Physics of non-autonomous systems in the life sciences: a new perspective on the time-variability of complex systems

尽管生命系统的复杂性令人望而生畏，但人们已经做出了很大努力，用现象学数学模型来描述它们功能的各个方面。一旦以这种方式实现了定量的理解，人们就可以希望根据描述系统的方程中的参数值来表征系统的状态，并预测它将如何演变到未来。在医学上有明显的潜在应用，例如在诊断病理条件、预后和评估治疗效果方面。然而，几乎所有试图用非常短的时间尺度以外的时间尺度来模拟生命系统动力学的尝试都遇到了同样的根本问题：它们的时间可变。生命系统从出生到生命，最后到死亡，都处于不断变化的状态。在整个过程中，它们处于一种持续变化的状态，在许多不同的时间尺度上。例如，心率随时间变化，即使是健康的受试者在休息时也是如此--这种现象被称为心率变异性(HRV)。由于心率变异性的幅值和频率可以用来衡量健康状况，因此引起了国际上的极大关注。鉴于目前已知的导致心率变异性的几种潜在的振荡过程，关于心血管系统最有希望的图景之一是耦合振荡器，并且已经提出了许多模型。但很明显，模型参数，例如特征频率，随时间变化。这不可避免地意味着传统建模的适用性是严格受限的，而且在许多情况下肯定注定要失败。因此，很明显，需要一种截然不同的方法。这就是我们现在提出的，基于近年来开发的处理非自治系统的思想和技术。非自治动力系统的概念认识到正在研究的系统受到外部影响，例如可能导致其参数变化，并提供了一种表征和量化结果现象的方法。它可能是描述热力学上开放的生命系统的理想选择，这些系统与周围环境以及不同部分之间的内部物质和能量进行持续的交换。生物体中的每个部分和过程都在某种程度上影响着其他每个子系统，因此，当人们在试图理解一个特定的子系统时，测量一个或两个变量时，观察到的行为是非常复杂的。例如，心率变异性的产生不仅来自呼吸对心率的影响，也来自于与肌源性、神经源性和内皮活动相对应的较慢的振荡过程的影响。非自治系统理论承诺量化非自治性的程度，并描述由此产生的现象，例如由所讨论的“外部影响”产生的额外吸引子(稳态)。我们提出的相当于反问题的一种新方法，寻求问题的答案：给定一个信号(一系列测量或时间序列)，产生它的系统是什么？这是一个在许多科学领域都能发现的难题，但在生理信号的情况下，由于它们的时间变异性，解决起来非常困难。因此，我们提出的工作如果成功，可能会产生深远的影响。我们的团队包括一名生物医学工程师(PI)和两名物理学家(CI和RCI)，他们在生物医学领域对自主动力系统拥有非常丰富的经验，一名数学家(VR)是非自主系统数学理论的世界领先专家，以及具有相关生理学专业知识的临床合作者。因此，我们将把相对深奥的、专题的、从物理学和数学到生理学的实际应用中，为临床实践中的创新铺平道路。

Despite the daunting complexity of living systems, great efforts have been made to describe aspects of their function in terms of phenomenological mathematical models. Once a quantitative understanding has been achieved in this way, one can hope to characterize the state of a system in terms of parameter values in the equation that describes it and to predict how it will evolve into the future. There are obvious potential applications in medicine, e.g. for the diagnosis of pathological conditions, prognosis, and assessment of the efficacy of treatment. However, almost all attempts to model the dynamics of living systems on other than very short timescales have run into the same fundamental problem: their time-variability.Living systems are in a state of continuous change, as they evolve from birth, through life, and finally to death. Throughout, they are in a state of continuous alteration, on many different timescales. For example, the heart rate varies in time, even for a healthy subject in repose - a phenomenon known as heart-rate variability (HRV). Because its amplitude and frequency content can be used as a measure of health, HRV has attracted enormous international attention. In view of the several underlying oscillatory processes now known to be responsible for HRV, one of the most promising pictures of the cardiovascular system is in terms of coupled oscillators, and a number of models have been proposed. But it is evident that the model parameters, e.g. characteristic frequencies, vary in time. This inevitably implies that conventional modeling is of strictly limited applicability, and must in many cases be doomed to failure. Thus it has become apparent that a radically different approach is needed. This is what we now propose, based on ideas and techniques developed in recent years for the treatment of nonautonomous systems.The notion of nonautonomous dynamical systems recognizes that a system under study is subject to outside influences that may e.g. cause its parameters to vary, and provides a way of characterising and quantifying the resultant phenomena. It is potentially ideal for the description of living systems which are thermodynamically open, subject to continuous exchange of matter and energy with their surroundings as well as internally between their different subsections. Every part and process within an organism to some extent influences every other subsystem, whence the extraordinary complexity of the observed behaviour when one measures one or two variables in attempting to understand a particular subsystem. For example HRV arises, not only from the influence of respiration on heart rate, but also through the influences of slower oscillatory processes corresponding to e.g. myogenic, neurogenic and endothelial activities. The theory of nonautonomous systems promises to quantify the degree of nonautonomicity and to describe resultant phenomena, e.g. extra attractors (steady states) created by the ``outside influences`` in question.What we propose amounts to a new approach to the inverse problem, seeking an answer to the question: given a signal (a sequence of measurements, or time series), what is the system that produced it? It is a conundrum found in many areas of science, but has been acutely difficult to tackle in the case of physiological signals on account of their time-variability. So the work we propose, if successful, is likely to have far-reaching consequences. Our team includes a biomedical engineer (PI) and 2 physicists (CI and RCI) who together have very extensive experience of autonomous dynamical systems in biomedicine, a mathematician (VR) who is a world-leading expert in the mathematical theory of non-autonomous systems, and clinical collaborators with expertise in the relevant physiology. We will thus bring relatively abstruse, topical, ideas from physics and mathematics to practical application in physiology, paving the way to innovation in clinical practice.