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Homotopy Theoretic and Categorical Models of Neural Information Networks

Yuri Manin, Matilde Marcolli

TL;DR

This work develops a cohesive, category-theoretic framework for neural information networks that carries metabolic, informational, and computational resources. By constructing configuration spaces of resource assignments via categories of summing functors and enriching them with Segal's $b3$-spaces and their endofunctor extensions, the authors define a dynamic, functorial Hopfield model compatible across subsystems. They connect these dynamics to topological representations through clique and nerve complexes, Gamma-spaces, and Gamma networks, and unify information measures via cohomological information theory, ultimately linking integrated information to topological and spectral invariants. The approach yields a versatile, multi-layered toolkit for analyzing neural computation under resource constraints, with potential impact on understanding consciousness-related information measures and the interplay between topology, information, and computation in brain-like networks.

Abstract

In this paper we develop a novel mathematical formalism for the modeling of neural information networks endowed with additional structure in the form of assignments of resources, either computational or metabolic or informational. The starting point for this construction is the notion of summing functors and of Segal's Gamma-spaces in homotopy theory. The main results in this paper include functorial assignments of concurrent/distributed computing architectures and associated binary codes to networks and their subsystems, a categorical form of the Hopfield network dynamics, which recovers the usual Hopfield equations when applied to a suitable category of weighted codes, a functorial assignment to networks of corresponding information structures and information cohomology, and a cohomological version of integrated information.

Homotopy Theoretic and Categorical Models of Neural Information Networks

TL;DR

This work develops a cohesive, category-theoretic framework for neural information networks that carries metabolic, informational, and computational resources. By constructing configuration spaces of resource assignments via categories of summing functors and enriching them with Segal's -spaces and their endofunctor extensions, the authors define a dynamic, functorial Hopfield model compatible across subsystems. They connect these dynamics to topological representations through clique and nerve complexes, Gamma-spaces, and Gamma networks, and unify information measures via cohomological information theory, ultimately linking integrated information to topological and spectral invariants. The approach yields a versatile, multi-layered toolkit for analyzing neural computation under resource constraints, with potential impact on understanding consciousness-related information measures and the interplay between topology, information, and computation in brain-like networks.

Abstract

In this paper we develop a novel mathematical formalism for the modeling of neural information networks endowed with additional structure in the form of assignments of resources, either computational or metabolic or informational. The starting point for this construction is the notion of summing functors and of Segal's Gamma-spaces in homotopy theory. The main results in this paper include functorial assignments of concurrent/distributed computing architectures and associated binary codes to networks and their subsystems, a categorical form of the Hopfield network dynamics, which recovers the usual Hopfield equations when applied to a suitable category of weighted codes, a functorial assignment to networks of corresponding information structures and information cohomology, and a cohomological version of integrated information.

Paper Structure

This paper contains 90 sections, 53 theorems, 114 equations, 4 figures.

Table of Contents

  1. Introduction and motivation
  2. Background motivation
  3. Cognition and computation
  4. Homotopical representations of stimulus spaces
  5. Homology and stimulus processing
  6. Informational complexity and integrated information
  7. Perception, representation, computation
  8. Structure of the paper and main results
  9. Comparison with other approaches
  10. Summing functors on networks
  11. The category of summing functors
  12. Networks and summing functors
  13. Conservation laws at vertices
  14. Vertex constraints by coequalizer
  15. Categories of summing functors on networks
  16. ...and 75 more sections

Key Result

Lemma 2.3

Let ${\mathcal{C}}$ denote a category with sums and zero object.

Figures (4)

  • Figure 1: Example of a morphism of transition systems.
  • Figure 2: A simple example of coproduct of two transition systems.
  • Figure 3: A simple example of the grafting operation of Lemma \ref{['acyclicgraft']}.
  • Figure 4: Grafting operation with matching external edges.

Theorems & Definitions (86)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Proposition 2.10
  • ...and 76 more