An impulsive model of endocrine regulation with two negative feedback loops

Whereas obtaining a global model of the human endocrine system remains a challenging problem, visible progress has been demonstrated in modeling its subsystems (axes) that regulate production of specific hormones. The axes are typically described by Goodwin-like cyclic feedback systems. Unlike the classical Goodwin oscillator, obeying a system of ordinary differential equations, the feedback mechanisms of brain-controlled hormonal regulatory circuits appear to be pulsatile, which, in particular, exclude the possibility of equilibrium solutions. The recent studies have also revealed that the regulatory mechanisms of many vital hormones (including e.g. testosterone and cortisol regulation) are more complicated than Goodwin-type oscillators and involve multiple negative feedback loops. Although a few “multi-loop” extensions of the classical Goodwin model have been studied in literature, the analysis of impulsive endocrine regulation models with additional negative feedbacks has remained elusive. In this paper, we address one of such models, obtained from the impulsive Goodwin-type oscillator by introducing an additional linear feedback. Since the levels of hormones’ concentrations oscillate periodically, examination of endocrine regulation circuits is primarily focused on periodic solutions. We prove the existence and uniqueness of periodic solutions of a special type, referred to as 1-cycles and featured by the unique discontinuous point in each period. Procedures for computing such a solution and testing its stability are discussed. The results are confirmed by numerical simulations.