Puberty has remained a longstanding biological puzzle for scientists. Much remains to be understood about mechanisms that trigger this important development. What is however established is that, the activity of a certain network of neurons called Gonadotropin-Releasing Hormone (GnRH) neurons lies at the very heart of this puzzle. For my thesis, I develop and analyze a deterministic mathematical model of a GnRH neuron consisting of a system of differential equations based on experimental observations and published results. Using the results of the model we test hypotheses of possible neurobiological mechanisms mediating the transitions in the activity patterns of GnRH neurons across puberty.
For a regime of parameter values the model neuron exhibits subthreshold (small-amplitude) membrane oscillations in conjunction with spikes (large-amplitude oscillations). Their intermixing can produce complicated patterns called mixed mode oscillations (MMOs). Mathematically, the occurrence of MMOs raises important questions pertaining to the geometry of invariant manifolds of the underlying dynamical system. Using geometric singular perturbation theory methods and mathematical software packages I analyze the MMOs arising in the GnRH model. This analysis is presented through a sequence of theorems that comprise the core results of my thesis.