Theoretical analysis of canard dynamics in a nonlinear neuron model with reset

We present a mathematics analysis showing that bursting oscillation and its complex transitions caused by reset-induced canard cycles are generated in a class of Izhikevich quadratic models. Using geometric singular perturbation theory and asymptotic expansion with boundary layer function, the expressions of the attracting and repelling parts of the slow manifold as well as the flow solution of the system are obtained, which is convenient to compute the time when the flow reaches the threshold line and the reset line. Based on the above results, it is proven that the system can support bursts of any period and canard cycles of any period as a function of model parameters, and the N - reset periodic cycles (for $N=2$) are asymptotically stable through constructing the Poincaré map. Finally, we prove that there is no chaos in the transition between N - and $(N+1)$ - reset periodic cycles when ε is in a certain small scope but it organized by canard cycles.