Modeling heart rate variability in healthy humans: a turbulence analogy

Multifractal foundations of biomarker discovery for heart disease and stroke

Any reliable biomarker has to be specific, generalizable, and reproducible across individuals and contexts. The exact values of such a biomarker must represent similar health states in different individuals and at different times within the same individual to result in the minimum possible false-positive and false-negative rates. The application of standard cut-off points and risk scores across populations hinges upon the assumption of such generalizability. Such generalizability, in turn, hinges upon this condition that the phenomenon investigated by current statistical methods is ergodic, i.e., its statistical measures converge over individuals and time within the finite limit of observations. However, emerging evidence indicates that biological processes abound with nonergodicity, threatening this generalizability. Here, we present a solution for how to make generalizable inferences by deriving ergodic descriptions of nonergodic phenomena. For this aim, we proposed capturing the origin of ergodicity-breaking in many biological processes: cascade dynamics. To assess our hypotheses, we embraced the challenge of identifying reliable biomarkers for heart disease and stroke, which, despite being the leading cause of death worldwide and decades of research, lacks reliable biomarkers and risk stratification tools. We showed that raw R-R interval data and its common descriptors based on mean and variance are nonergodic and non-specific. On the other hand, the cascade-dynamical descriptors, the Hurst exponent encoding linear temporal correlations, and multifractal nonlinearity encoding nonlinear interactions across scales described the nonergodic heart rate variability more ergodically and were specific. This study inaugurates applying the critical concept of ergodicity in discovering and applying digital biomarkers of health and disease.

Decomposing the complexity of heart-rate variability by the multifractal-multiscale approach to detrended fluctuation analysis: an application to low-level spinal cord injury

OBJECTIVE

While several studies have assessed autonomic cardiovascular control after a spinal cord lesion using heart-rate variability (HRV) indices in the frequency and time domains, complexity measures have rarely been used, even if detrended fluctuation analysis (DFA) appeared promising. Recent developments in DFA decompose the multifractal contributions using temporal scales. Our aim is to evaluate the potential of these new DFA tools, considering as an example application the decomposition of HRV complexity in individuals with spinal cord injury (SCI) at a low lesion level, for whom alterations in traditional indices are not expected.

APPROACH

We enrolled 14 subjects with SCI with a lesion below the eleventh thoracic vertebra and 34 able-bodied (AB) controls. We recorded the R-R intervals (RRI) for 10 min in supine and sitting postures. We applied the multifractal-multiscale (MFMS) DFA to derive scale coefficients, α(q,τ), with function of the multifractal order q and scale τ, and evaluated a scale-coefficient dispersion index, α _SD(τ), as the standard deviation of α(q,τ) over q. We calculated the RRI increments, their magnitude and sign, estimating the MFMS DFA coefficients for the series of magnitude α _m(q,τ) and sign α _s(q,τ).

MAIN RESULTS

While sitting, differences between SCI and AB groups depended on q for coefficients 16  <  τ  <  32 s, so that α _SD(τ) was lower in individuals with SCI at τ  =  25 s. In the supine condition, short-term scales were greater in individuals with SCI for all q, and α _SD(τ) did not differ between groups. Group differences were found in α _s(q,τ) and not in α _m(q,τ) or in traditional HRV indices. The surrogate analysis showed AB-SCI differences in linear HRV components at scales τ  <  16 s and nonlinear components at larger scales.

SIGNIFICANCE

Complexity decomposition by DFA describes autonomic alterations in HRV in low-level paraplegia better than traditional indices, probably pointing out a loss of system complexity in the sitting posture and an impaired sympatho/vagal modulation in the supine position.

Hypothetical Control of Heart Rate Variability

In the last three decades, the analysis of heart rate variability by nonlinear methods demonstrated the complexity of cardiovascular regulation. Additionally to the observations of periodic heart rate regulation by the autonomic nervous system, the long-term statistics of the heart rate has been determined to reminisce a tempered Lévy process. A number of heuristic arguments have previously been made to support a tempering conjecture, using exponentially truncated waiting times for the time intervals between heart beats. Herein we use the fractional probability calculus to frame our arguments and to parameterize the control process that tempers the Lévy process through a collective-induced potential. We also determine that the hypothesis of a self-induced nonlinear potential control resulting in such a tempered Lévy process is consistent with the hypothesis of disease being the loss of physiologic complexity made over 25 years ago.

On the Motion of Spikes: Turbulent-Like Neuronal Activity in the Human Basal Ganglia

Neuronal signals are usually characterized in terms of their discharge rate, a description inadequate to account for the complex temporal organization of spike trains. Complex temporal properties, which are characteristic of neuronal systems, can only be described with the appropriate, complex mathematical tools. Here, I apply high order structure functions to the analysis of neuronal signals recorded from parkinsonian patients during functional neurosurgery, recovering multifractal properties. To achieve an accurate model of such multifractality is critical for understanding the basal ganglia, since other non-linear properties, such as entropy, depend on the fractal properties of complex systems. I propose a new approach to the study of neuronal signals: to study spiking activity in terms of the velocity of spikes, defining it as the inverse function of the instantaneous frequency. I introduce a neural field model that includes a non-linear gradient field, representing neuronal excitability, and a diffusive term to consider the physical properties of the electric field. Multifractality is present in the model for a range of diffusion coefficients, and multifractal temporal properties are mirrored into space. The model reproduces the behavior of human basal ganglia neurons and shows that it is like that of turbulent fluids. The results obtained from the model predict that passive electric properties of neuronal activity, including ephaptic coupling, are far more relevant to the human brain than what is usually considered: passive electric properties determine the temporal and spatial organization of neuronal activity in the neural tissue.

Structure Function Revisited: A Simple Tool for Complex Analysis of Neuronal Activity

Neural systems are characterized by their complex dynamics, reflected on signals produced by neurons and neuronal ensembles. This complexity exhibits specific features in health, disease and in different states of consciousness, and can be considered a hallmark of certain neurologic and neuropsychiatric conditions. To measure complexity from neurophysiologic signals, a number of different nonlinear tools of analysis are available. However, not all of these tools are easy to implement, or able to handle clinical data, often obtained in less than ideal conditions in comparison to laboratory or simulated data. Recently, the temporal structure function emerged as a powerful tool for the analysis of complex properties of neuronal activity. The temporal structure function is efficient computationally and it can be robustly estimated from short signals. However, the application of this tool to neuronal data is relatively new, making the interpretation of results difficult. In this methods paper we describe a step by step algorithm for the calculation and characterization of the structure function. We apply this algorithm to oscillatory, random and complex toy signals, and test the effect of added noise. We show that: (1) the mean slope of the structure function is zero in the case of random signals; (2) oscillations are reflected on the shape of the structure function, but they don't modify the mean slope if complex correlations are absent; (3) nonlinear systems produce structure functions with nonzero slope up to a critical point, where the function turns into a plateau. Two characteristic numbers can be extracted to quantify the behavior of the structure function in the case of nonlinear systems: (1). the point where the plateau starts (the inflection point, where the slope change occurs), and (2). the height of the plateau. While the inflection point is related to the scale where correlations weaken, the height of the plateau is related to the noise present in the signal. To exemplify our method we calculate structure functions of neuronal recordings from the basal ganglia of parkinsonian and healthy rats, and draw guidelines for their interpretation in light of the results obtained from our toy signals.

Neural code alterations and abnormal time patterns in Parkinson's disease

OBJECTIVE

The neural code used by the basal ganglia is a current question in neuroscience, relevant for the understanding of the pathophysiology of Parkinson's disease. While a rate code is known to participate in the communication between the basal ganglia and the motor thalamus/cortex, different lines of evidence have also favored the presence of complex time patterns in the discharge of the basal ganglia. To gain insight into the way the basal ganglia code information, we studied the activity of the globus pallidus pars interna (GPi), an output node of the circuit.

APPROACH

We implemented the 6-hydroxydopamine model of Parkinsonism in Sprague-Dawley rats, and recorded the spontaneous discharge of single GPi neurons, in head-restrained conditions at full alertness. Analyzing the temporal structure function, we looked for characteristic scales in the neuronal discharge of the GPi.

MAIN RESULTS

At a low-scale, we observed the presence of dynamic processes, which allow the transmission of time patterns. Conversely, at a middle-scale, stochastic processes force the use of a rate code. Regarding the time patterns transmitted, we measured the word length and found that it is increased in Parkinson's disease. Furthermore, it showed a positive correlation with the frequency of discharge, indicating that an exacerbation of this abnormal time pattern length can be expected, as the dopamine depletion progresses.

SIGNIFICANCE

We conclude that a rate code and a time pattern code can co-exist in the basal ganglia at different temporal scales. However, their normal balance is progressively altered and replaced by pathological time patterns in Parkinson's disease.

A mathematics for medicine: The Network Effect

The theory of medicine and its complement systems biology are intended to explain the workings of the large number of mutually interdependent complex physiologic networks in the human body and to apply that understanding to maintaining the functions for which nature designed them. Therefore, when what had originally been made as a simplifying assumption or a working hypothesis becomes foundational to understanding the operation of physiologic networks it is in the best interests of science to replace or at least update that assumption. The replacement process requires, among other things, an evaluation of how the new hypothesis affects modern day understanding of medical science. This paper identifies linear dynamics and Normal statistics as being such arcane assumptions and explores some implications of their retirement. Specifically we explore replacing Normal with fractal statistics and examine how the latter are related to non-linear dynamics and chaos theory. The observed ubiquity of inverse power laws in physiology entails the need for a new calculus, one that describes the dynamics of fractional phenomena and captures the fractal properties of the statistics of physiological time series. We identify these properties as a necessary consequence of the complexity resulting from the network dynamics and refer to them collectively as The Network Effect.

Coarse-grained multifractality analysis based on structure function measurements to discriminate healthy from distressed foetuses

This paper proposes a combined coarse-grained multifractal method to discriminate between distressed and normal foetuses. The coarse-graining operation was performed by means of a coarse-grained procedure and the multifractal operation was based on a structure function. The proposed method was evaluated by one hundred recordings including eighty normal foetuses and twenty distressed foetuses. We found that it was possible to discriminate between distressed and normal foetuses using the Hurst exponent, singularity, and Holder spectra.

Non-gaussianity of low frequency heart rate variability and sympathetic activation: lack of increases in multiple system atrophy and Parkinson disease

The correlates of indices of long-term ambulatory heart rate variability (HRV) of the autonomic nervous system have not been completely understood. In this study, we evaluated conventional HRV indices, obtained from the daytime (12:00–18:00) Holter recording, and a recently proposed non-Gaussianity index (λ; Kiyono et al., 2008) in 12 patients with multiple system atrophy (MSA) and 10 patients with Parkinson disease (PD), known to have varying degrees of cardiac vagal and sympathetic dysfunction. Compared with the age-matched healthy control group, the MSA patients showed significantly decreased HRV, most probably reflecting impaired vagal heart rate control, but the PD patients did not show such reduced variability. In both MSA and PD patients, the low-to-high frequency (LF/HF) ratio and the short-term fractal exponent α₁, suggested to reflect the sympathovagal balance, were significantly decreased, as observed in congestive heart failure (CHF) patients with sympathetic overdrive. In contrast, the analysis of the non-Gaussianity index λ showed that a marked increase in intermittent and non-Gaussian HRV observed in the CHF patients was not observed in the MSA and PD patients with sympathetic dysfunction. These findings provide additional evidence for the relation between the non-Gaussian intermittency of HRV and increased sympathetic activity.

Fractal and nonlinear changes in the long-term baseline fluctuations of fetal heart rate

The interpretation of heart rate patterns obtained by fetal monitoring relies on the definition of a baseline, which is considered as the running average heart rate in the absence of external stimuli during periods of fetal rest. We present a study along gestation of the baseline's fluctuations, in relation to fractal and nonlinear properties, to assess these fluctuations according with time-varying attracting levels introduced by maturing regulatory mechanisms. A low-risk pregnancy was studied weekly from the 17th to 38th week of gestation during long-term recording sessions at night (>6 h). Fetal averaged pulse rate samples and corresponding baseline series were obtained from raw abdominal ECG ambulatory data. The fractal properties of these series were evaluated by applying detrended fluctuation analysis. The baseline series were also explored to evaluate nonlinear properties and time ordering by applying the scaling magnitude and sign analyses. Our main findings are that the baseline shows fractal and even nonlinear anticorrelated fluctuations. This condition was specially the case before mid-gestation, as revealed by α values near to unit, yet becoming significantly more complex after 30 weeks of gestation as indicated by α(mag) values >0.5. The structured (i.e. not random) fluctuations and particular nonlinear changes that we found thus suggest that the baseline provides on itself information concerning the functional integration of cardiac regulatory mechanisms.

Market dynamics immediately before and after financial shocks: Quantifying the Omori, productivity, and Bath laws

We study the cascading dynamics immediately before and immediately after 219 market shocks. We define the time of a market shock T{c} to be the time for which the market volatility V(T{c}) has a peak that exceeds a predetermined threshold. The cascade of high volatility "aftershocks" triggered by the "main shock" is quantitatively similar to earthquakes and solar flares, which have been described by three empirical laws-the Omori law, the productivity law, and the Bath law. We analyze the most traded 531 stocks in U.S. markets during the 2 yr period of 2001-2002 at the 1 min time resolution. We find quantitative relations between the main shock magnitude M≡log{10} V(T{c}) and the parameters quantifying the decay of volatility aftershocks as well as the volatility preshocks. We also find that stocks with larger trading activity react more strongly and more quickly to market shocks than stocks with smaller trading activity. Our findings characterize the typical volatility response conditional on M , both at the market and the individual stock scale. We argue that there is potential utility in these three statistical quantitative relations with applications in option pricing and volatility trading.

Singular behavior of slow dynamics of single excitable cells

In various kinds of cultured cells, it has been reported that the membrane potential exhibits fluctuations with long-term correlations, although the underlying mechanism remains to be elucidated. A cardiac muscle cell culture serves as an excellent experimental system to investigate this phenomenon because timings of excitations can be determined over an extended time period in a noninvasive manner through visualization of contractions, although the properties of beat-timing fluctuations of cardiac muscle cells at the single-cell level remains to be fully clarified. In this article, we report on our investigation of spontaneous contractions of cultured rat cardiac muscle cells at the single-cell level. It was found that single cells exhibit several typical temporal patterns of contractions and spontaneous transitions among them. Detrended fluctuation analysis on the time series of interbeat intervals revealed the presence of 1/f(beta) noise at sufficiently large timescales. Furthermore, multifractality was also found in the time series of interbeat intervals. These experimental trends were successfully explained using a simple mathematical model, incorporating correlated noise into ionic currents. From these findings, it was established that singular fluctuations accompanying 1/f(beta) noise and multifractality are intrinsic properties of single cardiac muscle cells.

Levels of complexity in scale-invariant neural signals

Many physical and physiological signals exhibit complex scale-invariant features characterized by 1/f scaling and long-range power-law correlations, indicating a possibly common control mechanism. Specifically, it has been suggested that dynamical processes, influenced by inputs and feedback on multiple time scales, may be sufficient to give rise to 1/f scaling and scale invariance. Two examples of physiologic signals that are the output of hierarchical multiscale physiologic systems under neural control are the human heartbeat and human gait. Here we show that while both cardiac interbeat interval and gait interstride interval time series under healthy conditions have comparable 1/f scaling, they still may belong to different complexity classes. Our analysis of the multifractal scaling exponents of the fluctuations in these two signals demonstrates that in contrast to the multifractal behavior found in healthy heartbeat dynamics, gait time series exhibit less complex, close to monofractal behavior. Further, we find strong anticorrelations in the sign and close to random behavior for the magnitude of gait fluctuations at short and intermediate time scales, in contrast to weak anticorrelations in the sign and strong positive correlation for the magnitude of heartbeat interval fluctuations-suggesting that the neural mechanisms of cardiac and gait control exhibit different linear and nonlinear features. These findings are of interest because they underscore the limitations of traditional two-point correlation methods in fully characterizing physiological and physical dynamics. In addition, these results suggest that different mechanisms of control may be responsible for varying levels of complexity observed in physiological systems under neural regulation and in physical systems that possess similar 1/f scaling.

Methods derived from nonlinear dynamics for analysing heart rate variability

Methods from nonlinear dynamics (NLD) have shown new insights into heart rate (HR) variability changes under various physiological and pathological conditions, providing additional prognostic information and complementing traditional time- and frequency-domain analyses. In this review, some of the most prominent indices of nonlinear and fractal dynamics are summarized and their algorithmic implementations and applications in clinical trials are discussed. Several of those indices have been proven to be of diagnostic relevance or have contributed to risk stratification. In particular, techniques based on mono- and multifractal analyses and symbolic dynamics have been successfully applied to clinical studies. Further advances in HR variability analysis are expected through multidimensional and multivariate assessments. Today, the question is no longer about whether or not methods from NLD should be applied; however, it is relevant to ask which of the methods should be selected and under which basic and standardized conditions should they be applied.

Scale invariance of human electroencephalogram signals in sleep

In this paper, we investigate the dynamical properties of electroencephalogram (EEG) signals of humans in sleep. By using a modified random walk method, we demonstrate that scale-invariance is embedded in EEG signals after a detrending procedure is applied. Furthermore, we study the dynamical evolution of the probability density function (PDF) of the detrended EEG signals by nonextensive statistical modeling. It displays a scale-independent property, which is markedly different from the usual scale-dependent PDF evolution and cannot be described by the Fokker-Planck equation.

Estimator of a non-Gaussian parameter in multiplicative log-normal models

We study non-Gaussian probability density functions (PDF's) of multiplicative log-normal models in which the multiplication of Gaussian and log-normally distributed random variables is considered. To describe the PDF of the velocity difference between two points in fully developed turbulent flows, the non-Gaussian PDF model was originally introduced by Castaing [Physica D 46, 177 (1990)]. In practical applications, an experimental PDF is approximated with Castaing's model by tuning a single non-Gaussian parameter, which corresponds to the logarithmic variance of the log-normally distributed variable in the model. In this paper, we propose an estimator of the non-Gaussian parameter based on the q th order absolute moments. To test the estimator, we introduce two types of stochastic processes within the framework of the multiplicative log-normal model. One is a sequence of independent and identically distributed random variables. The other is a log-normal cascade-type multiplicative process. By analyzing the numerically generated time series, we demonstrate that the estimator can reliably determine the theoretical value of the non-Gaussian parameter. Scale dependence of the non-Gaussian parameter in multiplicative log-normal models is also studied, both analytically and numerically. As an application of the estimator, we demonstrate that non-Gaussian PDF's observed in the S&P500 index fluctuations are well described by the multiplicative log-normal model.

Multifractality and scale invariance in human heartbeat dynamics

Human heart rate is known to display complex fluctuations. Evidence of multifractality in heart rate fluctuations in healthy state has been reported [Ivanov, Nature (London) 399, 461 (1999)]. This multifractal character could be manifested as the dependence of the probability density functions (PDFs) of the interbeat interval increments, which are the differences in two interbeat intervals that are separated by n beats, on n . On the other hand, "scale invariance in the PDFs of detrended healthy human heart rate increments" was recently reported [Kiyono, Phys. Rev. Lett. 93, 178103 (2004)]. In this paper, we clarify that the scale invariance reported is actually exhibited by the PDFs of the increments of the "detrended" integrated healthy interbeat interval and should, therefore, be more accurately referred as the scale invariance or n independence of the PDFs of the sum of n detrended interbeat intervals. Indeed, we demonstrate explicitly that the PDFs of detrended healthy interbeat interval increments are scale or n dependent in accord with its multifractal character. Our work also establishes that this n independence of the PDFs of the sum of n detrended interbeat intervals is a general feature of human heartbeat dynamics, shared by heart rate fluctuations in both healthy and pathological states.

Fractal scale-invariant and nonlinear properties of cardiac dynamics remain stable with advanced age: a new mechanistic picture of cardiac control in healthy elderly

Heart beat fluctuations exhibit temporal structure with robust long-range correlations, fractal and nonlinear features, which have been found to break down with pathologic conditions, reflecting changes in the mechanism of neuroautonomic control. It has been hypothesized that these features change and even break down also with advanced age, suggesting fundamental alterations in cardiac control with aging. Here we test this hypothesis. We analyze heart beat interval recordings from the following two independent databases: 1) 19 healthy young (average age 25.7 yr) and 16 healthy elderly subjects (average age 73.8 yr) during 2 h under resting conditions from the Fantasia database; and 2) 29 healthy elderly subjects (average age 75.9 yr) during approximately 8 h of sleep from the sleep heart health study (SHHS) database, and the same subjects recorded 5 yr later. We quantify: 1) the average heart rate (<R-R>); 2) the SD sigma(R-R) and sigma(DeltaR-R) of the heart beat intervals R-R and their increments DeltaR-R; 3) the long-range correlations in R-R as measured by the scaling exponent alpha(R-R) using the Detrended Fluctuation Analysis; 4) fractal linear and nonlinear properties as represented by the scaling exponents alpha(sgn) and alpha(mag) for the time series of the sign and magnitude of DeltaR-R; and 5) the nonlinear fractal dimension D(k) of R-R using the fractal dimension analysis. We find: 1) No significant difference in (P > 0.05); 2) a significant difference in sigma(R-R) and sigma(DeltaR-R) for the Fantasia groups (P < 10(-4)) but no significant change with age between the elderly SHHS groups (P > 0.5); and 3) no significant change in the fractal measures alpha(R-R) (P > 0.15), alpha(sgn) (P > 0.2), alpha(mag) (P > 0.3), and D(k) with age. Our findings do not support the hypothesis that fractal linear and nonlinear characteristics of heart beat dynamics break down with advanced age in healthy subjects. Although our results indeed show a reduced SD of heart beat fluctuations with advanced age, the inherent temporal fractal and nonlinear organization of these fluctuations remains stable. This indicates that the coupled cascade of nonlinear feedback loops, which are believed to underlie cardiac neuroautonomic regulation, remains intact with advanced age.

Multiscale probability density function analysis: non-Gaussian and scale-invariant fluctuations of healthy human heart rate

For a detailed characterization of intermittency and non-Gaussianity of human heart rate, we introduce an analysis method to investigate the deformation process of the probability density function (PDF) of detrended increments when going from fine to coarse scales. To characterize the scale dependence of the multiscale PDF, we use two methods: 1) calculation of Kullback-Leibler relative entropy; 2) parameter estimation based on Castaing's equation (B. Castaing et al, 1990). We compare scale-dependence of the increment PDFs between actual heart rate fluctuations and artificially generated Gaussian and non-Gaussian noise, including a widely used autoregressive model and a recently proposed multifractal model based on a random cascade process. Our analysis highlights an essential difference between heart rate fluctuations and those generated by other models. The outstanding feature of human heart rate is the robust scale-invariance of the non-Gaussian PDF, which is preserved not only in a quiescent condition, but also in a dynamic state during waking hours, in which the mean level of heart rate is dramatically changing. Our results strongly suggest the need for revising existing models of heart rate variability to incorporate the scale-invariance in the PDF.

Phase transition in a healthy human heart rate

A healthy human heart rate displays complex fluctuations which share characteristics of physical systems in a critical state. We demonstrate that the human heart rate in healthy individuals undergoes a dramatic breakdown of criticality characteristics, reminiscent of continuous second order phase transitions. By studying the germane determinants, we show that the hallmark of criticality--highly correlated fluctuations--is observed only during usual daily activity, and a breakdown of these characteristics occurs in prolonged, strenuous exercise and sleep. This finding is the first reported discovery of the dynamical phase transition phenomenon in a biological control system and will be a key to understanding the heart rate control system in health and disease.

Volatility of linear and nonlinear time series

Previous studies indicated that nonlinear properties of Gaussian distributed time series with long-range correlations, u(i), can be detected and quantified by studying the correlations in the magnitude series |u(i)|, the "volatility." However, the origin for this empirical observation still remains unclear and the exact relation between the correlations in u(i) and the correlations in |u(i)| is still unknown. Here we develop analytical relations between the scaling exponent of linear series u(i) and its magnitude series |u(i)|. Moreover, we find that nonlinear time series exhibit stronger (or the same) correlations in the magnitude time series compared with linear time series with the same two-point correlations. Based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series; the nonlinear multifractal time series is generated by multiplying a long-range correlated time series (that represents the magnitude series) with uncorrelated time series [that represents the sign series sgn (u(i))]. We apply our techniques on daily deep ocean temperature records from the equatorial Pacific, the region of the El-Ninõ phenomenon, and find: (i) long-range correlations from several days to several years with 1/f power spectrum, (ii) significant nonlinear behavior as expressed by long-range correlations of the volatility series, and (iii) broad multifractal spectrum.

1/f scaling in heart rate requires antagonistic autonomic control

We present systematic evidence for the origins of 1/f -type temporal scaling in human heart rate. The heart rate is regulated by the activity of two branches of the autonomic nervous system: the parasympathetic (PNS) and the sympathetic (SNS) nervous systems. We examine alterations in the scaling property when the balance between PNS and SNS activity is modified, and find that the relative PNS suppression by congestive heart failure results in a substantial increase in the Hurst exponent H towards random-walk scaling 1/ f(2) and a similar breakdown is observed with relative SNS suppression by primary autonomic failure. These results suggest that 1/f scaling in heart rate requires the intricate balance between the antagonistic activity of PNS and SNS.

Critical scale invariance in a healthy human heart rate

We demonstrate the robust scale-invariance in the probability density function (PDF) of detrended healthy human heart rate increments, which is preserved not only in a quiescent condition, but also in a dynamic state where the mean level of the heart rate is dramatically changing. This scale-independent and fractal structure is markedly different from the scale-dependent PDF evolution observed in a turbulentlike, cascade heart rate model. These results strongly support the view that a healthy human heart rate is controlled to converge continually to a critical state.

Hierarchical structure in healthy and diseased human heart rate variability

It is shown that the healthy and diseased human heart rate variability (HRV) possesses a hierarchical structure of the She-Leveque (SL) form. This structure, first found in measurements in turbulent fluid flows, implies further details in the HRV multifractal scaling. The potential of diagnosis is also discussed based on the characteristics derived from the SL hierarchy.

Robustness and perturbation in the modeled cascade heart rate variability

In this study, numerical experiments are conducted to examine the robustness of using cascade to describe the multifractal heart rate variability (HRV) by perturbing the hierarchical time scale structure and the multiplicative rule of the cascade. It is shown that a rigid structure of the multiple time scales is not essential for the multifractal scaling in healthy HRV. So long as there exists a tree structure for the multiplication to take place, a multifractal HRV and related properties can be captured by using the cascade. But the perturbation of the multiplicative rule can lead to a qualitative change. In particular, a multifractal to monofractal HRV transition can result after the product law is perturbed to an additive one at the fast time scale. We suggest that this explains the similar HRV scaling transition in the parasympathetic nervous system blockade.

Quantification of heart rate variability by discrete nonstationary non-Markov stochastic processes

We develop the statistical theory of discrete nonstationary non-Markov random processes in complex systems. The objective of this paper is to find the chain of finite-difference non-Markov kinetic equations for time correlation functions (TCF) in terms of nonstationary effects. The developed theory starts from careful analysis of time correlation through nonstationary dynamics of vectors of initial and final states and nonstationary normalized TCF. Using the projection operators technique we find the chain of finite-difference non-Markov kinetic equations for discrete nonstationary TCF and for the set of nonstationary discrete memory functions (MF's). The last one contains supplementary information about nonstationary properties of the complex system on the whole. Another relevant result of our theory is the construction of the set of dynamic parameters of nonstationarity, which contains some information of the nonstationarity effects. The full set of dynamic, spectral and kinetic parameters, and kinetic functions (TCF, short MF's statistical spectra of non-Markovity parameter, and statistical spectra of nonstationarity parameter) has made it possible to acquire the in-depth information about discreteness, non-Markov effects, long-range memory, and nonstationarity of the underlying processes. The developed theory is applied to analyze the long-time (Holter) series of RR intervals of human ECG's. We had two groups of patients: the healthy ones and the patients after myocardial infarction. In both groups we observed effects of fractality, standard and restricted self-organized criticality, and also a certain specific arrangement of spectral lines. The received results demonstrate that the power spectra of all orders (n=1,2, ...) MF m(n)(t) exhibit the neatly expressed fractal features. We have found out that the full sets of non-Markov, discrete and nonstationary parameters can serve as reliable and powerful means of diagnosis of the cardiovascular system states and can be used to distinguish healthy data from pathologic data.

Fractal dynamics in physiology: alterations with disease and aging

According to classical concepts of physiologic control, healthy systems are self-regulated to reduce variability and maintain physiologic constancy. Contrary to the predictions of homeostasis, however, the output of a wide variety of systems, such as the normal human heartbeat, fluctuates in a complex manner, even under resting conditions. Scaling techniques adapted from statistical physics reveal the presence of long-range, power-law correlations, as part of multifractal cascades operating over a wide range of time scales. These scaling properties suggest that the nonlinear regulatory systems are operating far from equilibrium, and that maintaining constancy is not the goal of physiologic control. In contrast, for subjects at high risk of sudden death (including those with heart failure), fractal organization, along with certain nonlinear interactions, breaks down. Application of fractal analysis may provide new approaches to assessing cardiac risk and forecasting sudden cardiac death, as well as to monitoring the aging process. Similar approaches show promise in assessing other regulatory systems, such as human gait control in health and disease. Elucidating the fractal and nonlinear mechanisms involved in physiologic control and complex signaling networks is emerging as a major challenge in the postgenomic era.

A phenomenology model of normal sinus rhythm in healthy humans

The fractal component in the daytime healthy heartbeat interval data is studied from the perspective of cascade in fluid turbulence. Based on the electrophysiology of the heart muscle cell, a bounded random cascade model is assumed and the scaling property of the model is derived. In application, a "cascade decomposition" is proposed to extract the model parameter based on the experimental data. Simulation results show that healthy heart rate variability (HRV) can be well captured by the multiplicative process and imply the significance of sympatho-vagal interaction in the fractal component of long-term healthy HRV.

From 1/f noise to multifractal cascades in heartbeat dynamics

We explore the degree to which concepts developed in statistical physics can be usefully applied to physiological signals. We illustrate the problems related to physiologic signal analysis with representative examples of human heartbeat dynamics under healthy and pathologic conditions. We first review recent progress based on two analysis methods, power spectrum and detrended fluctuation analysis, used to quantify long-range power-law correlations in noisy heartbeat fluctuations. The finding of power-law correlations indicates presence of scale-invariant, fractal structures in the human heartbeat. These fractal structures are represented by self-affine cascades of beat-to-beat fluctuations revealed by wavelet decomposition at different time scales. We then describe very recent work that quantifies multifractal features in these cascades, and the discovery that the multifractal structure of healthy dynamics is lost with congestive heart failure. The analytic tools we discuss may be used on a wide range of physiologic signals. (c) 2001 American Institute of Physics.