Fig. 2.

Exemplar connectivity matrices (left column) and their reordered versions in terms of optimal communities (right column). Matrices in the left column are diagonal-constant and symmetric and can be considered as perturbations to the matrix depicted in the bottom of the left column(exponential synaptic footprint) which is a standard matrix used in neural field theory; these matrices include both a translationally invariant part (around the main diagonal) and inhomogeneous two-point connections. Jirsa (2009) has shown that such matrices can provide a general formulation of neural field dynamics including both local homogeneous and long-range heterogeneous coupling. The matrices of the right column motivate a reordering of the realistic DSI matrix considered later that allows assessing homogeneity assumptions for whole-brain dynamics (see last section). In the bottom panel, we see an example of a translationally invariant connectivity matrix κ(x,y), with $\kappa \left(x,y\right)=\frac{s^{-\left|x-y\right|/\sigma }}{2\sigma }$. This kernel is called an exponential synaptic footprint and has often been used in the literature to account for excitatory and inhibitory interactions.