Spatiotemporal instabilities and pattern formation in systems of diffusively coupled Izhikevich neurons

TL;DR

This paper provides a mathematical and numerical analysis of spatiotemporal pattern formation in diffusively coupled Izhikevich neurons, revealing diverse structures like hexagonal patterns and oscillations relevant to brain activity.

Contribution

It introduces an explicit mathematical framework for analyzing pattern formation in coupled neuron systems, including bifurcation analysis and amplitude equations, with novel insights into pattern diversity.

Findings

Diverse spatial structures such as hexagonal patterns emerge.

Pattern stability and bifurcation points are analytically characterized.

Amplitude equations describe reaction-diffusion activity in neuron networks.

Abstract

Neurons are often connected, spatially and temporally, in phenomenal ways that promote wave propagation. Therefore, it is essential to analyze the emergent spatiotemporal patterns to understand the working mechanism of brain activity, especially in cortical areas. Here, we present an explicit mathematical analysis, corroborated by numerical results, to identify and investigate the spatiotemporal, non-uniform, patterns that emerge due to instability in an extended homogeneous 2D spatial domain, using the excitable Izhikevich neuron model. We examine diffusive instability and perform bifurcation and fixed-point analyses to characterize the patterns and their stability. Then, we derive analytically the amplitude equations that establish the activities of reaction-diffusion structures. We report on the emergence of diverse spatial structures including hexagonal and mixed-type patterns by providing a systematic mathematical approach, including variations in correlated oscillations, pattern variations and amplitude fluctuations. Our work shows that the emergence of spatiotemporal behavior, commonly found in excitable systems, has the potential to contribute significantly to the study of diffusively-coupled biophysical systems at large.