A Multiple Time Scale Coupling of Piecewise Linear Oscillators. Application to a Neuroendocrine System

Abstract : We analyze a four-dimensional slow-fast piecewise linear system consisting of two coupled McKean caricatures of the FitzHugh--Nagumo system. Each oscillator is a continuous slow-fast piecewise linear system with three zones of linearity. The coupling is one-way, that is, one subsystem evolves independently and is forcing the other subsystem. In contrast to the original FitzHugh--Nagumo system, we consider a negative slope of the linear nullcline in both the forcing and the forced system. In the forcing system, this lets us, by just changing one parameter, pass from a system having one equilibrium and a relaxation cycle to a system with three equilibria keeping the relaxation cycle. Thus, we can easily control the changes in the oscillation frequency of the forced system. The case with three equilibria and a linear slow nullcline is a new configuration of the McKean caricature, where the existence of the relaxation cycle was not studied previously. We also consider a negative slope of the $y$-nullcline in the forced system that enables us to reproduce a quasi-steady state called the surge. We analyze not only the qualitative behavior of the four-dimensional system, but also quantitative aspects such as the period, frequency, and amplitude of the oscillations. The system is used to reproduce all the features endowed in a former smooth model and reproduce the secretion pattern of the hypothalamic neurohormone GnRH along the ovarian cycle in different species.
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https://hal.inria.fr/hal-01146225
Contributor : Frederique Clement <>
Submitted on : Tuesday, April 28, 2015 - 8:35:20 AM
Last modification on : Friday, April 19, 2019 - 1:26:04 PM

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Soledad Fernández-García, Mathieu Desroches, Maciej Krupa, Frédérique Clément. A Multiple Time Scale Coupling of Piecewise Linear Oscillators. Application to a Neuroendocrine System. SIAM Journal on Applied Dynamical Systems, Society for Industrial and Applied Mathematics, 2015, 14 (2), pp.31. ⟨10.1137/140984464⟩. ⟨hal-01146225⟩

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