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 timevariability.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 heartrate 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 timevariability. So the work we propose, if successful, is likely to have farreaching 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 worldleading expert in the mathematical theory of nonautonomous 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.
