Abstract：
OBJECTIVE: It is known that long-term stress leads to trauma and very often to depression. Usually, the diagnosis of depression is dealt with by psychiatrists who, based on conversations and questions, diagnose the patient\'s illness and condition. Unfortunately, this diagnosis is not always reliable. To prevent the development of disease, it is necessary to detect illness in a timely manner. One of the indications of the possibility of the onset of disease is a disturbance in the level of hormones in the body, especially cortisol. The purpose of this study was to develop a mathematical model for cortisol variation resulting from stress which would be useful in making conclusions about depressive states.
METHODS: Rapid changes in cortisol concentration, according to ultradian rhythms, which are much faster than the daily circadian rhythm, is modelled as a truly nonlinear oscillator. The mathematical model contains two coupled first order differential equations. The stress is modeled as a pulsating action, described with a periodic trigonometric function, and cortisol production as a cubic nonlinear one. Three models for cortisol variation are considered: 1) the pure nonlinear model, 2) the periodically excited system, 3) and the chaotic system. The results from the study are supported with experimental measurements.
RESULTS: Without stress, cortisol variation is of an oscillatory type with a constant steady-state amplitude. Intensive stress causes a resonant phenomenon in cortisol oscillatory variation. The occasion is short and is usually without consequences. For long stress periods deterministic chaos occurs which permanently changes the levels of cortisol. This phenomenon is an indicator of depression. Results from the suggested models are compared with experimentally obtained ones and good quantitative agreement is obtained.
CONCLUSIONS: The nonlinear oscillator is a good model for indication of depression. The model provides not only general conclusions, but also individual ones, if personal characteristics are taken into consideration. Response of the model depends not only on the input data related to stress, but also on the system parameters that specify each individual. Findings obtained from this study have implications for the medical diagnosis and treatment of depression.
METHODS: Rapid changes in cortisol concentration, according to ultradian rhythms, which are much faster than the daily circadian rhythm, is modelled as a truly nonlinear oscillator. The mathematical model contains two coupled first order differential equations. The stress is modeled as a pulsating action, described with a periodic trigonometric function, and cortisol production as a cubic nonlinear one. Three models for cortisol variation are considered: 1) the pure nonlinear model, 2) the periodically excited system, 3) and the chaotic system. The results from the study are supported with experimental measurements.
RESULTS: Without stress, cortisol variation is of an oscillatory type with a constant steady-state amplitude. Intensive stress causes a resonant phenomenon in cortisol oscillatory variation. The occasion is short and is usually without consequences. For long stress periods deterministic chaos occurs which permanently changes the levels of cortisol. This phenomenon is an indicator of depression. Results from the suggested models are compared with experimentally obtained ones and good quantitative agreement is obtained.
CONCLUSIONS: The nonlinear oscillator is a good model for indication of depression. The model provides not only general conclusions, but also individual ones, if personal characteristics are taken into consideration. Response of the model depends not only on the input data related to stress, but also on the system parameters that specify each individual. Findings obtained from this study have implications for the medical diagnosis and treatment of depression.
摘要：
目的：众所周知，长期的压力会导致创伤，并经常导致抑郁。通常,抑郁症的诊断是由精神病医生处理的，基于对话和问题，诊断病人的病情。不幸的是,这种诊断并不总是可靠的。为了防止疾病的发展，有必要及时发现疾病。疾病发作的可能性的迹象之一是体内激素水平的紊乱，尤其是皮质醇.这项研究的目的是为压力引起的皮质醇变化建立数学模型，这将有助于得出有关抑郁状态的结论。
方法：皮质醇浓度的快速变化，根据Ultradian节奏,比每天的昼夜节律快得多，被建模为真正的非线性振荡器。该数学模型包含两个耦合的一阶微分方程。应力被建模为脉动作用，用周期性三角函数描述，皮质醇的产生是立方非线性的。考虑了皮质醇变化的三个模型：1）纯非线性模型，2）周期性激励系统，3)和混沌系统。该研究的结果得到了实验测量的支持。
结果：没有压力，皮质醇变化是振荡型，具有恒定的稳态幅度。强烈的压力会导致皮质醇振荡变化的共振现象。时间很短，通常没有后果。对于长时间的压力，会发生确定性的混乱，从而永久改变皮质醇的水平。这种现象是抑郁症的一个指标。将建议模型的结果与实验获得的结果进行比较，并获得了良好的定量一致性。
结论：非线性振荡器是抑郁症适应症的良好模型。该模型不仅提供了一般性结论，也包括个人，如果考虑到个人特征。模型的响应不仅取决于与压力相关的输入数据，而且还指定了每个人的系统参数。从这项研究中获得的发现对抑郁症的医学诊断和治疗具有重要意义。
方法：皮质醇浓度的快速变化，根据Ultradian节奏,比每天的昼夜节律快得多，被建模为真正的非线性振荡器。该数学模型包含两个耦合的一阶微分方程。应力被建模为脉动作用，用周期性三角函数描述，皮质醇的产生是立方非线性的。考虑了皮质醇变化的三个模型：1）纯非线性模型，2）周期性激励系统，3)和混沌系统。该研究的结果得到了实验测量的支持。
结果：没有压力，皮质醇变化是振荡型，具有恒定的稳态幅度。强烈的压力会导致皮质醇振荡变化的共振现象。时间很短，通常没有后果。对于长时间的压力，会发生确定性的混乱，从而永久改变皮质醇的水平。这种现象是抑郁症的一个指标。将建议模型的结果与实验获得的结果进行比较，并获得了良好的定量一致性。
结论：非线性振荡器是抑郁症适应症的良好模型。该模型不仅提供了一般性结论，也包括个人，如果考虑到个人特征。模型的响应不仅取决于与压力相关的输入数据，而且还指定了每个人的系统参数。从这项研究中获得的发现对抑郁症的医学诊断和治疗具有重要意义。
