# Macroscopic Coherent Structures in a Stochastic Neural Network: From Interface Dynamics to Coarse-Grained Bifurcation Analysis

## TL;DR

This paper explores how patterns like bumps and waves form in a stochastic neural network model using both analytical and numerical methods.

## Contribution

The study introduces lifting operators that use microscopic motifs to distinguish between stationary and travelling bumps in neural networks.

## Key findings

- Stationary and travelling bumps share similar meso- and macroscopic profiles but differ in microscopic structure.
- Waves are supported by high synaptic gain and long refractory times, while meandering bumps are elicited by short refractory times.
- Equation-free methods are used to compute patterns and their stability in the full stochastic model on a discrete lattice.

## Abstract

We study coarse pattern formation in a cellular automaton modelling a spatially-extended stochastic neural network. The model, originally proposed by Gong and Robinson [36], is known to support stationary and travelling bumps of localised activity. We pose the model on a ring and study the existence and stability of these patterns in various limits using a combination of analytical and numerical techniques. In a purely deterministic version of the model, posed on a continuum, we construct bumps and travelling waves analytically using standard interface methods from neural fields theory. In a stochastic version with Heaviside firing rate, we construct approximate analytical probability mass functions associated with bumps and travelling waves. In the full stochastic model posed on a discrete lattice, where a coarse analytic description is unavailable, we compute patterns and their linear stability using equation-free methods. The lifting procedure used in the coarse time-stepper is informed by the analysis in the deterministic and stochastic limits. In all settings, we identify the synaptic profile as a mesoscopic variable, and the width of the corresponding activity set as a macroscopic variable. Stationary and travelling bumps have similar meso‐ and macroscopic profiles, but different microscopic structure, hence we propose lifting operators which use microscopic motifs to disambiguate between them. We provide numerical evidence that waves are supported by a combination of high synaptic gain and long refractory times, while meandering bumps are elicited by short refractory times.