Medical Hypotheses (1998) 51, 367-376 © Harcourt Brace & Co. Ltd 1998

Fractal organization of the pointwise correlation dimension of the heart rate

E. NAHSHONI, E. ADLER*, S. LANIADO*, G. KEREN*

Department E, The Gehah Psychiatric Hospital, Petah-Tiqva, and Sackler School of Medicine, Tel Aviv, Israeb *Department of Cardiology, Tel Aviv Medical Center, Tel Aviv, and Sackler School of Medicine, Tel Aviv, Israel. Correspondence to: E. Nahshoni, POB 102, 49100 Petah-Tiqva, Israel (Phone: +972 3 9258258; Fax: + 972 3 9241041)

Abstract - - Objective: To depict and quantify the degree of organization of the heart rate variability (HRV) in normal subjects. Methods: A modified algorithm was created to estimate series of "point-dimensions" (PD2) from interbeat (R-R) interval series of 10 healthy subjects (21-56 years). Our innovation is twofold: (i) we quantified instances of low-dimensional chaos, random fluctuations, and those for which our method failed to provide either (due to poor statistics); (ii) consecutive subepochs of PD2s underwent a relative dispersion (RD) analysis, yielding an index (D) which quantifies the dynamical organization of the heart rate generator.

Results: The mean values of PD2 series varied between 4.58 and 5.88 (mean +_ SD= 5.21 +_ 0.41, n = 10). For group 1 (21-30 years, n = 6) we found an averaged PD2 of 5.49 _+ 0.27, while for group 2 (47-56 years, n = 4) PD2 averaged 4.79 +_. 0.17. The RD analysis performed for subepochs of PD2s yielded both instances obeying fractal scaling (D < 1.5) and stochasticity (D > 1.5). The average D for group 1 was 1.39 + 0.04 (14 subepochs) and for group 2, 1.20 _+ 0.008 (8 subepochs). Paired t-test and Hartley F-max test for comparison between D values and homogeneity of variance between the two groups were performed, yielding P-values 0.004 and 0.02, respectively.

Conclusions: The complexity of the HRV seems to be modulated by a non-random fractal mechanism of a 'hyperchaotic' system, i.e. it can be hypothesized to contain more than one attractor. Also, our results support the 'chaos hypothesis' put forth recently, namely, the complexity of the cardiovascular dynamics is reduced with aging. The index of relative dispersion of the dimensional complexity has to be tested in various clinico-pathological settings, in order to corroborate its value as a potential new physiological measure.

Introduction

Physiological systems have long been recognized to display complex temporal fluctuations, even during 'steady state' conditions. Attempts were made to attri- bute them to random influences, which perturb the

frequency and phases of biological oscillators, or to the coupling of various regulatory feedback loops, thus engaging nonlinear mechanisms for elucidation of the dynamics. Although, as in the physical sciences, solutions have resulted in 'linearizations', only during the last decade has a natural link been drawn between

Received 28 April 1997 Accepted 12 June 1997

367

368 MEDICAL HYPOTHESES'

the mathematico-physical field of nonlinear dynamics and physiology. This has triggered an ongoing trend of 'paradigm shift' in the medical sciences and in biological thinking in general.

Since the advent of digital processing, the heart rate became the most accessible and reliable signal for analysis among cardiovascular variables. The heart rate variability (HRV) is traditionally assessed using frequency (spectral analysis) and time (standard de- viations, interval occurrence histograms, etc.) domain techniques. Using such techniques a complex coupling with respiration, baroreceptors, the nervous system, body temperature, metabolic rate, hormones, sleep cycles, etc. was revealed. For example, spectral analysis, which exposed activity bands in the frequency domain comprising thermoregulation (~0.05Hz), baroreflex control of peripheral resistance (~ 0.1 Hz) and respiratory control (~ 0.2 Hz), was combined with pharmacological blockade to attribute the lower- band fluctuations (0.04--0.15 Hz) to the joint influence of the sympathetic and parasympathetic arms of the autonomic nervous system, while the higher fre- quency band (0.15-0.4 Hz) was shown to be purely parasympathetically mediated. The spectral signature of HRV was also related to various physiological and pathophysiological settings, such as standing, hemorrhage and hypotension, which enhance the low frequency fluctuations, while exercise and standing decrease the respiratory fluctuations. From the clinico- pathological viewpoint, patients with heart failure have diminished power spectrum at frequencies above ~ 0.02 Hz (1-7). The other arm of traditional analysis, the time domain analysis, has related decreased HRV in diabetes mellitus, ischemic heart disease, conges- tive heart failure (8), and also associated an increased mortality in patients after acute myocardial infarction (9).

Taken together, these techniques have several short- comings. For example, spectral analysis is a method mostly suited for linear systems, while physiological systems are inherently nonlinear. Also, time domain analysis, which is basically an averaging technique, overlooks the dynamical nature. Thus, it appears that these techniques are often insufficient to characterize the complex behavior of the heart rate generator.

Since the last decade, nonlinear methods of analysis, based on the paradigm of deterministic chaos (10), have permeated the realm of biomedical signal analysis (11). This was motivated by the observation of an inverse power-law scaling (also called 1/f spectrum), which some chaotic systems may display, and by its association to the fractal concept (mani- fested by self-similarity over multiple orders of temporal magnitude) (12-16). In the case of heart rate dynamics, these observations heralded new hypo-

theses, which motivated ongoing research efforts meant to quantify the dynamical characteristics of the heart rate dynamics under the assumption that it evolves on a low-dimensional 'strange attractor'. These attempts were based mainly on dimensional analysis, which resulted in correlation dimensions (interpreted as a static measure of the number of independent variables necessary to specify the state of the system under study), ranging between 3.6 and 5.2 in normal subjects (17). This was supported later, by introducing another measure of deterministic chaos, i.e. the largest Lyapunov exponent which yielded a finite positive value, thus demonstrating the property of sensitivity to initial conditions, which is the hall- mark of chaotic behavior (18). But later estimates of the correlation dimension were found to be much higher (-8.5) than previously reported, thus pre- cluding firm conclusions as to the true nature of the heart rate generator (19).

Recently, other modified measures of dynamical complexity, mostly suited for nonstationary, noisy, and limited record length signals, have been intro- duced. Among them is the estimation of the pointwise correlation dimension (PD2), which provides more information about the temporo-spatial evolution of the dominant complexity of the heartbeat (20,21). This technique was applied to a very limited number of subjects, from which no firm conclusions could be drawn, except for one clinical study which corre- lated a reduced dynamical complexity hours before the occurrence of lethal arrhythmias in high-risk patients (22).

In the light of the open questions and computa- tional restrictions in this growing field of research, we addressed the issue of the irregular nature of the HRV in 10 healthy subjects. We computed correlation dimensions and the series of pointwise dimensions. We also introduced a modified version of the point- wise dimension algorithm, which, we believe, can depict both instances of low-dimensional chaos and stochasticity. The complex relation between them was investigated using fractal techniques. The physio- logical and clinical importance of the measure we introduced is still unknown.

Methods

Subjects

Ten volunteers, aged 21-56 (6 males, 4 females) without symptoms or history of heart disease and under no medication, were recruited for the study. Their surface ECG, which showed no signs of patho- logy, was recorded at rest in a supine position, during quite spontaneous breathing (~ 15 breaths/min) for

FRACTAL ORGANIZATION IN HEART RATE 369

20 rain. All recordings were done between 10 and 12 a.m., and each subject was allowed to adjust comfortably for 10 min in a supine position before the data were collected. They all gave informed consent to the protocol.

Data aquisition

The ECG signals were continuously recorded using a laptop-based HIPEC ANALIZER HA-200/AH system (Aerotel - - computerized systems, Israel) with a sampling rate of 1000 Hz, and 16 bit signal resolu- tion. The ECG records were transferred to a personal computer for off-line analysis which started with a quality control procedure: visual inspection, baseline shift evaluation and a 'moving average' (four points averaging in succession along the record) for signal to noise ratio improvement. Then the interbeat intervals (R-R) were computed using an algorithm developed in our laboratory, with which an R wave threshold detection was combined with first derivative and QRS width considerations, for an accurate R wave detection.

Attractor reconstruction

Usually the experimentalist is confronted with in- ability to gain access to m simultaneous recordings necessary to describe the system's trajectory in m-dimensional phase space. Thus, only one scalar observable can be monitored as a function of time. Fortunately, it has been shown that certain properties of the dynamics are feasible through the method of time delays using Taken's theorem, as follows (23). Consider a single time series regularly spaced in time: xi = x%), i = 1 . . . . . N. Then a time lag "~ is intro- duced, such that m-dimensional vectors are created: xi= [x(ti), x(ti + "c) . . . . . x(ti+ (m-1)x)]. This process is termed embedding, and m is called the embedding dimension. Through this reconstruction a phase space is spanned and dynamical and metric measures (Lyapunov exponents, dimensions) may be accessible.

Correlation dimension of R-R intervals

The correlation dimension was calculated using the method of Grassberger and Procaccia (G-P) as follows (24). First, for each R-R interval series, the normal- ized autocorrelation function given by:

~g('~) = {(I/N) £[R-R)i - < (R-R) >][(R-R)i+x - < (R-R) >] }/{(I/N) Z[R-R)i - < (R-R) >]2}

where

< (R-R) > = (l/N) Z (R-R)/

was constructed, and its first zero crossing was calcu- lated to provide the time lag (x) in beats. Then, the

series were time-delayed for successive embedding dimensions (from m = 1 to m = 16). Within a given embedding dimension, the distance (r) of each point to every other point was calculated. Their absolute values were rank-ordered from the smallest to the largest, and the range from the smallest to the largest value was broken up into discrete intervals. Then, the number of times a distance fell within an interval was counted. A cumulative histogram was then formed by summing the number of instances for which a distance was less than or equal to the upper boundary of the interval. This is the correlation integral C(r). C(r) was then plotted as a function of r on a log-log representation, resulting in a sigmoid-shaped curve (in this case implying chaotic dynamics). The slope over the largest linear range (if there is one) was measured, using linear regression (with a regression coefficient R 2 > 0.98). In this scaling range the local exponent is constant and ~ d InC(r)/d In(r). Then, the embedding dimension was advanced and its corre- sponding slope was calculated. These slopes were then plotted versus the embedding dimension, looking for a saturation region, i.e. a region in which the slopes no longer grow. This plateau region was then considered as the correlation dimension (D2), and its value was calculated with a weighted average tech- nique (each value in the plateau region was weighted by the variance of its underlying slope calculation). This process was also performed for randomized versions of the R-R series (with similar mean and variance) in order to provide confidence limits for our calculations.

Pointwise dimension of R-R intervals

The 'point' estimate of the correlation dimension (PD2) begins with the time lag calculation, followed by the embedding procedure, as described before. Then, starting with the initial point in the series, its local correlation integral is calculated, i.e. the dis- tances are taken with respect to this point and ranked- ordered as usual, for each embedding dimension (m = 1 . . . . . 16). The slopes for each m were evaluated using a linear regression (R2> 0.98), and a slope values, corresponding to m = 8 . . . . . 16 were stored in a file. The algorithm steps to the next point in the series, and the whole procedure is repeated until the whole file is exhausted. Then comes the procedure that we call slope convergence, which calculates the slope of the 9 slopes versus the embedding dimensions (m = 8 . . . . . 16) using linear regression. Our innova- tion was to subject this to the imposition of three conditions as follows: Ill if the slope was less than 0.5 and larger than -0.23, we considered this as good convergence and the PD2 could be estimated using

370 MEDICAL HYPOTHESES

the weighted average technique as described before; (ii) if the slope was equal or larger than 0.5, we considered it as if no saturation existed, and at this point (or time), the system probably manifested a random fluctuation. In order to incorporate such a behavior into the sequence of PD2s, we decided, quite arbitrarily, to take the average of the two highest slope estimates, as the point-dimension, when such a condition appears; (iii) if the slope was equal to or less than --0.23, we considered it as if no slope convergence existed, and the dimensional estimate at this point was excluded, possibly because of poor statistics.

The results of the PD2 series were 'assigned' according to the three conditions mentioned above. This provided us with the ability to discriminate the points which manifested low-dimensional chaos and random behavior, from those for which a dimensional estimate could not be achieved. From the above output files we extracted sequences of dimensional subepochs, which were then suited for the relative dispersion analysis.

Dispersion analysis

There are three basic methods of dispersion analysis that can be applied to temporal observations (25). One of them, adopted in our study for each sequence of calculated pointwise dimensions, is called relative dispersion (RD) analysis. Our intention was to try and see if the temporal evolution of PD2 series obeys any scaling properties. Thus for each subject, this simple algorithm goes as follows: first, the mean, standard deviation (SD), and RD% (= 100 x SD/mean) of the original PD2 series were calculated. Then, pairs of adjacent PD2s were averaged and their RD% values were calculated, thus doubling the interval length. Recursive pairing with doubling of each previous interval length was carried out while its correspond- ing RD% was calculated. This was done until the whole record was exhausted. By plotting the RD% against the interval length on a logarithmic scale, the slope was estimated using a least-squares linear fit. The fractal dimension (D) could thus be extracted from the slope (slope --- l-D). In order to confirm the temporal organization of the PD2 series, randomized versions based on similar statistical characteristics (number of points, mean and standard deviation) were generated, and their RD analysis was also performed.

Statistical analysis

All data are expressed as mean ± SD. A paired t-test was performed when comparison between fractal

dimensions was needed. Homogeneity of variance was tested by the Hartley F-max test. Statistical significance was assumed if the null hypothesis could be rejected at the 0.05 probability level.

Results

Thc correlation dimension (D2) of R-R intervals varied from 3.29 to 5.16, with an overall mean of 4.01 ± 0.54 (Table 1). Fig. la illustrates one of the series of R-R intervals. This corresponding normal- ized autocorrelation function is shown in Fig. lb. The first zero crossing (x), in this case was equal to 6 beats. The correlation integral (C(r)) for embedding dimensions (m = 2,4,6,9,12,16) is shown in Fig. 2a, while the calculated slopes in the linear regions of the log-C(r) representation, versus the embedding dimension, is shown in Fig. 2b. Note the convergence towards a dimensional value of 4. Randomized ver- sions of the R-R intervals have demonstrated, as expected, non-convergence (Fig. 2c).

A sequence of pointwise dimensions (PD2s) versus the reference point is shown in Fig. 3a. Note three regions in the dimensional complexity plot, i.e. high values (PD2 > 6), low-dimensional region (3 > PD2 < 6) and zero-valued reference points, corresponding to non-convergence due to poor statistics. This can be seen from the histogram (Fig. 3b) showing the distrib- ution of the rounded dimensional values, including the points corresponding to stochasticity and to non- calculability at both ends of the figure. For the subject shown in the figure the average PD2 was 5.37 ± 0.93. In Fig. 4, four subepochs, each comprising ~150 PD2 values (corresponding to an average of about 2.5 minutes'-record-length each) are shown. In Fig. 5 the logarithmic plot of the RD(%) versus the interval

Table 1 Correlation dimension (D2) of 10 healthy subjects at rest

Gender Age (years) HR ± SD Correlation dimension (beat/rain) (D2 ± ZkD2)

M 21 69.1 ± 5.6 4.51 ± 0.13 F 25 65.5 ± 3.0 3.58± 0.07 M 26 65.2 ± 2.5 3.93 ± 0.09 F 28 67.6± 3.5 5.16 ± 0.02 M 30 57.6 ± 4.6 3.99 ± 0.16 M 30 54.2±2.3 4 .50±0.19 M 47 71.4± 3.9 3.87 ± 0.21 M 56 60.9 ± 3.0 3.29 ± 0.19 F 56 64.7 ± 2.3 3.48 ± 0.21 F 56 66.8 ± 2.6 3.77 ± 0.03

mean ± SD 37.5 ± 13.7 64.3 ± 5.0 4.01 ± 0.54

FRACTALORGANIZA~ONINHEARTRATE

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length (measured in beat number) for one of the subepochs is shown. Its slope provides the fractal dimension of the dimensional complexity at a parti- cular subepoch.

Table 2 summarizes the results of the fractal dimensions (D) of the subepochs of series of PD2s. The shortest subepoch consisted of 80 consecutive dimensional values, while the longest consisted of

850. The overall mean values of the PD2 series varied between 4.5 and 5.88 (mean = 5.21 ± 0.41, n = 10), but the mean PD2s of the various subepochs were smaller than the overall average, at least during one subepoch for each subject. We divided the subjects into two groups according to their age. For group 1 (21-30 years) the average PD2 varied between 5.19 and 5.88 (mean = 5.49 ± 0.27, n = 6), while the

372 MEDICAL HYPOTHESES

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The correlation integral C(r) versus r on a logarithmic plot. Seen are embedding dimensions m = 2, 4, 6, 9, 12, 16. (b) The slope of the scaling region as a function of the embedding dimension (m), for a healthy subject (26 years). Note the saturation towards a correlation dimension (D2) estimate of ~4. (c) For a randomly generated version of R-R intervals, the slope estimates of In C(r) versus lnr, as a function of the embedding dimension, do not saturate.

relative dispersion analysis of their consecutive PD2 series yielded both instances of fractal scaling (D < 1.5) and stochasticity (D > 1.5). The averaged fractal dimension for this group was 1.39 4-0.04 (14 subepochs). In group 2 (47-56 years), the PD2 mean values ranged between 4.58 and 5.03 (mean = 4.79 ± 0.17, n = 4). Note that in group 2 the fractal estimates ranged between 1.09 and 1.33 (mean = 1.20 ± 0.008, 8 subepochs), i.e. never exceeded 1.5.

The t-test and the F-test showed statistical signi- ficance when the means and variances of the fractal dimensions of the PD2 subepochs were compared (P values: 0.004 and 0.02, respectively). Note that the overall results of the RD analysis were indicative of fractal scaling (D < 1.5) in about 80% of all subepochs tested.

Discuss ion

The concept of fractals, first coined by B. Mandelbrot (26), and its association with chaos theory, heralded novel insights into the realm of structural and dynamical variability in the medical sciences (27) and biology in general (11). During the last decade, the intimate connection between deterministic chaos and fractal geometry has stimulated ongoing research efforts to quantify the dynamical aspects of the heart rate generator. Babloyantz and Destexhe were the first to quantify its dynamical measures using chaos theory techniques (17). Their results (correlation dimensions, Kolmogorov entropies and the largest Lyapunov exponent), were supportive to the contention that the heart rate generator evolves on a low-dimensional

FRACTAL ORGANIZATION IN HEART RATE

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Fig. 3 (a) Serial pointwise dimensions (PD2s) as a function of the beat number (nref) for one of the subjects. The zero-valued PD2s are only ' s ign ' of the instances for which a dimensional estimate could not be derived. (b) A histogram showing the distribution of the beat number as a function of the rounded dimensional estimates. At the two extremes of the diagram we note the number of points for which a random fluctuation is supposed to take place (slope > 0.5), and on the other side the number of point for which an estimate could not be found (slope < -0.23).

chaotic attractor. Later, other groups provided sup- portive evidence to this hypothesis (18,28), although recently Kanters et al found weak evidence in favor (19). Thus, the existence of low dimensional chaos in cardiac activity, at least at the whole organ level of activity, is still an open question. Most of the estima- tions were based on dimensional estimations of the widely used Grassbeger-Procaccia algorithm. Imple- mentation of this algorithm needs several precondi- tions to be observed (29,30): an adequate choice of embedding dimension, a suitable choice of the time delay needed to span the attractor, low level of noise present in the system, stationarity, and the data set should not be too short. Some of these requirements are not attainable in biology and in physiology in particular. Moreover, the G-P algorithm provides a dimensional estimate which averages out possible relevant dynamical features. Recently, the intro- duction of the pointwise dimension algorithm, which provides series of 'point' dimensions, has provided some solutions to the limitations of the G-P algo- rithm, namely, non-stationarity and record length. This method was implemented for heart transplant recipients and the dimensional complexity was found to oscillate almost periodically (20). Also, a roughly

periodic behavior was seen in normal subjects, with an increase in complexity during sleep (21). Recent studies found the dimensional complexity during experimental myocardial infarction in pigs to decline significantly prior to the occurrence of ventricular fibrillation (31). This has motivated Skinner et al to evaluate lethal arryhythmias in various groups of high-risk patients. It was found (with high specificity and sensitivity) that the dimensional complexity is reduced hours before the occurrence of lethal arrhythmias (22).

Our results support the contention of low- dimensional chaos, as proposed by others. The corre- lation dimension was found to vary between 3.29 and 5.16, with a mean of 4.01 + 0.54 (n = 10). The average pointwise dimension ranged between 4.58 and 5.88, with a mean of 5.21 + 0.41 (n = 10). Further- more, we noticed that our subjects could be divided into two groups according to age as follows: group 1 (21-30) years, n = 6) had a higher average, 5.49 + 0.27, than group 2 (47-56 years, n = 4 ) at 4.79+ 0.17. Our innovation in this study was twofold. First, we included in the dimensional complexity algorithm means to include both instances of low-dimensional chaos and stochastic bursts in a sequential manner,

374 MEDICAL HYPOTHESES

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i.e. as a function of the beat number. Second, this enabled us to apply a fractal technique (relative dispersion analysis) to explore the different subepochs of dimension series for scale independence. We found that the older group manifested fractal scaling (D < 1.5) in all subepochs tested. As for the younger group, only in 64% of tested subepochs did we find fractal scaling (D < 1.5), while the rest was indicative of a random control (D > 1.5). Moreover, the differ- ences in the averages and variances of the fractal dimensions between the two groups were found to be statistically significant. This is in contention with results from other chaos-derived techniques imple- mented by Kaplan et al on old versus young subjects (32). They found that the older group showed signifi-

cant decline in the complexity of the cardiovascular system (blood pressure and heart rate). Such findings may reflect the breakdown and decoupling of inte- grated physiologic regulatory systems with aging and may signal an impairment in the cardiovascular ability to adapt efficiently to internal and external perturbations. This is contradictory to the sacred principle of 'homeostasis', which was developed by Walter Cannon, and postulates that with disease and aging the body is less able to maintain a constant steady state, as a result of breakdown of its regulatory systems. Our findings support the chaos hypothesis of a 'homeokinetic' principle in physiology, namely, physiological systems in young healthy subjects tend to fluctuate between a set of metastable states, thus

FRACTAL ORGANIZATION IN HEART RATE 375

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making the system more adaptable to its internal and external surroundings (15,33).

We thus propose that the dimensional complexity

of the heart rate variability, which can be quantified using fractal techniques, seems to contain 'mixing' of both chaotic and random fluctuations. The nature of such a behavior is not yet understood, but one may hypothesize that an increase in the dimensional complexity (D2 can be thought of as a measure of independent variables necessary to describe the system), may correspond to recruitment of several subsystems influencing the heart rate generator, or to the activation of more independent control loops. A reduced complexity, on the other hand, may mani- fest deactivation of control loops, or maybe increased self-organization of some of these systems. Also, the abrupt changes in the dimensional complexity may represent shifts between different attractors of the system. Such hypotheses, may better be resolved by comparing the fractal dimensions of the dimensional complexity (and other measures of nonlinear tech- niques) under different physiological and clinico- pathological settings.

Currently, we are in the process of obtaining longer data records from heart transplant recipients, in order to gain more insight regarding the value of the fractal estimate of the dimensional complexity of the heart rate generator, as a potential new dynamical measure.

Table 2 Subepochs of pointwise dimension (PD2) series, averaged PD2s for each subepoch, fractal dimension (D) for each subepoch, and averaged PD2s for whole records

Gender Age Nref PD2 ± SD average over subepochs D(RD) ± SD PD2 ± SD average over total record

M 21 1-150 5.28 ± 0.88 1.42 + 0.07 5.37 ± 0.93 300--450 5.09 ± 0.09 1.27 + 0.03 570-720 5.61 ± 0.73 1.49 ± 0.06 850-1000 4.35 ± 0.69 1.09 ± 0.04

F 25 1-90 4.45 ± 1.05 1.21 ± 0.09 5.19 ± 1.04 200-350 4.86 + 0.75 1.54 ± 0.06

M 26 130-196 6.01 ± 0.97 1.72 ± 0.11 5.83 ± 1.02 250-850 5.62 ± 0.97 1.19 ± 0.10

F 28 1-300 5.90 ± 0.90 L28 + 0.09 5.34 ± 0.72 301--600 5.03 ± 0.35 1.36 ± 0.12

M 30 1-200 4.91 ± 0.69 1.55 ± 0.09 5.30 ± 0.11 600--995 5.09 ± 1.10 1.13 ± 0.06

M 30 30-110 5.92 ± 0.83 1.57 ± 0.09 5.88 ± 1.08 125-295 5.34 ± 0.82 1.63 ± 0.09

M 47 1-350 3.92 ± 0.51 1.11 ± 0.04 4.69 ± 0,77 400-1250 4.99 ± 0,63 1.21 ± 0.06

M 56 1-300 4.95 ± 0,68 1.18 ± 0.07 5.03 ± 0,71 500-1000 5.11 ± 0.69 1.32 ± 0.05

F 56 100-300 4.63 ± 0.72 1.24 ± 0.07 4.87 ± 0.96 500-750 4.32 ± 1.21 1.09 ± 0.05

F 56 1-80 4.39 ± 0.69 1.33 ± 0.08 4.58 ± 0.64 200-380 4.63 ± 0.79 1.14 ± 0.08

mean ± SD 5.02 ± 0.54 1.32 ± 0.18 5.21 ± 0.41

M, male; F, female; Nref, sequences of consecutive data points' subepochs; PD2 + SD, averaged pointwise dimension + standard deviation; D(RD), fractal dimension of each subepoch, derived from relative dispersion analysis.

376 MEDICAL HYPOTHESES

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