A Novel Threshold Across which the Negative Stimulation Evokes Action Potential Near a Saddle-Node Bifurcation in a Neuronal Model with Ih Current

Abstract

The negative or hyperpolarization pulse stimulation induces action potential, i.e. the post-inhibitory rebound spike, which has been widely observed in various single neurons with hyperpolarization-activated cation current ([Formula: see text]) in neuroscience and is suggested to be evoked from a focus near the Hopf bifurcation according to the traditional viewpoint of nonlinear dynamics. In the present paper, a novel viewpoint that post-inhibitory rebound spike can be evoked from a stable node near the saddle-node bifurcation on invariant circle (SNIC) is proposed, which can be well interpreted with hyperpolarization activation characteristic of [Formula: see text] current, bifurcation analysis, and threshold. Especially, the boundary between the subthreshold and suprathreshold initial values which respectively evoke subthreshold potential and action potential is acquired to be a threshold surface containing the saddle. [Formula: see text] current after the negative pulse stimulation for small conductance [Formula: see text] of [Formula: see text] is low enough to evoke just a subthreshold potential while for large [Formula: see text] is high enough to evoke a post-inhibitory rebound spike. For small [Formula: see text], the pulse induces the decrease of membrane potential [Formula: see text] and then the phase trajectory always stays within the subthreshold initial value region locating lower to the threshold surface with a nearly fixed [Formula: see text] value. For large [Formula: see text], the threshold surface changes and is composed of two parts: one part with a nearly fixed [Formula: see text] value and the other with a nearly fixed value of [Formula: see text] variable to describe [Formula: see text] inactivation probability. Although the negative pulse stimulation induces the decrease of [Formula: see text], [Formula: see text] increases to a level high enough and then the phase trajectory runs across the part with a nearly fixed [Formula: see text] value to form a post-inhibitory rebound spike. The appearance of the novel [Formula: see text] threshold is the internal dynamical mechanism for the generation of post-inhibitory rebound spike, and the external cause is that the negative pulse stimulation induces the phase trajectory to run across the [Formula: see text] threshold surface. The results present a novel nonlinear phenomenon and the corresponding dynamical mechanism related to post-inhibitory rebound spike induced by [Formula: see text] current near the SNIC bifurcation point.