arXiv:physics/0510215v1 [physics.geo-ph] 24 Oct 2005 ASnmes 13.k 54.a 54.p 87.19.Nn 05.45.Tp, 05.40.-a, 91.30.Dk, numbers: PACS electrocardiograms. their identify analysing behavior when the reminds which bursting, approaching The dynamics). (critical Signals Electric correlatio Seismic temporal and range long series ranged infinitely fBm from u stems in entropy simulations their numerical for by holds supported as same The distribution. “uniform” a nrp fsimceeti inl:Aayi nntrlti natural in Analysis signals: electric seismic of Entropy agrta hs ihrorpre.Teretoyi natur in entropy Their reported. hitherto those than larger lcrcsgashv enrcnl eodda h Earth’s the at recorded recently been have signals Electric nvriyo tes aeitmooi,Zgao 5 84, 157 Zografos Panepistimiopolis, Athens, of University .A Varotsos, A. P. 3 1 atqaePeito eerhCne,TkiUiest 3 University Tokai Center, Research Prediction Earthquake oi tt eto,PyisDprmn,Uiest fAth of University Department, Physics Section, State Solid 2 aeitmooi,Zgao 5 4 tes Greece Athens, 84, 157 Zografos Panepistimiopolis, oi at hsc nttt,PyisDepartment, Physics Institute, Physics Earth Solid hmz-rd,Siuk 2-60 Japan 424-8610, Shizuoka Shimizu-Orido, ,2, 1, ∗ .V Sarlis, V. N. time-reversal Abstract 1 1 .S Skordas, S. E. utain r on oices upon increase to found are fluctuations sadhneteesgasaeprobably are signals these hence and ns n udncricdahindividuals death cardiac sudden ing ltm ssalrta that, than smaller is time al ufc ihapiue appreciably amplitudes with surface ,2 1, o iervra.Ti behavior, This time-reversal. pon na no nemtec model, intermittency on-off an in n .K Tanaka K. H. and tes Greece Athens, -20-1, ens, eunder me 3 S u of , The time series analysis of various phenomena in complex systems (and especially those associated with impending catastrophic events, e.g.[1, 2]) in the framework of the newly defined time-domain[3, 4], termed natural time, reveals interesting features. Examples are electrocardiograms[5, 6], seismicity[3] and Seismic Electric Signals (SES) activities[3, 4, 7, 8, 9, 10]. This new time domain is optimal for enhancing the signals in time-frequency space when employing the Wigner function and measuring its localization property[11].

In a time series comprising N pulses, the natural time χk = k/N serves as the index[3, 4] for the occurrence of the k-th event. It is, therefore, smaller than, or equal to, unity.

In natural time analysis, the time evolution of the pair of the two quantities (χk, Qk) is

considered, where Qk denotes the duration of the k-th pulse. The entropy S in the natural time-domain is defined[8] as the derivative of the function hχqi−hχiq in respect to q, for q = 1, which gives [3, 8]: S ≡hχ ln χi−hχi lnhχi (1)

N N where hf(χ)i = Pk=1 pkf(χk) and pk = Qk/ Pn=1 Qn. It is a dynamic entropy depending on the sequential order of pulses[5, 6] and exhibits[9] concavity, positivity and Lesche[12, 13] stability. When the system enters the critical stage (infinitely ranged long range temporal

correlations[3, 4]), the S-value is smaller[8, 10] than the value Su(= 1/2 ln 2 − 1/4 ≈ 0.0966)

of a “uniform” distribution (defined in Refs. [3, 4, 6, 8], e.g. when when all pk are equal), i.e.,

S

The value of the entropy obtained upon considering the time reversal T , i.e., T pk = pN−k+1,

is labelled by S−. An important point emerged from the data analysis in Ref.[9] is the following: Although the study of the S values enables the distinction between SES activities and noises produced by nearby operating artificial (man made) electromagnetic sources

(AN), i.e., S

values. This is so, because for the SES activities we found that the S− values are smaller

than (or equal to) Su, while for AN the S− values are either smaller or larger than Su. Here, we provide more recent data on the SES activities, which strengthen the conclusion that

both S and S− are smaller than Su. In other words, the following key point seems to hold: In signals that exhibit critical dynamics (i.e., SES activities) upon time-reversal their entropy

values (though become different than those in forward time) still remain smaller than Su.

2 Why? The answer to this question is a challenge, because, if it is generally so, among similar looking signals we can distinguish those that exhibit critical dynamics. Since the latter signals are expected to exhibit infinitely ranged long range correlations, this might be the origin of the aforementioned behavior. To investigate this point, numerical simulations are presented here for fractional Brownian motion (fBm) time series as well as for an on-off intermittency model (for the selection of the latter see Ref.[14]). The simple case of an fBm was selected in view of the following suggestion forwarded in Ref.[7] concerning the Hurst exponent H. If we assume that, in general, H is actually a measure of the intensity of long range dependence, we may understand why in SES activities, when analyzed in the natural time domain, we find H values close to unity, while in AN (where the long range correlations are weaker[7]) the H-values are markedly smaller (e.g., around 0.7 to 0.8). We first present the recent experimental results. Figure 1 depicts five electric signals, labelled M1,M2, M3, M4 and V1, that have been recorded on March 21, 23 and April 7,

2005. Note that the first four (M1 to M4) have amplitudes that not only are one order of magnitude larger than the fifth one (V1, see Fig.1c), but also significantly exceed those hitherto reported[15]. The way they are read in natural time can be seen in Fig.2. The

analysis of these signals leads to the S and S− values given in Table I, an inspection of which

reveals that they are actually smaller than Su. Hence, on the basis of the aforementioned criterion, these signals cannot be classified as AN, but they are likely to be SES activities.

Note that, although in general S is different than S−, no definite conclusion can be drawn

on the sign of S − S− (see also Table I of Ref.[9]). We now present our results on fBm. We first clarify that Weron et al.[16] recently studied an algorithm distinguishing between the origins (i.e., the memory and the tails of the process) of the self-similarity of a given time series on the base of the computer test suggested in Ref.[17]. By applying it to the SES activities, they found the fBm as the appropriate type of modelling process. The fBm, which is H-self similar with stationary increments and it is the only Gaussian process with such properties for 0

sin(blt + d ) w(t)= c l , (3) X l blH l

where b > 1, cl normally distributed with mean 0 and standard deviation 1, and dl are

3 uniformly distributed in the interval [0, 2π] (cf. when using the increments of Eq.(3) one can also produce fractional Gaussian noise of a given H). By using Eq. (3), fBm for various values of H were produced, the one-signed segments of which were analyzed in the natural time domain (an example is given in Ref.[14]). The Monte-Carlo simulation results for each segment include not only the values of the S and S−, but also the exponent αDFA of the Detrended Fluctuation Analysis (DFA)[23, 24]. For segments of small number of points N (cf. only segments with N > 40 were considered), the value of αDFA may vary significanly from that expected for a given value of H; DFA was preferred, because it is one of the few well-defined and robust estimators of the scaling properties for such segments(e.g. [7], see also pp. 300-301 of Ref.[10]). The results are shown in Fig.3, in which we plot the S and S− values versus αDFA. Since the analysis of the SES activities in natural time result[7, 8] in DFA exponents αDFA around unity, hereafter we are solely focused in Fig.3 in the range 0.8 < αDFA < 1.2. An inspection of this figure reveals the following three conclusions: First, despite the large standard deviation, we may say that both S and S− are smaller than Su(≈ 0.0966) when αDFA ≈ 1. Second, S and S− are more or less comparable.

Third, comparing the computed S and S− values (≈ 0.08 for αDFA ≈ 1) with those resulting from the analysis of the SES activities (see Table I, see also Table I of Ref.[9]), we find a reasonable agreement. Note that these computations do not result in a definite sign for

S − S− in a similar fashion with the experimental results. In what remains, we present our results on a simple on-off intermittency model. We clarify that on-off intermittency is a phase-space mechanism that allows dynamical systems to undergo bursting (bursting is a phenomenon in which episodes of high activity are alternated with periods of inactivity likewise in Fig.2e). This mechanism is different from the well known Pomeau-Manneville scenario for the behavior of a system in the proximity of a saddle- node bifurcation[25]. Here, we use the simple model of the driven logistic map

Xt+1 = A(Yt)Xt[1 − Xt] (4) where we assume that the quantity A(Yt) is monotonic function of Yt and that 0 ≤ A ≤ 4 (cf. A is further specified below). The system has the invariant manifold X = 0 and the level of its activity is measured by Xt[26]. In order to have the on-off mechanism in action, we specialize to the case of a noise-driven logistic map, with

A(Yt)= A0 + αYt (5)

4 where Yt is δ-correlated noise which is uniformly-distributed in the interval [0,1] and A0 and

α are parameters. In order to have 0 ≤ A ≤ 4, we assume[26] A0 ≥ 0,α ≥ 0 and A0 +α ≤ 4. The relevant parameter plane for the noise-driven system of Eqs.(4) and (5) (as well as the parameter range for which the fixed point X = 0 is stable) can be found in Fig. 1 of Ref.[26], while the description of the intermittent dynamics is given in Refs.[27, 28, 29]. Bursting is observed in the temporal evolution of Xt as the stability of the fixed point X = 0 varies.

Following Ref.[28], for A0 = 0 there is a critical value αc > 1, below which the system asymptotically tends to the fixed point X = 0, without any sustained intermittent bursting.

For this case, i.e., A0 = 0, the value αc = e ≡ 2.71828 ... leads to on-off intermittency[26]. In the intermittent system under discusssion, both the signal amplitude and the power spectrum resulted[26] in power-law distributions (with low frequencies predominating in the power-spectrum) Several time-series have been produced for the above on-off intermittency model with the following procedure: The system was initiated at a time (tin = −200) with a uniformly distributed value Xtin in the region [0, 1], and then the mapping of Eqs.(4) and (5) was followed until N events will occur after t = 0. The results for Xt, t = 1, 2 ...N were

analyzed in natural time domain and the values of S and S− have been determined. This was repeated 103 (for a given number, N, of events) and the average values of S and 1/2 S− have been deduced. These values are plotted in Fig.4(a) versus (α − e)N (The factor N 1/2 stems from finite size scaling effects, since for large values of N, e.g., N >15000, a scaling -reminiscent of a 1st order phase transition- was observed, details on which will be published elsewhere). This figure reveals that as the critical value is approached from below

(i.e., α → e−) both S and S− are smaller than Su. Note that Fig.4(a) also indicates that S is probably larger than S−, while in the fBm time series no definite sign for S − S− could be obtained. Another interesting point resulted from the on-off intermittency model is depicted in

Fig.4(b), where we plot the fluctuations δS and δS− (i.e., the standard deviations of the 1/2 entropies S and S−, respectively) versus (α − e)N . It is clearly seen that these fluctu- ations are dramatically enhanced as the critical value is approached (i.e., α → e). This is strikingly reminiscent of our earlier results[5, 6] upon analyzing electrocardiograms (ECG) in natural time domain and studying the so called QT intervals. These results showed that the fluctuations of the entropy δS(QT ) are appreciably larger in sudden cardiac death (SD)

5 individuals than those in truly healthy (H) humans (see Fig. 2 of Ref.[6]). We emphasize that the aforementioned points should not be misinterpreted as stating that the simple logis- tic map model treated here can capture the complex heart dynamics, but only can be seen in the following frame: Since sudden (which may occur even if the electrocar- diogram looks similar to that of H) may be considered as a dynamic phase transition[5, 6], it is reasonable to expect that the entropy fluctuations significantly increase upon approaching the transition. In summary, recently recorded electric signals (having the largest amplitudes recorded to date) exhibit the property that both S and S− are smaller than Su and hence are likely to be SES activities (critical dynamics). This property seems to stem from their infinitely ranged long range correlations as supported by computational results in: (1) fBm time series and (2) a simple on-off intermittency model. The latter model also suggests that the

fluctuations (δS and δS−) significantly increase upon approaching the transition, which is strikingly reminiscent of the increased δS-values found for the QT-intervals for the sudden cardiac death individuals.

∗ Electronic address: [email protected] [1] R. Yulmetyev, P. H¨angii, and F. Gafarov, Phys. Rev. E 62, 6178 (2000). [2] R. Yulmetyev, F. Gafarov, P. H¨angii, R. Nigmatullin, and S. Kayamov, Phys. Rev. E 64, 066132 (2001). [3] P. A. Varotsos, N. V. Sarlis, and E. S. Skordas, Practica of Athens Academy 76, 294 (2001). [4] P. A. Varotsos, N. V. Sarlis, and E. S. Skordas, Phys. Rev. E 66, 011902 (2002). [5] P. A. Varotsos, N. V. Sarlis, E. S. Skordas, and M. S. Lazaridou, Phys. Rev. E 70, 011106 (2004). [6] P. A. Varotsos, N. V. Sarlis, E. S. Skordas, and M. S. Lazaridou, Phys. Rev. E 71, 011110 (2005). [7] P. A. Varotsos, N. V. Sarlis, and E. S. Skordas, Phys. Rev. E 67, 021109 (2003). [8] P. A. Varotsos, N. V. Sarlis, and E. S. Skordas, Phys. Rev. E 68, 031106 (2003). [9] P. A. Varotsos, N. V. Sarlis, H. K. Tanaka, and E. S. Skordas, Phys. Rev. E 71, 032102 (2005). [10] P. Varotsos, The Physics of Seismic Electric Signals (TERRAPUB, Tokyo, 2005).

6 [11] S. Abe, N. V. Sarlis, E. S. Skordas, H. K. Tanaka, and P. A. Varotsos, Phys. Rev. Lett. 94, 170601 (2005). [12] B. Lesche, J. Stat. Phys. 27, 419 (1982). [13] B. Lesche, Phys. Rev. E 70, 017102 (2004). [14] See EPAPS Document No. [. . . ] for additional information. This document may be re- trieved via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html) or from ftp.aip.org in the directory /epaps/. See the EPAPS homepage for more information. [15] P. Varotsos, N. Sarlis, and S. Skordas, Phys. Rev. Lett. 91, 148501 (2003). [16] A. Weron, K. Burnecki, S. Mercik, and K. Weron, Phys. Rev. E 71, 016113 (2005). [17] S. Mercik, K. Weron, K. Burnecki, and A. Weron, Acta Phys. Pol. B 34, 3773 (2003). [18] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Chapman & Hall/CRC, Florida, 1994). [19] B. Mandelbrot and J. R. Wallis, Water Resources Research 5, 243 (1969). [20] J. Szulga and F. Molz, J. Stat. Phys. 104, 1317 (2001). [21] B. B. Mandelbrot, Gaussian Self-Affinity and Fractals (Springer-Verlag, New York, 2001). [22] M. Frame, B. Mandelbrot, and N. Neger, fractal Geometry, Yale University, available from http://classes.yale.edu/fractals/, see http://classes.yale.edu/Fractals/RandFrac/fBm/fBm4.html. [23] C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 (1994). [24] S. V. Buldyrev, A. L. Goldberger, S. Havlin, R. N. Mantegna, M. E. Matsa, C.-K. Peng, M. Simons, and H. E. Stanley, Phys. Rev. E 51, 5084 (1995). [25] Y. Pomeau and P. Manneville, Commun. Math. Phys. 74, 189 (1980). [26] C. Toniolo, A. Provenzale, and E. A. Spiegel, Phys. Rev. E 66, 066209 (2002). [27] N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993). [28] J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994). [29] N. J. Balmforth, A. Provenzale, E. A. Spiegel, M. Martens, C. Tresser, and C. W. Wu, Proc. R. Soc. London, Ser. B 266, 311 (1999).

7 FIG. 1: Electric Signals recorded on March 21, 2005 (a), March 23, 2005 (b) and April 7, 2005 (c).

The signals in (a) and (c) are labeled hereafter M1 and V1 respectively, while that in (b) consists of the three signals’ activities labeled M2, M3, M4 (sampling frequency fexp=1Hz). The Universal Time (UT) is marked on the horizontal axis. Additional details for the two dipoles -records of which are shown here- as well as for the sites of the measuring stations are provided in Ref.[14].

8 60 (a) M1 50 40 30 Q [s] 20 10 0 0 0.2 0.4 0.6 0.8 1 χ (b) 30 M2 25 20 15 Q [s] 10 5 0 0 0.2 0.4 0.6 0.8 1 χ (c) 18 M3 16 14 12 10

Q [s] 8 6 4 2 0 0 0.2 0.4 0.6 0.8 1 (d) χ 70 M4 60 50 40

Q [s] 30 20 10 0 0 0.2 0.4 0.6 0.8 1 χ (e) 60 V1 50 40 30 Q [s] 20 10 0 0 0.2 0.4 0.6 0.8 1 χ

FIG. 2: How the signals depicted in Fig.1 are read in natural time. (a), (b), (c), (d), (e), correspond to the signals’ activities labeled M1, M2, M3, M4 and V1, respectively.

TABLE I: The values of S and S− together with the number of pulses N for the SES activities (the

3 4 original time series have lengths between 2 × 10 and 10 , compare Fig.1 with fexp=1Hz) shown in Fig.1.

Signal N S S−

M1 78 ± 9 0.094±0.005 0.078±0.003

M2 103 ± 5 0.089±0.003 0.084±0.003

M3 53 ± 3 0.089±0.004 0.093±0.004

M4 95 ± 3 0.080±0.005 0.086±0.006

V1 119 ± 14 0.078±0.006 0.092±0.005

9 0.1 0.095 0.09 0.085

- 0.08 0.075 S or 0.07 0.065 S 0.06 S- S 0.055 u 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 DFA

FIG. 3: Calculated values of S (squares) and S− (triangles) versus the DFA exponent αDFA. The error bars indicate the region of one standard deviation (±σ). The horizontal line corresponds to

Su.

(a) 0.1

0.08 - 0.06 S or 0.04

0.02

0 -40 -30 -20 -10 0 10 20 30 40 (α -e) N1/2

0.07 (b) 0.06

0.05 -

S 0.04 δ

S or 0.03 δ 0.02

0.01

0 -40 -30 -20 -10 0 10 20 30 40 (α -e) N1/2

FIG. 4: (Color Online) Calculated results for the on-off intermittency model discussed in the text:

(a) depicts the average values of S (closed symbols) and S− (open symbols) while (b) those of the

1/2 fluctuations δS and δS− versus the finite size scaling variable (α−αc)N . The quantity N stands for the number of the events considered in each sample time series; N=70000, 50000, 30000, 15000 correspond to squares, circles, triangles and inverted triangles, respectively. The horizontal line in

(a) corresponds to Su.

10