Department of Physiology, McGill University, Montreal, Quebec H3G 1Y6, Canada
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 ( This introductory essay summarizes the main issues and introduces the essays that respond to these questions. ©2009 American Institute of Physics

Chaos 19, 028501 (2009)
Published 30 June 2009
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Universidad Autonoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco No. 186, Col. Vicentina, 09340 Mexico D.F., Mexico
This paper explores the possibility of applying statistical nonlinear physics methods to elucidate the underlying mechanisms controlling the heart rate. In particular, the presence of delays in RR interval dynamics is studied by using a lagged detrended fluctuation analysis. The results indicate that patients with congestive heart failure (CHF) have a prolonged time delay in the baroreflex response. Some implications of large delays for the functioning of autonomic control in subjects with CHF are discussed. ©2009 American Institute of Physics

Chaos 19, 028502 (2009)
Published 30 June 2009
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1Departments of Economics and Finance, Michigan State University, East Lansing, Michigan 48824, USA and Department of Economics, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
2Department of Economics, Central Michigan University, Sloan 307, Mt. Pleasant, Michigan 48859, USA
3Department of Geology, Geography, and Physics, University of Tennessee, Martin, Tennessee 38238, USA

We present new evidence that normal heartbeat series are nonchaotic, nonlinear, and multifractal. In addition to considering the largest Lyapunov exponent and the correlation dimension, the results of the parametric and semiparametric estimation of the long memory parameter (long-range dependence) unambiguously reveal that the underlying process is nonstationary, multifractal, and has strong nonlinearity. ©2009 American Institute of Physics

Chaos 19, 028503 (2009)
Published 30 June 2009
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1Physics of Complex Systems Division, Faculty of Physics, Warsaw University of Technology, ul Koszykowa 75, 00-662 Warsaw, Poland
2Department of Hypertension, Institute of Cardiology, ul Alpejska 42, 04-628 Warsaw, Poland
3Chair of Psychiatry, Medical University of Warsaw, ul Nowowiejska 27, 00-665 Warsaw, Poland
Human heart rate is moderated by the autonomous nervous system acting predominantly through the sinus node (the main cardiac physiological pacemaker). One of the dominant factors that determine the heart rate in physiological conditions is its coupling with the respiratory rhythm. Using the language of stochastic processes, we analyzed both rhythms simultaneously taking the data from polysomnographic recordings of two healthy individuals. Each rhythm was treated as a sum of a deterministic drift term and a diffusion term (Kramers-Moyal expansion). We found that normal heart rate variability may be considered as the result of a bidirectional coupling of two nonlinear oscillators: the heart itself and the respiratory system. On average, the diffusion (noise) component measured is comparable in magnitude to the oscillatory (deterministic) term for both signals investigated. The application of the Kramers-Moyal expansion may be useful for medical diagnostics providing information on the relation between respiration and heart rate variability. This interaction is mediated by the autonomous nervous system, including the baroreflex, and results in a commonly observed phenomenon--respiratory sinus arrhythmia which is typical for normal subjects and often impaired by pathology. ©2009 American Institute of Physics 

Chaos 19, 028504 (2009)
Published 30 June 2009  
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1CORIA UMR 6614, Université de Rouen, Av. de l'Université, BP 12, F-76801 Saint-Etienne du Rouvray Cedex, France
2Service de Pneumologie et Soins Intensifs, UPRES EA 3830/IFR MP23, Centre Hospitalier Universitaire de Rouen, France
Providing a conclusive answer to the question "is this dynamics chaotic?" remains very challenging when experimental data are investigated. We showed that such a task is actually a difficult problem in the case of heart rates. Nevertheless, an appropriate dynamical analysis can discriminate healthy subjects from patients. ©2009 American Institute of Physics 

Chaos 19, 028505 (2009)
Published 30 June 2009 
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1PMB Intelligence LLC, P.O. Box 2077, West Lafayette, Indiana 47996, USA
2Affymetrix, Inc., 3380 Central Expressway, Santa Clara, California 95051, USA
3Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, Indiana 47907, USA
Previous studies on heart rate variability (HRV) using chaos theory, fractal scaling analysis, and many other methods, while fruitful in many aspects, have produced much confusion in the literature. Especially the issue of whether normal HRV is chaotic or stochastic remains highly controversial. Here, we employ a new multiscale complexity measure, the scale-dependent Lyapunov exponent (SDLE), to characterize HRV. SDLE has been shown to readily characterize major models of complex time series including deterministic chaos, noisy chaos, stochastic oscillations, random 1/f processes, random Levy processes, and complex time series with multiple scaling behaviors. Here we use SDLE to characterize the relative importance of nonlinear, chaotic, and stochastic dynamics in HRV of healthy, congestive heart failure, and atrial fibrillation subjects. We show that while HRV data of all these three types are mostly stochastic, the stochasticity is different among the three groups. ©2009 American Institute of Physics

Chaos 19, 028506 (2009)
Published 30 June 2009 
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1Dipartimento di Tecnologie dell'Informazione, Università degli studi di Milano, via Bramante 65, 26013 Crema, Italy
2Dipartimento di Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
In this paper, we participate to the discussion set forth by the editor of Chaos for the controversy, "Is the normal heart rate chaotic?" Our objective was to debate the question, "Is there some more appropriate term to characterize the heart rate variability (HRV) fluctuations?" We focused on the [approximate]24 h RR series prepared for this topic and tried to verify with two different techniques, generalized structure functions and wavelet transform modulus maxima, if they might be described as being multifractal. For normal and congestive heart failure subjects, the hq exponents showed to be decreasing for increasing q with both methods, as it should be for multifractal signals. We then built 40 surrogate series to further verify such hypothesis. For most of the series ([approximate]75%-80% of cases) multifractality stood the test of the surrogate data employed. On the other hand, series coming from patients in atrial fibrillation showed a small, if any, degree of multifractality. The population analyzed is too small for definite conclusions, but the study supports the use of multifractal series to model HRV. Also it suggests that the regulatory action of autonomous nervous system might play a role in the observed multifractality. ©2009 American Institute of Physics

Chaos 19, 028507 (2009)
Published 30 June 2009
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1Department of Physics, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
2Interdisciplinary Center for Dynamics of Complex Systems, University of Potsdam, 14476 Potsdam, Germany
3Potsdam Institute for Climate Impact Research, 14412 Potsdam, Germany
The incidence of cardiovascular diseases increases with the growth of the human population and an aging society, leading to very high expenses in the public health system. Therefore, it is challenging to develop sophisticated methods in order to improve medical diagnostics. The question whether the normal heart rate is chaotic or not is an attempt to elucidate the underlying mechanisms of cardiovascular dynamics and therefore a highly controversial topical challenge. In this contribution we demonstrate that linear and nonlinear parameters allow us to separate completely the data sets of the three groups provided for this controversial topic in nonlinear dynamics. The question whether these time series are chaotic or not cannot be answered satisfactorily without investigating the underlying mechanisms leading to them. We give an example of the dominant influence of respiration on heart beat dynamics, which shows that observed fluctuations can be mostly explained by respiratory modulations of heart rate and blood pressure (coefficient of determination: 96%). Therefore, we recommend reformulating the following initial question: "Is the normal heart rate chaotic?" We rather ask the following: "Is the normal heart rate `chaotic' due to respiration?" ©2009 American Institute of Physics  

Chaos 19, 028508 (2009)
Published 30 June 2009 
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1Biological Physics, University of Manchester, Manchester M13 9PL, United Kingdom
2College of Physics and Electronic Information, Anhui Normal University, Wuhu 241000, China
3Institute of Membrane and Systems Biology and Multidisciplinary Cardiovascular Research Centre, University of Leeds, Leeds LS2 9JT, United Kingdom
4Faculty of Medical and Human Sciences, University of Manchester, Manchester M13 9PL, United Kingdom
Fluctuations in the time interval between two consecutive R-waves of electrocardiogram during normal sinus rhythm may result from irregularities in the autonomic drive of the pacemaking sinoatrial node (SAN). We use a biophysically detailed mathematical model of the action potentials of rabbit SAN to quantify the effects of fluctuations in acetylcholine (ACh) on the pacemaker activity of the SAN and its variability. Fluctuations in ACh concentration model the effect of stochastic activity in the vagal parasympathetic fibers that innervate the SAN and produce varying rates of depolarization during the pacemaker potential, leading to fluctuations in cycle length (CL). Both the estimated maximal Lyapunov exponent and the noise limit of the resultant sequence of fluctuating CLs suggest chaotic dynamics. Apparently chaotic heart rate variability (HRV) seen in sinus rhythm can be produced by stochastic modulation of the SAN. The identification of HRV data as chaotic by use of time series measures such as a positive maximal Lyapunov exponent or positive noise limit requires both caution and a quantitative, predictive mechanistic model that is fully deterministic. ©2009 American Institute of Physics

Chaos 19, 028509 (2009)
Published 30 June 2009 
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Controversial Topics in Nonlinear Science is a new section of Chaos which appears from time to time to address timely and controversial topics related to nonlinear science. We hope this new section sparks thoughtful discussion on these topics.

Is the Normal Heart Rate Chaotic?
For this issue of Chaos, PhysioNet (, an NIH-funded resource that makes physiologic time series and related open-source software freely available to researchers) provided 15 time series, obtained from healthy subjects and from two groups of patients with heart disease, of time intervals between consecutive heart beats.

Each series is about 24 hours long (roughly 100,000 intervals). Several of the articles in this issue present analyses of the standard data sets provided by PhysioNet, which remain available at:

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