Spike-adding and reset-induced canard cycles in adaptive integrate and fire models

We study a class of planar integrate and fire models called adaptive (AIF) models, which possesses an adaptation variable on top membrane potential, whose subthreshold dynamics is piecewise linear . These AIF therefore have two reset conditions, enable bursting to emerge for suitable parameter values. Such can be thought as hybrid dynamical systems. consider particular slow within prove the existence cycles with N resets, any integer N. Furthermore, we transition between N- $$(N+1)$$ -reset upon vanishingly small variations (for $$N=2$$ ) that such transitions are organised by canard cycles. Finally, using numerical continuation compute branches cycles, including canard-explosive branches, in these suitably recasting periodic problem two-point boundary-value