Emergence of Turing patterns and dynamic visualization in excitable neuron model

Abstract

• The article focuses on the temporal and spatiotemporal dynamics of Hindmarsh–Rose (H-R) model. • Fixed point and bifurcation analysis presented for deterministic H-R model. • We also study Turing instability, Turing bifurcation and Hopf bifurcation for spatiotemporal model. • We explore Pattern formation in the Turing and Hopf–Turing regions. • We describe amplitude equations and study its stability analysis to validate numerical simulation results of Pattern formation. This article is focused on studying the spatially extended reaction-diffusion system with a diagonal diffusion matrix in a bounded domain for a biophysically motivated excitable model. Diffusion induces spontaneous stationary patterns in the spatially extended homogeneous medium. We investigate the dynamics of the diffusively coupled network modulated by a Hindmarsh–Rose prototype model that describes the emergence of self-excited spiking activities with certain parameters and a constant injected stimulus. The linear stability analysis in this framework around the homogeneous steady states illustrates the emergence of stationary patterns. Turing domains are reported in the parameter space where Hopf bifurcation is determined. The bifurcation diagram helps us in understanding the transition mechanism in the spatial system. We have investigated the existence of Turing–Hopf bifurcation and established how Hopf and Turing curves divide the parameter space into three different dynamically significant regions. We have also studied the existence of Hopf bifurcation in the spatiotemporal system. Theoretically, the amplitude equations are derived by means of nonlinear multiple-scale analysis method and analyzed near the Hopf and Turing instabilities in the system. In particular, we observe asymptotic expressions for a wide range of various patterns (stationary, hexagonal, mixed-state) sustained by the spatial system. We obtain the explicit conditions to establish the structural transitions and stability of the diverse forms of these Turing patterns. These results reveal how the diffusive network evolves. To establish the results, the analytical derivations are demonstrated that are corroborated by numerical simulations of the corresponding diffusion induced system. Finally, we observe that the coupled excitable systems participate in a collective behavior that may contribute significantly to irregular neural dynamics associated with certain brain pathologies.