Abstract
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.
Highlights
Homoeostasis refers to the body’s ability to maintain certain variables within a narrow range and is a result of the negative feedback loops which occur within the body (Cannon 1932; Thomas et al 1995)
While many biological systems can be modelled with ordinary differential equations (ODEs), delays can so often prove crucial in realistically replicating core aspects of these systems (Thomas et al 1995)
This paper focuses on the ultradian rhythms that occur within the glucose–insulin system
Summary
Homoeostasis refers to the body’s ability to maintain certain variables within a narrow range and is a result of the negative feedback loops which occur within the body (Cannon 1932; Thomas et al 1995). To answer this question, we introduce a perturbative scheme for the periodic solutions of a two-compartment delay differential equation (DDE) model of the ultradian rhythms in the glucose–insulin regulatory system based on the Poincaré–Lindstedt (P–L) method. Crossing curves can be described in this situation using geometric arguments (Gu et al 2005), we show that characteristic frequencies can be obtained by studying the zeroes of linear combinations of Chebyshev polynomials of the first type This is achieved by assuming the commensurateness of delays, that is, τ2 = κτ, where κ ∈ Z+ is a coupling parameter. The polynomial form of the system contrasts with previous models in which the hepatic and pancreatic secretions are represented by bounded sigmoidal functions (Bennett and Gourley 2004; Huard et al 2015; Li et al 2006), it is appropriate for studying dynamics in the neighbourhood of the limit cycle.
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