STABILITY AND BIFURCATION ANALYSIS OF A HINDMARSH-ROSE NEURONAL MODEL WITH TIME DELAY

STABILITY AND BIFURCATION ANALYSIS OF A HINDMARSH-ROSE NEURONAL MODEL WITH TIME DELAY

Abstract:
The Hindmarsh-Rose (HR) model is a widely used mathematical model for describing the dynamics of neuronal activity. In this study, we investigate the effects of time delay on the stability and bifurcation behavior of the HR neuronal model. Time delays are known to play a crucial role in neuronal systems, as they can give rise to various dynamical phenomena, such as oscillations and synchronization.

We begin by incorporating a time delay term into the HR model, which represents the delayed feedback between different components of the neuron. The resulting model is a set of nonlinear delay differential equations. We then analyze the stability of the equilibrium points and investigate the occurrence of bifurcations as the time delay parameter is varied.

Using analytical and numerical techniques, we examine the stability properties of the equilibrium points, including local stability and Hopf bifurcation. Specifically, we calculate the characteristic equation and study its roots to determine the conditions under which stability switches occur. By employing the delay-specific bifurcation theory, we explore the emergence of various bifurcation phenomena, such as Hopf bifurcation, period-doubling bifurcation, and torus bifurcation.

Our results reveal that the presence of time delay can significantly affect the stability and dynamics of the HR neuronal model. We observe that as the delay parameter is increased, the system can undergo a series of bifurcations, leading to the appearance of complex oscillatory behaviors. Furthermore, we find that the stability of the equilibrium points can be influenced by the delay, resulting in the destabilization or stabilization of the system.

Understanding the stability and bifurcation properties of neuronal models with time delay is crucial for gaining insights into the dynamics of real-world neuronal systems. The findings of this study contribute to advancing our knowledge of the HR model and provide a foundation for investigating the impact of time delay in more complex neural networks. These findings may have implications in the study of neurological disorders and the development of control strategies for neural systems.

Keywords: Hindmarsh-Rose model, neuronal dynamics, time delay, stability analysis, bifurcation analysis, Hopf bifurcation, delay differential equations.

STABILITY AND BIFURCATION ANALYSIS OF A HINDMARSH-ROSE NEURONAL MODEL WITH TIME DELAY, GET MORE PHYSICS PROJECT TOPICS AND MATERIALS

Sharing is caring!