Multitype Activity Coexistence in an Inertial Two-Neuron System with Multiple Delays

In this paper, an inertial two-neuron system with multiple delays is analyzed to exhibit the effect of time delays on system dynamics. The parameter region with multiple equilibria is obtained employing the pitchfork bifurcation of trivial equilibrium. The stability analysis illustrates that two nontrivial equilibria are both stable for any delays. It implies that the neural system exhibits a stability coexistence of two resting states. Further, due to the existence of multiple delays, the neural system has a periodic activity around the trivial equilibrium via Hopf bifurcation. Finally, numerical simulations are employed to illustrate many richness coexistence for multitype activity patterns. Employing the period-adding route and fold bifurcation of periodic orbit, the neural system may have multistability coexistence of two resting states, two ASP-3s (anti-symmetric periodic activity with period three), one SSP-1 (self-symmetric periodic activity with period one), and one quasi-periodic spiking. Additionally, with increasing delay, quasi-periodic spiking evolves into chaos behavior.