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

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 a 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.