- 1. 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 frequency and phases of biological oscillators, or to the coupling of various regulatory feedback loops, Physiological systems have long been recognized to thus engaging nonlinear mechanisms for elucidation display complex temporal fluctuations, even during of the dynamics. Although, as in the physical sciences, 'steady state' conditions. Attempts were made to attri- solutions have resulted in 'linearizations', only during bute them to random influences, which perturb the the last decade has a natural link been drawn between Received 28 April 1997 Accepted 12 June 1997 367
- 2. 368 MEDICAL HYPOTHESES' the mathematico-physical field of nonlinear dynamics theses, which motivated ongoing research efforts and physiology. This has triggered an ongoing trend meant to quantify the dynamical characteristics of of 'paradigm shift' in the medical sciences and in the heart rate dynamics under the assumption that biological thinking in general. it evolves on a low-dimensional 'strange attractor'. Since the advent of digital processing, the heart rate These attempts were based mainly on dimensional became the most accessible and reliable signal for analysis, which resulted in correlation dimensions analysis among cardiovascular variables. The heart (interpreted as a static measure of the number of rate variability (HRV) is traditionally assessed using independent variables necessary to specify the state of frequency (spectral analysis) and time (standard de- the system under study), ranging between 3.6 and 5.2 viations, interval occurrence histograms, etc.) domain in normal subjects (17). This was supported later, by techniques. Using such techniques a complex coupling introducing another measure of deterministic chaos, with respiration, baroreceptors, the nervous system, i.e. the largest Lyapunov exponent which yielded a body temperature, metabolic rate, hormones, sleep finite positive value, thus demonstrating the property cycles, etc. was revealed. For example, spectral analysis, of sensitivity to initial conditions, which is the hall- which exposed activity bands in the frequency mark of chaotic behavior (18). But later estimates domain comprising thermoregulation (~0.05Hz), of the correlation dimension were found to be much baroreflex control of peripheral resistance (~ 0.1 Hz) higher (-8.5) than previously reported, thus pre- and respiratory control (~ 0.2 Hz), was combined cluding firm conclusions as to the true nature of the with pharmacological blockade to attribute the lower- heart rate generator (19). band fluctuations (0.04--0.15 Hz) to the joint influence Recently, other modified measures of dynamical of the sympathetic and parasympathetic arms of the complexity, mostly suited for nonstationary, noisy, autonomic nervous system, while the higher fre- and limited record length signals, have been intro- quency band (0.15-0.4 Hz) was shown to be purely duced. Among them is the estimation of the pointwise parasympathetically mediated. The spectral signature correlation dimension (PD2), which provides more of HRV was also related to various physiological information about the temporo-spatial evolution of and pathophysiological settings, such as standing, the dominant complexity of the heartbeat (20,21). hemorrhage and hypotension, which enhance the low This technique was applied to a very limited number frequency fluctuations, while exercise and standing of subjects, from which no firm conclusions could decrease the respiratory fluctuations. From the clinico- be drawn, except for one clinical study which corre- pathological viewpoint, patients with heart failure lated a reduced dynamical complexity hours before have diminished power spectrum at frequencies above the occurrence of lethal arrhythmias in high-risk ~ 0.02 Hz (1-7). The other arm of traditional analysis, patients (22). the time domain analysis, has related decreased HRV In the light of the open questions and computa- in diabetes mellitus, ischemic heart disease, conges- tional restrictions in this growing field of research, tive heart failure (8), and also associated an increased we addressed the issue of the irregular nature of the mortality in patients after acute myocardial infarction HRV in 10 healthy subjects. We computed correlation (9). dimensions and the series of pointwise dimensions. Taken together, these techniques have several short- We also introduced a modified version of the point- comings. For example, spectral analysis is a method wise dimension algorithm, which, we believe, can mostly suited for linear systems, while physiological depict both instances of low-dimensional chaos and systems are inherently nonlinear. Also, time domain stochasticity. The complex relation between them analysis, which is basically an averaging technique, was investigated using fractal techniques. The physio- overlooks the dynamical nature. Thus, it appears that logical and clinical importance of the measure we these techniques are often insufficient to characterize introduced is still unknown. the complex behavior of the heart rate generator. Since the last decade, nonlinear methods of analysis, based on the paradigm of deterministic chaos (10), Methods have permeated the realm of biomedical signal analysis (11). This was motivated by the observation Subjects of an inverse power-law scaling (also called 1/f Ten volunteers, aged 21-56 (6 males, 4 females) spectrum), which some chaotic systems may display, without symptoms or history of heart disease and and by its association to the fractal concept (mani- under no medication, were recruited for the study. fested by self-similarity over multiple orders of Their surface ECG, which showed no signs of patho- temporal magnitude) (12-16). In the case of heart rate logy, was recorded at rest in a supine position, during dynamics, these observations heralded new hypo- quite spontaneous breathing (~ 15 breaths/min) for
- 3. FRACTAL ORGANIZATION IN HEART RATE 369 20 rain. All recordings were done between 10 and series were time-delayed for successive embedding 12 a.m., and each subject was allowed to adjust dimensions (from m = 1 to m = 16). Within a given comfortably for 10 min in a supine position before the embedding dimension, the distance (r) of each point data were collected. They all gave informed consent to every other point was calculated. Their absolute to the protocol. values were rank-ordered from the smallest to the largest, and the range from the smallest to the largest Data aquisition value was broken up into discrete intervals. Then, the number of times a distance fell within an interval was The ECG signals were continuously recorded using a counted. A cumulative histogram was then formed laptop-based HIPEC ANALIZER HA-200/AH system by summing the number of instances for which a (Aerotel - - computerized systems, Israel) with a distance was less than or equal to the upper boundary sampling rate of 1000 Hz, and 16 bit signal resolu- of the interval. This is the correlation integral C(r). tion. The ECG records were transferred to a personal C(r) was then plotted as a function of r on a log-log computer for off-line analysis which started with a representation, resulting in a sigmoid-shaped curve quality control procedure: visual inspection, baseline (in this case implying chaotic dynamics). The slope shift evaluation and a 'moving average' (four points over the largest linear range (if there is one) was averaging in succession along the record) for signal to measured, using linear regression (with a regression noise ratio improvement. Then the interbeat intervals coefficient R 2 > 0.98). In this scaling range the local (R-R) were computed using an algorithm developed exponent is constant and ~ d InC(r)/d In(r). Then, the in our laboratory, with which an R wave threshold embedding dimension was advanced and its corre- detection was combined with first derivative and QRS sponding slope was calculated. These slopes were width considerations, for an accurate R wave detection. then plotted versus the embedding dimension, looking for a saturation region, i.e. a region in which the Attractor reconstruction slopes no longer grow. This plateau region was then Usually the experimentalist is confronted with in- considered as the correlation dimension (D2), and its ability to gain access to m simultaneous recordings value was calculated with a weighted average tech- necessary to describe the system's trajectory in nique (each value in the plateau region was weighted m-dimensional phase space. Thus, only one scalar by the variance of its underlying slope calculation). observable can be monitored as a function of time. This process was also performed for randomized Fortunately, it has been shown that certain properties versions of the R - R series (with similar mean and of the dynamics are feasible through the method of variance) in order to provide confidence limits for our time delays using Taken's theorem, as follows (23). calculations. Consider a single time series regularly spaced in time: xi = x%), i = 1..... N. Then a time lag "~is intro- Pointwise dimension of R-R intervals duced, such that m-dimensional vectors are created: xi= [x(ti), x(ti + "c)..... x(ti+ (m-1)x)]. This process is The 'point' estimate of the correlation dimension termed embedding, and m is called the embedding (PD2) begins with the time lag calculation, followed dimension. Through this reconstruction a phase by the embedding procedure, as described before. space is spanned and dynamical and metric measures Then, starting with the initial point in the series, its (Lyapunov exponents, dimensions) may be accessible. 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 Correlation dimension of R-R intervals (m = 1..... 16). The slopes for each m were evaluated The correlation dimension was calculated using the using a linear regression (R2> 0.98), and a slope method of Grassberger and Procaccia (G-P) as follows values, corresponding to m = 8 ..... 16 were stored in (24). First, for each R - R interval series, the normal- a file. The algorithm steps to the next point in the ized autocorrelation function given by: series, and the whole procedure is repeated until the whole file is exhausted. Then comes the procedure ~g('~) = {(I/N) £[R-R)i - < (R-R) >][(R-R)i+x - that we call slope convergence, which calculates the < (R-R) >] }/{(I/N) Z[R-R)i - < (R-R) >]2} slope of the 9 slopes versus the embedding dimensions where (m = 8 ..... 16) using linear regression. Our innova- tion was to subject this to the imposition of three < (R-R) > = (l/N) Z (R-R)/ conditions as follows: Ill if the slope was less than was constructed, and its first zero crossing was calcu- 0.5 and larger than -0.23, we considered this as good lated to provide the time lag (x) in beats. Then, the convergence and the PD2 could be estimated using
- 4. 370 MEDICAL HYPOTHESES the weighted average technique as described before; dimensions was needed. Homogeneity of variance (ii) if the slope was equal or larger than 0.5, we was tested by the Hartley F-max test. Statistical considered it as if no saturation existed, and at significance was assumed if the null hypothesis could this point (or time), the system probably manifested be rejected at the 0.05 probability level. 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 Results slope estimates, as the point-dimension, when such a condition appears; (iii) if the slope was equal to Thc correlation dimension (D2) of R-R intervals or less than --0.23, we considered it as if no slope varied from 3.29 to 5.16, with an overall mean of convergence existed, and the dimensional estimate 4.01 ± 0.54 (Table 1). Fig. la illustrates one of the at this point was excluded, possibly because of poor series of R-R intervals. This corresponding normal- statistics. ized autocorrelation function is shown in Fig. lb. The The results of the PD2 series were 'assigned' first zero crossing (x), in this case was equal to 6 according to the three conditions mentioned above. beats. The correlation integral (C(r)) for embedding This provided us with the ability to discriminate the dimensions (m = 2,4,6,9,12,16) is shown in Fig. 2a, points which manifested low-dimensional chaos and while the calculated slopes in the linear regions of random behavior, from those for which a dimensional the log-C(r) representation, versus the embedding estimate could not be achieved. From the above dimension, is shown in Fig. 2b. Note the convergence output files we extracted sequences of dimensional towards a dimensional value of 4. Randomized ver- subepochs, which were then suited for the relative sions of the R-R intervals have demonstrated, as dispersion analysis. expected, non-convergence (Fig. 2c). A sequence of pointwise dimensions (PD2s) versus the reference point is shown in Fig. 3a. Note three Dispersion analysis regions in the dimensional complexity plot, i.e. high values (PD2 > 6), low-dimensional region (3 > PD2 < 6) There are three basic methods of dispersion analysis and zero-valued reference points, corresponding to that can be applied to temporal observations (25). One non-convergence due to poor statistics. This can be of them, adopted in our study for each sequence of seen from the histogram (Fig. 3b) showing the distrib- calculated pointwise dimensions, is called relative ution of the rounded dimensional values, including dispersion (RD) analysis. Our intention was to try and the points corresponding to stochasticity and to non- see if the temporal evolution of PD2 series obeys any calculability at both ends of the figure. For the subject scaling properties. Thus for each subject, this simple shown in the figure the average PD2 was 5.37 ± 0.93. algorithm goes as follows: first, the mean, standard In Fig. 4, four subepochs, each comprising ~150 PD2 deviation (SD), and RD% (= 100 x SD/mean) of the values (corresponding to an average of about 2.5 original PD2 series were calculated. Then, pairs of minutes'-record-length each) are shown. In Fig. 5 the adjacent PD2s were averaged and their RD% values logarithmic plot of the RD(%) versus the interval were calculated, thus doubling the interval length. Recursive pairing with doubling of each previous interval length was carried out while its correspond- Table 1 Correlation dimension (D2) of 10 healthy ing RD% was calculated. This was done until the subjects at rest whole record was exhausted. By plotting the RD% against the interval length on a logarithmic scale, the Gender Age (years) HR ± SD Correlation dimension slope was estimated using a least-squares linear fit. (beat/rain) (D2 ± ZkD2) The fractal dimension (D) could thus be extracted from the slope (slope --- l-D). In order to confirm the M 21 69.1 ± 5.6 4.51 ± 0.13 temporal organization of the PD2 series, randomized F 25 65.5 ± 3.0 3.58± 0.07 versions based on similar statistical characteristics M 26 65.2 ± 2.5 3.93 ± 0.09 F 28 67.6± 3.5 5.16 ± 0.02 (number of points, mean and standard deviation) were M 30 57.6 ± 4.6 3.99 ± 0.16 generated, and their RD analysis was also performed. 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 Statistical analysis F 56 66.8 ± 2.6 3.77 ± 0.03 All data are expressed as mean ± SD. A paired t-test mean ± SD 37.5 ± 13.7 64.3 ± 5.0 4.01 ± 0.54 was performed when comparison between fractal
- 5. FRACTALORGANIZA~ONINHEARTRATE 371 1.25 0.75 e- •- 0.5- I e¢- 0.25 0 1 500 1000 Beat number 0.7 0.6 0.5 0.4-- 0.3-- 0.2 0.1- 0 -0.1 .'v , , .... 'IW ,v,' ' V V_ -0.2 m 100 200 300 b T Fig. 1 (a) R - R intervals for one of the subjects (1296 intervals, 20 min). (b) The normalized autocorrelation function of R - R intervals shown in (a). The first zero crossing was found to be 6 beats. length (measured in beat number) for one of the 850. The overall mean values of the PD2 series varied subepochs is shown. Its slope provides the fractal between 4.5 and 5.88 (mean = 5.21 ± 0.41, n = 10), dimension of the dimensional complexity at a parti- but the mean PD2s of the various subepochs were cular subepoch. smaller than the overall average, at least during one Table 2 summarizes the results of the fractal subepoch for each subject. We divided the subjects dimensions (D) of the subepochs of series of PD2s. into two groups according to their age. For group 1 The shortest subepoch consisted of 80 consecutive (21-30 years) the average PD2 varied between 5.19 dimensional values, while the longest consisted of and 5.88 (mean = 5.49 ± 0.27, n = 6), while the
- 6. 372 MEDICAL HYPOTHESES 0 -1, -2. -3 to t- -4 -5 -6 -7 1'5 ' ' 2~0 . . . . 2~5 . . . . Inr Emb 9 --K--- Emb 12 ~ Emb 16 5 8 C C 0 O r r 4 r ? ?6 i 3 . i 0 O n n 4 d i 2 ? m m e e 2 n 1 [I 0 O n 0 n . . . . . . . . . . . . . . 0 . . . . . . . . 110 111 I i I4 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 3 4 5 6 7 8 9 1 13 1 1 16 b Embedding dimension c Embedding dimension Fig. 2 The correlation integral C(r) versus r on a logarithmicplot. Seen are embeddingdimensions m = 2, 4, 6, 9, 12, 16. (b) The slope of the scaling region as a function of the embeddingdimension (m), for a healthy subject (26 years). Note the saturation towards a correlation dimension (D2) estimate of ~4. (c) For a randomlygenerated version of R-R intervals, the slope estimates of In C(r) versus lnr, as a functionof the embeddingdimension, do not saturate. relative dispersion analysis of their consecutive PD2 Discussion series yielded both instances of fractal scaling (D < 1.5) and stochasticity (D > 1.5). The averaged The concept of fractals, first coined by B. Mandelbrot fractal dimension for this group was 1.39 4-0.04 (26), and its association with chaos theory, heralded (14 subepochs). In group 2 (47-56 years), the PD2 novel insights into the realm of structural and mean values ranged between 4.58 and 5.03 (mean = dynamical variability in the medical sciences (27) 4.79 ± 0.17, n = 4). Note that in group 2 the fractal and biology in general (11). During the last decade, estimates ranged between 1.09 and 1.33 (mean = the intimate connection between deterministic chaos 1.20 ± 0.008, 8 subepochs), i.e. never exceeded 1.5. and fractal geometry has stimulated ongoing research The t-test and the F-test showed statistical signi- efforts to quantify the dynamical aspects of the heart ficance when the means and variances of the fractal rate generator. Babloyantz and Destexhe were the first dimensions of the PD2 subepochs were compared to quantify its dynamical measures using chaos theory (P values: 0.004 and 0.02, respectively). Note that the techniques (17). Their results (correlation dimensions, overall results of the RD analysis were indicative Kolmogorov entropies and the largest Lyapunov of fractal scaling (D < 1.5) in about 80% of all exponent), were supportive to the contention that the subepochs tested. heart rate generator evolves on a low-dimensional
- 7. FRACTAL ORGANIZATION IN HEART RATE 373 10 I N of points round(D2)- alp-)0.50 0 N o! p- 4z m 0 | round(O2)- I N of p- 0 i I round(D2}- round(D2)- 2 H o f p- 3 H of rouncl(D2)m 4 N o f roun4(D2)- S N of roarl4(D2)- 6 N o f p- pm 246 p- Se2 pm 146 0 "k roun4(D2)- 7 X o f p - 64 round(D2)- 8 g of p,, ]. round(D2)- 9 N o f p- 0 round(D2)-20 N of p- 0 i roun4(D2)-22 N of p- 0 round(D2)-12 N of p- 0 round(D2)-13 N of p- 0 round(D2)-14 N of p- 0 round(D2)-25 N of p- 0 round(D2)-26 H of p" 0 round(D2)-17 N of p,, 0 round(D2)-28 N of p- 0 round(D2)-lg N Of p- 0 round(D2)-20 N of p- @ H of exclu sip<-0.23 2v • a 0 250 500 750 1000 1250 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 'sign' 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- periodic behavior was seen in normal subjects, with portive evidence to this hypothesis (18,28), although an increase in complexity during sleep (21). Recent recently Kanters et al found weak evidence in favor studies found the dimensional complexity during (19). Thus, the existence of low dimensional chaos in experimental myocardial infarction in pigs to decline cardiac activity, at least at the whole organ level of significantly prior to the occurrence of ventricular activity, is still an open question. Most of the estima- fibrillation (31). This has motivated Skinner et al tions were based on dimensional estimations of the to evaluate lethal arryhythmias in various groups of widely used Grassbeger-Procaccia algorithm. Imple- high-risk patients. It was found (with high specificity mentation of this algorithm needs several precondi- and sensitivity) that the dimensional complexity tions to be observed (29,30): an adequate choice of is reduced hours before the occurrence of lethal embedding dimension, a suitable choice of the time arrhythmias (22). delay needed to span the attractor, low level of noise Our results support the contention of low- present in the system, stationarity, and the data set dimensional chaos, as proposed by others. The corre- should not be too short. Some of these requirements lation dimension was found to vary between 3.29 are not attainable in biology and in physiology in and 5.16, with a mean of 4.01 + 0.54 (n = 10). The particular. Moreover, the G-P algorithm provides a average pointwise dimension ranged between 4.58 dimensional estimate which averages out possible and 5.88, with a mean of 5.21 + 0.41 (n = 10). Further- relevant dynamical features. Recently, the intro- more, we noticed that our subjects could be divided duction of the pointwise dimension algorithm, which into two groups according to age as follows: group 1 provides series of 'point' dimensions, has provided (21-30) years, n = 6) had a higher average, 5.49 + 0.27, some solutions to the limitations of the G-P algo- than group 2 (47-56 years, n = 4 ) at 4.79+ 0.17. rithm, namely, non-stationarity and record length. Our innovation in this study was twofold. First, we This method was implemented for heart transplant included in the dimensional complexity algorithm recipients and the dimensional complexity was found means to include both instances of low-dimensional to oscillate almost periodically (20). Also, a roughly chaos and stochastic bursts in a sequential manner,
- 8. 374 MEDICAL HYPOTHESES 10 2O g tJL, JJl| RF' Jd l I0 10 i1. D ill I'r ,I Fig. 4 Four subepochs of sequential PD2s from the series shown in Fig. 3a. Each subepoch comprises about 150 PD2s. i.e. as a function of the beat number. Second, this cant decline in the complexity of the cardiovascular enabled us to apply a fractal technique (relative system (blood pressure and heart rate). Such findings dispersion analysis) to explore the different subepochs may reflect the breakdown and decoupling of inte- of dimension series for scale independence. We found grated physiologic regulatory systems with aging that the older group manifested fractal scaling and may signal an impairment in the cardiovascular (D < 1.5) in all subepochs tested. As for the younger ability to adapt efficiently to internal and external group, only in 64% of tested subepochs did we find perturbations. This is contradictory to the sacred fractal scaling (D < 1.5), while the rest was indicative principle of 'homeostasis', which was developed by of a random control (D > 1.5). Moreover, the differ- Walter Cannon, and postulates that with disease and ences in the averages and variances of the fractal aging the body is less able to maintain a constant dimensions between the two groups were found to steady state, as a result of breakdown of its regulatory be statistically significant. This is in contention with systems. Our findings support the chaos hypothesis of results from other chaos-derived techniques imple- a 'homeokinetic' principle in physiology, namely, mented by Kaplan et al on old versus young subjects physiological systems in young healthy subjects tend (32). They found that the older group showed signifi- to fluctuate between a set of metastable states, thus
- 9. FRACTAL ORGANIZATIONIN HEART RATE 375 of the heart rate variability, which can be quantified 20 using fractal techniques, seems to contain 'mixing' of both chaotic and random fluctuations. The nature 10 of such a behavior is not yet understood, but one 5 may hypothesize that an increase in the dimensional complexity (D2 can be thought of as a measure --,.. of independent variables necessary to describe the 2 system), may correspond to recruitment of several 1 subsystems influencing the heart rate generator, or to the activation of more independent control loops. 0.50 A reduced complexity, on the other hand, may mani- 0.20 fest deactivation of control loops, or maybe increased self-organization of some of these systems. Also, the abrupt changes in the dimensional complexity may 0.10 represent shifts between different attractors of the system. Such hypotheses, may better be resolved by 140 74 37 19 10 5 3 comparing the fractal dimensions of the dimensional Fig. 5 Plot of RD% (relative dispersion) versus interval length on complexity (and other measures of nonlinear tech- a logarithmic scale. The fractal dimension (D) is derived from the niques) under different physiological and clinico- slope (slope = l-D). pathological settings. Currently, we are in the process of obtaining longer data records from heart transplant recipients, in order making the system more adaptable to its internal and to gain more insight regarding the value of the fractal external surroundings (15,33). estimate of the dimensional complexity of the heart We thus propose that the dimensional complexity rate generator, as a potential new dynamical measure. T a b l e 2 Subepochs o f pointwise dimension (PD2) series, averaged P D 2 s for each subepoch, fractal dimension (D) for each subepoch, and averaged P D 2 s for w h o l e 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.
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