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.Comment: Technical report 2011-March-15, The KEY Institute for Brain-Mind Research Zurich, KMU Osak
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