Symmetric coupling of multiple timescale systems with Mixed-Mode Oscillations and synchronization

Abstract

We analyze a six-dimensional slow–fast system consisting of two symmetrically coupled identical oscillators. Each oscillator is a three-dimensional slow–fast system formed by a FitzHugh–Nagumo model with an additional slow variable. Individually, each three-dimensional subsystem admits an attractive Mixed-Mode Oscillations limit cycle, featuring small oscillations due to the presence of a folded singularity. We consider a linear symmetric coupling of the fast variable of each oscillator upon one of the slow equations of the other one and study the synchronization properties between oscillators according to the value of the coupling gain parameter.We examine theoretically and numerically both excitatory and inhibitory cases associated with positive and negative values of the coupling gain. Apart from in-phase locking synchronization and antiphase synchronization patterns, the coupled system generates phase-locking almost-in-phase synchronization, relaxation loss of one of the oscillators and total oscillation death, intertwined with complex transitions involving period doubling cascades, period adding phenomena and chaos. We present a theoretical proof of the behavior repartition according to the coupling gain parameter both in inhibitory and excitatory cases. In the case of antiphase synchronization, we prove the increase in the oscillation frequency of the 6D system asymptotic orbit as the negative coupling parameter value increases. In the case of excitatory coupling, we point out the role of Mixed-Mode Oscillations in the birth of almost-in-phase and in-phase locking synchronization patterns and the transitions between them.