Nonlinear mechanisms for opposite responses of bursting activities induced by inhibitory autapse with fast and slow time scale

Fast and slow inhibitory synapses, respectively, induce anti- and in-phase synchronizations, showing different roles of synapse with different time scales. In the present paper, the opposite roles of inhibitory autapses with fast and slow time scales on modulating bursting activities are investigated in a theoretical model, which present a novel viewpoint on the inhibitory autapse and bursting observed in the brain neurons. Firstly, the inhibitory autapses with fast and slow decay time constant are identified to induce enhancement and reduction of bursting activities, respectively, which presents a paradoxical phenomenon and a common phenomenon. Secondly, the bifurcation mechanisms for the paradoxical and common responses are acquired. As autaptic conductance increases, for fast autapse, saddle-node (SN) bifurcation point of the fast subsystem is nearly fixed, while saddle-homoclinic orbit (SH) bifurcation point moves to right to widen the burst region and enhance the bursting activities, while more left shift of SN bifurcation curve than SH bifurcation curve to shorten the burst region and reduce the bursting activities, which is acquired by a novel fast–slow variable dissection method with two slow variables (slow variable C of the model and gating variable of autapse S ). Finally, fast and slow dynamics of autaptic current underlying the bifurcation mechanisms are acquired. For the fast autapse, the autaptic current is negative during spike to reduce the spike amplitude to shift the SH point, while it exhibits fast decay to zero between spikes (i.e., node), which cannot influence the SN point. For the slow autapse, the autaptic current exhibits negative value during both the burst and quiescent (between burst) due to slow decay, which induces left shift of both the SN and SH bifurcation curves. The results of the opposite response of bursting to fast and slow autaptic currents present potential functions and diverse dynamics of autapse with different time scales in brain neurons, a novel case of counterintuitive phenomenon related to inhibitory self-feedback, and a novel performance process of analysis method to complex bursting with two slow variables.