Eur. Phys. J. Special Topics 227, 591–601 (2018)
https://doi.org/10.1140/epjst/e2018-00129-2

Regular Article

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

¹ Department of Mathematics, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, Devon EX4 4QF, UK
² Living Systems Institute, University of Exeter, Stocker Road, Exeter EX4 4QD, UK
³ EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter EX4 4QJ, UK

^a e-mail: K.Tsaneva-Atanasova@exeter.ac.uk

Received: 30 November 2017
Received in final form: 8 March 2018
Published online: 4 October 2018

Abstract

Fenichel’s geometric singular perturbation theory and the blow-up 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 blow-up 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, blow-up method and projection onto the Poincaré sphere to understand the return mechanism of the periodic orbits in the singular case, ϵ = 0.