  We investigate a model of the GnRH pulse and surge generator, with the
definite aim of constraining the model GnRH output with respect to a
physiologically relevant list of specifications. The alternating pulse and
surge pattern of secretion results from the interaction between a GnRH
secreting system and a regulating system exhibiting fast-slow dynamics. The
mechanisms underlying the behavior of the model are reminded from the study of
the Boundary-Layer System according to the "dissection method" principle. Using
singular perturbation theory, we describe the sequence of bifurcations
undergone by the regulating (FitzHugh-Nagumo) system, encompassing the rarely
investigated case of homoclinic connexion. Basing on pure dynamical
considerations, we restrict the space of parameter search for the regulating
system and describe a foliation of this restricted space, whose leaves define
constant duration ratios between the surge and the pulsatility phase in the
whole system. We propose an algorithm to fix the parameter values to also meet
the other prescribed ratios dealing with amplitude and frequency features of
the secretion signal. We finally apply these results to illustrate the dynamics
of GnRH secretion in the ovine species and the rhesus monkey.
