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

Preksha Girish, Rachana Mysore, Mahanthesha U, Shrey Kumar, Shipra Prashant

TL;DR

The paper addresses the challenge of preserving intrinsic topological structure in neural data during representation learning. It proposes a brain-inspired framework that unifies persistent homology, sheaf cohomology, and cohomological flows over evolving simplicial complexes, with differentiable topological vectorizations. Key contributions include a rigorous formalization of neural computation as cohomological flows, integration of topology-aware regularization, and comprehensive benchmarking showing improved topology preservation, noise robustness, and interpretability over baselines. The approach offers a principled, topology-grounded foundation for robust brain-inspired representation learning with potential broad impact in AI and neuroscience.

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.

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

TL;DR

The paper addresses the challenge of preserving intrinsic topological structure in neural data during representation learning. It proposes a brain-inspired framework that unifies persistent homology, sheaf cohomology, and cohomological flows over evolving simplicial complexes, with differentiable topological vectorizations. Key contributions include a rigorous formalization of neural computation as cohomological flows, integration of topology-aware regularization, and comprehensive benchmarking showing improved topology preservation, noise robustness, and interpretability over baselines. The approach offers a principled, topology-grounded foundation for robust brain-inspired representation learning with potential broad impact in AI and neuroscience.

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.

Paper Structure

This paper contains 38 sections, 31 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Mathematical Background
  4. Simplicial Complexes
  5. Filtrations and Persistent Homology
  6. Cochains and Coboundary Operators
  7. Cohomological Flows
  8. Topological Vectorizations
  9. Proposed Brain-Inspired Architecture
  10. Input Data and Preprocessing
  11. Simplicial Complex Embedding Layer
  12. Persistent Homology Extraction
  13. Cohomological Flow Layers
  14. Topological Embedding Aggregation
  15. Loss Function and Training
  16. ...and 23 more sections

Figures (2)

  • Figure 1: Overview of the brain-inspired representation learning architecture. Input data (synthetic or neural recordings) is mapped to simplicial complexes, followed by persistent homology extraction, cohomological flow layers, and topological embeddings, producing invariant-rich representations.
  • Figure 2: Comprehensive results of the proposed cohomological flow architecture, showing topological robustness, EEG classification accuracy, Betti correlation, and t-SNE embeddings across synthetic and neural datasets.