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In this study, we propose a new information theoretic measure to quantify the complexity of biological systems based on time-series data. We demonstrate the potential of our method using two distinct applications to human cardiac dynamics. Firstly, we show that the method clearly discriminates between segments of electrocardiogram records characterized by normal sinus rhythm, ventricular tachycardia and ventricular fibrillation. Secondly, we investigate the multiscale complexity of cardiac dynamics with respect to age in healthy individuals using interbeat interval time series and compare our findings with a previous study which established a link between age and fractal-like long-range correlations. The method we use is an extension of the symbolic mapping procedure originally proposed for permutation entropy. We build a Markov chain of the dynamics based on order patterns in the time series which we call an ordinal network, and from this model compute an intuitive entropic measure of transitional complexity. A discussion of the model parameter space in terms of traditional time delay embedding provides a theoretical basis for our multiscale approach. As an ancillary discussion, we address the practical issue of node aliasing and how this effects ordinal network models of continuous systems from discrete time sampled data, such as interbeat interval time series.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'., (© 2017 The Author(s).)

Recently, methods have been developed to model low-dimensional chaotic systems in terms of stochastic differential equations. We tested such methods in an electronic circuit experiment. We aimed to obtain reliable drift and diffusion coefficients even without a pronounced time-scale separation of the chaotic dynamics. By comparing the analytical solutions of the corresponding Fokker-Planck equation with experimental data, we show here that crisis-induced intermittency can be described in terms of a stochastic model which is dominated by state-space-dependent diffusion. Further on, we demonstrate and discuss some limits of these modelling approaches using numerical simulations. This enables us to state a criterion that can be used to decide whether a stochastic model will capture the essential features of a given time series., (This journal is © 2010 The Royal Society)

In 1963, the mathematician and meteorologist Edward Lorenz published a paper (Lorenz 1963 J. Atmos. Sci. 20, 130-141) that changed the way scientists think about the prediction of geophysical systems, by introducing the ideas of chaos, attractors, sensitivity to initial conditions and the limitations to forecasting nonlinear systems. Three years earlier, the mathematician and engineer Rudolf Kalman had published a paper (Kalman 1960 Trans. ASME Ser. D, J. Basic Eng. 82, 35-45) that changed the way engineers thought about prediction of electronic and mechanical systems. Ironically, in recent years, geophysicists have become increasingly interested in Kalman filters, whereas engineers have become increasingly interested in chaos. It is argued that more often than not the tracking and forecasting of nonlinear systems has more to do with the nonlinear dynamics that Lorenz considered than it has to do with statistics that Kalman considered. A position with which both Lorenz and Kalman would appear to agree., (This journal is © 2010 The Royal Society)