Bifurcation Analysis in a Neuronal Model without/with Delay

In this study, we provide a global picture of the bifurcation scenario of a two-dimensional Hindmarsh-Rose (HR) type model. We present all of the possible classifications based on the following results: first, the number and stability of the equilibrium are analyzed in detail with a table built to show not only how to change the stability of the equilibrium but also which two equilibria collapse through the saddle-node bifurcation; secondly, sufficient conditions for an Andronov-Hopf bifurcation and a saddle-node bifurcation are mathematically confirmed; and finally, we provide sufficient conditions for a Bogdanov-Takens (BT) bifurcation and a Bautin bifurcation. Finally, we present characteristic equation for the HR type model with delay. These results provide us a diversity of behaviors for the model. The results in the paper should be helpful when choosing suitable parameters for fitting experimental observations