Mixed-mode oscillations in a multiple time scale phantom bursting system

TL;DR

This paper analyzes mixed mode oscillations in a complex mathematical model of hormone secretion, revealing local dynamics near fold points and proving the existence of a stable limit cycle in a multi-time scale system.

Contribution

It introduces a novel multi-time scale phantom burster model for GnRH secretion and provides rigorous analysis of local and global dynamics, including canard phenomena and limit cycle existence.

Findings

Characterization of small canards and sectors of rotation near folded nodes.

Proof of a unique attracting limit cycle in certain parameter regimes.

Demonstration of stability of oscillations during canard explosions.

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 nullcline 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 nullcline through a fold point of the fast nullcline. 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.