Bumps and Oscillons in Networks of Spiking Neurons

TL;DR

This paper analyzes localized patterns, called bumps and oscillons, in a mean-field model of spiking neurons, revealing their existence, stability, and complex bifurcation structures under various conditions.

Contribution

It provides the first detailed bifurcation analysis of localized solutions in a mean-field model of spiking neurons, including analytical estimates and numerical methods.

Findings

Localized bump solutions exist within specific parameter ranges.

Multiple bump patterns form complex bifurcation structures like snake-and-ladder.

Time-periodic oscillons exhibit period doubling and chaos.

Abstract

We study localized patterns in an exact mean-field description of a spatially-extended network of quadratic integrate-and-fire (QIF) neurons. We investigate conditions for the existence and stability of localized solutions, so-called bumps, and give an analytic estimate for the parameter range where these solutions exist in parameter space, when one or more microscopic network parameters are varied. We develop Galerkin methods for the model equations, which enable numerical bifurcation analysis of stationary and time-periodic spatially-extended solutions. We study the emergence of patterns composed of multiple bumps, which are arranged in a snake-and-ladder bifurcation structure if a homogeneous or heterogeneous synaptic kernel is suitably chosen. Furthermore, we examine time-periodic, spatially-localized solutions (oscillons) in the presence of external forcing, and in autonomous, recurrently coupled excitatory and inhibitory networks. In both cases we observe period doubling cascades leading to chaotic oscillations.