A minimalist model of calcium-voltage coupling in GnRH cells

We present a minimalist model to describe the interplay between burst firing and calcium dynamics in Gonadotropin-releasing hormone (GnRH) cells. This model attempts to give a qualitative representation of Duan's model [3], and it comprises two FithzHugh-Nagumo (FHN) coupled systems describing the dynamics of the membrane potential and calcium concentration in the GnRH cells. Within the framework of our minimalist model, we find that the calcium subsystem drives burst firing by making the voltage subsystem to undergo a Hopf bifurcation. Specifically, fast relaxation oscillations occur in a specific region of the c-z plane (c being the calcium concentration, and z a calcium-dependent gating variable). Slow calcium oscillations, instead, are carried by the voltage subsystem by successive shifts of the calcium steady state, and have the net effect of an external perturbation. The full comprehension of the phase-plane of the voltage subsystem and the 3-dimensional phase-space of the calcium subsystem ultimately allows us to study the behaviours of the entire model under the change of certain parameters. Those special parameters do not necessarily follow realistic assumptions, but merely intend to mimic some pharmacological tests which have been performed experimentally and also simulated by Duan's model under the corresponding physiological considerations.