BrainFlow: A Holistic Pathway of Dynamic Neural System on Manifold

A fundamental challenge in cognitive neuroscience is understanding how cognition emerges from the interplay between structural connectivity (SC) and dynamic functional connectivity (FC) in the brain. Network neuroscience has emerged as a powerful framework to understand brain function through a holistic perspective on structure-function relationships. In this context, current machine learning approaches typically seek to establish direct mappings between structural connectivity (SC) and functional connectivity (FC) associated with specific cognitive states. However, these state-independent methods often yield inconsistent results due to overlapping brain networks across cognitive states. To address this limitation, we conceptualize to uncover the dendritic coupling mechanism between one static SC and multiple FCs by solving a flow problem that bridges the distribution of SC to a mixed distribution of FCs, conditioned on various cognitive states, along a Riemannian manifold of symmetric positive-definite (SPD) manifold. We further prove the equivalence between flow matching on the SPD manifold and on the computationally efficient Cholesky manifold. Since a spare of functional connections is shared across cognitive states, we introduce the notion of consensus control to promote the shared kinetic structures between multiple FC-to-SC pathways via synchronized coordination, yielding a biologically meaningful underpinning on SC-FC coupling mechanism. Together, we present BrainFlow, a reversible generative model that achieves state-of-the-art performance on not only synthetic data but also large-scale neuroimaging datasets from UK Biobank and Human Connectome Project.