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

Takao Inoué

TL;DR

The paper addresses how to interpret the relation between local neural functions and global cognition within a precise semantic framework. It models local functions as sections of a neural sheaf over a brain-related space, with global coherence requiring a global section and obstructions captured by $H^1(\mathcal{B},\mathcal{F})$. The main contributions include a rigorous sheaf-theoretic interpretation of neural manifolds, a cohomological classification of brain pathology as global integration failures, and a Leibnizian philosophical framing that links monads, atlases, and pre-established harmony. This framework provides a mathematically principled language for discussing cognitive unity and its disruption, promoting interdisciplinary dialogue among neuroscience, mathematics, and philosophy of mind.

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.

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

TL;DR

The paper addresses how to interpret the relation between local neural functions and global cognition within a precise semantic framework. It models local functions as sections of a neural sheaf over a brain-related space, with global coherence requiring a global section and obstructions captured by . The main contributions include a rigorous sheaf-theoretic interpretation of neural manifolds, a cohomological classification of brain pathology as global integration failures, and a Leibnizian philosophical framing that links monads, atlases, and pre-established harmony. This framework provides a mathematically principled language for discussing cognitive unity and its disruption, promoting interdisciplinary dialogue among neuroscience, mathematics, and philosophy of mind.

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.
Paper Structure (24 sections, 4 theorems, 11 equations, 2 figures)

This paper contains 24 sections, 4 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Brain-Related Spaces and Neural Manifolds
  4. Sheaf Semantics of Brain Function
  5. Pathology as Cohomological Obstruction
  6. Correspondence Table
  7. From Open Covers to Cohomological Obstruction
  8. Clinical Case Studies and Literature Alignment
  9. Aphasia.
  10. Agnosia.
  11. Schizophrenia.
  12. Leibnizian Interpretation
  13. Methodological Remark
  14. Leibnizian Interpretation: Monads, Atlases, and Brain
  15. Connection to Leibnizian Monadology
  16. ...and 9 more sections

Key Result

Theorem 4.1

For the local-to-global principle formalized here, see standard treatments of sheaf theory such as Mac Lane and Moerdijk (1992, Chapter III) MacLaneMoerdijk or Kashiwara and Schapira (2006, Section 2) KashiwaraSchapira. A globally coherent brain function corresponds to the existence of a global sect

Figures (2)

  • Figure 1: From an open cover $\{U_i\}$ of a brain-related space $\mathcal{B}$ to a Čech 1-cocycle. Local sections $s_i$ agree on overlaps in the healthy case. Pathology corresponds to a nontrivial cocycle obstructing global gluing.
  • Figure 2: Brain modeled as a manifold with an open cover. (a) Consistent cocycle conditions yield coherent global integration. (b) Local degeneration of transitions, corresponding to aphasia or agnosia. (c) Global inconsistency of cocycles, corresponding to schizophrenia-like disorders.

Theorems & Definitions (7)

  • Definition 3.1: Brain-Related Space
  • Definition 3.2: Neural Sheaf
  • Theorem 4.1: Global Brain Function
  • Remark 4.1: Standard Sheaf-Theoretic Fact
  • Theorem 5.1: Pathological Obstruction
  • Theorem 5.2: Cohomological Classification
  • Theorem 8.1