A biomathematical model of time-delayed feedback in the human male hypothalamic-pituitary-Leydig cell axis

We develop, implement, and test a feedback and feedforward biomathematical construct of the male hypothalamic [gonadotropin-releasing hormone (GnRH)]-pituitary [luteinizing hormone (LH)]-gonadal [testosterone (Te)] axis. This stochastic differential equation formulation consists of a nonstationary stochastic point process responsible for generating episodic release of GnRH, which is modulated negatively by short-loop (GnRH) and long-loop (Te) feedback. Pulsatile GnRH release in turn drives bursts of LH secretion via an agonistic dose-response curve that is partially damped by Te negative feedback. Circulating LH stimulates (feedforward) Te synthesis and release by a second dose response. Te acts via negative dose-responsive feedback on GnRH and LH output, thus fulfilling conditions of a closed-loop control system. Four computer simulations document expected feedback performance, as published earlier for the human male GnRH-LH-Te axis. Six other simulations test distinct within-model coupling mechanisms to link a circadian modulatory input to a pulsatile control node so as to explicate the known 24-h variations in Te and, to a lesser extent, LH. We conclude that relevant dynamic function, internodal dose-dependent regulatory connections, and within-system time-delayed coupling together provide a biomathematical basis for a nonlinear feedback-feedforward control model with combined pulsatile and circadian features that closely emulate the measurable output activities of the male hypothalamic-pituitary-Leydig cell axis.