Nonlinear mechanism for enhanced and reduced bursting activity respectively induced by fast and slow excitatory autapse

Paradoxical phenomena that inhibitory modulations enhance neuronal firing activity or excitatory modulations reduce firing activity have attracted much attention in recent studies on neurodynamics. The essential mechanism for these paradoxical phenomena is still an open problem. In the present paper, fast and slow excitatory autapses are identified to induce the opposite responses of a “Fold/Homoclinic” bursting activity in a neuronal model. On one hand, the fast decay of the excitatory autapse is identified as the essential factor to induce the reduced bursting activity. In the fast subsystem, the autaptic current is positive only in a short duration around the peak of a spike and is zero in the remaining duration of the spike. With increasing conductance of the autapse, the positive autaptic current around the peak induces an increased maximal membrane potential, which induces the potassium current during the downstroke of the spike increased, resulting in a decreased minimal membrane potential. Such reduced minimal membrane potential induces left shift of the homoclinic bifurcation point of the fast subsystem, leading to a shortened duration of the burst and less spikes per burst. These present the nonlinear and current mechanisms for the paradoxical response of the bursting. On the other hand, for an autapse with slow decay, the autaptic current is positive in the whole duration of a spike of the fast subsystem. Within increasing conductance of the autapse, the autaptic current in the whole duration (containing the downstroke) increases. Then, the total current during the downstroke increases, resulting in the elevated minimal membrane potential. Such elevated minimal membrane potential induces right shift of the homoclinic bifurcation, which then results in a widened burst duration and more spikes per burst. These underlie the nonlinear and current mechanisms for the common response of the bursting. Our results present deep insight and comprehensive recognition to autapse dynamics, enrich the content of nonlinear dynamics, and provide a potential strategy to modulate the neuronal bursting behavior.