Narrow Results

A nonlinear recurrence involving a piecewise constant McCulloch–Pitts function and two 2-periodic coefficient sequences is investigated. By allowing the threshold parameter to vary from 0⁺ to +∞, we work out a complete bifurcation analysis for the asymptotic behaviors of the corresponding solutions. We show that there are four steady state solutions and that all solutions will tend to one of them. We hope that our results will be useful in further investigating neural networks involving the McCulloch–Pitts function with threshold and more general periodic coefficients.

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 ($I_{\mathrm{\text{h}}}$) 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 $I_{\mathrm{\text{h}}}$ 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. $I_{\mathrm{\text{h}}}$ current after the negative pulse stimulation for small conductance $g_{\mathrm{\text{h}}}$ of $I_{\mathrm{\text{h}}}$ is low enough to evoke just a subthreshold potential while for large $g_{\mathrm{\text{h}}}$ is high enough to evoke a post-inhibitory rebound spike. For small $g_{\mathrm{\text{h}}}$, the pulse induces the decrease of membrane potential $V$ and then the phase trajectory always stays within the subthreshold initial value region locating lower to the threshold surface with a nearly fixed $V$ value. For large $g_{\mathrm{\text{h}}}$, the threshold surface changes and is composed of two parts: one part with a nearly fixed $V$ value and the other with a nearly fixed value of $H$ variable to describe $I_{\mathrm{\text{h}}}$ inactivation probability. Although the negative pulse stimulation induces the decrease of $V$, $H$ increases to a level high enough and then the phase trajectory runs across the part with a nearly fixed $H$ value to form a post-inhibitory rebound spike. The appearance of the novel $H$ 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 $H$ threshold surface. The results present a novel nonlinear phenomenon and the corresponding dynamical mechanism related to post-inhibitory rebound spike induced by $I_{\mathrm{\text{h}}}$ current near the SNIC bifurcation point.

In this article, we consider a SIV infectious disease control system with two-threshold guidance, in which infection rate and vaccination rate are represented by a piecewise threshold function. We analyze the global dynamics of the discontinuous system using the theory of differential equations with discontinuous right-hand sides. We find some dynamical behaviors, including the disease-free equilibrium and endemic equilibria of three subsystems, a globally asymptotically stable pseudo-equilibrium and two endemic equilibria bistable, one of the two pseudo-equilibria or pseudo-attractor is stable. Conclusions can be used to guide the selection of the most appropriate threshold and parameters to achieve the best control effect under different conditions. We hope to minimize the scale of the infection so that the system can eventually stabilize at the disease-free equilibrium, pseudo-equilibrium or pseudo-attractor, corresponding to the disease disappearing or becoming endemic to a minimum extent, respectively.