Excitatory autapse induces different cases of reduced neuronal firing activities near Hopf bifurcation

Abstract Excitatory autapse with short time delay can lower the firing frequency of spiking behaviors near subcritical Hopf bifurcation, which is interpreted with type II phase response curve wherein an excitatory impulse current applied at the phase following a spike can induce the next spike delayed. Such a nonlinear phenomenon is different from the common viewpoint that excitatory effect typically induces enhancement of neuronal firing activities. In this paper, more cases of reduction of firing activities induced by excitatory autapse than the one mentioned above are simulated in the Hodgkin-Huxley model with relatively short time delay, and the dynamical mechanism is provided with phase trajectories modulated by excitatory autaptic current and bifurcations. For the monostable spiking near the subcritical Hopf bifurcation, one novel case is induced. The delayed autaptic current pulse evoked by a spike induces the trajectory moved to the neighborhood of the unstable focus, and after the pulse, the subthreshold oscillations with ascending amplitude appear until they evolve to the next spike, which results in mixed subthreshold oscillations and a single spike. For the spiking coexisting with the resting state, which locates between the subcritical Hopf bifurcation and saddle-node bifurcation of limit cycles, two novel cases are induced. One is the mixed-mode oscillations, and the trajectory of subthreshold oscillations begins from the neighborhood outside of the unstable limit cycle, which is induced by the delayed autaptic current pulse. The other is induced by the autaptic current pulse that can cause the trajectory to run into the unstable limit cycle, which leads to the appearance of subthreshold oscillations with descending amplitude after the pulse, and finally evolves to the resting state corresponding to the coexisting stable focus. The results present novel cases that excitatory effects can suppress rather than enhance firing activities and the corresponding dynamical mechanism related to bifurcations, which enrich the study of nonlinear dynamics and neurodynamics, and provide the potential functions of excitatory autapse.

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