Bifurcations of Time-Delay-Induced Multiple Transitions Between In-Phase and Anti-Phase Synchronizations in Neurons with Excitatory or Inhibitory Synapses

Abstract

Time-delay-induced synchronous behaviors and synchronization transitions have been widely investigated for coupled neurons, and they play important roles for physiological functions. In the present study, time-delay-induced synchronized subthreshold oscillations were simulated, and the bifurcations underlying the synchronized behaviors were identified for a pair of coupled FitzHugh–Nagumo neurons. Multiple transitions between in-phase and anti-phase synchronizations induced by the time delay were simulated for the inhibitory and excitatory couplings. Subcritical or supercritical Hopf bifurcations and the stability of the Hopf-bifurcating periodic subthreshold oscillations were acquired using center manifold reduction and normal form theory. The in-phase or anti-phase synchronizations of the stable periodic subthreshold oscillations, which appear for multiple values of the time delay, were interpreted with the related eigenspace. The distributions of the different dynamical behaviors, including the synchronizations and bifurcations in the two-parameter plane of the time delay and coupling strength, were acquired for both types of synapses, and the different roles of the inhibitory and excitatory couplings on the synchronization transitions were compared.