On Brain as a Mathematical Manifold: Neural Manifolds, Sheaf Semantics, and Leibnizian Harmony

TL;DR

This paper proposes a novel mathematical framework using sheaf theory to model brain functions, emphasizing local-global coherence and interpreting pathologies as obstructions within this structure.

Contribution

It introduces a sheaf-theoretic model of brain function that links neural states, coherence, and pathologies, integrating philosophical insights from Leibniz.

Findings

Sheaf sections represent local neural functions.

Global sections correspond to coherent brain states.

Pathologies are modeled as obstructions to global sections.

Abstract

We present a mathematical and philosophical framework in which brain function is modeled using sheaf theory over neural state spaces. Local neural or cognitive functions are represented as sections of a sheaf, while global coherence corresponds to the existence of global sections. Brain pathologies are interpreted as obstructions to such global integration and are classified using tools from sheaf cohomology. The framework builds on the neural manifold program in contemporary neuroscience and on standard results in sheaf theory, and is further interpreted through a Leibnizian lens \cite{Churchland2012, Leibniz1714, MacLaneMoerdijk, Perich2025}. This paper is intended as a conceptual and formal proposal rather than a complete empirical theory.