Homotopy Theoretic and Categorical Models of Neural Information Networks

TL;DR

This paper introduces a new mathematical framework using homotopy theory and category theory to model neural networks with resource assignments, capturing their dynamics and information structures.

Contribution

It develops a categorical formalism for neural networks, including functorial models of network dynamics, information cohomology, and a cohomological approach to integrated information.

Findings

Functorial assignment of network architectures and codes

Categorical formulation of Hopfield network dynamics

Cohomological model of integrated information

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.