Spike-Adding Canard Explosion in a Class of Square-Wave Bursters
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Spike-Adding Canard Explosion in a Class of Square-Wave Bursters Paul Carter1 Received: 20 May 2018 / Accepted: 18 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper examines a spike-adding bifurcation phenomenon whereby smallamplitude canard cycles transition into large-amplitude bursting oscillations along a single continuous branch in parameter space. We consider a class of three-dimensional singularly perturbed ODEs with two fast variables and one slow variable and singular perturbation parameter ε 1 under general assumptions which guarantee such a transition occurs. The primary ingredients include a cubic critical manifold and a saddle homoclinic bifurcation within the associated layer problem. The continuous transition from canard cycles to N -spike bursting oscillations up to N ∼ O(1/ε) spikes occurs upon varying a single bifurcation parameter on an exponentially thin interval. We construct this transition rigorously using geometric singular perturbation theory; critical to understanding this transition are the existence of canard orbits and slow passage through the saddle homoclinic bifurcation, which are analyzed in detail. Keywords Bursting oscillations · Spike-adding · Canards · Geometric singular perturbation theory · Saddle-homoclinic bifurcation Mathematics Subject Classification 34C25 · 34E17 · 34D15 · 37G15 · 92B25
1 Introduction The phenomenon of bursting has been widely studied in models of neurons and neuroendocrine cells, as well as other excitable media, including physical systems such as semiconductor lasers (Al-Naimee et al. 2009; Ruschel and Yanchuk 2017), or in chemical reactions (Rinzel and Troy 1982). These solutions are characterized by alternation
Communicated by Paul Newton.
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Paul Carter [email protected] School of Mathematics, University of Minnesota, Minneapolis, USA
123
Journal of Nonlinear Science
between slow quiescent phases and active bursting phases comprised of a sequence of action potentials or spikes and can be time periodic or aperiodic. One of the earliest models was introduced by Chay and Keizer (1983) to describe bursting dynamics in pancreatic beta cells, which formed the basis in Rinzel (1985), Rinzel (1987) for analyzing the bursting phenomenon in the context of singularly perturbed, or fast–slow, ordinary differential equations. This has since been a primary mathematical formulation for understanding bursting in numerical and analytical studies. In this context, bursting solutions can frequently arise as periodic orbits, in which the active phase is governed by oscillations on the fast timescale, and the quiescent phase is associated with drift along a slow manifold. A feature which is prevalent in many bursting models is that of spike-adding, in which variation in system parameters can result in additional spikes during the bursting phase. This has been demonstrated and analyzed numerically in a variety of bursting models (Desroches et al. 2016a, 2013; Guckenheimer and Kuehn 2009; Linaro et al. 2012; Nowac
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