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 functional connectivity (FC). Current machine learning approaches typically seek to establish direct mappings from SC to FC associated with specific cognitive states. However, these methods often treat SC and FC as distinct endpoints, failing to capture the coupling relationship throughout the progressive transformation between them. To address this limitation, we propose BrainFlow, a reversible generative model designed to parametrize flows between the distribution of SC and the mixed distribution of FCs from different cognitive tasks. Our method explicitly models the SC-FC coupling by training on interpolated states along the symmetric positive-definite (SPD) manifold. We further prove the equivalence between flow matching on the SPD manifold and on the computationally efficient Cholesky manifold, enhancing numerical stability. To mitigate cumulative errors during reverse-flow simulation, we introduce a consensus control mechanism that utilizes complementary information across multiple FC-to-SC pathways, yielding a biologically meaningful reconstruction of the underlying structural scaffold. Together, BrainFlow achieves state-of-the-art performance on both synthetic data and large-scale neuroimaging datasets from the UK Biobank and Human Connectome Project.