Inhibitory autapse with time delay induces mixed-mode oscillations related to unstable dynamical behaviors near subcritical Hopf bifurcation

Mixed-mode oscillations (MMOs) consisting of spikes alternating with a series of subthreshold oscillations have been observed in various neurons related to some physiological functions. In the present paper, inhibitory-autapse-induced MMOs are simulated by using the Hodgkin-Huxley neuron model, and the underlying dynamical mechanism is identified to be related to dynamics of unstable behaviors near subcritical Hopf bifurcation. For the monostable spiking, a delayed inhibitory current pulse activated by a spike can suppress the phase trajectory corresponding to depolarization phase of the next spike to the unstable focus nearby or the neighborhood outside of unstable limit cycle, respectively. Then the trajectory rotates multiple cycles away and converges to the stable limit cycle, resulting in an evolution process of membrane potential from small-amplitude subthreshold oscillations to a large-amplitude spike, i.e., MMOs. For the spiking coexisting with the resting state, inhibitory autapse induces MMOs and resting state from the spiking. The difference in the MMOs from those induced by the excitatory autapse is identified. The result presents the underlying nonlinear mechanisms of inhibitory autapse to suppress the neuronal firing and reveals the potential role to control the neuronal firing patterns near subcritical Hopf bifurcation.