Figure 2
From: Spectral mapping of brain functional connectivity from diffusion imaging
Spectral mapping method. In the individual spectral mapping problem (a–d), we map the functional connectivity matrix F j of an individual j at rest (d) from his/her structural connectivity (a). Our mapping is based on a two-stage process. In the first stage, we use a polynomial transformation of order k (characterized by the coefficients \({a}_{j0},\ldots ,{a}_{jk}\)) to map the eigenvalues of F j from those of S j . In (b), we include the histogram of the eigenvalues of S j and F j for the j-th individual (in linear and log-linear scale, respectively). In the second stage, we use a rotation matrix R j , specific to individual j, to infer the eigenmodes of F j from those of S j (as illustrated in (c), for the first eigenmode). In the lower part of the figure (e–l), we illustrate the group spectral mapping problem, in which we find a ‘universal’ mapping, valid for a whole group of individuals. For that purpose, we specify a training set composed by structural connectivity graphs (e) and their corresponding functional connectivity matrices (h). In the first stage (f), we find a common polynomial transformation (characterized by \({c}_{0},\ldots ,{c}_{k}\)), using the eigenvalues of structural and functional connectivity matrices in the training set. In the second stage (g), we find a common set of eigenmodes described by a matrix Q for all the individuals in the training set. Finally, using both the polynomial transformation (j) and the rotation (k), we estimate the functional connectivity matrix F j of a subject j (l) from his/her structural graph (i).
