Ultradian rhythms in glucose regulation: a mathematical assessment

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.