Mixed-Mode Oscillations in a Multiple Time Scale Phantom Bursting System

Abstract

In this work we study mixed mode oscillations in a model of secretion of GnRH (Gonadotropin Releasing Hormone). The model is a phantom burster consisting of two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The forcing system (Regulator) evolves on the slowest scale and acts by moving the slow null-cline of the forced system (Secretor). There are three modes of dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady state) and small oscillations related to the passage of the slow null-cline through a fold point of the fast null-cline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain two results; one on the local dynamics near the fold in the parameter regime corresponding to the presence of small oscillations and the other on the global dynamics, more specifically on the existence of an attracting limit cycle. Our local result is a rigorous characterization of small canards and sectors of rotation in the case of folded node with an additional time scale, a feature allowing for a clear geometric argument. The global result gives the existence of an attracting unique limit cycle, which, in some parameter regimes, remains attracting and unique even during passages through a canard explosion.

Modeling the Dynamics of Gonadotropin‐Releasing Hormone (GnRH) Secretion in the Course of an Ovarian Cycle

“…In particular, during the transition from a surge back to a pulsatile phase, a pause consisting of small oscillations superimposed on a long-duration pulse may occur (Krupa et al, 2012) for some parameter values. As most experimental protocols have focused on the transition corresponding to the GnRH surge triggering rather than the reverse transition corresponding to the resumption of the pulsatile regime, the biological mechanisms underlying the transition from the surge mode back to the pulsatile mode are still poorly understood, and the possible existence of a pause in the model is quite an intriguing observation.…”

Nonlinear oscillatory mixing in the generalized Landau scenario

“…Mixed-mode oscillations are typically observed in fast-slow systems involving local phasespace phenomena like folded singularities, canard orbits, or singular Hopf bifurcations, through which the structure in the oscillatory behavior is originated. With such a kind of mechanisms, structured periodic oscillations with three components of different amplitudes and frequencies have been numerically obtained in two feedforward coupled FitzHugh-Nagumo systems [8] and in a four-dimensional piecewise linear system [9]. However, the combination of such a kind of mechanisms to provide a generic way for the achievement of really complex oscillations with arbitrarily large numbers of oscillatory modes seems not obvious.…”

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

“…The smooth model, with three different time scales, can display Mixed-Mode Oscillations (MMOs) of three different types [7]. Apart from the pulses and surge, a pause may exist between the end of the surge and the resumption of the pulses, where small oscillations take place, see Fig.…”

“…Apart from the pulses and surge, a pause may exist between the end of the surge and the resumption of the pulses, where small oscillations take place, see Fig. 1 in [7]. However, the McKean caricatures do not capture the existence of such small oscillations.…”

“…In the smooth model, such a compensation mechanism has been explored numerically [7] it turned out to be too complicated to be studied analytically. The PWL nature of the system proposed in this work allows us to better understand the dynamical mechanisms than in the case of the smooth model.…”