From nodes to pathways: an edge-centric model of brain structure-function coupling via constrained Laplacians

Understanding how functional interactions emerge from the brain structure remains a central challenge in network neuroscience. Structural connectivity (SC), derived from diffusion MRI tractography, provides the anatomical scaffold, whereas functional connectivity (FC), measured with fMRI, EEG, or MEG, reflects dynamic neural interactions that often extend beyond direct anatomical links. Existing spectral graph theoretical models successfully capture indirect interactions but largely remain node-centric, leaving the contributions of individual structural pathways implicit.

This work introduces an edge-centric framework for structure-function coupling that reformulates the graph Laplacian as a constrained flow problem grounded in spectral graph theory and realized numerically through Modified Nodal Analysis (MNA). Functional relationships are modeled as nodal constraints (voltage sources), structural connections as conductances, and the resulting nodal potentials induce edge-wise flows (currents) that explicitly quantify how functional signals are distributed in anatomical network. This formulation preserves the theoretical foundations of Laplacian eigenmodes while extending them to compute interpretable, pathway-specific measures of functional signal flow.

As a proof-of-concept, the framework is demonstrated in three settings: (i) a single-subject Default Mode Network example illustrating tractography filtering and streamline-wise functional currents, (ii) a ground-truth diffusion phantom showing accurate recovery of both direct and indirect pathways, and (iii) a group-level analysis of 207 Human Connectome Project subjects yielding a sparse, current-based connectome complementary to conventional SC and FC matrices. By bridging spectral graph theory and circuit analysis, this approach provides a mathematically consistent and physically interpretable method for mapping functional interactions onto structural brain networks, with practical applications in tractography filtering, network analysis, and multimodal connectomics.