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.

中文翻译：

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周期性扰动的多延迟非线性葡萄糖动力学的振幅和频率变化

表征对葡萄糖刺激的血糖反应是检测人类糖尿病等缺陷的重要工具。已知在健康个体中存在恒定葡萄糖输注的情况下，由于器官和组织水平的反馈机制，这种控制导致振荡缓慢。在本文中，我们提供了这种振荡反应对生理功能的依赖性的新型定量描述。这是通过研究葡萄糖-胰岛素调节中的超店面振荡模型来实现的，该模型采用具有两个离散延迟的非线性方程组的形式。由于此类系统的非线性结构和非局部性质，因此研究其解的行为在数学上可能具有挑战性，特别注意模型的周期解。这些起因于霍普夫分叉，霍普夫分叉是由外部葡萄糖刺激以及胰腺胰岛素释放延迟和肝糖原形成的延迟共同作用引起的。每个生理子系统对振荡幅度和周期的影响通过对其周期解进行扰动分析来体现。结果表明，假设时延相当，可以通过研究切比雪夫多项式线性组合的根来表征霍普夫分叉曲线。所得的表达式为研究生理功能与产生振荡状态的延迟之间的相互作用以及血糖控制策略的相关信息提供了宝贵的工具。这些起因于霍普夫分叉，霍普夫分叉是由外部葡萄糖刺激以及胰腺胰岛素释放延迟和肝糖原形成的延迟共同作用引起的。每个生理子系统对振荡的幅度和周期的影响通过对其周期解进行扰动分析来体现。结果表明，假设时延相当，可以通过研究切比雪夫多项式线性组合的根来表征霍普夫分叉曲线。所得的表达式为研究生理功能与产生振荡状态的延迟之间的相互作用以及血糖控制策略的相关信息提供了宝贵的工具。这些起因于霍普夫分叉，霍普夫分叉是由外部葡萄糖刺激以及胰腺胰岛素释放延迟和肝糖原形成的延迟共同作用引起的。每个生理子系统对振荡幅度和周期的影响通过对其周期解进行扰动分析来体现。结果表明，假设时延相称，可以通过研究切比雪夫多项式线性组合的根来表征霍普夫分叉曲线。所得的表达式为研究生理功能与产生振荡状态的延迟之间的相互作用以及血糖控制策略的相关信息提供了宝贵的工具。