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生畏，但人们已经做出了巨大的努力来描述它们在现象学数学模型方面的功能。一旦以这种方式实现了定量的理解，人们就可以希望用描述系统的方程中的参数值来描述系统的状态，并预测它将如何演变到未来。在医学中有明显的潜在应用，例如用于病理状况的诊断、预后和治疗功效的评估。然而，几乎所有试图在非非常短的时间尺度上模拟生命系统动力学的尝试都遇到了同样的基本问题：它们的时间可变性。生命系统处于一种持续变化的状态，因为它们从出生，经过生命，最后到死亡。在整个过程中，它们在许多不同的时间尺度上处于持续变化的状态。例如，心率随时间变化，即使对于处于休息状态的健康受试者也是如此-这种现象称为心率变异性（HRV）。由于其振幅和频率含量可以作为健康的衡量标准，HRV引起了国际上的极大关注。鉴于几个潜在的振荡过程，现在已知的是负责HRV，心血管系统的最有前途的图片之一是在耦合振荡器，并提出了一些模型。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的建模是严格有限的适用性，并在许多情况下注定要失败。因此，显然需要一种完全不同的方法。基于近年来发展起来的处理非自治系统的思想和技术，我们现在提出非自治动力系统的概念，认为所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。它是潜在的理想的生命系统的描述是开放的，不断交换的物质和能量与他们的周围环境，以及内部之间的不同子部分。有机体中的每一部分和过程都在某种程度上影响着其他的子系统，因此，当人们试图理解一个特定的子系统时，测量一两个变量时，观察到的行为异常复杂。例如，HRV不仅由呼吸对心率的影响产生，而且还通过对应于例如肌源性、神经源性和内皮活动的较慢振荡过程的影响产生。非自治系统的理论承诺量化的非线性程度和描述所产生的现象，例如，额外的吸引子（稳态）所产生的“外部影响”的问题。我们提出的金额是一个新的方法来反问题，寻求一个答案的问题：给定一个信号（一系列的测量，或时间序列），是什么系统，产生了它？这是一个在许多科学领域都存在的难题，但在生理信号的情况下，由于其时间可变性，很难解决。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），他们在生物医学中的自主动力系统方面拥有非常丰富的经验，一名数学家（VR）是世界领先的非自主系统数学理论专家，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，从物理和数学的想法，在生理学的实际应用，铺平了道路，在临床实践中的创新。

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.