To understand the sophisticated control mechanisms of the human's endocrine system is a challenging task that is a crucial step towards precise medical treatment of many dysfunctions and diseases. Although mathematical models describing the endocrine system as a whole are still elusive, recently some substantial progress has been made in analyzing theoretically its subsystems (or axes) that regulate the production of specific hormones. Secretion of many vital hormones, responsible for growth, reproduction and metabolism, is orchestrated by feedback mechanisms that are similar in structure to the model of simple genetic oscillators, proposed first by B.C. Goodwin. Unlike the celebrated Goodwin's model, the endocrine regulation mechanisms are in fact known to have non-cyclic structures and involve multiple feedbacks; a Goodwin-type model thus represents only a part of such a complicated mechanism. In this paper, we examine a non-cyclic feedback system of hormonal regulation, obtained from the classical Goodwin's oscillator by introducing an additional negative feedback. We establish global properties of this model and show, in particular, that the local instability of its unique equilibrium implies that almost all system's solutions oscillate; furthermore, under additional restrictions these solutions converge to periodic or homoclinic orbits.@en

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.@en

Whereas development of mathematical models describing the endocrine system as a whole remains a challenging problem, visible progress has been demonstrated in modeling its subsystems, or axes. Models of hormonal axes portray only the most essential interactions between the hormones and can be described by low-order systems of differential equations. This paper analyzes the properties of a novel model of a hypothalamic-pituitary axis, portraying the interactions in a chain of a release hormone (secreted by the hypothalamus), a tropic hormone (produced by the pituitary gland) and an effector hormone (secreted by a target gland). This model, unlike previously published ones, captures two prominent features of neurohormonal regulation systems, namely, the pulsatile (episodic) production of the release hormone and a complex non-cyclic feedback mechanism that maintains the involved hormone concentrations within certain biological limits. At the same time, the discussed model is analytically tractable; in particular, the existence of a so-called 1-cycle featured by a single pulse over one period is proven mathematically.@en

Whereas development of mathematical models describing the endocrine system as a whole remains a challenging problem, visible progress has been demonstrated in modeling its subsystems, or axes. Models of hormonal axes portray only the most essential interactions between the hormones and can be described by low-order systems of differential equations. This paper analyzes the properties of a novel model of a hypothalamic-pituitary axis, portraying the interactions in a chain of a release hormone (secreted by the hypothalamus), a tropic hormone (produced by the pituitary gland) and an effector hormone (secreted by a target gland). This model, unlike previously published ones, captures two prominent features of neurohormonal regulation systems, namely, the pulsatile (episodic) production of the release hormone and a complex non-cyclic feedback mechanism that maintains the involved hormone concentrations within certain biological limits. At the same time, the discussed model is analytically tractable; in particular, the existence of a so-called 1-cycle featured by a single pulse over one period is proven mathematically.@en