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.

The bursting and subthreshold resonance of mesencephalic trigeminal nucleus neurons play a critical role in bite force control and orofacial pain processing, yet their underlying dynamical mechanisms remain poorly understood. The nonlinear and ionic mechanisms are presented in the present paper. First, bifurcations from the resting state to bursting, modulated by persistent sodium current (I_NaP) and applied current (I_app), are obtained, including a Hopf bifurcation. Second, three typical bursting patterns are distinguished via codimension-2 bifurcations, and coexisting behaviors are acquired through fast/slow analysis. Modulated by I_NaP, the most common bursting pattern, similar to experimental observations, exhibits bursts interrupted by long quiescent states. In the fast subsystem, the limit cycle for the burst is far from the coexisting equilibrium for the quiescent state. Then, stochastic transitions between the two behaviors cannot be evoked, resulting in robustness to noise. Two other bursting patterns manifest transitions between bursts for a large limit cycle and subthreshold oscillation for a small limit cycle. Proximity between the two limit cycles allows noise to drive stochastic transitions, leading to noise-sensitive dynamics. The sensitivity is determined by the distance between the two cycles. Finally, subthreshold resonance evoked from the resting state is reproduced, which is mediated by potassium current. Enhanced I_NaP and I_app amplify the resonance amplitude and frequency by driving the resting state toward the Hopf bifurcation. The dynamics of bursting and resonance advance the understanding of sensorimotor processing and present potential targets for treating orofacial motor disorders.

A high spike-timing precision characterized by a small variation in interspike intervals of neurons is important for information processing in various brain functions. An experimental study on fast-spiking interneurons has shown that inhibitory autapses functioning as negative self-feedback can enhance spike-timing precision. In the present paper, bifurcation and negative self-feedback mechanisms for the enhanced spike-timing precision to stochastic modulations are obtained in two theoretical models, presenting theoretical explanations to the experimental finding. For stochastic spikes near both the saddle-node bifurcation on an invariant cycle (SNIC) and the subcritical Hopf (SubH) bifurcation with classes 1 and 2 excitabilities, respectively, enhanced spike-timing precision appears in large ranges of the conductance and the decaying rate of inhibitory autapses, closely matching the experimental observation. The inhibitory autaptic current reduces the membrane potential after a spike to a level lower than that in the absence of inhibitory autapses and the threshold to evoke the next spike, making it more difficult for stochastic modulations to affect spike timings, and thereby enhancing spike-timing precision. In addition, firing frequency near the SubH bifurcation is more robust than that near the SNIC bifurcation, resulting in a higher spike-timing precision for the SubH bifurcation. The bifurcation and negative self-feedback mechanisms for the enhanced spike-timing precision present potential measures to modulate the neuronal dynamics or the autaptic parameters to adjust the spike-timing precision.