Unifying Brain Geometry and Connectivity

Explaining how brain structure gives rise to emerging dynamics is a central goal in neuroscience. We present an anatomical constraint showing that rare long-range (LR) connections are essential for capturing both spontaneous and task-evoked functional organization. This framework unifies cortical geometry and local connectivity via the exponential distance rule (EDR), while explicitly incorporating LR exceptions identified in the structural connectome. Using this enriched structural description, we demonstrate that task-evoked activity lies on a low-dimensional manifold composed of a small set of modes, suggesting that efficient information processing relies on a compact dynamical repertoire.While cortical geometry and EDR-based local connectivity explain much of large-scale functional structure, they miss the computational importance of rare LR projections. These connections, though few, markedly enhance information integration and are not predictable from cortical folding alone. We show that harmonic modes combining EDR with LR connectivity (EDR+LR) provide a markedly superior basis for reconstructing functional dynamics - including LR functional connectivity and task-evoked responses - compared to geometry - or EDR-only models. Overall, our results reveal that rare LR connections crucially shape a low-dimensional manifold of fundamental modes that efficiently captures the complexity of human brain activity.