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Fractals, Vol. 13, No. 4 (2005) 253–263 c World Scientific Publishing Company


Articles

SOME CASES OF CROSSOVER BEHAVIOR IN HEART INTERBEAT AND ELECTROSEISMIC TIME SERIES ˜ A. MUNOZ-DIOSDADO Unidad Profesional Interdisciplinaria de Biotecnolog´ıa, Instituto Polit´ecnico, Nacional Av. Acueducto s/n, L. Ticom´ an, M´exico, D. F. 07340, M´exico [email protected] ´ L. GUZMAN-VARGAS Unidad Profesional Interdisciplinaria en Ingenier´ıa y Tecnolog´ıas Avanzadas Instituto Polit´ecnico Nacional, Av. IPN No. 2580, L. Ticom´ an M´exico D. F. 07340, M´exico A. RAM´IREZ-ROJAS Departamento de Ciencias B´ asicas, Universidad Aut´ onoma Metropolitana Azcapotzalco, M´exico D. F., 02200, M´exico J. L. DEL R´IO-CORREA Universidad Auton´ oma Metropolitana Iztapalapa M´exico D. F., 09340, M´exico F. ANGULO-BROWN 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 Received January 25, 2005 Accepted March 23, 2005

Abstract Fractal time series with scaling properties expressed through power laws appear in many contexts. These properties are very important from several viewpoints. For instance, they reveal the nature of the correlations present in the fractal signals. It is common that the scaling properties characterized by means of invariant quantities suffer changes along with the dynamical

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evolution of the studied systems. One of these changes is a crossover in the scaling properties reflecting an important change in the system dynamical behavior. In this article, we present two cases of crossover behavior corresponding to interbeat and electroseismic time series, we observe the crossovers in time series of experimental data and their corresponding simulation with simple models. We suggest a possible explanation of the observed crossovers in terms of the models considered. Keywords: Fractal Dimension; Time Series; Multifractals; Heart Rate Dynamics; Crossover Behavior; Electroseismic Time Series.

1. INTRODUCTION In recent years, the so-called fractal time series have received a lot of attention in many fields of science. In fractal fluctuating signals is common to find power law scaling behavior in several statistical measures of the time series. Among the most usual methods for scaling studies of time series are the Hurst rescaled-ranged analysis,1 the power spectrum analysis,2 the detrended fluctuation analysis (DFA)3,4 and the Higuchi fractal method.5,6 Two of the fields where those methods have been applied are both in some physiological signals7,8 as the heartbeat time series and in the so-called seismic electric signals (SES).9,10 In the first case, the healthy heartbeat is traditionally thought to be regulated according to the principle of homeostasis whereby physiological systems operate to reduce variability and achieve an equilibrium-like state.11 However, some recent studies12 show that even under resting conditions the human heartbeat fluctuations display the kind of long-range correlations typically exhibited by dynamical systems far from equilibrium.13 Nevertheless, in some cases, such as patients with severe congestive heart failure and healthy elderly subjects the long-range correlation behavior shows a breakdown manifested as a crossover phenomenon in some statistical measure of the time series, such as, fractal dimension D;8,14 DFA exponent α4,7 and the spectral exponent β.15 In the case of SES, crossover phenomena have also been reported — which are possibly relevant for the question of discriminating between a true signal (SES activity) and an artificial noise.16 The solution of this question is considered a crucial test for the feasibility of the so-called VAN method for searching electric precursors of seisms.17 Relevant crossover phenomena in scaling properties have also been reported in the study of correlations in sea-level elevations of the North Sea;15 in

systems exhibiting continuous phase transitions;18 in particle flow through a single pore;19 in simplified models of a limit order market20 and in jerky flows.21 In the present work, we study some interesting crossover phenomena occurring in both heart interbeat time series and SES arising from electric self-potentials measured in an electro-seismic station located at Guerrero state near the Mexican subduction zone.22 The article is organized as follows. In Sec. 2, we introduce a brief resume of the four methods we use for the time series analysis (DFA, Higuchi method, power spectrum and multifractal analysis). In Sec. 3, we show the crossovers appearing in both cardiac and electroseismic series when they are analyzed by means of the methods of Sec. 2. In Sec. 4, we propose some numerical models reproducing the crossover behavior observed in Sec. 2. Finally, in Sec. 5, we present some conclusions of our study.

2. METHODS 2.1. Spectral Method A time series can be either described in the time domain as x(t) or in the frequency domain in terms of the amplitude X(f, T ) where f is the frequency. This amplitude can be calculated by means of the Fourier transform applied to x(t) in the interval 0 < t < T.
T x(t)e2πif t dt. (1) X(f, T ) = 0

The quantity |X(f, t)|2 df is the contribution to the total energy of x(t) from those components with frequencies between f and f + df . The power spectral density is obtained by dividing by the period T , thus, S(f ) =

1 |X(f, T )|2 , T

(2)

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in the limit T → ∞. Then, the product S(f )df is the power in the time series associated with the frequency range f and f + df . A fractal property of a time series is reflected as a power law dependence between the spectral power and the frequency by means of an exponent, termed the spectral exponent S(f ) ∼

1 . fβ

(3)

The value of β, is the slope of the best-fit straight line to log(S(f )) versus log(f ) and is a measure of the strength of the persistence or anti-persistence in a time series.2 It is well known that the power spectrum can be considered as the Fourier cosine transform of the correlation function as is stated by the Weiner-Khintchine theorem.23 In this sense, also, the value of β is strongly related to the type of correlations present in a time series. For example, the uncorrelated white noise has a β ≈ 0, Brownian motion, the typical example of short correlated noise has a β ≈ 2, and the 1/f noise, that exhibits long-range correlations, has β ≈ 1. For calculation purposes, it is more convenient to use a more efficient method as the well-known Fast Fourier Transform algorithm.

2.2. DFA Analysis New methods arising from the nonlinear dynamics have been converted in important tools to obtain relevant information from physiological time series. In 1993, Peng et al.3 introduced a new method to detect long-range correlations called Detrended Fluctuation Analysis. This method is based in the classical random walk variations and has been used to detect long-range correlations in highly heterogeneous DNA sequences and other physiological signals.3 To illustrate the DFA method we depart from an initial time series (of total length N ), first,  this series is integrated, y(k) = N i=1 [B(i) − Bave ], after this, the resulting series is divided into boxes of size n. In each box of length n, a straight line is fitted to the points, termed the local trend, yn (k). Next, the line points are subtracted from the integrated series y(k), in each box. The root mean square fluctuation of the integrated and detrended series is calculated by means of   N 1  [y(k) − yn (k)]2 . (4) F (n) =  N k=1

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This process is taken over several scales (box sizes) to obtain a power law behavior between α F (n) and n , with α an exponent which reflects self-similar properties of the signal. The exponent α gives the information about the type of signal, α = 0.5 corresponds to white noise (noncorrelated signal), α = 1 means 1/f noise and α = 1.5 represents a Brownian motion. There exist also a well-known relationship between α and the spectral exponent β, α = (1 + β)/2.24 The DFA analysis has been used in the analysis of correlations of interbeat time series, for example, in 1996, Iyengar et al.24 suggested that aging is associated with disruption in the fractal-like longrange correlation that characterizes healthy sinus rhythm cardiac interval dynamics. They showed by means of DFA and spectral analysis that healthy young subjects have interbeat time series with fractal scaling expressed through a unique scaling exponent (in the case of DFA, α ≈ 1). On the other hand, in the case of healthy old individuals, the interbeat interval time series had two scaling regions, over the short range, interbeat interval fluctuations resembled a random walk process (α ≈ 1.5) whereas over the long-range, they resembled white noise (α ≈ 0.5). According to Iyengar et al.,24 this age-related loss of fractal organization (the pass of one exponent to two exponents) in heartbeat dynamics may reflect the degradation of integrated physiological regulatory systems and may impair an individual’s ability to adapt to stress.

2.3. The Higuchi’s Method Fractal dimension of self-similar objects in the plane is defined in terms of the isotropic distribution of their parts which can be scaled by a unique scale factor. This property changes in the case of selfaffine fractals because their spatial distribution is not isotropic and the scaling factor is different for each direction. Higuchi5 proposed a method to calculate fractal dimension of self-similar curves in terms of the slope of a straight line that fits the length of the curve versus the time interval (the lag) in a double log plot. The method consists in considering a finite set of data taken at a regular interval,

v(1), v(2), v(3), . . . , v(N ).

(5)

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From this series, we construct a new time series defined as

  N −k ·k v(m), v(m + k), v(m + 2k), . . . , v m + k (6) k , vm

with m = 1, 2, 3, . . . , k, where [ ] denotes Gauss’s notation and k, m are integers that indicate the initial time and the interval time, respectively. The k , is defined as: length of the curve vm  [ N−m ] k 1   |v(m + ik) Lm (k) =  k i=1





 N −1  − v(m + (i − 1)k|  N −m   k k

(7)

N −1 represents a normalization [ N−m ]k.k k factor. Then, the length of the curve for the interval k, is given by L(k), the average value over k sets Lm (k). Finally, if

and the term

L(k) ∝ k−D

(8)

then the curve is fractal with dimension D.5 For the case of self-affine curves, this fractal dimension is related with the spectral exponent β by means of β = 5 − 2D, where if D is in the range 1 < D < 2 then 1 < β < 3.5 Higuchi showed that this method provides an accurate estimation of the fractal dimension.

2.4. Multifractal Analysis Multifractal structures have been found in a growing number of physical problems. Recently, multifractal analysis has also been used for the study of physiological time series.11,25 For instance, Ivanov et al.11 established the relevance of the multifractal formalism for the description of a physiological signal. They stated that it is natural to expect a need for multifractal concepts to describe heartbeats since they are a result of the interaction of many physiological components operating in different scales. Therefore, the output of the signal has a nonlinear and inhomogeneous character. Multifractal distributions are characterized by the function f (α), the fractal dimension, plotted against α, the H¨ older exponent, or the multifractal spectrum.11,26 For the analysis of interbeat time series, we used the method proposed by Chhabra

and Jensen27,28 that provides a highly accurate, practical and efficient method for the direct computation of the singularity spectrum f (α). We can consider a time series as a singular measure P (x) if we normalize it. We calculate the f (α)-curve, first covering the measure with boxes of length L = 2−n and computing the probabilities Pi (L) in each of the boxes. We then construct the one-parameter family of normalized measures with [Pi (L)]q . µi (q, L) =  [Pj (L)]q

(9)

j

Finally, for each value of q we evaluate the numerators on the right-hand-sides of the equations:  µi (q, L) ln [µi (q, L)] i (10) f (q) = lim L→0 ln L  µi (q, L) ln [Pi (L)] i α(q) = lim (11) L→0 ln L for decreasing box sizes (increasing N ). We extract f (q) and α(q) from the numerators slopes versus ln L. The parameter q provides a microscope for exploring different regions of the singular measure. For q > 1, µ(q) amplifies the more singular regions of P , while for q < 1 it accentuates the less singular regions, and for q = 1 the measure replicates the original measure. Equations (10) and (11) provide a relationship between a Haussdorf dimension f and an average singularity strength α as implicit functions of the parameter q. The plot of f versus α is the multifractal spectrum. The mass exponent1 τ (q) is given in terms of the α(q) and the fractal dimension f (α(q)) by τ (q) = qα(q) − f (α(q)).

(12)

The curve f (α) characterizes the measure and is equivalent to the sequence of mass exponents τ (q). We can also describe the measure with the generalized fractal dimensions Dq which are given by Dq =

τ (q) . 1−q

(13)

3. OBSERVED CROSSOVERS A well-established example of crossover behavior in power law type relationships is the so-called b-value of the Gutenberg-Richter law for frequency magnitude distribution of earthquakes.29 This fact implies a break in self-similarity, from small to large

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earthquakes occurring at a point where the dimension of the event equals the down-dip width of the seismological layer (see Fig. 1 of Pacheco et al.29 ). Small earthquakes grow in both length (L) and width (W ); their rupture dimensions have no bounds. On the other hand, large earthquakes, have no bounds in rupture length, but their down-dip width is limited by thickness of the region capable of generating earthquakes.29 Another phenomenon where dimensionality plays an important role in crossover behavior of power laws is the case of sealevel elevations, in the North Sea. Dimon et al.15 analyzed a 105-year time record of hourly sea-level elevations for the port of Esbjerg, Denmark. These authors calculated the power spectrum of this time series finding that, in addition to well-known periodic components, the power spectrum has a lowfrequency broadband structure with three regimes behaving approximately as f 0 , f −1.2 and f −2.4 as frequency increases. They attributed this behavior to driven, damped Kelvin waves. The crossover to a nearly white noise spectrum occurs at fwn ∼ 5 × 10−7 Hz ∼ (20d)−1 (d is days). Above this frequency, there is a region where S(f ) ∼ f −1.2 (1/f like noise) and at a frequency fc ∼ 5 × 10−6 Hz ∼ (2d)−1 , the spectrum crosses over rather sharply into a regime where S(f ) ∼ f −2.4 underneath the lunar tidal peaks. Under a model based in the so-called telegraph equation, they interpret the crossover frequency fc as that where the penetration depth of the waves is of the order of h (the unperturbed water depth). At high frequencies the penetration depth is small and dissipation at the bottom is not important. At low frequencies the penetration

Fig. 1 The log-log of S versus f of a healthy young individual (32 years old) displays a 1/f -like behavior.

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depth becomes comparable to the depth, dissipation becomes dominant, and waves can no longer propagate. That is, in the Dimon et al. model the role of the fc has a clear physical interpretation in terms of a change in water depths. Another case of crossover behavior in power law relationships is that reported by Berge et al.19 by means of a rescaledrange analysis and power spectra of particle flow trough a single pore. In their experiment (and simulation), particles suspended in an electrolyte are caused to flow trough a pore by pressure difference. Each particle in the pore gives rise to a pulse in the measured pore resistance. The pulse height is proportional to the particle volume and the pulse width is given by the particle velocity. In dilute solutions, individual particle pulses can be detected and analyzed, but at higher concentrations the pulses overlap and the signal looks like noise. The signals were analyzed by Berge et al. by means of the Hurst’s exponent and the power spectrum and they find a clear crossover from persistent behavior for short times corresponding to the fact that particles reside a finite time in the pore to an independent process corresponding to the uncorrelated entry of particles into the pore. Here, we focused our attention in the study of crossover phenomena occurring in two systems: (1) heart interbeat time series and (2) signals arising from electric self-potentials measured in an electroseismic station near the Mexican subduction zone in the state of Guerrero. Heart rate dynamics is related to a large number of control mechanisms. Heartbeat fluctuations are very complex manifestations of regulatory neuroautonomic feedback loops.30 The output of healthy living systems, under certain parameter conditions, reveals a type of complex variability associated with long-range (fractal) correlations. The irregularity of the beat-to-beat time series for the case of healthy human heartbeat exhibits an absence of characteristic time scales. This is compatible with the concept of adaptability understood as a system ability to respond to environmental stimuli.14 Heart variability has been proposed as an important marker of changes of neuroautonomic control.7,24 The degradation of the neuroautonomic control of heart as a process occurring with aging and some heart failures has been proposed.7 An important question related with aging is to quantify the loss of 1/f -like behavior (long-range correlations), a synonym of healthy heart variability towards degraded regimes as those associated with aging. In Fig. 1,

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we depict a typical log-log plot of the power spectrum corresponding to a healthy young individual (32 years old) displaying an 1/f -like behavior. In Fig. 2, we show the power spectrum of a representative healthy elderly subject (77 years old) with an evident crossover around f = 0.03. Both previous time series were taken from Physionet31 corresponding to short segments of ECGs (two hours) equivalent to approximately 8000 beats. The observed crossover power spectra can also be observed by means of both DFA analysis (Fig. 3) and the fractal Higuchi’s method (Fig. 4). We shall discuss this crossover behavior in the next section. Between 1992 and 1998, the fluctuations of the electric self-potential of the ground in several sites

Fig. 2 The log-log of S versus f of a healthy elderly individual (77 years old) has a crossover around f = 0.03 Hz.

Fig. 3 DFA plot of log F (n) versus log n for data from one elderly healthy subject (Fig. 2) and its simulation, the crossover is evident. The agreement between experimental and simulated data is reasonable.

Fig. 4 Plot of logL(k) versus log k of heart interbeat sequences of the same data of Fig. 2. The crossover in both experimental and simulated data is also evident.

of Mexico were recorded. These measurements were taken at electroseismic stations whose details are described by Y´epez et al.22 Some stations were located along the coast of the state of Guerrero, near the Middle American trench, which is the border between the Cocos and the North American tectonic plates. In a typical electroseismic station, data were taken with sampling intervals of two and four seconds during periods in the scale of months and years. Several studies of those electroseismic time series in the context of searching for electric patterns possibly linked to impending earthquakes have been published elsewhere.22,32 The data collecting system consists of two electrodes buried 2 m separated approximately 50 m for each direction North-South (NS) and EastWest (EW). The self-potential was filtered with a cut-off frequency f c = 3 Hz (ULF range). One of the most interesting behaviors of these electroseismic time series was observed in the Acapulco station (16◦ 50’ N, 99◦ 47’ W) during recordings from August 1994 until July 1996 associated to an earthquake (EQ), Ms = 7.4, which occurred on 14 September 1995, approximately 110 km away from the station.32 In that reference, the problem of possible correlations between changes in the spectral exponent, β, and the Ms = 7.4 EQ were discussed. The β-exponent remained correlated for short segments of the electroseismic time series (∼6-hour files with 10,600 points), which showed a weak stationarity in the sense of Isliker and Kurths.33 In Figs. 5 and 6, we depict some very short segments of typical electroseismic time series. In Figs. 7 to 9, we show how during the considered period

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Fig. 5 Details observed in a short segment corresponding to NS channel monitored through an electroseismic station at Acapulco, Mexico.

Fig. 8 Representative power spectra behavior in the NS channel: 1/f -like behavior.

Fig. 6 Details observed in a short segment corresponding to EW channel monitored in the station mentioned in Fig. 5.

Fig. 9 Representative power spectra behavior in the NS channel: electroseismic crossover.

Fig. 7 Representative power spectra behavior in the NS channel: white noise.

(August 1994–July 1996) we find several behaviors: white noise (Fig. 7), 1/f -like noise (Fig. 8) and evident crossovers (Fig. 9). For a discussion on these different behaviors in relation to the Ms = 7.4 EQ, see Ram´ırez-Rojas et al.32 When the electric signal file corresponding to Fig. 7 is analyzed by means of the DFA and the Higuchi’s method, we obtain Figs. 10 and 11 respectively, where the crossover phenomena are also evident.

Fig. 10 The electroseismic crossover calculated by using the DFA method.

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1.035 1.03 1.025

o old

1.02 * young

Dq

1.015 1.01 1.005 1 0.995 0.99 0.985 -30

Fig. 11 The electroseismic crossover calculated by using the Higuchi method.

-10

0 q

10

20

30

Fig. 13 Spectrum of generalized fractal dimensions Dq as a function of the moment order q for real data (young and elderly healthy subjects).

1 0.9 0.8 0.7 0.6

f(α)

For the case of the multifractal analysis, the crossover behavior corresponding to Figs. 2 to 4 is depicted in Figs. 12 to 14. The multifractal spectrum does not show the crossover in a clear way, but it differentiates the dynamics of the interbeat time series of young healthy persons and elderly healthy persons. This difference is displayed as a loss of multifractality in the interbeat time series, it means that the width of the spectrum of elderly persons is small in comparison with the width of the spectrum of young persons (Fig. 13). The narrow multifractal spectrum for the elderly persons in Fig. 12

-20

0.5 0.4 * young (simulated)

0.3

o old (simulated)

0.2 0.1

1

0 0.94

0.9

0.8

0.96

Fig. 14

0.98

1

1.02 α

1.04

1.06

1.08

1.1

Multifractal spectra of simulated data.

f(α)

0.7

0.6

0.5

* young o old

0.4

0.3 0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

α

Fig. 12 Multifractal spectra of real data (young healthy and elderly healthy subjects). Interbeat time series of healthy young subjects and their simulation (see Fig. 14) display a broad multifractal spectra whereas the series of elderly healthy subjects and their simulation (Fig. 14) show narrow spectra.

is reflected in the flat Dq spectrum in Fig. 13. It seems to be that a crossover is reflected as a change in the width of the multifractal spectrum. Interestingly, this narrowness of the multifractal spectrum is also observed when we take the iterated function system rules for the Sierpinski gasket34 and applying them with unequal probabilities gives rise to a f (α) curve that becomes more narrow as the probabilities become more equal. What is becoming more narrow is the range of local H¨ older exponents throughout the multifractal. This narrowness process is consistent with the loss of repertoire of frequencies and time constants observed with aging in interbeat time series.14,35

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4. SOME NUMERICAL MODELS Recently, a simple model of the aging effect in heart interbeat time series was proposed.14 This model is based on combinations of noisy first-order autoregressive series. A simple model of 1/f noise is a stochastic process composed of a superposition of many modes with different time constants.13 One time constant can be obtained from a single firstorder autoregressive process, Xt+τ +1 = aXt−τ + εt−τ

(14)

where ε is a Gaussian distributed random variable and a is a coefficient that is related to correlation of events. Correlations between different events can τ be calculated as C(τ ) = Aa = Aeτ ln a , with A a constant. Time constants are related to the correlation function as the characteristic time in which the correlation has decayed 1/e. Thus, the autocorrelation function for a stochastic process with a single characteristic time is C(τ ) = Ae−τ /τ0 , with τ0 = − ln1a , clearly τ0 goes from zero to infinite while a varies from 0 to 1. As was reported by Iyengar et al.,24 healthy elderly heart rate dynamics can be resembled by a single first-order autoregressive relation with a single characteristic time. In fact, this result was obtained for healthy elderly subjects (three of them the oldest in their sample, 76, 77 and 81 years). In Guzman-Vargas and Angulo-Brown,14 the aging of the interbeat time series (RR-series) from an 1/f -like behavior corresponding to young individuals up to distinct degrees of deviation from this behavior expressed by crossovers in log-log plots of the Higuchi fractal dimension was mimicked by means of a loss of repertoire of responses to environmental stimuli. This was accomplished by a progressive diminution of the number of autoregressive processes [Eq. (14)] participating in the superposition of modes with exponential decay corresponding to a-values in the interval 0 ≤ a ≤ 1.14 In Figs. 3 and 4, a good agreement between the experimental and the simulated data is observed in the DFA α-exponents and in the Higuchi fractal dimension D. In Fig. 14, we see a reasonable agreement between experimental and simulated data under the multifractal analysis. A clear loss of multifractality is also observed in the simulated data of elderly subjects. For the numerical simulation of electroseismic time series (see Figs. 5 and 6), we use the fact that many of our short segment files has a dichotomous

Fig. 15

Liebovitch and Toth map.

nature, such as in Varotsos et al.9,10 Another phenomenon where time series with a dichotomous behavior have been reported is in ion current fluctuations in membrane channels.9,10 For a qualitative description of ion current recordings, Liebovitch and Toth (LT)36 proposed a linear map from the unit interval onto itself. This map divides the plane space in three regions (see Fig. 15). The closed states (CS) region corresponds to the interval [0, d1 ), where the LT map is given by f1 ; the open states (OS) region is given by f3 in the interval (d2 , 1] and the intermediate region is specified by f2 in the interval [d1 , d2 ]. All the previous functions are defined by  if 0 ≤ x ≤ d1 f1 = a1 x    d2 − x if d1 ≤ x ≤ d2 (15) f (x) = f2 = d2 − d1    f3 = a2 (x − 1) + 1 if d2 ≤ x ≤ 1. For both the CS and OS regions, the parameters a1 and a2 must be bigger than one, d1 is the threshold value separating the CS and OS regions. The LT map has three fixed points, i.e. points that satisfy the condition f (x∗ ) = x∗ , which are: d2 and x∗3 = 1. The three x∗1 = 0, x∗2 = d1 −d 2 +1 fixed points satisfy |f  (x∗i )| > 1, therefore they are repulsive points. As usual, the orbits of the LT map are generated by means of the iterative process xn+1 = f (xn ) beginning with the initial condition x0 . In Fig. 16, we can see the power spectrum of a signal generated by means of the LT map [Eq. (15)], including the crossover phenomenon.

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5. CONCLUDING REMARKS

Fig. 16 Electroseismic crossover (S versus f ) simulated with the LT-map with a1 = a2 = 1.2, d1 = 0.2 and d2 = 0.8.

In Figs. 17 and 18, we see how under DFA and Higuchi methods the crossover behavior is also observed in the LT map simulation of electroseismic behavior which is reminiscent of Fig. 9.

Fig. 17 Electroseismic crossover simulated with LT map by using the DFA method.

In the present paper, we report two crossover behavior corresponding to both interbeat and electroseismic time series, respectively. In the first case, we observe that interbeat time series of young healthy individuals can be described by only one exponent for the three employed methods (power spectra, Higuchi and DFA). However, for elderly healthy individuals a crossover appears. This crossover can be mimicked by means of a simple statistical model based on first order autoregressive processes. This model suggests that the pass of one to two exponents when the crossover appears is due to a loss of time constants repertoire with aging. For the case of the crossover behavior in electroseismic time series, a transition from white noise to 1/f -like noise is observed. That is, from an uncorrelated signal to a signal with long-range correlations. This behavior was observed few weeks before a main shock (Ms = 7.4) at the Pacific Mexican Coast in 14 September 1995.32 Here, a possible explanation has to do with the fact that the permanent white noise behavior of the electric signal between the two electrodes begins to be affected by a new signal probably linked to the preparation process of the impeding earthquake. The model used here indicates that such a signal could be a dichotomous signal similar to the one observed in ionic channels (as suggested by Varotsos et al.16 ). In summary, the crossover cases presented in this work are examples of the importance of these phenomena in some complex systems whose dynamics can be approached by means of nonlinear methods.

ACKNOWLEDGMENTS The authors wish to thank EDI, EDD and COFAA-IPN for partial support. We also thank an anonymous reviewer who brought our attention to the Sierpinski gasket issue.

REFERENCES

Fig. 18 Electroseismic crossover simulated with LT map by using the Higuchi method.

1. J. Feder, Fractals (Plenum Press, New York, 1988). 2. B. D. Malamud and D. L. Turcotte, Adv. Geophys. 40 (1999) 1–90. 3. C.-K. Peng, S. V. Buldyrev, A. L. Goldberger, S. Havlin, S. Simons and H. E. Stanley, Phys. Rev. E 47(5) (1993) 3730–3733. 4. A. L. Goldberger, L. A. Nunez-Amaral, J. M. Hausdorff, P. Ch. Ivanov, C.-K. Peng and H. E. Stanley, Proc. Nat. Acad. Sci. USA 99 (2002) 2466–2472.

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