Theoretical Analysis of Synchronization Phenomena in Two Pulse-Coupled Resonate-and-Fire Neurons

We address the synchronization properties of two pulse-coupled resonate-and-fire neuron (RFN) models. The RFN model is a spiking neuron model that has second-order membrane dynamics with a threshold and a reset value. Due to such dynamics, the RFN model exhibits subthreshold oscillation of the membrane potential, and is sensitive to the timing of stimuli. So far the existence of anti-phase synchronization states and their stability in a system of two pulse-coupled RFN models have been reported. However, the effects of the reset value after firing on such synchronization states have not been considered. The reset value may affect the sensitivity to the input timing, leading to change in synchronization properties in the pulse-coupled RFN models. We newly found out-of-phase burst synchronization states and related bifurcation phenomena depending on the coupling strength in the system as the reset value was changed. Focusing on the symmetry of the system, we analyzed the stability of such phenomena by using a firing time difference map constructed from ID return maps with respect to firing time difference between two neurons. The analyses revealed the global stability of the out-of-phase synchronization states and the existence of the type I intermittency chaotic behavior.