T Ax t ie t ii

The vector x (t) contains the time series for each region, the vector e (t) is a zero-mean uncorrelated noise process and the coefficient matrices A. can be found by solving the multivariate Yule-Walker equations. If transformed to the frequency domain, the MVAR equation takes on the more simple form x (f) = H (f) • E (f), where H(f) is the transfer function matrix (Kus, et al., 2004). From the frequency domain representation we can compute the Directed Transfer Function (DTF), a frequency dependent measure of the interaction between nodes in the network (Kaminski, et al., 2001). Similarly to phase-locking and coherence measures, the DTF is a measure of interaction at a single frequency, although these interactions are computed simultaneously across the entire frequency range of interest. An advantage of MVAR models is that they allow one to determine the direction of interaction by analogy to the concept of Granger Causality. Furthermore, unlike the SEM models, no a priori knowledge is required about connectivity in the network. The wide sense stationarity requirement does limit the utility of MVAR models in event related studies, but since they are parametric they can be fit using relatively few temporal samples (with the covariance computed by averaging across epochs) and nonstationarity then detected using a sliding window. Since the method is linear, crosstalk will have a significant effect on DTF. If the mixed signals are given by y = M • x, then instead of the 'true' coefficient matrices A one gets the mixed coefficient matrices AiM, but there is no way to identify the mixing matrix M from the estimated coefficient matrices.

Neural Mass Models

Neural mass models (David et al., 2004, David and Friston, 2003, Jansen and Rit, 1995) are physiologically motivated models of cortical activity, which attempt a mathematical description of the network of neurons that generate the signals measured by MEG/EEG. The purpose of these models is to provide a realistic network simulation of cortical activity with controllable connectivity parameters, but they can also be used to estimate network parameters from real data (David and Friston, 2003). The basic elements of the model are excitatory and inhibitory columns of neurons (Jansen and Rit, 1995). The state of each element is described by an average membrane potential and a mean firing rate. The membrane potential and average firing rate are related by nonlinear input/output functions for each element and the parameters of these functions are thus the parameters of the element. A number of interaction constants between the ensembles can be used to describe the network structure. If no external input signal is used, the model is typically driven by random noise. David and Friston (2003) expanded the double column model proposed by Jansen and Rit (1995) to a meta-ensemble or -mass model, which instead of containing only two columns is comprised of N columns. These neuronal mass models can be thought of as cortical areas and by introduction of multiple areas and coupling between those, macroscopic behavior of the brain, as can be measured by MEG/EEG, can be simulated. Mass models can exhibit a broad range of frequencies and their non-linear nature and controllable macroscopic coupling make them good candidates for testing connectivity measures (David et al., 2004). Also by defining meta parameters of the model such as the number of areas and their connectivity, the models themselves can be used to fit real measurements and thus provide an estimate of functional connectivity (David and Friston, 2003).


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