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.
Highlights
The Jirsa–Kelso excitator model is a class of excitable planar systems proposed as a minimal model to describe generation of rhythmic and discrete human movement [7]
As in the case of the FitzHugh–Nagumo model (FHN) model the relaxation oscillations in the Jirsa–Kelso excitator (JKE) appear through canard explosion, a rapid growth of periodic orbits’ amplitude that happens in an exponentially small range of the control parameter [10]
In this paper we presented a detailed analysis of the mechanism that leads to appearance of relaxation oscillations in the Jirsa–Kelso excitator
Summary
The Jirsa–Kelso excitator model is a class of excitable planar systems proposed as a minimal model to describe generation of rhythmic and discrete human movement [7]. In (1) and (2), u(t) is interpreted as a position and v(t) as a velocity of the movement, a and b are intrinsic model parameters, T and are a time scale constants and I(t) is an external stimulus input (in the rest of the paper we consider the autonomous systems, i.e. I = 0). The modelling approach introduced in [7] is motivated by excitable systems with time-scale separation. Since the transformation (5) is homeomorphism (for > 0) the intrinsic dynamics of the JKE (4) and FHN (6) systems are equivalent for > 0 [7]. The singularity in (5) for = 0 implies that the global mechanism responsible for the canard cycles and relaxation oscillations in the JKE is different than the one in the FHN; i.e. the two systems have different critical manifolds. In the rest of the paper we present a detailed study of the critical manifold C0 of the uncoupled JKE model
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
