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下丘脑-垂体-肾上腺轴(HPA)的数学建模,包括海马机制。

Mathematical modeling of the hypothalamic-pituitary-adrenal gland (HPA) axis, including hippocampal mechanisms.

机构信息

Technical University of Denmark, DTU Compute, Matematiktorvet 303B, 2800 Kongens Lyngby, Denmark; Roskilde University, Building 27.1, NSM, IMFUFA, 4000 Roskilde, Denmark.

出版信息

Math Biosci. 2013 Nov;246(1):122-38. doi: 10.1016/j.mbs.2013.08.010. Epub 2013 Sep 4.

Abstract

This paper presents a mathematical model of the HPA axis. The HPA axis consists of the hypothalamus, the pituitary and the adrenal glands in which the three hormones CRH, ACTH and cortisol interact through receptor dynamics. Furthermore, it has been suggested that receptors in the hippocampus have an influence on the axis. A model is presented with three coupled, non-linear differential equations, with the hormones CRH, ACTH and cortisol as variables. The model includes the known features of the HPA axis, and includes the effects from the hippocampus through its impact on CRH in the hypothalamus. The model is investigated both analytically and numerically for oscillating solutions, related to the ultradian rhythm seen in data, and for multiple fixed points related to hypercortisolemic and hypocortisolemic depression. The existence of an attracting trapping region guarantees that solution curves stay non-negative and bounded, which can be interpreted as a mathematical formulation of homeostasis. No oscillating solutions are present when using physiologically reasonable parameter values. This indicates that the ultradian rhythm originate from different mechanisms. Using physiologically reasonable parameters, the system has a unique fixed point, and the system is globally stable. Therefore, solutions converge to the fixed point for all initial conditions. This is in agreement with cortisol levels returning to normal, after periods of mild stress, in healthy individuals. Perturbing parameters lead to a bifurcation, where two additional fixed points emerge. Thus, the system changes from having a unique stable fixed point into having three fixed points. Of the three fixed points, two are stable and one is unstable. Further investigations show that solutions converge to one of the two stable fixed points depending on the initial conditions. This could explain why healthy people becoming depressed usually fall into one of two groups: a hypercortisolemic depressive group or a hypocortisolemic depressive group.

摘要

本文提出了一个 HPA 轴的数学模型。HPA 轴由下丘脑、垂体和肾上腺组成,其中三种激素 CRH、ACTH 和皮质醇通过受体动力学相互作用。此外,有人认为海马中的受体对轴有影响。提出了一个具有三个耦合非线性微分方程的模型,其中激素 CRH、ACTH 和皮质醇作为变量。该模型包括 HPA 轴的已知特征,并包括通过其对下丘脑 CRH 的影响来自海马的影响。该模型从分析和数值两个方面研究了与超低频节律相关的振荡解,以及与高皮质醇血症和低皮质醇血症相关的多个固定点。吸引捕获区域的存在保证了解曲线保持非负和有界,这可以解释为体内平衡的数学公式。当使用生理合理的参数值时,不存在振荡解。这表明超低频节律源自不同的机制。使用生理合理的参数,系统具有唯一的固定点,并且系统是全局稳定的。因此,对于所有初始条件,解都收敛到固定点。这与健康个体在轻度应激后皮质醇水平恢复正常的情况一致。扰动参数导致分岔,其中出现两个附加的固定点。因此,系统从具有唯一稳定的固定点变为具有三个固定点。在这三个固定点中,有两个是稳定的,一个是不稳定的。进一步的研究表明,根据初始条件,解收敛到两个稳定固定点之一。这可以解释为什么健康人变得抑郁通常分为两类:高皮质醇血症抑郁组或低皮质醇血症抑郁组。

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