Mathematical description of the ultradian oscillations in the glucose-insulin regulatory system

Bridgewater, Adam (2018) Mathematical description of the ultradian oscillations in the glucose-insulin regulatory system. Doctoral thesis, Northumbria University.

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Abstract

Nonlinear delay-differential equations represent infinite-dimensional systems for which characterising the behaviour of solutions can be technically challenging. Their appearance in many applied fields, such as population biology, is in great part due to their ability to model dynamics with non-instantaneous effects. These phenomena often exhibit behaviour which does not occur in the associated non-delayed counterpart. In particular, it is known that delays can trigger the occurrence of oscillatory behaviour and can, in several cases, bring evidence for mechanisms underlying oscillations in biological systems. Oscillatory dynamics play an important role in the accurate regulation of hormones. Such rhythms in the ultradian regime have been observed in multiple physiological areas such as the HPA-axis, phases of sleep, and the glucose regulatory system.

In this contribution, the effect of diabetic deficiencies on the production of an oscillatory ultradian regime is studied 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. 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.

Furthermore, a simplified nonlinear polynomial model of the ultradian oscillations in glucose-insulin regulation, at the organ and tissue level, is studied. Particular attention is given to its periodic solutions, arising from a Hopf bifurcation which is induced by delays in pancreatic insulin release, hepatic glycogenesis and a glucose infusion. The model also includes terms accounting for insulin independent and dependent glucose utilisation as well as insulin clearance.

The effect of each of these functions on the amplitude and period of the oscillations is exhibited by performing a Poincaré perturbative analysis of its periodic solutions.

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