Exploiting the information content of noise in complex systems: Bayesian inference of nonlinear stochastic models and applications to human blood flow

An enduring problem in many branches of science and engineering is that of identifying and characterising a nonlinear stochastic system from the signals it produces. Often, the system itself is inaccessible (e.g. remanent quantum vortex loops in superfluid helium or the inversion population in a semiconductor laser), or difficult to measure directly (e.g. the cardiovascular system, where measurements must usually be non-invasive). Furthermore, in addition to any dynamical fluctuations in the system itself (dynamical noise), the results will always be to some extent corrupted by external noise (measurement noise). Examples also arise in ecology, and in other scientific areas as diverse as molecular motors and coupled matter-radiation systems in astrophysics. The chief difficulty stems from the fact that, in a great number of important problems, it is not possible to derive a suitable model from first principles, and one is therefore faced with a rather broad range of possible parametric models to consider. Furthermore, experimental data can often be highly skewed, so that important hidden features of a model (e.g. coupling parameters) can be very difficult to extract due to the intricate interplay between noise and nonlinearity. There is still no reliable method of analysis, despite intensive effort by many scientists.A solution to this problem is sorely is needed, not only to gain physical insight into the complex dynamics of the system under consideration, but also to facilitate the development of realistic models and thus to improve the reliability and accuracy with which the future can be predicted.We now propose to solve the problem. Although it has been one of the most challenging in statistical physics, we can now tackle it - with a high expectation of success - by developing a full theory of Bayesian inference for stochastic nonlinear dynamical systems. It will exploit the information content of dynamical noise and will be robust in the face of measurement noise. The proposed approach is based on novel ideas developed in our collaboration with Dr. V.N. Smelyanskiy. The technique involves path-integral calculations of the likelihood of dynamical events, and it is applicable to complex stochastic nonlinear systems quite generally. As well as developing the fundamental ideas, we propose to apply them to a particular example of a complex system that is in practice inaccessible: the human cardiovascular system (CVS). Here, the measured time series data (a sequence of measurements equally spaced in time) are attributable to physiological processes occurring deep within the body.Enough initial work has already been completed between Lancaster and the NASA/Ames Research Center to demonstrate the feasibility of the research. Exploiting the huge amount of information about the originating system that is contained within the seemingly random fluctuations themselves, and following a number of innovations, we have succeeded in reconstructing stochastic nonlinear models from measurements of time series data.Our exemplary application to the CVS will not only provide the practical experience needed for iterative improvement in the basic technique, but the enterprise will also be intrinsically extremely useful. Suitable CVS data are currently being recorded as part of a separate research project in collaboration with the Royal Lancaster Infirmary. If a good stochastic nonlinear model of the CVS can be reconstructed, there are potentially important implications for medicine because we anticipate that it will prove possible to relate parameter values in the model to the state of the system. There potential for both early diagnosis of cardiovascular disease and for quantitative assessment of the effect of treatment. We emphasize that the range of applicability of our new inference method is potentially very broad, encompassing the wide range of problems mentioned above and many others.