Functional connectivity is defined in Friston et al. (1993) as the “temporal correlation of spatially remote neurophysiogical events.” Neurophysiological signals simultaneously obtained from different regions of the brain give rise to data in the form of a multivariate time series. Here, we characterize functional connectivity via partial coherence between each pair of signals in a multivariate time series and develop a novel method for modeling and estimating the cross-dependence structure.
Frequency domain metrics have been used successfully for investigating the dependency structure of multivariate neurophysiological signals (Timmer et al., 2000
; Mirski et al., 2003
; Sun et al., 2004
; Salvador et al., 2005
). The most common frequency domain cross-dependency metric is coherence, which is a time-invariant metric of pair-wise linear association. Coherence is the frequency domain analog of cross-correlation (Brillinger, 2001
) and is derived from the spectral density matrix. In a P
-channel multivariate time series, the spectral density matrix is a P
semipositive Hermitian matrix which is approximately equal to the covariance matrix of the P
-dimensional vector of Fourier coefficients computed for each frequency. However, in multivariate time series, simply using cross-correlation or coherence between the components can lead to misleading conclusions on the cross-dependence structure of the signals (Kus et al., 2004
). Neither can distinguish between a pair of components that are directly linked versus a pair that is indirectly linked through a third component. Thus, to have a better understanding on how two components directly interact with each other, it is necessary to remove the effects of the other components.
In the time domain, a metric for direct linear association is partial cross-correlation. In fMRI studies, partial cross-correlation has been used for removing the temporal effects of experimental designs (McIntosh et al., 1994
) and for removing the effects of other brain signals of interest (Marrelec et al., 2006
). The frequency domain analog of partial cross-correlation is partial coherence. It has been successfully implemented in analyzing scalp EEG (Timmer et al., 2000
), intracortical EEG (Mirski et al., 2003
), and fMRI (Sun et al., 2004
, Zhou et al, 2009
). As we will describe in further detail, an efficient approach to estimating partial coherence involves inversion of the spectral density matrix. This approach requires estimates of the spectral density matrix to be numerically stable because otherwise, even small perturbations in the estimates of the spectral density matrix will result in large changes in the entries of its inverse, consequently giving highly variable partial coherence estimates.
We develop a novel shrinkage estimation method for estimating functional interconnectivity across brain sites. The shrinkage estimator is a weighted average of a mildly-smoothed periodogram matrix and the scaled identity matrix. The resulting shrinkage estimates have lower condition numbers than the classical smoothed periodogram and hence are more numerically stable. Böhm (2008)
developed shrinkage estimation for the spectral density matrix of multivariate time series in a single-trial setting. We extend the shrinkage estimator to handle multiple-trial multivariate time series and, via numerical experiments, demonstrate that it does well in estimating both the spectral density matrix and partial coherence. We also develop a randomization procedure for testing for differences in functional connectivity between experimental conditions. We apply these new methods to an electroencephalogram (EEG) data set acquired from an experiment to study the brain network that mediates voluntary movement.