Routes to Chaos Induced by a Discontinuous Resetting Process in a Hybrid Spiking Neuron Model.

Several hybrid spiking neuron models combining continuous spike generation mechanisms and discontinuous resetting processes following spiking have been proposed. The Izhikevich neuron model, for example, can reproduce many spiking patterns. This model clearly possesses various types of bifurcations and routes to chaos under the effect of a state-dependent jump in the resetting process. In this study, we focus further on the relation between chaotic behaviour and the state-dependent jump, approaching the subject by comparing spiking neuron model versions with and without the resetting process. We first adopt a continuous two-dimensional spiking neuron model in which the orbit in the spiking state does not exhibit divergent behaviour. We then insert the resetting process into the model. An evaluation using the Lyapunov exponent with a saltation matrix and a characteristic multiplier of the Poincar'e map reveals that two types of chaotic behaviour (i.e. bursting chaotic spikes and near-period-two chaotic spikes) are induced by the resetting process. In addition, we confirm that this chaotic bursting state is generated from the periodic spiking state because of the slow- and fast-scale dynamics that arise when jumping to the hyperpolarization and depolarization regions, respectively.