INTRODUCTION

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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 beat-to-beat 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, 32-34), the role for low-dimensional 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 beat-to-beat 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 beat-to-beat 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 beat-to-beat 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

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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 60-min sequences from supine subjects who were off all medications by using a system that incorporates a synchronization pulse to reduce variability due to tape-speed 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 2,000-beat sequence of R-R intervals from each corrected 60-min 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 2,000-element control sequences (see Table 1). Nonlinear systems of equations were solved using 5th-order Runge-Kutta-Fehlberg integration, and the associated control sequences were obtained by cubically splining the solution of a single variable to an appropriate time step.

Reconstruction of phase-space dynamics. An essential step in the nonlinear analysis of a single-variable time series is the reconstruction of the phase-space dynamics. Because the variable of interest in this study is heart period, we followed the method of Shaw (30) and reconstructed the phase-space 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 the E-dimensional embedding space has E 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 = 5-dimensional space as a five-dimensional point. A time-ordered set of E-dimensional points, representing the evolution of the system in that particular E-dimensional state space, is made by moving the reference point along the heart period sequenceand constructing an E-dimensional point indexed to each heart period value.

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 two-dimensional 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. For k = 1, each side length of a box is (1/2)^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 = µ_klog₂(µ_k). I_k is then plotted as a function of the logarithmic inverse scale log₂(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 and k.

View larger version (23K): [in this window] [in a new window]   Fig. 1.   Explanation of embedding dimension and scale definitions using reconstructed Henon attractor as an example. Attractor points are from x(n) plotted as a function of x(n + 1) for Henon system (see Table 1). For each value of k, the 2-dimensional embedding space is partitioned into boxes with side lengths of a2k and b2k. Largest scale boxes correspond to k = 1, where each box side length is one-half the length of the corresponding axis. Boxes for k = 1 and k = 2 are outlined with bold rules. At each scale k, the points in each box are counted and normalized, forming the set of natural measures µk. The information content (Ik) at each scale is calculated from Ik = µklog2(µk).

	RESULTS

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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. 2 to 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 larger-scale fluctuations (in the order of 10 beats and longer). Both heart period sequences 8 and 10 (HPS8 and HPS10) in Fig. 2B show clear examples of the two different scales of structure combined in one signal.

View larger version (54K): [in this window] [in a new window]   View larger version (41K): [in this window] [in a new window]   Fig. 2.   A: plots of first 200 elements of control sequences. See Table 1 for description of equations and solutions used for control sequences. B: plots of first 200 interbeat intervals of 10 heart period sequences (HPS). Vertical scales are R-R intervals in ms.

View larger version (31K): [in this window] [in a new window]   Fig. 3.   Plots of nonlinear predictability (r) as a function of embedding dimension (E) for 10 resting heart period sequences (curves 1-10). Embedding dimensions of greatest predictability (rmax) are used as embedding dimensions for reconstructing putative attractors for each heart period sequence. These attractors are used for determining information scaling characteristics.

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. Low-dimensional periodic and chaotic attractors show a linear increase in the information content I_k as a function of k, as expected for deterministic attractors (9, 12, 23, 26, 30). Figure 4, A and B, shows I_k plotted as a function of k 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 lower k values). For example, the high-dimensional MG100 sequence, as embedded in E = 17-dimensional space, attains a peak information content at k = 3 and remains unaltered for progressively finer scales. Alternatively, the Henon sequence embedded in an E = 2-dimensional 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 2² boxes for E = 2 (see Fig. 1) and 2¹⁷ boxes for E = 17, it is clear that higher-dimensional 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.

View larger version (29K): [in this window] [in a new window]   Fig. 4.   Plot of information content (Ik) as a function of scale (k) for control sequences and heart period sequences. A: information scaling plot for periodic control sequences. Note that plots have their steepest slopes over the coarsest scales (k = 1 to k = 2). B: similar plots for chaotic control sequences. Note that points obtained for RösslerC sequence () and quantized RösslerCQ sequence (+) are nearly identical. Also note that plots have their steepest slopes over the coarsest scales (k = 1 to k = 2) and that plots for high-dimensional chaotic controls (MG30 and MG100) flatten at coarser scales than do plots for low-dimensional chaotic controls (Henon, Rössler, Lorenz, and MG17). See text for discussion. C: similar plots for 10 heart period sequences. Note that, unlike periodic and chaotic controls, these plots do not have their steepest slopes over the coarsest scales.

	DISCUSSION

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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 free-roaming animals interacting with an ever-changing 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 phase-randomized surrogates, several investigators have shown that certain nonlinear methods can statistically differentiate chaotic sequences from linearly correlated noise (15, 18, 32-35). For example, in one study (35), the capacity-dimension characteristics of heart period sequences and their phase-randomized 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 capacity-dimension 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 low-dimensional 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 low-dimensional chaotic or periodic control sequences.

Our results are consistent with other studies in suggesting the low likelihood that any low-dimensional 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 low-dimensional 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 0.25-Hz spectral band, and various forms of evanescent hemodynamic regulation, corresponding to the 0.1-Hz 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. 5A. This is the same as the Periodic1 control sequence except that we have included one of several similar 20-beat tachycardic events from HPS1. Figure 5B displays the reconstructed attractor in a two-dimensional embedding space. Note the looplike structure of the periodic structure and the excursion from this structure due to the tachycardic event. Figure 5C 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 between k = 1 and k = 2. This can be explained by considering how the local excursion from the periodic loop affects the first two sets of natural measures. Figure 5B 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 two-dimensional space. From Fig. 5B and the inscribed boxes corresponding to the scales k= 1 and k = 2, it is clear that the set of natural measures for k = 1 and k = 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 zero-dimensional 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.

View larger version (30K): [in this window] [in a new window]   Fig. 5.   Model heart period sequence demonstrating a way to obtain information scaling characteristics similar to those obtained for heart period sequences. A: plot of first 200 intervals for 2,000-element model sequence. Sequence x(n) is similar to Periodic1 sequence except that a local 24-beat tachycardic event extracted from HPS1 was inserted at element 100 of sequence. B: plot of reconstructed attractor in E = 2-dimensional embedding space. Note that tachycardic event has caused embedding space to expand, thus isolating periodic loop in top right corner. Boxes for calculating information content at scales k = 1, 2 are shown. C: information scaling plot for 2,000-element model sequence. Note that this plot is similar to those obtained for heart period sequences (see Fig. 4) in that information content increases slowly from scale k = 1 to k = 2. See text for full discussion.

Second, consider the model sequence of Fig. 6A. This model sequence is band-limited 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. Figure 6B shows the sequence embedded in a two-dimensional space. Note the many concentric arc segments circling the center of the structure at different radii. Figure 6C 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 zero-dimensional point at the coarsest scales and more like a two-dimensional disk at finer scales. This example shows how nondeterministic components can lead to information scaling characteristics quite unlike those for purely deterministic systems.

View larger version (44K): [in this window] [in a new window]   Fig. 6.   Model heart period sequence demonstrating another way to obtain information scaling characteristics similar to those obtained for heart period sequences. A: plot of first 200 intervals of 2,000-element model sequence. Sequence x(n) is similar to Periodic1 control sequence except that oscillations are neither perfectly periodic nor of constant amplitude. Sequence is band-limited noise centered at 0.25 Hz. B: plot of reconstructed attractor in E = 2-dimensional embedding space. Note that "attractor" is made up of many concentric arclike segments at different radii. C: information scaling plot for 2,000-element model sequence. Note that this plot is similar to those obtained for heart period sequences (see Fig. 4) in that information content increases slowly from scale k = 1 to k = 2. See text for full discussion.

Finally, consider the three-dimensional trajectory plotted in Fig. 7. This is the reconstructed phase space for HPS1 in an E = 3-dimensional embedding space. The most predictable phase space for HPS1 was E = 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 band-limited noise presented in Fig. 6B. Thus the information scaling characteristics of HPS1 are governed by imperfectly repeating events of various durations and amplitude.

View larger version (28K): [in this window] [in a new window]   Fig. 7.   Plot of first 700 points of HPS1 sequence embedded in E = 3-dimensional embedding space. Note that most points are clustered in 1 small section of embedding space, and there are only a few imperfectly repeating excursions from this main cluster to other sections of embedding space. Also note that the main cluster of points is not a perfectly periodic loop but instead appears more akin to band-limited noise presented in Fig. 6B.

In summary, the information scaling properties of the heart period sequences are significantly different from those obtained for sequences derived from low-dimensional 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 events-based approach to modeling heart rate variability. Furthermore, recent documentation (27) of abrupt, localized, hypotensive-bradycardic events (i.e., "faintlets") in rabbits and humans attests to the existence of such events. Moreover, the events-based 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.