Abstract
Many chaos detection methods have proven inherently ambiguous in that they yield similar results for chaotic signals and correlated noise. The purpose of this work was to determine whether human resting heart period sequences have global properties characteristic of chaotic systems. We investigated the inherent global organization of heart period sequences by quantifying how the information content of the embedded sequences varied as a function of scale. We compared the information scaling characteristics of 60min heart period sequences obtained from 10 healthy resting volunteers with those obtained from numerous periodic and chaotic control sequences. The information scaling properties of the heart period sequences were significantly different from those obtained for the controls, particularly at the coarsest scales (P = 0.0003 vs. lowdimensional periodic controls; P = 0.0005 vs. lowdimensional chaotic controls;P = 0.0003 vs. lowdimensional periodic and chaotic controls). We also showed that nondeterministic components, such as large tachycardic (or bradycardic) events or aperiodic fluctuations, can lead to scaling characteristics similar to those observed for the resting heart period sequences. This, in addition to previous evidence from spectral, nonlinear predictability and lexical studies, favors an eventsbased approach to understanding heart rate variability.
 chaos
 modeling
 determinism
 information content
 predictability analysis
much has been written about the organization of the heart period sequences that form the basis of heart rate variability. In part, this wealth of research has been fueled by studies that demonstrate how various heart rate variability measures are predictive of survival characteristics for certain cardiac patient populations (2,4, 5, 24, 31, 36). Other heart rate variability studies are more concerned with the dynamics of the system(s) that govern the beattobeat changes in heart rate. In particular, since chaotic determinism for cardiac rhythm was first postulated (3, 10, 11, 17), there has been much discussion regarding the role of nonlinear dynamics and chaos in heart rate variability. As these new nonlinear analytical techniques were refined, and as their limitations became increasingly understood (7, 8, 16, 18, 3234), the role for lowdimensional chaotic determinism in normal resting heart rate variability has come into serious doubt (14, 15, 19, 35). Unfortunately, many of the chaos detection methods have proven inherently ambiguous in that they yield similar results for chaotic signals and colored noise (37). Thus it is likely that the role of chaotic determinism will not be fully assessed with the use of a single test but rather with a body of related evidence (37).
In addition to addressing the chaos hypothesis, the many chaos detection algorithms each have the potential to provide fresh insight into some particular aspect of the organization of heart period sequences (37). Unfortunately, this potential has not been fully realized in many studies. Furthermore, only a few studies (e.g., Ref.19) attempt to relate analytical results to actual beattobeat structures observed in heart period sequences. Moreover, although some studies relate results to global aspects of the physiological control systems, very few attempt to relate their findings to the physiological control of actual beattobeat structures. At the most basic level, the paucity of published heart period sequence data in many of the heart rate variability chaos studies demonstrates that numerical results are often analytically detached from actual beattobeat structures.
In an effort to address both this sense of detachment and the chaos hypothesis, we investigated the information scaling properties of heart period sequences obtained from 10 resting human subjects. More specifically, we calculated the information content of the embedded heart period sequences as a function of scale (9, 12, 23, 26, 30) and then compared these information scaling characteristics with those obtained for numerous periodic and chaotic (both low and high dimensional) control sequences. The results from this analysis prompted us to use this method to determine whether an alternative paradigm of heart rate variability might account for our findings. We proposed and tested the hypothesis that heart period sequences are assemblages of imperfectly repeating discrete events. Using actual and model heart period sequences, we showed that this notion of imperfectly repeating events can lead to information scaling characteristics similar to those observed for resting heart period sequences.
METHODS
Study subjects.
There were four male and six female subjects, all without history of structural heart disease, ventricular tachycardia, diabetes, autonomic neuropathy, syncope, or hypertension. The mean age was 38 ± 6 yr. All subjects were interviewed by the senior author (R. S. Sheldon), and all gave informed consent.
Data acquisition.
We recorded 60min sequences from supine subjects who were off all medications by using a system that incorporates a synchronization pulse to reduce variability due to tapespeed fluctuations. The Holter recordings were then digitized at 125 Hz and analyzed with the use of the Marquette 8000 Scanner with version 5.7 of the Marquette Arrhythmia Analysis Program to identify and label each QRS. Unclassified beats were corrected manually and verified. None of the patients experienced any ectopic beats during the sampling period. A 2,000beat sequence of RR intervals from each corrected 60min recording was transferred into MATLAB for further analysis.
Control sequences.
A variety of periodic and chaotic (low and high dimensional) systems were used to produce the 2,000element control sequences (see Table1). Nonlinear systems of equations were solved using 5thorder RungeKuttaFehlberg integration, and the associated control sequences were obtained by cubically splining the solution of a single variable to an appropriate time step.
Reconstruction of phasespace dynamics.
An essential step in the nonlinear analysis of a singlevariable time series is the reconstruction of the phasespace dynamics. Because the variable of interest in this study is heart period, we followed the method of Shaw (30) and reconstructed the phasespace dynamics using integer delays instead of uniform time delays. For time series from chaotic systems of unknown dimension, Sugihara and May (32) demonstrated that the dimension of the underlying attractor can be estimated from the embedding dimension (E), which yields the greatest predictability. Recall from Ref. 32 that, for a given embedding dimension E, each point in theEdimensional embedding space hasE components, where each component is assigned the value of the heart period sequence as measured after successive delays. For example, a subsequence of five successive heart period values is projected in an E = 5dimensional space as a fivedimensional point. A timeordered set ofEdimensional points, representing the evolution of the system in that particularEdimensional state space, is made by moving the reference point along the heart period sequenceand constructing an Edimensional point indexed to each heart period value.
Here, nonlinear predictability refers to the ability to predict, using reconstructed state spaces, the next heart period value (HP_{n+1}) for each heart period value (HP_{n}) of the sequence. Predictability arises in deterministic systems because any two points that are close to each other in the reconstructed state space evolve along nearly identical paths and thus can be used to predict each other’s temporal evolution. Recall that each element of the heart period sequence (HP_{n}) is indexed to anEdimensional point (Z_{n} ) in the reconstructed state space. To predict HP_{n+1}, the prediction algorithm finds an Edimensional point (Z_{m} ) in the reconstructed state space that is the nearest neighbor to the pointZ_{n} . The heart period value at HP_{m+1} is then used as a predictor of the heart period value HP_{n+1}. The set of predicted values {HP_{m+1}} is plotted as a function of the set of actual next values {HP_{n+1}} for all elements of the heart period sequence. The correlation coefficient (r) between these two sets is a measure of the predictability of the heart period sequence as projected in the Edimensional embedding space.
To maximize the likelihood of finding nearest neighbors, we permitted any nearestneighbor pair of points to predict each other, provided their corresponding heart period subsequences did not overlap. This allowed for the identification of similar points despite any timedependent changes in the system. Sequences of 2,000 beats from each subject were analyzed for nonlinear predictability usingE = 2–15. From these, we determined the maximum predictability (r _{max}) and theE dimension of maximum predictability for each heart period sequence.
Calculation of information scaling.
For each heart period sequence and control sequence, the information content of the reconstructed phase space was calculated as a function of scale. Figure 1 demonstrates the notion of scale as it applies to the twodimensional embedding space containing the reconstructed Henon attractor. Scale refers to the side length of individual boxes that partition the embedding space. The largest boxes represent the coarsest scale of partitioning; this scale is designated as 2^{−k}, where k = 1. Fork = 1, each side length of a box is (½)^{k} the linear distance along the respective axis of the embedding space. At each scale, i.e., for k = 1, ... 6, the natural measure (μ_{k}) is calculated for each box, where μ_{k} is the number of attractor points within each box; μ_{k} is normalized such that the sum of all μ_{k} at a given scale 2^{−k} is unity. For chaotic systems, the sets of natural measures vary in a systematic way as the scale of the box is changed. To assess how the information content (I_{k}; or entropy) of these sets varies as a function of scale, we first calculate I_{k} from I_{k} = −Σμ_{k}log_{2}(μ_{k}). I_{k} is then plotted as a function of the logarithmic inverse scale −log_{2}(2^{−k}) = k, for the reconstructed phase spaces of the heart period sequences and control sequences. From information theory, long sequences derived from chaotic and periodic systems should show a linear relationship between I_{k} andk.
RESULTS
Embedding dimension and predictability.
Figure 2 displays the first 200 elements of the control sequences and the first 200 intervals of the heart period sequences. The sequences were limited to 200 beats for Fig. 2to depict as much as possible the different scales and types of structures within the sequences. In particular, for the heart period sequences, note the many fine scale fluctuations (in the order of 3–5 beats) and the occasional largerscale fluctuations (in the order of 10 beats and longer). Both heart periodsequences 8 and 10 (HPS8 and HPS10) in Fig.2 B show clear examples of the two different scales of structure combined in one signal.
The embedding dimension E was then determined for each of the 10 heart period sequences as described inmethods. The embedding dimensions for the controls were obtained from previously published material (6, 20). Figure 3 displays the predictabilityr of the reconstructed phase spaces plotted as a function of E for the sequences. In general, the heart period sequences demonstrate a broad peak of predictability centered near E= 4, followed by decreasing predictability asE increases. The optimum predictability coefficientr _{max} ranged from 0.50 to 0.85, and the optimum E ranged from 4 to 8. The optimal embedding dimensions and the nonlinear predictability values for the 10 heart period sequences are listed in Table 2.
Information scaling plots.
The optimum embedding dimensions were used to reconstruct the putative attractors for each of the sequences, and the information content of these attractors was calculated at each scale. Lowdimensional periodic and chaotic attractors show a linear increase in the information content I_{k} as a function ofk, as expected for deterministic attractors (9, 12, 23, 26, 30). Figure 4,A andB, shows I_{k} plotted as a function ofk for the periodic and chaotic control sequences. Note that, for the controls, the slopes are steepest at the coarsest scale (for 1 ≤ k ≤ 2) and decrease at finer scales. This flattening of slope is an artifact of the relatively short sequence lengths. At progressively finer scales, the boxes contain fewer points and the range of possible natural measures is reduced, as is the information content. Note also that the higher dimensional systems flatten at coarser scales (i.e., at lowerk values). For example, the highdimensional MG100 sequence, as embedded inE = 17dimensional space, attains a peak information content atk = 3 and remains unaltered for progressively finer scales. Alternatively, the Henon sequence embedded in an E = 2dimensional space shows very little evidence of flattening in its information scaling plot. Given that, for k = 1, the 2,000 attractor points are partitioned into 2^{2} boxes forE = 2 (see Fig. 1) and 2^{17} boxes forE = 17, it is clear that higherdimensional attractors require much more data to accurately estimate the natural measures at finer scales. Thus the information scaling properties of the control sequences behave as expected from information theory.
The information scaling plots for heart period sequences are presented in Fig. 4 C. The most noticeable difference between these plots and those obtained for the control sequences occurs at the coarsest scales. All of the control sequences have their steepest slopes between the coarsest and next coarsest scales (i.e., between k = 1 andk = 2). In contrast, 9 of 10 heart period sequence plots are not at their steepest over this range of scales. Tables 3 and4 list the values of the change in information content in going from a scale ofk = 1 to a scale ofk = 2 (slope 1), and from a scale ofk = 2 to a scale ofk = 3 (slope 2) for the control sequences and heart period sequences, respectively. The ratio of the slopes at the coarsest to next coarsest scales (i.e., slope 1/slope 2) is 1.158 ± 0.118 for the lowdimensional periodic controls and 1.151 ± 0.104 for the lowdimensional chaotic control sequences. In contrast, this ratio is 0.64 ± 0.30 for the heart period sequences (P = 0.0003 vs. lowdimensional periodic controls, P = 0.0005 vs. lowdimensional chaotic controls, andP = 0.0003 vs. lowdimensional periodic and chaotic controls using Welch’s unpaired 2sidedttest). Thus the information scaling plots for the heart period sequences are inconsistent with a lowdimensional periodic or chaotic origin. Figure4 B indicates that the highdimensional chaotic sequences contain insufficient elements to accurately depict the scaling behavior. Nevertheless, the information scaling plot for MG30 (see Fig. 4 B) shows a shape similar to that of the lowdimensional controls in that its steepest slope spans the coarsest scales. Furthermore, its ratio of slopes at the coarsest to next coarsest scale is 1.251, which is significantly different from that for the resting heart period sequences (P < 0.0001).
Figure 4 B also shows the information scaling plot for the RösslerCQ control sequence. This sequence is identical to the RösslerC sequence except that it has been quantized to the same extent as the heart period sequences. That is, the RösslerCQ sequence is the RösslerC sequence rounded such that there are only 70 possible heart period values. This is equivalent to the average heart period sequence range, 560 ms, divided by the 8ms heart period measurement precision. Note that the information scaling plot is almost identical to that obtained for the nonquantized sequence. Thus it is unlikely that the measuring error inherent in the heart period sequences would account for the difference in the information scaling properties.
DISCUSSION
The possible contribution of periodic or chaotic dynamics to the organization of heart period sequences has been of considerable interest over the past decade. However, discriminating between deterministic models and other causes of heart rate variability has been difficult, mainly because many of the mathematical tools lacked the power to discriminate between chaos and locally correlated noise (18, 37). This has led to a search for analytic techniques that can distinguish between deterministic models, such as chaos and periodicity, and nondeterministic models, such as colored noise and assemblages of events. Several techniques may be sufficiently robust to address this problem. First, however, it is necessary to obtain recordings from situations as close to stationary as possible. Although it is unlikely that stationarity exists for freeroaming animals interacting with an everchanging environment, stationary sequences might be approximated by recording sequences from resting, unstressed subjects. By comparing results obtained for the original sequences with results obtained for sets of phaserandomized surrogates, several investigators have shown that certain nonlinear methods can statistically differentiate chaotic sequences from linearly correlated noise (15, 18, 3235). For example, in one study (35), the capacitydimension characteristics of heart period sequences and their phaserandomized surrogates were found to be similar. From these results, it was concluded that temporal correlation, rather than the presence of an attractor, was responsible for the capacitydimension characteristics. In another study, Kanters et al. (15) showed that the correlation dimensions differed only slightly between the original and surrogate sequences, indicating the lack of lowdimensional chaotic determinism. In the same study, the nonlinear predictability of the heart period sequences was demonstrated to significantly decrease following phase randomization. This was interpreted to indicate the presence of some nonlinear determinism. Thus the degree to which chaotic determinism and/or nonlinear determinism influence heart rate variability remained an open issue.
These results led us to use information scaling analysis to further test the chaos hypothesis. This technique does not require surrogates as control sequences, because it simply assesses the information content as a function of scale. For chaotic and periodic control sequences the technique performed as expected on the basis of information theory. With this technique, we showed that the scaling relationships in 9 of 10 heart period sequences were significantly different from those of the lowdimensional chaotic or periodic control sequences.
Our results are consistent with other studies in suggesting the low likelihood that any lowdimensional chaotic system underlies resting heart rate variability. Although the information scaling characteristics preclude a global attractor, they do not preclude the existence of locally organized (nontrivial) structures. Not surprisingly, there is indirect evidence from other studies that suggests the importance of local structures in these sequences. First, spectral analyses have demonstrated the presence of characteristic bands with center frequencies of 0.1 and 0.25 Hz, consistent with recurrent structures lasting 10 and 4 s, respectively (22, 25, 29). Second, there is some loss of predictability when heart period sequences are phase randomized, suggesting the presence of some characteristic (i.e., phase dependent) structures (15). Third, a lexical approach to the study of mouse cardiac interbeat intervals (19) reveals that occurrence rates of repeated subsequences are unlike those for lowdimensional chaotic systems or colored noise. This implies that some other type of local organization is responsible for the observed subsequence occurrence rates.
We suggest that results from our study and from others (15, 19, 22, 25,29, 35, 37) can be explained if heart period sequences are modeled as neither continuous deterministic functions nor linearly correlated noise. Instead, we suggest that heart period sequences be modeled as concatenated assemblages of characteristically scaled and characteristically shaped events. These events include breathing, corresponding to the 0.25Hz spectral band, and various forms of evanescent hemodynamic regulation, corresponding to the 0.1Hz spectral band (1, 22, 25, 29). These two kinds of events would have durations of ∼4 and 10 s, respectively. If these events recur, albeit imperfectly and aperiodically, they will cause limited and extinguishing predictability as well as characteristic spectral bands. Conversely, because these events are not members of a continuous function, they will yield information scaling results unlike those for chaotic or periodic functions. We test two aspects of this model below.
First, consider the model sequence displayed in Fig.5 A. This is the same as the Periodic1 control sequence except that we have included one of several similar 20beat tachycardic events from HPS1. Figure 5 B displays the reconstructed attractor in a twodimensional embedding space. Note the looplike structure of the periodic structure and the excursion from this structure due to the tachycardic event. Figure5 C shows the corresponding information scaling plot. Note that the information content increases in a manner similar to those of the scaling plots of the heart period sequences. In particular, the information content does not increase rapidly betweenk = 1 andk = 2. This can be explained by considering how the local excursion from the periodic loop affects the first two sets of natural measures. Figure5 B shows that the local event has expanded the embedding space beyond the scale of the periodic loop, causing the periodic loop to be isolated in one corner of the twodimensional space. From Fig. 5 Band the inscribed boxes corresponding to the scalesk= 1 andk = 2, it is clear that the set of natural measures for k = 1 andk = 2 will be quite similar. That is, one box will contain most of the points, thus keeping the information content (or entropy) of the set low. Thus, in a sense, the expanded embedding space caused the looplike structure to appear as a zerodimensional point at scales k= 1 and k = 2. In this way, nonstationary events can cause the information scaling characteristics to be quite unlike those for the deterministic controls.
Second, consider the model sequence of Fig.6 A. This model sequence is bandlimited noise centered around the respiratory frequency (0.25 Hz). This sequence is similar to the Periodic1 sequence except that the amplitudes of the individual fluctuations are not consistent and the oscillations are not perfectly periodic. Figure6 B shows the sequence embedded in a twodimensional space. Note the many concentric arc segments circling the center of the structure at different radii. Figure6 C shows the corresponding information scaling plot. Again, note that the information content does not increase most rapidly at the coarsest scales, as did the Periodic1 sequence. In this case, for the coarsest scales (k = 1, 2), the set of natural measures is dominated by the higher density of points near the center of the structure. Thus, in a sense, the structure behaves as a zerodimensional point at the coarsest scales and more like a twodimensional disk at finer scales. This example shows how nondeterministic components can lead to information scaling characteristics quite unlike those for purely deterministic systems.
Finally, consider the threedimensional trajectory plotted in Fig.7. This is the reconstructed phase space for HPS1 in an E = 3dimensional embedding space. The most predictable phase space for HPS1 wasE = 5 (see Table 2); thus Fig. 7 is a projection of this most predictable reconstructed attractor. Nevertheless, Fig. 7 does show that the reconstructed phase space displays both of the elements modeled in Figs. 5 and 6. First, most of the points are clustered in one small section of embedding space, and there are only a few imperfectly repeating excursions from this main cluster to other sections of the embedding space. This is similar to the model presented in Fig. 5. Second, the main cluster of points is not a perfectly periodic loop; instead, it appears more akin to the bandlimited noise presented in Fig.6 B. Thus the information scaling characteristics of HPS1 are governed by imperfectly repeating events of various durations and amplitude.
In summary, the information scaling properties of the heart period sequences are significantly different from those obtained for sequences derived from lowdimensional chaotic and periodic systems. We show that nondeterministic components, such as large tachycardic (or bradycardic) events or aperiodic fluctuations, can lead to scaling characteristics like those observed for the heart period sequences. This, in addition to evidence from spectral, nonlinear predictability, and lexical studies, leads us to favor a local eventsbased approach to modeling heart rate variability. Furthermore, recent documentation (27) of abrupt, localized, hypotensivebradycardic events (i.e., “faintlets”) in rabbits and humans attests to the existence of such events. Moreover, the eventsbased approach, in contrast to approaches based on putative attractors, is also likely to be more amenable to the study of transient events that precede physiological and clinical events such as arrhythmia or syncope.
Acknowledgments
This work was supported by grants from the Medical Research Council of Canada, Ottawa, ON, Canada (PG11188) and the Calgary General Hospital Research and Development Committee, Calgary, AB, Canada. D. E. Roach was a Fellow of the Heart and Stroke Foundation of Canada, Ottawa, ON, Canada.
Footnotes

Address for reprint requests: R. S. Sheldon, Faculty of Medicine, Univ. of Calgary, Health Sciences Center, 3330 Hospital Dr. N.W., Calgary, AB, Canada T2N 4N1.
 Copyright © 1998 the American Physiological Society