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. Introduction.Piecewise linear (PWL) systems are a family of nonsmooth systems well known for reproducing the dynamics of models coming from applications. The first examples of PWL systems were developed for the modeling of engineering problems (such as mechanical, electronic, and control device problems) [1]. Since then, the applicability of these systems has been fully demonstrated. They have been used to model not only dynamical processes coming from engineering, but also, for instance, social behaviors and financial or biological problems [3,12,13,25]. The analysis of PWL systems revealed that they exhibit rich dynamics as smooth systems, in particular limit cycles and periodic orbits [15,17], homoclinic and heteroclinic connections [5], and strange attractors [25].The celebrated FitzHugh-Nagumo system [14,27] models, in first approximation, the behavior of an excitable system, for example, a neuron. Basically, it is assumed that a neuron behaves as an electronic circuit. The van der Pol oscillator [2] can be considered as a special

Estimation of Synaptic Conductance in the Spiking Regime for the McKean Neuron Model

“…As we have mentioned in the Introduction, related works such as Tonnelier (2003) and Fernández-García et al (2015), among others, make approximations of the period T of the limit cycle in system (1), where g syn is considered to be identically zero. In this section we present a more accurate approximation of T , which also takes into account the case when g syn is not identically zero.…”

“…We note that this value coincides with the constant term of the power series expansion in C of the period T . More recently, in Fernández-García et al (2015), the authors provide an approximate expression of the period T by taking advantage of the slow invariant manifolds for 0 < C 1. In this case, T is approximated by computing the flight time of the limit cycle in each lateral region, i.e.…”

Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling

“…In [1], we considered a piecewise linear (PWL) version of the system initially proposed in [2,3], a model introduced in a context of mathematical neuroendocrinology. The original smooth model in [2,3] consists of a four dimensional (4D) system made of two coupled FitzHugh-Nagumo subsystems running on different timescales, in which the fast subsystem is forced by the slow subsystem.…”

“…The original smooth model in [2,3] consists of a four dimensional (4D) system made of two coupled FitzHugh-Nagumo subsystems running on different timescales, in which the fast subsystem is forced by the slow subsystem. We replaced in [1] the original FitzHugh-Nagumo subsystems [4,5] by two PWL equivalents (McKean caricatures [6]). This change allowed us to obtain more information on the dynamics as well as a more direct control of the quantitative features of the system output.…”

“…However, the McKean caricatures do not capture the existence of such small oscillations. Here, we propose an extension of the model considered in [1]. In the new extended model, small oscillations can happen, see Fig.…”