Linking time-lagged functional dynamics to spatial constraints in resting-state fMRI

Linear state-space models have been shown to effectively reproduce large-scale brain dynamics. We applied this approach to resting-state fMRI data acquired from 20 mice, focusing on the system's Jacobian matrix, i.e. the effective connectivity, and specifically on its component encoding nonzero-lag interactions: the differential covariance matrix. Within this matrix, we concentrated on the off-diagonal component (dC-Cov), which reflect endogenous time-lagged correlations. Our aim was to identify a decomposition of the Jacobian matrix that facilitates its interpretation from a mechanistic perspective. Since the dC-Cov captures the rotational component of signal trajectories, we employed Schur decomposition to extract 2D rotational modes, each characterized by a pair of orthogonal vectors, and an associated angular frequency. This provides a more generative formulation of the modeling framework, thereby reducing the interpretability gap between this approach and connectome-based network models of coupled neural masses. Within this framework, the precision matrix governs the coupling between different Schur modes, while we hypothesize that the dC-Cov reflects spatial constraints imposed by inter-regional distances. By examining the relationship between dC-Cov and structural constraints imposed by the spatial placement of brain areas, we found a consistent alignment between the faster Schur modes across mice and the leading eigenvectors of the structural distance matrix.