Identifying chaos from heart rate: the right task?

Beyond HRV: attractor reconstruction using the entire cardiovascular waveform data for novel feature extraction

Advances in monitoring technology allow blood pressure waveforms to be collected at sampling frequencies of 250-1000 Hz for long time periods. However, much of the raw data are under-analysed. Heart rate variability (HRV) methods, in which beat-to-beat interval lengths are extracted and analysed, have been extensively studied. However, this approach discards the majority of the raw data.

OBJECTIVE

Our aim is to detect changes in the shape of the waveform in long streams of blood pressure data.

APPROACH

Our approach involves extracting key features from large complex data sets by generating a reconstructed attractor in a three-dimensional phase space using delay coordinates from a window of the entire raw waveform data. The naturally occurring baseline variation is removed by projecting the attractor onto a plane from which new quantitative measures are obtained. The time window is moved through the data to give a collection of signals which relate to various aspects of the waveform shape.

MAIN RESULTS

This approach enables visualisation and quantification of changes in the waveform shape and has been applied to blood pressure data collected from conscious unrestrained mice and to human blood pressure data. The interpretation of the attractor measures is aided by the analysis of simple artificial waveforms.

SIGNIFICANCE

We have developed and analysed a new method for analysing blood pressure data that uses all of the waveform data and hence can detect changes in the waveform shape that HRV methods cannot, which is confirmed with an example, and hence our method goes 'beyond HRV'.

Generation and dynamics analysis of N-scrolls existence in new translation-type chaotic systems

In this paper, we propose two kinds of translation type chaotic systems for creating 2 N + 1-and 2(N + 1)-scrolls chaotic attractors from a simple three-dimensional system, which are named the translation-2 chaotic system (a₁₂a₂₁ < 0) and the translation-3 chaotic system (a₁₂a₂₁ > 0). We also propose the successful design criterion for constructing 2 N + 1-and 2(N + 1)-scrolls, respectively. Then, the dynamics property of the translation-2 chaotic system is studied in detail. MATLAB simulation results show that very sophisticated dynamical behaviors and unique chaotic behaviors of the system. Finally, the definition and criterion of multi-scroll attractors for the translation-3 chaotic system is obtained. Three representative examples are shown in some classical chaotic systems that can be equally obtained via the set parameters of the translation type chaotic system. Furthermore, we show that the translation type chaotic systems have similar but topologically non-equivalent chaotic attractors, and they are the three-dimensional ordinary differential equations.

An easy-to-use technique to characterize cardiodynamics from first-return maps on ΔRR-intervals

Heart rate variability analysis using 24-h Holter monitoring is frequently performed to assess the cardiovascular status of a patient. The present retrospective study is based on the beat-to-beat interval variations or ΔRR, which offer a better view of the underlying structures governing the cardiodynamics than the common RR-intervals. By investigating data for three groups of adults (with normal sinus rhythm, congestive heart failure, and atrial fibrillation, respectively), we showed that the first-return maps built on ΔRR can be classified according to three structures: (i) a moderate central disk, (ii) a reduced central disk with well-defined segments, and (iii) a large triangular shape. These three very different structures can be distinguished by computing a Shannon entropy based on a symbolic dynamics and an asymmetry coefficient, here introduced to quantify the balance between accelerations and decelerations in the cardiac rhythm. The probability P111111 of successive heart beats without large beat-to-beat fluctuations allows to assess the regularity of the cardiodynamics. A characteristic time scale, corresponding to the partition inducing the largest Shannon entropy, was also introduced to quantify the ability of the heart to modulate its rhythm: it was significantly different for the three structures of first-return maps. A blind validation was performed to validate the technique.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Electrocardiogram classification using delay differential equations

Time series analysis with nonlinear delay differential equations (DDEs) reveals nonlinear as well as spectral properties of the underlying dynamical system. Here, global DDE models were used to analyze 5 min data segments of electrocardiographic (ECG) recordings in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. The number of terms and delays in the model as well as the order of nonlinearity of the model have to be selected that are the most discriminative. The DDE model form that best separates the three classes of data was chosen by exhaustive search up to third order polynomials. Such an approach can provide deep insight into the nature of the data since linear terms of a DDE correspond to the main time-scales in the signal and the nonlinear terms in the DDE are related to nonlinear couplings between the harmonic signal parts. The DDEs were able to detect atrial fibrillation with an accuracy of 72%, congestive heart failure with an accuracy of 88%, and normal heart beat with an accuracy of 97% from 5 min of ECG, a much shorter time interval than required to achieve comparable performance with other methods.

Individuality of breathing patterns in patients under noninvasive mechanical ventilation evidenced by chaotic global models

Autonomous global models based on radial basis functions were obtained from data measured from patients under noninvasive mechanical ventilation. Some of these models, which are discussed in the paper, turn out to have chaotic or quasi-periodic solutions, thus providing a first piece of evidence that the underlying dynamics of the data used to estimate the global models are likely to be chaotic or, at least, have a chaotic component. It is explicitly shown that one of such global models produces attractors characterized by a Horseshoe map, two models produce toroidal chaos, and one model produces a quasi-periodic regime. These topologically inequivalent attractors evidence the individuality of breathing profiles observed in patient under noninvasive ventilation.

Working conditions for safe detection of nonlinearity and noise titration

Even if noise titration cannot be satisfactorily used to prove the presence of chaos, it can still be used to detect nonlinear component in dynamics. Nevertheless, since the technique have the use of nonlinear models for one-step-ahead predictions, it requires an acute choice of modeling parameters, i.e., the number of terms and the nonlinearity degree of the models. Based on illustrative examples, we propose conditions under which the method of noise titration can be reliably applied to characterize nonlinearity in the time series. It is thus possible to compare different time series and state which one is governed by the strongest nonlinearity. For instance, it is shown that, when there is a single nonlinear term in the equations describing the system, the variable on which it acts can be identified among the others.

Computing the multifractal spectrum from time series: an algorithmic approach

We show that the existing methods for computing the f(alpha) spectrum from a time series can be improved by using a new algorithmic scheme. The scheme relies on the basic idea that the smooth convex profile of a typical f(alpha) spectrum can be fitted with an analytic function involving a set of four independent parameters. While the standard existing schemes [P. Grassberger et al., J. Stat. Phys. 51, 135 (1988); A. Chhabra and R. V. Jensen, Phys. Rev. Lett. 62, 1327 (1989)] generally compute only an incomplete f(alpha) spectrum (usually the top portion), we show that this can be overcome by an algorithmic approach, which is automated to compute the D(q) and f(alpha) spectra from a time series for any embedding dimension. The scheme is first tested with the logistic attractor with known f(alpha) curve and subsequently applied to higher-dimensional cases. We also show that the scheme can be effectively adapted for analyzing practical time series involving noise, with examples from two widely different real world systems. Moreover, some preliminary results indicating that the set of four independent parameters may be used as diagnostic measures are also included.

Introduction to controversial topics in nonlinear science: is the normal heart rate chaotic?

In June 2008, the editors of Chaos decided to institute a new section to appear from time to time that addresses timely and controversial topics related to nonlinear science. The first of these deals with the dynamical characterization of human heart rate variability. We asked authors to respond to the following questions: Is the normal heart rate chaotic? If the normal heart rate is not chaotic, is there some more appropriate term to characterize the fluctuations (e.g., scaling, fractal, multifractal)? How does the analysis of heart rate variability elucidate the underlying mechanisms controlling the heart rate? Do any analyses of heart rate variability provide clinical information that can be useful in medical assessment (e.g., in helping to assess the risk of sudden cardiac death)? If so, please indicate what additional clinical studies would be useful for measures of heart rate variability to be more broadly accepted by the medical community. In addition, as a challenge for analysis methods, PhysioNet [A. L. Goldberger et al., "PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals," Circulation 101, e215-e220 (2000)] provided data sets from 15 patients of whom five were normal, five had heart failure, and five had atrial fibrillation (http://www.physionet.org/challenge/chaos/). This introductory essay summarizes the main issues and introduces the essays that respond to these questions.