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Evolution in time and scales of the stability of heart interbeat rate ´ndez-Pe ´rez1 , L. Guzma ´n-Vargas2 , I. Reyes-Ram´ırez2,3 and F. Angulo-Brown3 R. Herna 1

SATMEX. Av. de las Telecomunicaciones S/N CONTEL Edif. SGA-II. M´exico, D.F. 09310, M´exico. Unidad Profesional Interdisciplinaria en Ingenier´ıa y Tecnolog´ıas Avanzadas, Instituto Polit´ecnico Nacional, Av. IPN No. 2580, Col. Ticom´ an, M´exico D.F. 07340, M´exico. 3 Departamento de F´ısica, Escuela Superior de F´ısica y Matem´ aticas, Instituto Polit´ecnico Nacional, Edif. No. 9 U.P. Zacatenco, M´exico D. F., 07738, M´exico. 2

PACS PACS PACS

87.19.Hh – Cardiac dynamics 89.20.-a – Interdisciplinary applications of physics 89.75.Da – Systems obeying scaling laws

Abstract. - We approach heart interbeat rate by observing the evolution of its stability on scales and time, using tools for the analysis of frequency standards. In particular, we employ the dynamic Allan variance, which is used to characterize the time-varying stability of an atomic clock, to analyze heart interbeat time series for normal subjects and patients with congestive heart failure (CHF). Our stability analysis shows that healthy dynamics is characterized by at least two stability regions along different scales. In contrast, diseased patients exhibit at least three different stability regions; over short scales the fluctuations resembled white noise behavior whereas for large scales a drift is observed. The inflection points delimiting the first two stability regions for both groups are located around the same scales. Moreover, we find that CHF patients show lower variation of the stability in time than healthy subjects.

Introduction. – Analysis of heart rate variability using nonlinear tools have shown that the heart rate is related to different frequency components derived from a large number of control mechanisms from the autonomous nervous system [1]. Recent evidence of studies of heart rate variability indicates that healthy systems even at rest display highly irregular dynamics with multifractal character [2], and subjects with congestive heart failure (CHF) exhibit changes in this multifractality with time [2]. Moreover, different studies conducted on heart interbeat have revealed quantitative differences in the fractal organization and correlation properties between healthy and diseased subjects, by using different tools from statistical physics and nonlinear dynamics [2, 3]. Among the methods that have been used for the analysis of heartbeat signals are: detrended fluctuations analysis [4, 5], power spectral density [6], fractal dimension [7,8], multifractality [2, 9], Allan and Fano factors [6, 10], wavelets [1, 11] and excursions [12]. In this work, we are interested in studying the stability of the heartbeat interval rate by using the Allan variance statistics. The Allan variance was introduced in the field of time and frequency metrology to quantitatively characterize the frequency fluctuations observed in pre-

cise frequency standards, which exhibit nonstarionarities that are not satisfactorily treated with conventional statistical tools, such as the classical standard deviation [13]; and it has become a standard to define quantitatively the frequency instabilities of an oscillator; that consist of any unwanted departure from its nominal frequency value ν0 ; i. e., frequency stability is the degree to which an oscillator produces a constant frequency over a specified time interval [14]. Recently, the dynamic Allan variance was developed, which is a representation of the time-varying stability of an atomic clock and it has been reported its effectiveness in tracking common nonstationary features of an oscillator [15]. Previous studies in the statistical features of heartbeat dynamics, such as using the Allan factor to count the number of beats in different time windows [6] or those using the segmentation method [16] or excursions [12], analyze the statistics and scale invariance of the heart interbeat signals for multiple scales. In the present work we map the heart beat to the “tick” of a precise oscillator (which is subject to noise) to take advantage of statistical and conceptual tools developed in time and frequency metrology for the quantitative characterization of frequency instabilities in an oscillator. Therefore, we study the stability of the heart interbeat interval rate, ap-

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R. Hern´andez-P´erez et al. plying the dynamic Allan variance to: (i) identify nonsta- motion2 . This scaling property allows the identification of tionarities in the heart interbeat time series; (ii) quantify certain stochastic components in a signal (Fig. 1). the evolution of stability with scales and time; and, (iii) asσy (τ) y(t) sess how these properties change for healthy and diseased 10 5 subjects. −0.5 Our stability analysis shows that healthy dynamics is 10 0 characterized by at least two stability regions along differ−5 ent scales. In contrast, diseased patients exhibit at least 10 200 400 600 800 1000 10 10 10 three different stability regions; over short scales the flucτ t σy (t,τ) tuations resembled white noise behavior whereas for large 10 scales a drift is observed. The inflection points delimiting the first two stability regions for both groups are located around the same scales. Interestingly, we find that CHF 10 1000 800 τ 600 400 200 patients show lower variation in time of the stability than 10 0 t healthy subjects. 0

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Methods. – Allan Variance. In time and frequency metrology, the normalized frequency deviation y(t) of the oscillators (most commonly atomic clocks) is defined in terms of the nominal oscillator frequency ν0 and the instantaneous frequency ν(t) as: y(t) = (ν(t) − ν0 ) /ν0 [13]. The Allan variance (AVAR) is used as a standard to define quantitatively the stability of an oscillator. The definition of AVAR is given by the expression σy2 (τ ) = 1 yt+τ − y¯t )2 ⟩ [13], where τ is the observation interval, 2 ⟨(¯ the operator ⟨· ⟩ denotes time averaging and the average ∫ t+τ frequency deviation y¯t is defined as y¯t = τ1 t y(t′ ) dt′ . In discrete time, the AVAR is computed from frequency deviation data with the following estimator: σ ˆy2 [k] = ∑ M −2k+1 2 1 (¯ yk [i + k] − y¯k [i]) [13], where M is j=1 2(M −2k+1) the total number of data points, τ0 is the minimum observation time interval, and the integer k = τ /τ0 represents the discrete-time observation interval typically taking values k = 1, 2, . . . , ⌊N/3⌋1 (where ⌊·⌋ denotes integer part), and the averaged frequency values are given by ∑i+k−1 y¯k [i] ≡ k1 j=i y[j]. The AVAR at a particular scale is directly related to the variance of wavelet coefficients at that scale when Haar wavelet filters are used [17]. The frequency deviations of frequency standards are either systematic or stochastic. The latter are often well described by power-law spectral processes: Sy (f ) ∼ f β ; for which AVAR has an interesting property: it exhibits a power-law behavior : σy2 (τ ) ∼ τ η , where the exponents are related by η = −β −1 for −2 ≤ β ≤ 0 [13]. Another way to write the scaling relation is through the Allan deviation:

Fig. 1: (a) Synthetic nonstationary time series representing the frequency deviation of an oscillator (N = 1024); (b) its AVAR, σy (τ ) vs. τ (stability plot); and, (c) its DAVAR (Nw = N/8). According to AVAR, the signal seems a white noise (µ = −0.5); while, DAVAR captures the nonstationarities and the changes in the variability of the signal.

Dynamic Allan Variance. The dynamic Allan variance (DAVAR) was recently developed for the study of nonstationarities in atomic clocks [15]. It is obtained by calculating the estimator σ ˆy2 [k] of AVAR for the data contained in a window of certain length that slides through the data. Thus, DAVAR provides a way to quantify the evolution in time of the stability. The DAVAR is a deterministic quantity defined as the expected value of the estimator [15]: σ ˆy2 [n, k] = ∑ n+N /2−2k 2 w 1 yk [m + k] − y¯k [m]) , n = t/τ0 m=n−Nw /2 (¯ 2(Nw −2k+1) is the discrete time, Nw is the length of the analysis window3 , the∑ averaged frequency values are computed with i+k−1 y¯k [i] ≡ k1 j=i y[j], k = 1, 2, . . . , ⌊Nw /3⌋ and the sum runs from m = n − Nw /2 to m = n + Nw /2 − 2k − 1. The DAVAR is represented by a 3-D graph showing the variation of the oscillator’s stability (Fig. 1(c)). Thus, DAVAR at time t can be interpreted as a representation of the instantaneous stability of the oscillator; which is the result of an averaging process on the analysis window of length Nw [15]. Since noisy signals with power-law spectra follow a scaling law in the AVAR (Eq. 1), understanding whether there are nonstationarities in the clock behavior depends on identifying the changes in the scaling exponent µ.

Heart Rate Dynamics. – There is a debate about whether it is preferable to analyze heartbeat with referσy (τ ) ∼ τ η/2 ∼ τ µ , (1) ence to real time or to beat number. In Ref. [10], two methods are adapted for analyzing the data from each with −0.5 ≤ µ ≤ 0.5, where µ = −0.5 corresponds to 2 to the relation between the spectral (β) and Hurst white noise, µ = 0 to 1/f noise and µ = 0.5 to Brownian (H) According exponents [6], the following relations hold: µ = H − 1 for −1 < 1 It

is a convention extensively used in the time and frequency metrology field, and it is related to the uncertainty in the estimation of the Allan variance, which for a given averaging time is proportional to the number of differences that contribute to it.

β ≤ 0 and µ = H, for −2 ≤ β ≤ −1 3 Selecting the window length N w is a trade-off of two opposite goals [15]: (i) the window must be short for tracking quickly the fast variations in the signal; and (ii) the window must be long for having a good confidence in the estimation of DAVAR.

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perspective. The first approach treats the heartbeat as the event of interest and assumes that there is an objective “clock” for the timing of the event and that a heartbeat time record can be treated as a point process, i.e., a sequence of events (beats) distributed on the time axis. This approach was followed in Ref. [6], using AVAR to quantify changes in heartbeat time series for heart-failure and normal patients. The second approach emphasizes the interbeat interval and uses the beat number as an index of the biological time. In the identification of the heartbeat with the frequency deviations of an atomic clock, we use an intermediate approach: what we consider as the input signal is the duration of the interbeat interval (time between two consecutive beats). This signal is indexed with the interval number, on which is based the definition of the “scales” τ in the AVAR. Thus, the stability is referred to the variation in the duration of the interbeat interval for different “scales”. Then, instead of talking about the stability in a certain time scale, we talk about the stability at a certain interbeat scale (certain number of interbeat intervals). The advantage of this approach is that it allows studying the evolution of the important quantity: the interbeat rate, i.e., the frequency of the oscillator (the heart in this case).

over scales for a fixed time window; and it is defined as ( ) ∂ log σy (t, τ ) µt (τ ) ≡ , (2) ∂ log τ t which can be estimated by: µ ˆt (τ ) =

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Notice that µt (τ ) represents the instantaneous exponent of the scaling law in the Allan deviation (Eq. 1). This parameter is similar to the one used in Ref. [10] to estimate the deviation from power-law scaling by applying the Allan factor to the counts of the number of beats. On the other hand, the parameter γτ (t) quantifies the local slope, at scale τ fixed, of DAVAR with respect to time t, i. e., changes in DAVAR over time (different analysis windows) for a fixed scale. It is defined as: ( ) ∂σy (t, τ ) γτ (t) ≡ , (4) ∂t τ which is estimated by: γˆτ (t) =

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Both parameters µt (τ ) and γτ (t) will produce a threedimensional surface, representing the variation of the Stability of Heart Oscillations. The concept of homeheart interbeat rate stability in scale and in time, respecostasis, that refers to the tendency of biological systems to tively. maintain a relative constancy of the internal environment (blood sugar, blood pressure and the like) after perturbaResults. – We analyze RR interval sequences from tion [18], led to the proposal that physiological variables, two groups: 16 healthy subjects and 12 CHF patients [19], such as the cardiac interbeat interval T (n), maintain an with 6 hours of diurnal ECG records each with length approximately constant value in spite of continual pertur- N ∼ 30000 beats (Fig. 2). Notice that the normal subbations. Thus one can write in general T (n) = T0 + ξ, ject exhibits a higher variability than the CHF patient, where T0 is the preferred level for the interbeat interval although the latter exhibits more spikes and jumps than and ξ is a white noise with standard deviation σ [18]. for the normal subject. This database is an extended set Therefore, homeostasis is strongly related to the stability of records which we have used in previous studies [9, 12]. of the heart interbeat intervale rate. Since the heart can We compute DAVAR for each time series, considering winbe thought of as an oscillator, the applicability of AVAR dows with Nw = ⌊N/30⌋ ∼ 1000 points, with steps of size seems very natural. Therefore, mapping the heart inter- ⌊Nw /4⌋ ∼ 250 points and for scales k = 1, 2, . . . , kmax , beat to a frequency standard is relevant, and then, apply- with kmax = ⌊Nw /3⌋ ∼ 333 points. This selection, afing DAVAR to the heart interbeat time series for healthy ter computing other options, allows having enough data and unhealthy subjects would draw some conclusions on points to have good confidence in the estimation of AVAR the stability of the heart signals; for instance, it allows and tracking quick variations in the signals [15]. studying the evolution of the stability of the interbeat rate The stability analysis plots illustrating the results of for both groups. DAVAR for records of a normal subject and a CHF patient are shown in Figs. 3 and 4, respectively. The DAVAR Evaluating Changes in the Stability of Interbeat Interval profile for the normal subject (Fig. 3(a)) shows a less Rate. We are interested in quantifying the evolution of changing behavior than for the CHF patient (Fig. 4(a)), stability in scales and time of the heart interbeat rate. For suggesting that for the latter there are more nonstationary this task we introduce two parameters, namely µt (τ ) and features in the signals. The more changing behavior for γτ (t), which are computed from the DAVAR, and repre- the CHF patient is captured by µt (τ ) (Fig. 4(b)). Moresent the change of DAVAR with respect to scale and time, over, γτ (t) provides the evolution in time of the value of respectively. DAVAR for a given scale τ , and for the CHF patient it is The parameter µt (τ ) quantifies the local slope, at time t observed a smoother surface (Fig. 4(c)) than for the norfixed, of DAVAR with respect to the scale τ , i. e., changes mal subject (Fig. 3(c)), although with some noticeable in the slope of DAVAR (in the plane log σy (t, τ ) vs log τ ) peaks. p-3

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Since both parameters µt (τ ) and γτ (t) provide a surface, thus, to summarize the results for each group, we averaged them over time, denoted by ⟨·⟩t , and over scales, denoted by ⟨·⟩τ ; such that averaging over time(scales) gives the mean value of the parameter for a certain scale(time). Figures 5(a) and (b) show the average values for µt (τ ) for records of normal and CHF subjects, respectively, with time-averages shown in the main plots, and scale-averages in the insets. The behaviour of the averages for each group of subjects is remarkably different. For normal subjects, two inflection points are observed for the average scaling exponent µ over the scales (resulting in three regions), while for the CHF patients there are three inflection points (leading to four regions). The first two inflection points for each group occur approximately at the same scales: ∼ 10 and ∼ 60 interbeat intervals, respectively, while the third inflection point for the CHF group occurs at ∼ 120 interbeat intervals. Inflection points at ∼ 60 beats is consistent with previous studies [20]. Moreover, scale-averages for normal subjects are negative, while for CHF subjects they fluctuate around zero. The time average values for µ¯t (τ ) for the CHF patients have higher dispersion than for normal subjects. Moreover, for both groups the time-average for the shortest scale, i. e., few consecutive interbeat intervals, has a significant different value and the largest dispersion than for other scales. On the other hand, Figs. 5(c) and (d) show that both average values of γ¯τ (t) for normal subjects have higher dispersion than for CHF patients, indicating that on average the variation of σy (t, τ ) from one time(scale) to another, for a given scale(time), is more dependent on the value of scale(time). In contrast, the corresponding average values for CHF patients have lower dispersion. Thus, the local stability for heart interbeat of normal subjects tends to change in time and scales differently with respect to the time and scales, than for CHF patients for which a more regular evolution of stability in time and scales is observed. Furthermore, for analyzing the scales and time evolution, we compare the value of both µt (τ ) and γτ (t) for selected combinations of scales and time. For instance, for the short-term (standing at the first analysis window) and the long-term (last analysis window), we pick values of µt (τ ) for short (µτS ) and large (µτL ) scales (Figs. 6(a) and (b), respectively). Similarly, for the shortest and the

Fig. 3: Stability analysis for a normal subject (Fig. 2(a)): (a) the DAVAR profile; (b) µt (τ ), where each line corresponds to an analysis window (time fixed); and, (c) γτ (t), where each line corresponds to one scale.

Fig. 4: Stability analysis for a CHF patient (Fig. 2(b)): (a) the DAVAR profile; (b) µt (τ ); and, (c) γτ (t).

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largest scales, we pick values for the short (µtS ) and the long (µtL ) terms (Fig. 6(c) and (d), respectively). The same approach is applied to γτ (t) (Fig. 7). For the scaling exponent µt (τ ) (Figs. 6(a) and (b)), we observe that for healthy subjects the values for the small scales are slightly larger than for large scales; while for CHF patients the values from small scales are quite larger than for large ones. The values of µt (τ ) for healthy subjects are significantly different than for CHF patients (p-value < 0.05 by t-Student test). For small-scales (Fig. 6(c)), the scaling exponents are almost over the identity line for both groups indicating low variation with time. Moreover, for the healthy group the scaling exponents are positive, while for the CHF group is the opposite. For large scales (Fig. 6(d)), the scaling exponents are more spread. For the variation in time for DAVAR, expressed by γτ (t), we observe for short- and long-terms (Fig. 7(a) and (b)) a cluster around zero for CHF patients (specially for the shortterm), while for normal subjects the values are spread. In this case, we have not found statistical difference between both groups. A similar clustering around zero is observed also for CHF patients for values corresponding to small and large scales (Figs. 7(c) and (d)). For healthy subjects, the values for small scales are also clustered although at a less extent, however, at large scales they are spread out at a higher extent than for CHF group. Discussion. – Some previous studies have approached heart interbeat by using the so-called Allan factor, based in AVAR, calculated over the number of beats counted in boxes of length T , considering the heart beat dynamics as a point process [6,10]. However, the approach followed in these studies does not take advantage of the property of the AVAR to quantify the frequency stability of an oscillator. Moreover, the previous studies did not performed a dynamic analysis, i.e., an investigation of the time evolution of stability. The stability of a frequency

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Fig. 6: Scatter plots of the DAVAR scaling exponent, represented by µt (τ ), for normal (circles) and CHF (diamonds) subjects: (a) short-term, (b) long-term, (c) short-scale, and (d) large-scale.

standard, as quantified by AVAR, tells us the deviation of the oscillator’s frequency from its nominal value, at a certain observation period (scale) τ . Thus, a low value of AVAR indicates a more precise frequency standard (more stable oscillator), and also the variation of AVAR for different scales τ indicates the behavior of the stability. We found three scaling regions for the DAVAR for the healthy group and four for the CHF group, where the first two regions are delimited by inflection points at around the same scale (number of interbeat intervals). However, the values of the time-averaged scaling exponent ⟨µ⟩t in each of these regions are remarkably different between the groups. The behavior is opposite between the groups in the first two regions, while in the third one, the values of ⟨µ⟩t for the normal group decreases linearly, while for the CHF group they increase up to an inflection point, after which they remain approximatly constant. After the inflection point at k ∼ 60, the values of ⟨µ⟩t are negative and decrease monotonically, indicating that the stability of the interbeat rate for the normal group increases. On the contrary, for the CHF group the values of ⟨µ⟩t increase to reach positive values, indicating a degradation in stability of the interbeat rate. Previous studies have reported different scaling regimes for healthy and heartfailure subjects in sleep and wake phases, with a noticeable cross-over at ≈ 60 beats [20]; strongly suggesting that the scaling characteristics in the heartbeat fluctuations during sleep and wake phases are related to intrinsic mechanisms of neuroautonomic control. The several scaling regions reported in the present work support the hypothesis of the presence of intrinsic mechanisms of neuroautonomic control, discarding that these scaling differences are due to environmental noise, which can be treated as a trend and distinguished from the more subtle fluctuations that may reveal intrinsic correlation properties of the dynamics [20].

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We thank L. Galleani for helpul advice and discussion; and the anonymous referees whose suggestions led to improvement of the manuscript. This work was partially supported by CONACYT (Grant 49128-F-26020), COFAAIPN, EDI-IPN, M´exico.

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Moreover, the evolution of the DAVAR profile for both groups is captured by the scatter plots for µt (τ ) and γτ (t). For a stationary signal, we would expect to see a cluster around a certain value of µt (τ ). However, if the signal had different nonstationary features or correlated components, there would be larger dispersions or deviations from the identity line in the scatter plots. However, regarding parameter µt (τ ), we observe that for healthy subjects the scaling exponent for the small scales is slightly larger than for large scales; while for CHF patients we identify two regions with opposite behaviors. Thus, CHF signals have stochastic components close to white noise. These components dominate for small scales for which it corresponds a lower value of the scaling exponent (close to −0.5), which is consistent with other approaches [7]. This is not the case for healthy subjects, for which more long-correlated components are present, whose scaling exponent is larger. Conclusions. – We map the heart to an oscillator and then we perform a stability analysis of the heart interbeat interval rate, using tools developed for the characterization of the frequency stability of frequency standards. In particular, we employ the dynamic Allan variance (DAVAR) to analyze heart interbeat time series for normal subjects and patients with CHF. We find that DAVAR can discriminate the heart interbeat dynamics between both groups. Moreover, the evolution of stability in scales and time is revealed by DAVAR and its numerical partial derivatives. We consider that the DAVAR can contibute to the study of heart interbeat, not only for its computational simplicity but also for its ability to quantify the local stability and to reveal the nonstationarities in the signals. For making it useful for clinical usage, further work has to be performed to characterize the DAVAR profiles for several cardiopathies.

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