We study the effect of diabetic deficiencies on the production of an oscillatory ultradian regime using a deterministic nonlinear model which incorporates two physiological delays. It is shown that insulin resistance impairs the production of oscillations by dampening the ultradian cycles. Four strategies for restoring healthy regulation are explored. Through the introduction of an instantaneous glucose-dependent insulin response, explicit conditions for the existence of periodic solutions in the linearised model are formulated, significantly reducing the complexity of identifying an oscillatory regime. The model is thus shown to be suitable for representing the effect of diabetes on the oscillatory regulation and for investigating pathways to reinstating a physiological healthy regime.

Glucose regulation is an essential function of the human body which enables energy to be effectively utilized by the brain, organs and muscles. This regulation operates in a cyclic manner, in different periodic regimes. Indeed, ultradian rhythms with a period of 70 to 150 minutes have been clinically observed in healthy patients under various glucose stimulation patterns. Various models of these oscillations in plasma glucose and insulin have shown that the presence of two delays in hepatic glycogenesis and pancreatic insulin secretion provide a pathway for explaining these oscillations. The efficacy of this control is typically reduced in the presence of diabetes. In this contribution, we adopt the presence and the accurate tuning of ultradian rhythms as a criterion for healthy glucose regulation. We then investigate a model with two delays of these ultradian rhythms which incorporates parameters accounting for insulin sensitivity and insulin secretion. Additionally, the effect of diabetic deficiencies on this feedback loop is explored by quantifying the joint contribution of delays and diabetic parameters on the limit cycle of this model, which is generated through a Hopf bifurcation. Strategies for restoring an oscillatory regime in a physiologically appropriate range are discussed. Finally, a simple polynomial model of the oscillations is introduced to give further insight into the influence of each physiological subsystem. The approach provides a quantified relationship between diabetic impairments and the plasma glucose-insulin feedback loop.

Characterising the glycemic response to a glucose stimulus is an essential tool for detecting deficiencies in humans such as diabetes. In the presence of a constant glucose infusion in healthy individuals, it is known that this control leads to slow oscillations as a result of feedback mechanisms at the organ and tissue level. In this paper, we provide a novel quantitative description of the dependence of this oscillatory response on the physiological functions. This is achieved through the study of a model of the ultradian oscillations in glucose-insulin regulation which takes the form of a nonlinear system of equations with two discrete delays. While studying the behaviour of solutions in such systems can be mathematically challenging due to their nonlinear structure and non-local nature, a particular attention is given to the periodic solutions of the model. These arise from a Hopf bifurcation which is induced by an external glucose stimulus and the joint contributions of delays in pancreatic insulin release and hepatic glycogenesis. The effect of each physiological subsystem on the amplitude and period of the oscillations is exhibited by performing a perturbative analysis of its periodic solutions. It is shown that assuming the commensurateness of delays enables the Hopf bifurcation curve to be characterised by studying roots of linear combinations of Chebyshev polynomials. The resulting expressions provide an invaluable tool for studying the interplay between physiological functions and delays in producing an oscillatory regime, as well as relevant information for glycemic control strategies.