Toward an Effective Theory of Neurodynamics: Topological Supersymmetry Breaking, Network Coarse-Graining, and Instanton Interaction

TL;DR

This paper develops a topological and supersymmetric framework for understanding neurodynamics, proposing a coarse-graining method based on simplicial complexes to model neural network interactions and instanton dynamics.

Contribution

It introduces a novel coarse-graining approach using simplicial complexes and the enveloping lattice to construct effective theories of neurodynamics within the supersymmetric stochastic framework.

Findings

Neural networks can be represented by high-dimensional topological spaces.

The effective neurodynamics are modeled as instanton interactions in the coarse-grained space.

The approach links neurodynamics to topological invariants like de Rham cohomology.

Abstract

Experimental research has shown that the brain's fast electrochemical dynamics, or neurodynamics (ND), is strongly stochastic, chaotic, and instanton (neuroavalanche)-dominated. It is also partly scale-invariant which has been loosely associated with critical phenomena. It has been recently demonstrated that the supersymmetric theory of stochastics (STS) offers a theoretical framework that can explain all of the above ND features. In the STS, all stochastic models possess a topological supersymmetry (TS), and the "criticality" of ND and similar stochastic processes is associated with noise-induced, spontaneous breakdown of this TS (due to instanton condensation near the border with ordinary chaos in which TS is broken by non-integrability). Here, we propose a new approach that may be useful for the construction of low-energy effective theories of ND. Its centerpiece is a coarse-graining procedure of neural networks based on simplicial complexes and the concept of the "enveloping lattice." It represents a neural network as a continuous, high-dimensional base space whose rich topology reflects that of the original network. The reduced one-instanton state space is determined by the de Rham cohomology classes of this base space, and the effective ND dynamics can be recognized as interactions of the instantons in the spirit of the Segal-Atiyah formalism.