Relaxation oscillations and canards in the Jirsa–Kelso excitator model: global flow perspective

Abstract

Fenichel’s geometric singular perturbation theory and the blowup method have been very successful in describing and explaining global non-linear phenomena in systems with multiple time-scales, such as relaxation oscillations and canards. Recently, the blowup method has been extended to systems with flat, unbounded slow manifolds that lose normal hyperbolicity at infinity. Here, we show that transition between discrete and periodic movement captured by the Jirsa-Kelso excitator is a new example of such phenomena. We, first, derive equations of the Jirsa-Kelso excitator with explicit time scale separation and demonstrate existence of canards in the systems. Then, we combine the slow-fast analysis, blowup method and projection onto the the Poincar´e sphere to understand the return mechanism of the periodic orbits in the singular case, € = 0.