the estimation procedure to achieve a deconvolutionof fMRI time series and obtain estimates of the underlying neuronal states. DCMhas also been applied to EEG/MEG, in which case the observation model encapsulatesthe lead-field matrix from neuronal sources to EEG electrodes or MEG sensors (Kiebelet al., 2009).Third, the approach to estimating the hidden state trajectories (i.e. filtering andsmoothing) and parameter values in DCM is cast in a Bayesian framework. In short,Bayes’ theorem is used to combine priors p(Θ|M)and likelihood p(y|Θ, M)into the89