A mathematical modelling approach to the prediction of circadian phase under conditions of entrainment

人类的昼夜节律是身体在一天中发生的生理和行为变化的循环。如果在昼夜节律的特定阶段进行医疗干预，可能会更有效，但为了正确地确定干预的时间，我们需要一种估计昼夜节律阶段的方便方法。暗光褪黑素发作(DLMO)是昼夜节律的准确生理标志，但测量DLMO的程序昂贵且不切实际，不适合临床使用。昼夜节律的数学模型提供了一种非侵入性的方法来预测昼夜节律，但现有的模型仅在实验室环境中进行了验证，目前尚不清楚这些模型是否能在现实世界中产生准确的预测。本文将回顾昼夜节律的关键属性，如内生周期性和对光的敏感性，并探索如何使用一维或二维振荡器对它们进行数学建模。Juett-Forger-Kronauer(JFK)模型是一种流行的基于二维极限环振荡器的人体昼夜节律系统模型，它的设计目的是在实验室条件下尝试并重现强光刺激对昼夜节律系统的相位和幅度的影响。JFK模型并不是专门设计来预测在真实世界条件下携带的受试者的携带阶段。该模型将被详细审查，以期了解振荡器是如何引入周期性光刺激的。该模型的预测将使用在现实世界条件下佩戴监控设备的受试者的光数据进行测试。该模型对相位的预测将与DLMO在这些受试者中测量的相位进行比较。JFK模型有两个主要部分：(I)过程C，它是一个数学振子；(Ii)过程L，在这个过程中，光信号被转换为作用于振子的强迫项。模型的这两个部分将分别考虑。在许多情况下，JFK模型振荡器部分的角动力学可以很好地用一个一维的‘纯相位’模型来近似。仅位相模型有助于更好地理解振子在不同条件下如何缠绕到周期性光刺激中。可以写下一个简单的纯相位时钟模型，该模型具有解析解。这为提高我们对夹带的理解提供了一个很好的起点。本文将研究简单时钟模型的动力学行为，包括微扰的影响。JFK模型中的L流程是在1990年代末开发的，但该模型的这一部分自开始以来一直没有重大开发。与此同时，在这段时间里，关于光对昼夜节律系统的影响已经有了一些重要的发现，例如，我们知道它是由视网膜中固有的光敏视网膜神经节细胞介导的，与视杆细胞和视锥细胞相比，这些细胞具有不同的光谱响应性。特别是，很明显，测量明视勒克斯的光强度(如在JFK模型中)不是对昼夜节律系统接收到的信号强度的适当测量。本文将回顾对光信息处理和传递到昼夜节律系统的最新理解，并尝试将这一知识纳入光信号转导的新模型中。

The circadian rhythm in humans is a cycle of physiological and behavioural changes in the body that takes place over the course of the day. Medical interventions may be more effective if they are delivered at particular phases of the circadian cycle but in order to time the interventions correctly we require a convenient method of estimating circadian phase. Dim-light melatonin onset (DLMO) is an accurate physiological marker of circadian phase but the procedure to measure DLMO is costly and impractical for use in a clinical setting. Mathematical models of the circadian rhythm provide a non-invasive method to predict circadian phase but the existing models have only been validated in the laboratory setting and it is not known if the models produce accurate predictions in real world conditions. The thesis will review key properties of the circadian rhythm such as endogenous periodicity and sensitivity to light, and explore how they can be modelled mathematically using one-dimensional or two-dimensional oscillators. The Jewett-Forger-Kronauer (JFK) model is a popular model of the human circadian system based on a two-dimensional limit cycle oscillator, and it was designed to try and reproduce the effects of bright light stimuli on the phase and amplitude of the circadian system in laboratory conditions. The JFK model was not specifically designed to predict the phase of entrainment in subjects entrained in real world conditions. The model will be reviewed in detail with a view to understanding how the oscillator entrains to a periodic light stimulus. The predictions of the model will be tested using light data from subjects wearing monitoring devices in real world conditions. The model's predictions of phase will be compared with measurements of phase in these subjects by DLMO. The JFK model has two principal parts: (i) Process C, which is a mathematical oscillator, and (ii) Process L, whereby a light signal is transduced to a forcing term that acts on the oscillator. These two parts of the model will be considered separately. In many circumstances the angular dynamics of the oscillator part of the JFK model can be well-approximated by a one-dimensional 'phase only' model. Phase only models can facilitate a better understanding of how oscillators entrain to periodic light stimuli under different conditions. A simple phase only clock model can be written down which has analytic solutions. This provides a good starting point to improve our understanding of entrainment. The thesis will investigate the dynamical behaviour of the simple clock model including the effects of perturbations. Process L in the JFK model was developed in the late 1990s but there has been no significant development of this part of the model since its inception. Meanwhile in that time there have been a number of important discoveries about the effect of light on the circadian system, for example we know that it is mediated by intrinsically-photosensitive retinal ganglion cells in the retina which have a different spectral responsivity compared to rods and cones. In particular it is clear that measuring the light intensity in photopic lux (as in the JFK model) is not an appropriate measure of the intensity of the signal received by the circadian system. The thesis will review the latest understanding of the processing and transmission of light information to the circadian system, and try and incorporate this knowledge in a new model of the transduction of a light signal.