We consider exploratory methods for the discovery of cortical functional
connectivity. Typically, data for the i-th subject (i=1...NS) is represented as
an NVxNT matrix Xi, corresponding to brain activity sampled at NT moments in
time from NV cortical voxels. A widely used method of analysis first
concatenates all subjects along the temporal dimension, and then performs an
independent component analysis (ICA) for estimating the common cortical
patterns of functional connectivity. There exist many other interesting
variations of this technique, as reviewed in [Calhoun et al. 2009 Neuroimage
45: S163-172]. We present methods for the more general problem of discovering
functional connectivity occurring at all possible time lags. For this purpose,
brain activity is viewed as a function of space and time, which allows the use
of the relatively new techniques of functional data analysis [Ramsay &
Silverman 2005: Functional data analysis. New York: Springer]. In essence, our
method first vectorizes the data from each subject, which constitutes the
natural discrete representation of a function of several variables, followed by
concatenation of all subjects. The singular value decomposition (SVD), as well
as the ICA of this new matrix of dimension [rows=(NT*NV); columns=NS] will
reveal spatio-temporal patterns of connectivity. As a further example, in the
case of EEG neuroimaging, Xi of size NVxNW may represent spectral density for
electric neuronal activity at NW discrete frequencies from NV cortical voxels,
from the i-th EEG epoch. In this case our functional data analysis approach
would reveal coupling of brain regions at possibly different frequencies.