Post inhibitory rebound spike related to nearly vertical nullcline for small homoclinic and saddle-node bifurcations

A spike induced by inhibitory stimulation instead of excitatory stimulation, called post-inhibitory rebound (PIR) spike, has been found in multiple neurons with important physiological functions, which presents counterintuitive behavior mainly related to focus near Hopf bifurcation. In the present paper, the condition for the PIR spike is extended to small homoclinic orbit (SHom) and saddle-node (SN) bifurcations, and the underlying mechanism is acquired in a neuron model. Firstly, PIR spike is evoked from a stable node near the SHom or SN bifurcation by a strong inhibitory stimulation. Then, the dynamics of threshold curve for a spike, vector fields, and nullcline of recovery variable are used to well explain the cause for the PIR spike. The shape of threshold curve for the node resembles that of focus. The nullcline plays an important role in forming PIR spike, which is analytically identified at last. Besides, a sufficient condition is acquired from the integration to a differential equation, and the range of parameters for the PIR spike is presented. The extended bifurcation types and the underlying mechanisms for the PIR spike such as the nullcline present comprehensive and deep understandings for the PIR spike, which also provides potential strategy to modulate the PIR phenomenon and even related physiological functions of neurons.