The human ECG - nonlinear deterministic versus stochastic aspects

We discuss aspects of randomness and of determinism in electrocardiographic signals. In particular, we take a critical look at attempts to apply methods of nonlinear time series analysis derived from the theory of deterministic dynamical systems. We will argue that deterministic chaos is not a likely explanation for the short time variablity of the inter-beat interval times, except for certain pathologies. Conversely, densely sampled full ECG recordings possess properties typical of deterministic signals. In the latter case, methods of deterministic nonlinear time series analysis can yield new insights.Comment: 6 pages, 9 PS figure

Nonlinear time-series analysis revisited

In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data---typically univariate---via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems

Differential Landauer's principle

Landauer's principle states that the erasure of information must be a dissipative process. In this paper, we carefully analyze the recording and erasure of information on a physical memory. On the one hand, we show that in order to record some information, the memory has to be driven out of equilibrium. On the other hand, we derive a differential version of Landauer's principle: We link the rate at which entropy is produced at every time of the erasure process to the rate at which information is erased.Comment: 11 pages, 6 figure

Recurrence time analysis, long-term correlations, and extreme events

The recurrence times between extreme events have been the central point of statistical analyses in many different areas of science. Simultaneously, the Poincar\'e recurrence time has been extensively used to characterize nonlinear dynamical systems. We compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series. In particular, we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function, on the interval whereto the recurrences are observed, and on the temporal correlations of time series. In the case of long-term correlations, we verify the validity of the stretched exponential distribution, which is uniquely defined by the exponent $\gamma$, at the same time showing that it is restricted to the class of linear long-term correlated processes. Simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different $\gamma$, which shows a lack of invariance under the change of observables.Comment: 9 pages, 7 figure

Crooks' fluctuation theorem for the fluctuating lattice-Boltzmann model

We probe the validity of Crooks' fluctuation relation on the fluctuating lattice-Boltzmann model (FLBM), a highly simplified lattice model for a thermal ideal gas. We drive the system between two thermodynamic equilibrium states and compute the distribution of the work performed. By comparing the distributions of the work performed during the forward driving and time reversed driving, we show that the system satisfies Crooks' relation. The results of the numerical experiment suggest that the temperature and the free energy of the system are well defined.Comment: To be published in J. Stat. Mec

Scale invariant Green-Kubo relation for time averaged diffusivity

In recent years it was shown both theoretically and experimentally that in certain systems exhibiting anomalous diffusion the time and ensemble average mean squared displacement are remarkably different. The ensemble average diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale invariant non-stationary velocity correlation function with the transport coefficient. Here we obtain the relation between time averaged diffusivity, usually recorded in single particle tracking experiments, and the underlying scale invariant velocity correlation function. The time averaged mean squared displacement is given by $\overline{\delta^2} \sim 2 D_\nu t^{\beta}\Delta^{\nu-\beta}$ where $t$ is the total measurement time and $\Delta$ the lag time. Here $\nu>1$ is the anomalous diffusion exponent obtained from ensemble averaged measurements $\langle x^2 \rangle \sim t^\nu$ while $\beta\ge -1$ marks the growth or decline of the kinetic energy $\langle v^2 \rangle \sim t^\beta$. Thus we establish a connection between exponents which can be read off the asymptotic properties of the velocity correlation function and similarly for the transport constant $D_\nu$. We demonstrate our results with non-stationary scale invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. This is the case, for example, if averaged kinetic energy is finite, i.e. $\beta=0$, where $\langle \overline{\delta^2}\rangle \neq \langle x^2\rangle$

Practical implementation of nonlinear time series methods: The TISEAN package

Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from http://www.mpipks-dresden.mpg.de/~tisean . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.Comment: 27 pages, 21 figures, downloadable software at http://www.mpipks-dresden.mpg.de/~tisea