Relaxation oscillations and canards in the Jirsa–Kelso excitator model: global flow perspective
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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.
- 12.Y.A. Kuznetsov, in Elements of applied bifurcation theory (Springer Science & Business Media, Dordrecht, Netherlands, 2013), Vol. 112 Google Scholar
- 13.L. Perko, in Differential equations and dynamical systems (Springer Science & Business Media, Dordrecht, Netherlands, 2013), Vol. 7 Google Scholar
- 14.C. Rocsoreanu, A. Georgescu, N. Giurgiteanu, in The FitzHugh–Nagumo model: bifurcation and dynamics (Springer Science & Business Media, Dordrecht, Netherlands, 2012), Vol. 10 Google Scholar
- 15.W.E. Sherwood, Fitzhugh–Nagumo model, in Encyclopedia of Computational Neuroscience, edited by D. Jaeger, R. Jung (Springer, New York, NY, 2013), pp. 1–11 Google Scholar