Application of a two-dimensional hindmarsh-rose type model for bifurcation analysis

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.