Synchrony due to parametric averaging in neurons coupled by a shared signal

Gonadotropin-releasing hormone (GnRH) is a decapeptide secreted by GnRH neurons located in the hypothalamus. It is responsible for the onset of puberty and the regulation of hormone release from the pituitary. There is a strong evidence suggesting that GnRH exerts an autocrine regulation on its own release via three types of G-proteins [L.Z. Krsmanovic, N. Mores, C.E. Navarro, K.K. Arora, K.J. Catt, An agonist-induced switch in G protein coupling of the gonadotropin-releasing hormone receptor regulates pulsatile neuropeptide secretion, Proc. Natl. Acad. Sci. 100 (2003) 2969–2974]. A mathematical model based on this proposed mechanism has been developed and extended to explain the synchrony observed in GnRH neurons by incorporating the idea of a common pool of GnRH [A. Khadra, Y.X. Li, A model for the pulsatile secretion of gonadotropin-releasing hormone from synchronized hypothalamic neurons, Biophys. J. 91 (2006) 74–83]. This type of coupling led to a very robust synchrony between these neurons. We aim in this paper to reduce the one cell model to a two-variable model using quasi-steady state (QSS) analysis, to further examine its dynamics analytically and geometrically. The concept of synchrony of a heterogeneous population will be clearly defined and established for certain cases, while, for the general case, two different types of phases are introduced to gain more insight on how the model behaves. Bifurcation diagrams for certain parameters in the one cell model are also shown to explain some of the phenomena observed in a coupled population. A comparison between the population model and an averaged two-variable model is also conducted.