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 projective filtering I: Background in chaos theory

We derive a locally projective noise reduction scheme for nonlinear time series using concepts from deterministic dynamical systems, or chaos theory. We will demonstrate its effectiveness with an example with known deterministic dynamics and discuss methods for the verification of the results in the case of an unknown deterministic system.Comment: 4 pages, PS figures, needs nolta.st

Nonlinear projective filtering I: Application to real time series

We discuss applications of nonlinear filtering of time series by locally linear phase space projections. Noise can be reduced whenever the error due to the manifold approximation is smaller than the noise in the system. Examples include the real time extraction of the fetal electrocardiogram from abdominal recordings.Comment: 4 pages, PS figures, needs nolta.st

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

Testing for Chaos in Deterministic Systems with Noise

Recently, we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal Lyapunov exponent. In this paper, we investigate the capability of the test to cope with moderate amounts of noisy data. Comparisons are made between an improved version of our test and both the ``tangent space'' and ``direct method'' for computing the maximal Lyapunov exponent. The evidence of numerical experiments, ranging from the logistic map to an eight-dimensional Lorenz system of differential equations (the Lorenz 96 system), suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods

Divergence Measure Between Chaotic Attractors

We propose a measure of divergence of probability distributions for quantifying the dissimilarity of two chaotic attractors. This measure is defined in terms of a generalized entropy. We illustrate our procedure by considering the effect of additive noise in the well known H\'enon attractor. Comparison of two H\'enon attractors for slighly different parameter values, has shown that the divergence has complex scaling structure. Finally, we show how our approach allows to detect non-stationary events in a time series.Comment: 9 pages, 6 figure

Nonlinear analysis of bivariate data with cross recurrence plots

We use the extension of the method of recurrence plots to cross recurrence plots (CRP) which enables a nonlinear analysis of bivariate data. To quantify CRPs, we develop further three measures of complexity mainly basing on diagonal structures in CRPs. The CRP analysis of prototypical model systems with nonlinear interactions demonstrates that this technique enables to find these nonlinear interrelations from bivariate time series, whereas linear correlation tests do not. Applying the CRP analysis to climatological data, we find a complex relationship between rainfall and El Nino data

Assessing coupling dynamics from an ensemble of time series

Finding interdependency relations between (possibly multivariate) time series provides valuable knowledge about the processes that generate the signals. Information theory sets a natural framework for non-parametric measures of several classes of statistical dependencies. However, a reliable estimation from information-theoretic functionals is hampered when the dependency to be assessed is brief or evolves in time. Here, we show that these limitations can be overcome when we have access to an ensemble of independent repetitions of the time series. In particular, we gear a data-efficient estimator of probability densities to make use of the full structure of trial-based measures. By doing so, we can obtain time-resolved estimates for a family of entropy combinations (including mutual information, transfer entropy, and their conditional counterparts) which are more accurate than the simple average of individual estimates over trials. We show with simulated and real data that the proposed approach allows to recover the time-resolved dynamics of the coupling between different subsystems

Measuring Information Transfer

An information theoretic measure is derived that quantifies the statistical coherence between systems evolving in time. The standard time delayed mutual information fails to distinguish information that is actually exchanged from shared information due to common history and input signals. In our new approach, these influences are excluded by appropriate conditioning of transition probabilities. The resulting transfer entropy is able to distinguish driving and responding elements and to detect asymmetry in the coupling of subsystems.Comment: 4 pages, 4 Figures, Revte

Quantification of depth of anesthesia by nonlinear time series analysis of brain electrical activity

We investigate several quantifiers of the electroencephalogram (EEG) signal with respect to their ability to indicate depth of anesthesia. For 17 patients anesthetized with Sevoflurane, three established measures (two spectral and one based on the bispectrum), as well as a phase space based nonlinear correlation index were computed from consecutive EEG epochs. In absence of an independent way to determine anesthesia depth, the standard was derived from measured blood plasma concentrations of the anesthetic via a pharmacokinetic/pharmacodynamic model for the estimated effective brain concentration of Sevoflurane. In most patients, the highest correlation is observed for the nonlinear correlation index D*. In contrast to spectral measures, D* is found to decrease monotonically with increasing (estimated) depth of anesthesia, even when a "burst-suppression" pattern occurs in the EEG. The findings show the potential for applications of concepts derived from the theory of nonlinear dynamics, even if little can be assumed about the process under investigation.Comment: 7 pages, 5 figure