Application of a Two-Dimensional Hindmarsh–rose Type Model for Bifurcation Analysis

Abstract

In this study, we examine the bifurcation scenarios of a two-dimensional Hindmarsh–Rose type model [Tsuji et al., 2007] with four parameters and simulate some resemblances of neurophysiological features for this model using spike-and-reset conditions. We present possible classifications based on the results of the following assessments: (1) the number and stability of the equilibria are analyzed in detail using a table to demonstrate the matter in which the stability of the equilibrium changes and to determine which two equilibria collapse through the saddle-node bifurcation; (2) the sufficient conditions for an Andronov–Hopf bifurcation and a saddle-node bifurcation are mathematically confirmed; and (3) we elaborately evaluate the sufficient conditions for the Bogdanov–Takens (BT) and Bautin bifurcations. Several numerical simulations for these conditions are also presented. In particular, two types of bistable behaviors are numerically demonstrated: the BT and Bautin bifurcations. Notably, all of the bifurcation curves in the domain of the remaining parameters are similar when the time scale is large. Additionally, to show the potential for a limit cycle, the existence of a trapping region is demonstrated. These results present a variety of diverse behaviors for this model. The results of this study will be helpful in assessing suitable parameters for fitting the resemblances of experimental observations.

Delay-induced mixed-mode oscillations in a 2D Hindmarsh-Rose-type model

“…For k = 1, the system is the same as in [42,10]. We can identify the number of equilibria and their stability and several codimension-one and codimension-two bifurcations, such as the SN, Hopf, Bautin and Bogdanov-Takens bifurcations.…”

“…However, to the best of our knowledge, few studies have considered a two-dimensional, single-neuron model in a differential-difference form. Therefore, we chose to study the dynamics of the 2DHR-type model [42,10] in a differential-difference form.Large-scale oscillatory behavior in a cortex often involves large numbers of intrinsic single-neuron oscillations. The oscillatory dynamics involving oscillations of various amplitudes, such as MMOs, are important in neuron models [19] and…”

“…However, to the best of our knowledge, few studies have considered a two-dimensional, single-neuron model in a differential-difference form. Therefore, we chose to study the dynamics of the 2DHR-type model [42,10] in a differential-difference form.…”

“…2, we introduce a general class of nonlinear neuron models in a differential-difference form. The model can be considered a generalization of the 2DHR-type model [42,10]. From the characteristic equation of this model, we analyze the local stability of an equilibrium as a function of the time delay.…”

“…The delay Hindmarsh-Rose-type model. Let us consider that the simplest possible modification of the systems described in [42,10] to simulate synaptic feedback is the following system of differential-difference equations:…”