Inflection, Canards and Folded Singularities in Excitable Systems: Application to a 3D FitzHugh–Nagumo Model

Specific kinds of physical and biological systems exhibit complex Mixed-Mode Oscillations mediated by folded-singularity canards in the context of slow-fast models. The present manuscript revisits these systems, specifically by analysing the dynamics near a folded singularity from the viewpoint of inflection sets of the flow. Originally, the inflection set method was developed for planar systems [Brons and Bar-Eli in Proc R Soc A 445(1924):305–322, 1994; Okuda in Prog Theor Phys 68(6):1827–1840, 1982; Peng et al. in Philos Trans R Soc A 337(1646):275–289, 1991] and then extended to N-dimensional systems [Ginoux et al. in Int J Bifurc Chaos 18(11):3409–3430, 2008], although not tailored to specific dynamics (e.g. folded singularities). In our previous study, we identified components of the inflection sets that classify several canard-type behaviours in 2D systems [Desroches et al. in J Math Biol 67(4):989–1017, 2013]. Herein, we first survey the planar approach and show how to adapt it for 3D systems with an isolated folded singularity by considering a suitable reduction of such 3D systems to planar non-autonomous slow-fast systems. This leads us to the computation of parametrized families of inflection sets of one component of that planar (non-autonomous) system, in the vicinity of a folded node or of a folded saddle. We then show that a novel component of the inflection set emerges, which approximates and follows the axis of rotation of canards associated to folded-node and folded-saddle singularities. Finally, we show that a similar inflection-set component occurs in the vicinity of a delayed Hopf bifurcation, a scenario that can arise at the transition between folded node and folded saddle. These results are obtained in the context of a canonical model for folded-singularity canards and subsequently we show it is also applicable to complex slow-fast models. Specifically, we focus the application towards the self-coupled 3D FitzHugh–Nagumo model, but the method is generically applicable to higher-dimensional models with isolated folded singularities, for instance in conductance-based models and other physical-chemical systems.