Bifurcation mechanism of not increase but decrease of spike number within a neural burst induced by excitatory effect

Nonlinear dynamics is identified to play very important roles in identifying the complex phenomenon, dynamical mechanism, and physiological functions of neural electronic activities. In the present paper, a novel viewpoint that the excitatory stimulus cannot enhance but reduce the number of the spikes within a burst, the novel viewpoint which is different from the traditional viewpoint, is proposed and is explained with the nonlinear dynamics. When the impulse current or the autaptic current with suitable strength is used in the suitable phase within the quiescent state of the bursting pattern of the Rulkov model, a novel firing pattern with reduced number of spikes within a burst is evoked. The earlier the application phase of the current within the quiescent state, the higher the threshold of the current strength to evoke the novel firing pattern is and the less the number of the spikes within a burst of the novel firing pattern. Moreover, such a novel phenomenon can be explained by the intrinsic nonlinear dynamics of the bursting combined with the characteristics of the current. The nonlinear behaviors of the fast subsystem of the Rulkov model are acquired by the fast and slow variable dissection method, respectively. For the fast subsystem, there exist a stable node with lower membrane potential, a stable limit cycle with higher membrane potential, a saddle serving as the border between the stable node and limit cycle, a saddle-node bifurcation, and a homoclinic orbit bifurcation. When external simulation is not received, the bursting pattern of the Rulkov model exhibits behavior alternating between the spikes corresponding to the limit cycle of the fast subsystem and quiescent state of the fast subsystem, which is located within the parameter region between the saddle-node bifurcation point and the homoclinic orbit bifurcation point of the fast subsystem. The spikes begin with the saddle-node bifurcation and end with the homoclinic orbit bifurcation. As the bifurcation parameter turns close to the homoclinic orbit bifurcation, the disturbation or stimulus that can induce the transition from the quiescent state to the spikes becomes strong. Therefore, as the application phase of the current within the quiescent state becomes earlier, the strength threshold of the current that can induce the transition from the quiescent state to the spikes becomes stronger, and the initial phase of the spikes becomes closer to the homoclinic orbit bifurcation, which leads the parameter region of the spikes to become shorter and then leads the number of spikes within a burst to turn less. It is the dynamical mechanism of the decrease of the spike number induced by the excitatory currents. The results enrich the nonlinear phenomenon and dynamical mechanism, present a novel viewpoint for the excitatory effect, and provide a new approach to modulating the neural bursting patterns.