A mathematical model of adult GnRH neurons in mouse brain and its bifurcation analysis

GnRH neurons are hypothalamic neurons that secrete gonadotropin-releasing hormone (GnRH) which stimulates the release of gonadotropins, one of the crucial hormones for sexual development, fertility and maturation. A mathematical model was built to help elucidate the mechanisms underlying electrical bursting and synchronous Ca²⁺] transients in GnRH neurons (Lee et al., 2010). The model predicted that bursting in GnRH neurons (at least of the short-bursting type) requires the existence of a Ca²⁺]-dependent slow after-hyperpolarisation current (sI_AHP-UCL), and this predicted current was found experimentally. GnRH behaviour under a wide range of conditions (inhibition of Na⁺ channels, IP₃ receptors, Ca²⁺]-dependent K⁺ channels, or Ca²⁺ pumps, or in the presence of zero extracellular Ca²⁺]) is successfully reproduced by the model. In this paper, a simplified version of the previous model, with the same qualitative behaviour, is constructed and studied using timescale separation techniques and bifurcation analysis.