Multiscale Poincaré plots for visualizing the structure of heartbeat time series
 Teresa S. Henriques^{1, 2, 3},
 Sara Mariani^{1, 4},
 Anton Burykin^{1},
 Filipa Rodrigues^{1, 5},
 Tiago F. Silva^{1, 5} and
 Ary L. Goldberger^{1, 3}Email author
DOI: 10.1186/s1291101602520
© Henriques et al. 2016
Received: 9 October 2015
Accepted: 27 January 2016
Published: 9 February 2016
Abstract
Background
Poincaré delay maps are widely used in the analysis of cardiac interbeat interval (RR) dynamics. To facilitate visualization of the structure of these time series, we introduce multiscale Poincaré (MSP) plots.
Methods
Starting with the original RR time series, the method employs a coarsegraining procedure to create a family of time series, each of which represents the system’s dynamics in a different time scale. Next, the Poincaré plots are constructed for the original and the coarsegrained time series. Finally, as an optional adjunct, color can be added to each point to represent its normalized frequency.
Results
We illustrate the MSP method on simulated Gaussian white and 1/f noise time series. The MSP plots of 1/f noise time series reveal relative conservation of the phase space area over multiple time scales, while those of white noise show a marked reduction in area. We also show how MSP plots can be used to illustrate the loss of complexity when heartbeat time series from healthy subjects are compared with those from patients with chronic (congestive) heart failure syndrome or with atrial fibrillation.
Conclusions
This generalized multiscale approach to Poincaré plots may be useful in visualizing other types of time series.
Keywords
Atrial fibrillation Complexity Congestive heart failure Fractal Heart rate Multiscale Nonlinear dynamics Poincaré plot Time series VisualizationBackground
The use of delay (also called return) maps is central to the qualitative and quantitative analysis of dynamical systems [1, 2]. The phase space realization with dimension of two and delay of one is referred to as a Poincaré plot [1–3]. This graphical method is widely used to visualize and quantify short and longerterm properties of heart rate variability (HRV) [3–11].
Here we propose a multiscale generalization of the Poincaré plot method, prompted by the observation that physiologic systems generate fluctuations over a broad range of scales. These fluctuations are a marker of the complexity of biologic dynamics, especially in healthy organisms under “freerunning” conditions [12–15]. A variety of computational tools, including fractal and multifractal methods [16–18], multiscale entropy [19–22], and multiscale time irreversibility [23, 24] have been proposed to probe the temporal richness of physiologic signals and of their dynamical alterations with senescence and pathology.
We sought to develop a complementary graphical method to aid in visualizing the multiscale properties of cardiac interbeat interval and other types of time series, in conjunction with these computational analyses. We were further motivated by the pedagogic need for graphical techniques to assist students and trainees in developing an intuitive sense for concepts and terms such as multiscale, selfsimilarity (fractality) and complexity loss. To this end, we adapted and extended the methodology of delay (Poincaré) maps. Classical Poincaré maps are singlescale, since they graph the value of one data point of the original time series against the next. The novelty of our method consists in the generation of multiscale Poincaré (MSP) plots. This multiscale implementation is accomplished via a simple coarsegraining procedure [19, 21] that produces multiple rescaled “copies” of the original signal. For each coarsegrained time series, we create a Poincaré plot, which is then assembled into the final montage. Furthermore, as a potentially useful, but optional adjunct, the data points in each plot are colorcoded using an estimated normalized probability density function to further enhance visualization of time series properties.
To introduce and illustrate the MSP method, we first apply it to synthetic Gaussian white and 1/ftype noise time series. The technique is then applied to RR interval time series obtained in health, chronic (congestive) heart failure and atrial fibrillation. The primary goal here is to introduce this method as a simpletoimplement visualization tool.
Methods
The MSP technique consists of three steps: i) construction of the coarsegrained time series; ii) construction of a Poincaré plot for the original and each of the coarsegrained time series, and iii) colorization of the Poincaré plots based on an estimated normalized probability density function.
Coarsegraining technique and construction of MSP montage
Considering a time series X of length N, X = {x _{1}, x _{2}, x _{3}, …, x _{ N − 1}, x _{ N }}, its Poincaré plot is the scatter plot representing the set of points: (x _{1}, x _{2}), (x _{2}, x _{3}), …, (x _{ N − 1}, x _{ N }) [4–6, 8].
Here, the Poincaré plots for the original and the coarsegrained time series are constructed and assembled into the MSP montage.
Colorization of MSP plots
Results and discussion
For illustrative purposes, we applied the MSP method to synthetic white and 1/f noise time series and to RR intervals time series in health and selected pathologic states.
MSP plots for synthetic white and 1/f noise time series
MSP plots for RR interval time series
The MSP technique was then applied to recordings from an openaccess dataset of deidentified cardiac interbeat interval time series from Holter monitor (~24 h) recordings (http://www.physionet.org/challenge/chaos/) [33]. This database includes RR interval time series from ostensibly healthy subjects, as well as patients with congestive (chronic) heart failure (CHF) syndrome, and patients with permanent atrial fibrillation (AF). Here we describe the geometry of the MSP plots from one subject in each of these three groups, representing the extremes of health and heart disease. The MSP plots for the other subjects in each group showed similar characteristics.
Healthy heartbeat dynamics
Chronic Heart Failure (CHF) and Atrial Fibrillation (AF) Dynamics
Previous reports [3–5] have shown that the area of the Poincaré plot of RR interval time series is markedly smaller for patients with severe CHF (but still in sinus rhythm) than healthy subjects. Here, we extend this finding by showing that the area is invariant under the coarsegraining operation. Our results (Fig. 5  middle panels) are consistent with an overall reduction in multiscale complex variability with heart failure [17, 21].
Prior studies [25, 39] have shown that the Poincaré plots (scale 1) of RR interval time series from subjects with AF are reminiscent of those derived from white noise signals. MSP plots (Fig. 5  bottom panels) highlight these results. In addition, they show that the resemblance between the Poincaré plots of the AF subject and those of white noise signals is most apparent for relatively short time scales (in this example, scales < 10, approximately lower than 15 s). In both white noise and AF cases, the areas of the Poincaré plots decrease with scale. Such behavior is attributable to the uncorrelated structure of the time series fluctuations. However, for larger time scales, the Poincaré plots for the subject with AF show the classical elliptical shape indicative of longrange correlations [40]. This finding is consistent with previous studies [40, 41] reporting that the absolute value of the scaling exponents derived from loglog power spectral plots of RR intervals time series from subjects with AF are closer to 1 (fractal noise) than to 0.5 (white noise) or 2 (brown noise) across the lower frequency bands, with a “crossover” toward those of white noise at higher frequencies.
The persistence of correlated behavior at higher scales (lower frequencies) in AF may be related to the degree to which the atrioventricular (AV) junction and autonomic nervous system function are preserved in this common arrhythmia. Whether AF associated with the most severe derangements of AV nodal conduction (and concomitant myocardial disease) shows a complete breakdown of correlations is an intriguing question with basic and clinical implications. The MSP method may be of use in screening “big datasets” in order to gain some intuition about the multiscale behavior of RR intervals in AF in different clinical subsets. We hypothesize that permanent AF associated with heart failure would have a less complex structure by this method than socalled “lone” AF, which is not associated with clinically apparent heart disease.
Use of colorization
Finally, we note that the colorization of the MSP plots, an optional feature of the multiscale renderings, is intended to facilitate rapid assessment of the values of the most frequently observed pairs of RR intervals (mode) as well as of the shape of the probability density function. For example, Fig. 5 shows that the most frequently observed values are ~1 s for the healthy subject, ~0.75 s for the patient with CHF and ~ 0.5 s for the patient with AF. In addition, Fig. 5 also shows that the probability density function is skewed to the left in the case of the healthy subject and to the right in case of the CHF patient and the subject with AF on short time scales. Whether quantitative analyses developed for traditional Poincaré plots [5, 6, 8, 11, 35, 44] can be usefully extended to MSP plots is of interest but outside the scope of this brief methodological note. We also emphasize that these plots are intended to complement current quantitative methods of time series analysis (e.g., Fourier, fractal/multifractal, and entropyrelated analyses, to name but a few).
Conclusions
We introduce a novel delay map implementation termed multiscale Poincaré (MSP) plots, to facilitate visualization of multiscale structure of cardiac interbeat interval time series. The method comprises: i) a coarsegraining operation that generates a family of time series; ii) delay map construction for the original and the coarsegrained time series; and iii) colorization of the delay maps based on an estimated normalized probability density function. The method appears to be useful in depicting concepts such as scaling behavior in health and disease and contrasting “realworld” and simulated data. Future studies are needed to evaluate its use in heart rate dynamics, as well as its potential utility in studying other types of time series.
Abbreviations
 AF:

Atrial fibrillation
 AV:

Atrioventricular
 CHF:

Congestive (chronic) heart failure
 HRV:

Heart rate variability
 MSP:

Multiscale Poincaré
Declarations
Acknowledgments
This work was supported by the Wyss Institute, the G. Harold & Leila Y. Mathers Foundation, the National Institutes of Health grants R24HL114473 and 2R01GM104987. We gratefully acknowledge the advice and input of Dr. Madalena D. Costa.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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