Fast excitatory modulations paradoxically reduce spiking activity in the network and single neuron with autapse: complex bifurcations and unstable limit cycles

The paradoxical phenomenon that excitatory modulations sometimes reduce neuronal activity has attracted increasing attention. In this paper, reduced activity induced by fast excitatory modulations and the corresponding mechanisms are investigated around the subcritical Hopf bifurcation of the Hodgkin–Huxley (HH) model. In networks with different topologies, two cases of reduced activity with strong synchronization, including spiking annihilation and spiking delay that behaves as mixed-mode oscillations, occur respectively at weak and strong conductances of excitatory synapses with fast decay, whereas two cases of enhanced activity appear for excitatory synapses with slow decay. Due to the strong synchronization, the network dynamics can be reproduced and explained using a single HH neuron modulated by an excitatory autapse. Six bifurcation curves that separate different reduced and enhanced activities are derived in the parameter plane of autaptic conductance and decay rate. Furthermore, the relationship between the afterpotential of reduced activity and the unstable limit cycle separating the stable spiking and resting state is obtained. As the excitatory modulations with weak and strong strength induce the afterpotential to run across the unstable limit cycle to locate at and oscillate around the stable resting state, spiking annihilation and spiking delay appear, respectively. These complex dynamics provide insights for understanding paradoxical functions of excitatory modulations in the nervous system.