https://hal.inria.fr/hal-01146225Fernández-García, SoledadSoledadFernández-GarcíaMycenae - Multiscale dYnamiCs in neuroENdocrine AxEs - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en AutomatiqueDesroches, MathieuMathieuDesrochesMycenae - Multiscale dYnamiCs in neuroENdocrine AxEs - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en AutomatiqueKrupa, MaciejMaciejKrupaMycenae - Multiscale dYnamiCs in neuroENdocrine AxEs - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en AutomatiqueClément, FrédériqueFrédériqueClémentMycenae - Multiscale dYnamiCs in neuroENdocrine AxEs - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en AutomatiqueA Multiple Time Scale Coupling of Piecewise Linear Oscillators. Application to a Neuroendocrine SystemHAL CCSD2015[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS][SDV.BDLR] Life Sciences [q-bio]/Reproductive BiologyClement, Frederique2015-04-28 08:35:202022-01-21 03:22:312015-04-28 08:35:20enJournal articles10.1137/1409844641We 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.