Deviations from uniform power law scaling in nonstationary time series

A classic problem in physics is the analysis of highly nonstationary time series that typically exhibit long-range correlations. Here we test the hypothesis that the scaling properties of the dynamics of healthy physiological systems are more stable than those of pathological systems by studying beat-to-beat fluctuations in the human heart rate. We develop techniques based on the Fano factor and Allan factor functions, as well as on detrended fluctuation analysis, for quantifying deviations from uniform power-law scaling in nonstationary time series. By analyzing extremely long data sets of up to N = 10(5) beats for 11 healthy subjects, we find that the fluctuations in the heart rate scale approximately uniformly over several temporal orders of magnitude. By contrast, we find that in data sets of comparable length for 14 subjects with heart disease, the fluctuations grow erratically, indicating a loss of scaling stability.     

Short-term forecasting of life-threatening cardiac arrhythmias based on symbolic dynamics and finite-time growth rates

Ventricular tachycardia or fibrillation (VT-VF) as fatal cardiac arrhythmias are the main factors triggering sudden cardiac death. The objective of this study is to find early signs of sustained VT-VF in patients with an implanted cardioverter-defibrillator (ICD). These devices are able to safeguard patients by returning their hearts to a normal rhythm via strong defibrillatory shocks; additionally, they store the 1000 beat-to-beat intervals immediately before the onset of a life-threatening arrhythmia. We study these 1000 beat-to-beat intervals of 17 chronic heart failure ICD patients before the onset of a life-threatening arrhythmia and at a control time, i.e., without a VT-VF event. To characterize these rather short data sets, we calculate heart rate variability parameters from the time and frequency domain, from symbolic dynamics as well as the finite-time growth rates. We find that neither the time nor the frequency domain parameters show significant differences between the VT-VF and the control time series. However, two parameters from symbolic dynamics as well as the finite-time growth rates discriminate significantly both groups. These findings could be of importance in algorithms for next generation ICD's to improve the diagnostics and therapy of VT-VF.     

Long-range anticorrelations and non-Gaussian behavior of the heartbeat

We find that the successive increments in the cardiac beat-to-beat intervals of healthy subjects display scale-invariant, long-range anticorrelations (up to 10(4) heart beats). Furthermore, we find that the histogram for the heartbeat intervals increments is well described by a Lévy stable distribution. For a group of subjects with severe heart disease, we find that the distribution is unchanged, but the long-range correlations vanish. Therefore, the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat.     

Scaling Behaviour and Memory in Heart Rate of Healthy Human

We investigate a set of complex heart rate time series from healthy human in different behaviour states with the detrended fluctuation analysis and diffusion entropy （DE） method. It is proposed that the scaling properties are influenced by behaviour states. The memory detected by DE exhibits an approximately same pattern after a detrending procedure. Both of them demonstrate the long-range strong correlations in heart rate. These findings may be helpful to understand the underlying dynamical evolution process in the heart rate control system, as well as to model the cardiac dynamic process.     

Detrended fluctuation analysis of heart intrabeat dynamics

We investigate scaling properties of electrocardiogram (ECG) recordings of healthy subjects and heart failure patients based on detrended fluctuation analysis (DFA). While the vast majority of scaling analysis has focused on the characterization of the long-range correlations of interbeat (i.e., beat-to-beat) dynamics, in this work we consider instead the characterization of intrabeat dynamics. That is, here we use DFA to study correlations for time scales smaller than one heart beat period (about 0.75 s). Our results show that intrabeat dynamics of healthy subject are less correlated than for heart failure dynamics. As in the case of interbeat dynamics, the DFA scaling exponents can be used to discriminate healthy and pathological data. It is shown that 0.5 h recordings suffices to characterize the ECG correlation properties.     

Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series

The healthy heartbeat is traditionally thought to be regulated according to the classical principle of homeostasis whereby physiologic systems operate to reduce variability and achieve an equilibrium-like state [Physiol. Rev. 9, 399-431 (1929)]. However, recent studies [Phys. Rev. Lett. 70, 1343-1346 (1993); Fractals in Biology and Medicine (Birkhauser-Verlag, Basel, 1994), pp. 55-65] reveal that under normal conditions, beat-to-beat fluctuations in heart rate display the kind of long-range correlations typically exhibited by dynamical systems far from equilibrium [Phys. Rev. Lett. 59, 381-384 (1987)]. In contrast, heart rate time series from patients with severe congestive heart failure show a breakdown of this long-range correlation behavior. We describe a new method--detrended fluctuation analysis (DFA)--for quantifying this correlation property in non-stationary physiological time series. Application of this technique shows evidence for a crossover phenomenon associated with a change in short and long-range scaling exponents. This method may be of use in distinguishing healthy from pathologic data sets based on differences in these scaling properties.     

Alteration in scaling behavior of short-term heartbeat time series for professional shooting athletes from rest to exercise

Scaling analysis of heartbeat time series has emerged as a useful tool for assessing the autonomic cardiac control under various physiologic and pathologic conditions. We study the heartbeat activity and scaling behavior of heartbeat fluctuations regulated by autonomic nervous system for professional shooting athletes under two states: rest and exercise, by applying the detrended fluctuation analysis method. We focus on alteration in correlation properties of heartbeat intervals for the shooters from rest to exercise, which may have a potential value in monitoring the quality of training and evaluating the sports capacity of the athletes. The result shows that scaling exponents of short-term heart rate variability signals from the shooters get significantly larger during exercise compared with those obtained at rest. It demonstrates that during exercise stronger correlations appear in the heartbeat series of shooting athletes in order to satisfy the specific requirements for high concentration and better control on their heart beats.     

Age-related changes in the "complexity" of cardiovascular dynamics: A potential marker of vulnerability to disease

Healthy physiologic control of cardiovascular function is a result of complex interactions between multiple regulatory processes that operate over different time scales. These include the sympathetic and parasympathetic nervous systems which regulate beat-to-beat heart rate (HR) and blood pressure (BP), as well as extravascular volume, body temperature, and sleep which influence HR and BP over the longer term. Interactions between these control systems generate highly variable fluctuations in continuous HR and BP signals. Techniques derived from nonlinear dynamics and chaos theory are now being adapted to quantify the dynamic behavior of physiologic time series and study their changes with age or disease. We have shown significant age-related changes in the 1/f(x) relationship between the log amplitude and log frequency of the heart rate power spectrum, as well as declines in approximate dimension and approximate entropy of both heart rate and blood pressure time series. These changes in the "complexity" of cardiovascular dynamics reflect the breakdown and decoupling of integrated physiologic regulatory systems with aging, and may signal an impairment in cardiovascular ability to adapt to external and internal perturbations. Studies are currently underway to determine whether the complexity of HR or BP time series can distinguish patients with fainting spells due to benign vasovagal reactions from those due to life-threatening cardiac arrhythmias. Thus, measures of the complexity of physiologic variability may provide novel methods to monitor cardiovascular aging and test the efficacy of specific interventions to improve adaptive capacity in old age. (c) 1995 American Institute of Physics.     

Quantification of fetal heart rate regularity using symbolic dynamics

Fetal heart rate complexity was examined on the basis of RR interval time series obtained in the second and third trimester of pregnancy. In each fetal RR interval time series, short term beat-to-beat heart rate changes were coded in 8 bit binary sequences. Redundancies of the 2(8) different binary patterns were reduced by two different procedures. The complexity of these sequences was quantified using the approximate entropy (ApEn), resulting in discrete ApEn values which were used for classifying the sequences into 17 pattern sets. Also, the sequences were grouped into 20 pattern classes with respect to identity after rotation or inversion of the binary value. There was a specific, nonuniform distribution of the sequences in the pattern sets and this differed from the distribution found in surrogate data. In the course of gestation, the number of sequences increased in seven pattern sets, decreased in four and remained unchanged in six. Sequences that occurred less often over time, both regular and irregular, were characterized by patterns reflecting frequent beat-to-beat reversals in heart rate. They were also predominant in the surrogate data, suggesting that these patterns are associated with stochastic heart beat trains. Sequences that occurred more frequently over time were relatively rare in the surrogate data. Some of these sequences had a high degree of regularity and corresponded to prolonged heart rate accelerations or decelerations which may be associated with directed fetal activity or movement or baroreflex activity. Application of the pattern classes revealed that those sequences with a high degree of irregularity correspond to heart rate patterns resulting from complex physiological activity such as fetal breathing movements. The results suggest that the development of the autonomic nervous system and the emergence of fetal behavioral states lead to increases in not only irregular but also regular heart rate patterns. Using symbolic dynamics to examine the cardiovascular system may thus lead to new insight with respect to fetal development.     

Preservation of long range temporal correlations under extreme random dilution

Many natural time series exhibit long range temporal correlations that may be characterized by power-law scaling exponents. However, in many cases, the time series have uneven time intervals due to, for example, missing data points, noisy data, and outliers. Here we study the effect of randomly missing data points on the power-law scaling exponents of time series that are long range temporally correlated. The Fourier transform and detrended fluctuation analysis (DFA) techniques are used for scaling exponent estimation. We find that even under extreme dilution of more than 50%, the value of the scaling exponent remains almost unaffected. Random dilution is also applied on heart interbeat interval time series. It is found that dilution of 70%-80% of the data points leads to a reduction of only 8% in the scaling exponent; it is also found that it is possible to discriminate between healthy and heart failure subjects even under extreme dilution of more than 90%.     

Effects of Cardiac Resynchronization Therapy on Cardio-Respiratory Coupling

In this study, the effect of cardiac resynchronization therapy (CRT) on the relationship between the cardiovascular and respiratory systems in heart failure subjects was examined for the first time. We hypothesized that alterations in cardio-respiratory interactions, after CRT implantation, quantified by signal complexity, could be a marker of a favorable CRT response. Sample entropy and scaling exponents were calculated from synchronously recorded cardiac and respiratory signals 20 min in duration, collected in 47 heart failure patients at rest, before and 9 months after CRT implantation. Further, cross-sample entropy between these signals was calculated. After CRT, all patients had lower heart rate and CRT responders had reduced breathing frequency. Results revealed that higher cardiac rhythm complexity in CRT non-responders was associated with weak correlations of cardiac rhythm at baseline measurement over long scales and over short scales at follow-up recording. Unlike CRT responders, in non-responders, a significant difference in respiratory rhythm complexity between measurements could be consequence of divergent changes in correlation properties of the respiratory signal over short and long scales. Asynchrony between cardiac and respiratory rhythm increased significantly in CRT non-responders during follow-up. Quantification of complexity and synchrony between cardiac and respiratory signals shows significant associations between CRT success and stability of cardio-respiratory coupling.     

Mechanisms for initiation of cardiac discordant alternans

Electrical alternans, defined as a beat-to-beat change in the duration of the excited phase of cardiac cells, is among the known precursors of sudden cardiac death. It may appear as concordant (all the tissue presenting the same phase of oscillation) or discordant (with out-of-phase regions distributed among tissue). Spatially discordant alternans can lead to unidirectional block that initiates reentry and ventricular fibrillation. The role played by tissue heterogeneities and heart rate changes in their initiation remains, however, unclear. We study the mechanisms for initiation of spatially discordant alternans by numerical simulations of an ionic model spatially distributed in a one-dimensional cable and in an anatomical model of the rabbit heart. The effects of CV-restitution, ectopic beats, and the role of spatial gradients of electrical restitution properties are investigated. In homogeneous tissue, the origin of discordant alternans may be dynamical, through CV-restitution, or due to a localized change in the pacing period. We also find that a sudden change of stimulation rate can initiate discordant alternans in the presence of a spatial gradient of APD-restitution without necessitating CV-restitution. The mechanism of, and the conditions for, initiation are determined based on an iterated map analysis of beat to beat changes of APD. This analysis leads to the definition of a vulnerable window for initiation of discordant alternans. Moreover, the pattern of spatially discordant alternans is found to change slowly over several beats following initiation, as reflected in ECG recordings.     

Unidirectional pinning and hysteresis of spatially discordant alternans in cardiac tissue

Spatially discordant alternans is a widely observed pattern of voltage and calcium signals in cardiac tissue that can precipitate lethal cardiac arrhythmia. Using spatially coupled iterative maps of the beat-to-beat dynamics, we explore this pattern's dynamics in the regime of a calcium-dominated period-doubling instability at the single-cell level. We find a novel nonlinear bifurcation associated with the formation of a discontinuous jump in the amplitude of calcium alternans at nodes separating discordant regions. We show that this jump unidirectionally pins nodes by preventing their motion away from the pacing site following a pacing rate decrease but permitting motion towards this site following a rate increase. This unidirectional pinning leads to strongly history-dependent node motion that is strongly arrhythmogenic.     

Respiratory modulation of heart beat-to-beat interval in diabetic patients

We have analyzed simultaneous recordings of respiration and heartbeat intervals in diabetic patients and control subjects. Our main findings are that in diabetic patients the heart beat-to-beat interval variability and cardiorespiratory crosscorrelation are decreased, the autocorrelation time of the interval series is increased, and the phase relation of the respiration with the heartbeat interval oscillations is often reversed in comparison with the control subjects. We have been able to reproduce the data using a biophysical model in which the time dependent input signal to the sinoatrial node was constituted of quasiperiodic and aperiodic components. The quasiperiodic input was obtained from the recording of the respiratory signal and the aperiodic input was obtained from selected realizations of correlated noise. Our study indicates that both input components to the sinoatrial node are modified in diabetic patients.     

What does measure the scaling exponent of the correlation sum in the case of human heart rate?

It is shown that in the case of human heart rate, the scaling behavior of the correlation sum (calculated by the Grassberger-Procaccia algorithm) is a result of the interplay of various factors: finite resolution of the apparatus (finite-size effects), a wide dynamic range of mean heart rate, the amplitude of short-time variability being a decreasing function of the mean heart rate. This is done via constructing a simple model of heart rhythm: a signal with functionally modulated Gaussian noise. This model reproduces the scaling behavior of the correlation sum of real medical data. The value of the scaling exponent depends on all the above-mentioned factors, and is a certain measure of short-time variability of the signal.     

Detection of changes in the fractal scaling of heart rate and speed in a marathon race

The aim of this study was to detect changes in the fractal scaling behavior of heart rate and speed fluctuations when the average runner’s speed decreased with fatigue. Scaling analysis in heart rate (HR) and speed (S) dynamics of marathon runners was performed using the detrended fluctuation analysis (DFA) and the wavelet based structure function. We considered both: the short-range (α₁) and the long-range (α₂) scaling exponents for the DFA method separated by a change-point, (box length), the same for all the races. The variability of HR and S decreased in the second part of the marathon race, while the cardiac cost time series (i.e. the number of cardiac beats per meter) increased due to the decreasing speed behavior. The scaling exponents α₁ and α₂ of HR and α₁ of S, increased during the race () as did the HR wavelet scaling exponent (τ). These findings provide evidence of the significant effect of fatigue induced by long exercise on the heart rate and speed variability.     

Physiologic time: A hypothesis

The scaling of respiratory metabolism with body size in animals is considered by many to be a fundamental law of nature. One apparent consequence of this law is the scaling of physiologic time with body size, implying that physiologic time is separate and distinct from clock time. Physiologic time is manifest in allometry relations for lifespans, cardiac cycles, blood volume circulation, respiratory cycle, along with a number of other physiologic phenomena. Herein we present a theory of physiologic time that explains the allometry relation between time and total body mass averages as entailed by the hypothesis that the fluctuations in the total body mass are described by a scaling probability density.     

用晚钠电流终止心脏中的螺旋波和时空混沌

采用人类心脏模型研究了用晚钠电流控制二维心脏组织中的螺旋波和时空混沌,我们提出这样的控制策略来产生晚钠电流:让慢失活门变量j始终等于0.7,同时实时调节钠电流的快失活门变量h的阈值电压V_I,即先让阈值电压V_I经过T_1时间从71.55 mV均匀减少到50.55 mV,然后经过T_2时间再从50.55 mV均匀增加到71.55 mV,当阈值电压V_I回到71.55 mV,钠电流的快、慢失活门变量恢复正常变化.数值模拟结果表明:只要适当选择控制时间,不论心肌细胞是否存在自发的晚钠电流,控制产生的晚钠电流都可以有效抑制螺旋波和时空混沌,而且需要的晚钠电流都很小,且控制时间都很短,因为螺旋波和时空混沌消失主要是通过传导障碍消失,少数情况下时空混沌是通过转变为靶波消失.我们希望这种控制方法能为室颤控制提供新的思路.     

Progress toward controlling in vivo fibrillating sheep atria using a nonlinear-dynamics-based closed-loop feedback method

We describe preliminary experiments on controlling in vivo atrial fibrillation using a closed-loop feedback protocol that measures the dynamics of the right atrium at a single spatial location and applies control perturbations at a single spatial location. This study allows investigation of control of cardiac dynamics in a preparation that is physiologically close to an in vivo human heart. The spatial-temporal response of the fibrillating sheep atrium is measured using a multi-channel electronic recording system to assess the control effectiveness. In an attempt to suppress fibrillation, we implement a scheme that paces occasionally the cardiac muscle with small shocks. When successful, the inter-activation time interval is the same and electrical stimuli are only applied when the controller senses that the dynamics are beginning to depart from the desired periodic rhythm. The shock timing is adjusted in real time using a control algorithm that attempts to synchronize the most recently measured inter-activation interval with the previous interval by inducing an activation at a time projected by the algorithm. The scheme is "single-sided" in that it can only shorten the inter-activation time but not lengthen it. Using probability distributions of the inter-activation time intervals, we find that the feedback protocol is not effective in regularizing the dynamics. One possible reason for the less-than-successful results is that the controller often attempts to stimulate the tissue while it is still in the refractory state and hence it does not induce an activation. (c) 2002 American Institute of Physics.     

Analysis of heartbeat asymmetry based on multi-scale time irreversibility test

We investigate the asymmetry of heart rate control system and suggest a simple index to quantify this asymmetry by performing high-dimensional time irreversibility tests to heartbeat interval time series over multiple scales. The results provide strong evidence to the concept that the asymmetry is an intrinsic property of heart rate control system. As a simple and visual method, it is proved to be effective in classifying physiologic and synthetic subjects while the maximum scale is selected within a proper range, and also provides a new way to analyze the time irreversibility for other high-dimensional systems.