Accurately Modeling the Resting Brain Functional Correlations Using Wave Equation With Spatiotemporal Varying Hypergraph Laplacian

How spontaneous brain neural activities emerge from the underlying anatomical architecture, characterized by structural connectivity (SC), has puzzled researchers for a long time. Over the past decades, much effort has been directed toward the graph modeling of SC, in which the brain SC is generally considered as relatively invariant. However, the graph representation of SC is unable to directly describe the connections between anatomically unconnected brain regions and fail to model the negative functional correlations. Here, we extend the static graph model to a spatiotemporal varying hypergraph Laplacian diffusion (STV-HGLD) model to describe the propagation of the spontaneous neural activity in human brain by incorporating the Laplacian of the hypergraph representation of the structural connectome (
${h}$
SC) into the regular wave equation. Theoretical solution shows that the dynamic functional couplings between brain regions fluctuate in the form of an exponential wave regulated by the spatiotemporal varying Laplacian of
${h}$
SC. Empirical study suggests that the cortical wave might give rise to resonance with SC during the self-organizing interplay between excitation and inhibition among brain regions, which orchestrates the cortical waves propagating with harmonics emanating from the
${h}$
SC while being bound by the natural frequencies of SC. Besides, the average statistical dependencies between brain regions, normally defined as the functional connectivity (FC), arises just at the moment before the cortical wave reaches the steady state after the wave spreads across all the brain regions. Comprehensive tests on four extensively studied empirical brain connectome datasets with different resolutions confirm our theory and findings.