Delay-induced mixed-mode oscillations in a 2D hindmarsh-rose-type model

In this study, we investigate a Hindmarsh-Rose-type model with the structure of recurrent neural feedback. The number of equilibria and their stability for the model with zero delay are reviewed first. We derive condi- tions for the existence of a Hopf bifurcation in the model and derive equations for the direction and stability of the bifurcation with delay as the bifurcation parameter. The ranges of parameter values for the existence of a Hopf bifurca- tion and the system responses with various levels of delay are obtained. When a Hopf bifurcation due to delay occurs, canard-like mixed-mode oscillations (MMOs) are produced at the parameter value for which either the fold bifur- cation of cycles or homoclinic bifurcation occurs in the system without delay. This behavior can be found in a planar system with delay but not in a planar system without delay. Therefore, the results of this study will be helpful for determining suitable parameters to represent MMOs with a simple system with delay.