Analysis of Spatially Extended Excitable Izhikevich Neuron Model near Instability

Abstract The article focuses on the issue of a spatiotemporal excitable biophysical model that describes the propagation of electrical potential called spikes to model the diffusion induced dynamics based on an analytical development of amplitude equations. Considering the Izhikevich neuron model consisting of coupled systems of ODEs , we demonstrate various results of spatiotemporal architecture ( PDEs ) using a suitable parameter regime. We analytically perform the saddle node bifurcation and Hopf bifurcation analysis with bifurcating periodic solutions that show the transition phases in the system dynamics. We study different types of firing patterns both analytically and numerically by the formation of Riccati differential equation. To examine the characteristics of diffusive instabilities, we use Turing amplitude equations by multiscaling method and then expansion in powers of a small control parameter. The instabilities and Turing bifurcation are established using the theoretical analysis and numerical simulations. The spatial system has potential effects on the deterministic system as a result of the diffusive matrices with various couplings and the coupled oscillators with this nearest neighbor coupling show synchronization measured by the synchronization factor analysis. Our results qualitatively reproduce different phenomena of the extended excitable system based with an efficient analytical scheme.