Persistent Topological Structures and Cohomological Flows as a Mathematical Framework for Brain-Inspired Representation Learning

Abstract

This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the evolution of cochain maps over dynamic simplicial complexes, enabling representations that capture invariants across temporal, spatial, and functional brain states. The proposed architecture integrates algebraic topology with differential geometry to construct cohomological operators that generalize gradient-based learning within a homological landscape. Synthetic data with controlled topological signatures and real neural datasets are jointly analyzed using persistent homology, sheaf cohomology, and spectral Laplacians to quantify stability, continuity, and structural preservation. Empirical results demonstrate that the model achieves superior manifold consistency and noise resilience compared to graph neural and manifold-based deep architectures, establishing a coherent mathematical foundation for topology-driven representation learning.