Paradoxical reduction and the bifurcations of neuronal bursting activity modulated by positive self-feedback

Paradoxical enhancement rather than reduction in firing activity induced by inhibitory effect is very important for both nonlinear dynamics and neuroscience. In the present paper, to build a more comprehensive viewpoint related to excitatory effect, a paradoxical phenomenon modulated by the excitatory self-feedback of autapse is extended to bursting activity in a modified Morris–Lecar model. With increasing the strength of excitatory autapse, the bursting patterns exhibit inverse period-adding bifurcations to spiking pattern via chaos, and the burst duration, the spike number per burst, and the firing frequency of bursting decrease, which is different from the traditional view that firing frequency should be enhanced. Furthermore, the paradoxical reduction in bursting activity is well explained with the nonlinear dynamics of the fast subsystem. The burst begins from neighborhood of a fold bifurcation of equilibrium point and terminates at a saddle homoclinic (SH) bifurcation. With increasing conductance of autapse, the fold bifurcation point remains unchanged, which is due to too small autaptic current near the fold point to influence the dynamics, and the SH point shifts left, which follows from the enhanced potassium current at the minimal value of the stable limit cycle. Therefore, the range between the fold and SH point becomes narrower to shorten the burst duration to reduce spike number per burst and firing frequency. The novel example of paradoxical reduction modulated by positive self-feedback and the associated bifurcation mechanism enrich the phenomena of nonlinear science and present the potential functions of excitatory autapse in the brain neurons with bursting behavior.