A nonlinear mathematical model for pulsatile discharges of luteinizing hormone mediated by hypothalamic and extra-hypothalamic pathways

Abstract

A mathematical model of hormone secretion in the hypothalamo-pituitary-gonadal axis in man is extended to incorporate two different neural pathways, which have been suggested by clinical data to be capable of stimulating pulsatile discharges of LH (luteinizing hormone) independently of each other. Analysis of the nonlinear model is carried out through the use of geometric singular perturbation methods. In this way, existence of a limit cycle is proved for certain ranges of the system parameters. When the LH secretion rate independent of the hypothalamus is assumed constant, dropping the hypothalamus stimulated secretion term from the model blocks the hypothalamus pathway, implying that sustained oscillations in the hormone levels may not be attainable. Therefore, a sinusoidal term is incorporated into the model so that the system can still exhibit pulsatile LH secretion independent of the hypothalamus mediation. It is shown, by a construction of a bifurcation diagram, that the pulsatile hormone secretion can develop into chaotic dynamics when the amplitude of oscillation stimulated by extra-hypothalamic structures is high enough to disturb the synchrony of hypothalamic control. The resulting numerical simulation is found to compare well with the clinically observed data.