Vol. 278, Issue 6, H2163-H2172, June 2000
SPECIAL COMMUNICATION
Entropies of short binary sequences in heart period
dynamics
D.
Cysarz1,
H.
Bettermann1, and
P.
van
Leeuwen2
1 Department of Clinical Research, Gemeinschaftskrankenhaus,
D-58313 Herdecke; 2 Research and Development Center for
Microtherapy, D-44799 Bochum, Germany
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ABSTRACT |
Dynamic aspects of R-R intervals have often been analyzed by means of
linear and nonlinear measures. The goal of this study was to analyze
binary sequences, in which only the dynamic information is retained, by
means of two different aspects of regularity. R-R interval sequences
derived from 24-h electrocardiogram (ECG) recordings of 118 healthy
subjects were converted to symbolic binary sequences that coded the
beat-to-beat increase or decrease in the R-R interval. Shannon entropy
was used to quantify the occurrence of short binary patterns (length
N = 5) in binary sequences derived from 10-min
intervals. The regularity of the short binary patterns was analyzed on
the basis of approximate entropy (ApEn). ApEn had a linear dependence
on mean R-R interval length, with increasing irregularity occurring at
longer R-R interval length. Shannon entropy of the same sequences
showed that the increase in irregularity is accompanied by a decrease
in occurrence of some patterns. Taken together, these data indicate
that irregular binary patterns are more probable when the mean R-R
interval increases. The use of surrogate data confirmed a nonlinear
component in the binary sequence. Analysis of two consecutive 24-h ECG
recordings for each subject demonstrated good intraindividual
reproducibility of the results. In conclusion, quantification of binary
sequences derived from ECG recordings reveals properties that cannot be found using the full information of R-R interval sequences.
heart period dynamics; symbolic dynamics; approximate entropy; Shannon entropy; nonlinear dynamics; surrogate data
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INTRODUCTION |
IN RECENT YEARS linear measures of
heart rate variability (HRV) have been applied in a wide range of
contexts, leading to a well-established diagnostic tool with more or
less accepted standards (16, 17, 30). Today, HRV is applied not only in cardiac diseases but in diseases that generally affect the autonomic nervous system (ANS). However, the influence of the sympathetic and
parasympathetic branch of the ANS on linear measures of HRV, as well as
the independent prognostic value of these measures with respect to
high-risk patients with cardiac diseases, is still a matter of
investigation (6, 9, 12). On the other hand, assessing HRV with
nonlinear measures may supply information different from that of linear
measures with the promise of better risk stratification (13,
32-34). However, in most cases it is difficult to interpret these
complementary findings in one unifying picture. In this study we
examine the dynamic properties of heart periods with the use of two
different nonlinear approaches that can be regarded as two
complementary aspects of dynamic properties. The results also shed new
light on the interpretation of power spectral measures of HRV.
Different approaches lead to nonlinear measures of HRV. In nonlinear
dynamics theory, the so-called state space is reconstructed from
sequences of heartbeat periods that are generally defined as the time
duration between successive R waves in the electrocardiogram (ECG), the
R-R tachogram. In a second step, the state space and the dynamic
behavior of the reconstructed dynamics can be quantified (e.g., with
measures of dimension or Lyapunov exponents). For an overview, see Ref.
10. Practically, the sequences of heart periods are short, noisy, and
often nonstationary. Thus the application of nonlinear measures to ECG
recordings may lead to spurious indications of chaos (3, 7). However,
one may guardedly say that this approach has yielded evidence of
nonlinearities. Indeed, powerful quantities for describing heart period
dynamics and for stratification of high-risk patients are still lacking
(17).
Another approach to nonlinear measures of HRV is the quantification of
complexity from the point of view of information theory. To this end,
the sequence of heart periods can be analyzed with the help of entropy
measures such as Shannon entropy or renormalized entropy (11, 25).
These are often used in conjunction with the concept of symbolic
dynamics or coding theory, i.e., reducing the amount of information by
transforming the original time series into a symbolic sequence with a
small set of symbols (8). These measures proved to be useful in
detection of patients at high risk for sudden cardiac death (34).
Another entropy measure for quantification of regularity in a time
series is the approximate entropy (ApEn) (18, 24). ApEn has the ability
to detect subtle differences in heart period dynamics that cannot be
observed with commonly used linear measures (14, 15). Recently, the
evaluation of ApEn for R-R tachograms derived from 24-h ECG recordings
led to the suggestion of phase transitions, in the notion of
synergetics, between daytime and nighttime heart period dynamics (2).
It has also been shown that changes in fetal heart period complexity during pregnancy can be documented using ApEn (31). Though approximate entropy has been introduced for symbolic dynamics (20), its application
to symbolic dynamics derived from physiological data has not been
performed yet.
The goal of this study was to examine binary sequences derived from
Holter recordings of healthy subjects to determine their pure dynamic
properties. To this end, a "dynamic" or differential symbolization
was used (1). Such a transformation into binary sequences is of
particular interest because this method extracts solely dynamic
properties of the R-R series, disregarding all information influenced
by the absolute values of the R-R intervals, e.g., mean R-R interval,
R-R standard deviation, and other measures of R-R interval variability.
ApEn was used as a nonlinear measure of irregularity of short binary
sequences to quantify their dynamic properties. Shannon entropy
quantifies regularity on a larger scale of the symbolic dynamics under
consideration and thus helped to make the results more precise. It is
still unknown whether binary coding preserves nonlinear properties of
the original R-R tachogram. To test the hypothesis that the binary
representation of R-R dynamics still contains some important nonlinear
properties, we made use of surrogate data. To demonstrate the
intraindividual reproducibility of the binary approximate entropy, two
consecutive 24-h ECG recordings for each subject were analyzed.
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METHODS |
Subjects.
The subjects for this study were drawn from a previous study in which
121 healthy subjects were included (5). Three subjects were excluded
from this analysis because of missing data. Two consecutively recorded
24-h ECGs (ECGs A and B) were available for the
remaining 118 subjects (age: 20-40 yr, mean ± SD: 27 ± 6 yr; 78 females). The 24-h ECGs were recorded with Oxford FD3 solid-state recorders (Oxford Instruments, Abingdon, UK) with simultaneous R wave detection and a maximum sampling rate of 1,024 Hz
during the QRS complex. This permitted a maximum resolution of 1 ms for
the detection of the R waves. An Oxford Excel ECG analyzer allowed
visual inspection of the automatically detected R waves. Generally, the
number of ectopic or unrecognized beats was small (<1%), and thus
such beats were not replaced or inserted. For further analysis the R
times were written to a binary data file that was exported to a
personal computer for further analysis.
Construction of symbolic sequences.
For each 10-min interval in the 24-h ECG (maximum 144 intervals/recording), the times between subsequent R waves (R-R
intervals or heart periods) formed the corresponding R-R tachogram.
Transformation of each 10-min R-R tachogram into a binary sequence was
done as follows (see Fig. 1):
beat-to-beat differences R-Rn+1 
R-Rn > 0, i.e., a decrease in heart rate,
were set to a value of 1, and differences
R-Rn+1 R-Rn 
0, i.e., an increase in heart rate, were
set to a value of 0. The binary sequences are quantified by estimation
of two different entropies: ApEn and Shannon entropy. Each entropy
reveals different aspects of the binary sequence under consideration:
ApEn is a nonlinear measure of irregularity in a time series (24),
whereas Shannon entropy quantifies the amount of information in a time series (28).
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Fig. 1.
Example of construction of symbolic sequences from
electrocardiogram (ECG) recordings, keeping the dynamic aspects. RR,
R-R interval.
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Approximate entropy.
The goal of ApEn is to quantify irregularity or fluctuations in a time
series on the basis of Kolmogorov-Sinai entropy (21, 23). It quantifies
dynamic aspects of the time series under consideration in a statistical
manner. A short description of the formal implementation of ApEn
follows (for further details, see Refs. 18 and 22).
Given a time series (e.g., R-R tachogram) with N data points
u(1), u(2), ... , u(N),
sequences of vectors x(1), ... ,
x(N m + 1) are formed by
defining x(i) = [u(i),
u(i + 1), ... ,
u(i + m 1)]. The
parameter m, the number of components in each vector, has to
be fixed. In nonlinear dynamics theory this would be interpreted as an
"m-dimensional state space reconstruction." Next the
distance d[x(i),
x(j)] between two vectors x(i) and x(j) is
defined by the maximum difference of all their scalar components as
The "correlation sum" of vector x(i)
is
The parameter r acts like a filter value:
within resolution r, the numerator counts the number of
vectors that are approximately the same as a given reference vector
x(i). The quantity Cim(r) is called the
correlation sum because it quantifies the summed (or global)
correlation of vector x(i) with all other vectors.
Next, the mean logarithmic correlation sum of all vectors is defined as
and ApEn is represented as
ApEn(m, r, N)(u) measures the logarithmic
frequency with which vectors with m components that are
close (within resolution r) remain close when the number of
vector components is increased by one. This is the key to a measure of
irregularity: small values of ApEn indicate regularity, and large
values imply substantial fluctuations or irregularity in a time series
u.
This concept can also be applied to short binary sequences or other
symbolic dynamics. To understand the notion of irregularity in binary
sequences, consider the sequences 00000, 11111, 01010, and 10110. The
first two sequences are easily identified as very regular sequences. In
the third sequence, the 0's and 1's alternate, and thus it is
suitable to call this sequence regular, too. Only the last sequence
does not contain any symmetries or periodically recurring subsequences;
in other words, this sequence is more irregular. This concept of
irregularity for binary sequences can be quantified using ApEn.
Formally, if ApEn is applied to binary sequences consisting of 1's and
0's, the distance d[x(i),
x(j)] will be either 0 or 1. Thus it only
makes sense to set the resolution r < 1. To keep
things as easy as possible, we restricted ourselves to
m = 1. Next, the optimal length of binary sequences to
be quantified with ApEn had to be found. As pointed out in Ref. 20, the
evaluation of ApEn with m = 1 is based on the
calculation of the frequencies of the subsequences {0, 1, 00, 01, 10, 11} in the binary sequence under consideration. In a random binary
pattern, the longer the binary sequence, the higher the probability
that the subsequences occur with almost the same frequency. This would
always lead to approximately the same values of ApEn. Thus short binary
patterns would be better suited to produce ApEn values that can be
distinguished from one another. In this work, we analyzed very short
binary sequences (N = 5), permitting a good
differentiation of the values of ApEn for the distinct binary patterns.
We referred to these very short sequences as "binary patterns,"
distinguishing them from the 10-min "binary sequences" of heart
period dynamics.
To distinguish this use of approximate entropy from the normal use, we
called this quantity "binary approximate entropy" (BinApEn). Practically, BinApEn was evaluated for each binary pattern of length
N = 5 in the whole binary sequence generated from the
10-min R-R tachogram. The average of all BinApEn values was used to
quantify heart period irregularity of the binary patterns.
Shannon entropy.
In contrast to BinApEn, Shannon entropy considers the whole binary
sequence generated from the 10-min R-R tachogram. Shannon entropy gives
a number that characterizes the probability that different binary
patterns of length N occur. For a very regular binary
sequence, only few distinct patterns occur. Thus Shannon entropy would
be small because the probability for these patterns is high and only
little information is contained in the whole sequence. For a random
binary sequence, all possible patterns of length N occur
with the same probability, and the content of information is maximal.
This case is indicated by maximal values of Shannon entropy.
To formalize this concept, first the probabilities of each pattern of
length N are estimated from the whole binary sequence (28)
where
ns1, ..., sN
is the number of occurrences of the pattern s1,
s2, ... , sN and
ntot is the total number of patterns. Next, the
entropy estimation S(N) is defined as
For a better comparison when using different pattern
lengths N, S(N) is divided by N. Thus
the maximal estimation of Shannon entropy is always 1. The properties
of this measure are as follows. If only one binary pattern occurs in
the whole sequence, S(N) = 0. If all
2N patterns are equally distributed in the
sequence, i.e., the probability is 
= 1/2N for all patterns, and then
S(N) = 1. This means that all N
bits are needed to describe the whole binary sequence properly.
According to the pattern length of the BinApEn algorithm, a length
(i.e., embedding dimension) of N = 5 symbols for the
subsequences is used. Keeping in mind that each 10-min interval
contains ~800 heartbeats, this guarantees a proper estimation of the
probabilities of all 25 = 32 binary subsequences.
Deviations from identical distribution of all binary patterns are
observed more easily than for shorter or longer pattern lengths. This
entropy estimation is referred to as BinShan.
Surrogate data.
The properties of binary sequences generated from heart period dynamics
are still unknown. It is not known whether nonlinear properties can be
found in such binary sequences or whether these can be fully described
with the help of linear methods. In other words, does the sequence of
acceleration and deceleration of heart periods already contain
nonlinearities, or is the nonlinear information only revealed if the
absolute R-R intervals are taken into account? To answer this question,
we used an iterative scheme introduced by Schreiber and Schmitz (27) to
produce surrogate data. At the moment, this method seems to be the best
choice of all randomization techniques, preserving almost all linear
properties of the original time series with relatively low
computational costs. In contrast to other techniques, the iterative
scheme not only retains the mean and the standard deviation (i.e., the
distribution) but also maintains the power spectrum (i.e., the
autocorrelations) of the original time series (relative error < 0.1%). All other properties are randomized. Thus the surrogate data
cannot be distinguished from original data with any linear measure of HRV.
In this study surrogate data were constructed for each 10-min interval
of all 24-h ECGs, and in a second step the binary sequences were
generated as described above. If the binary sequences derived from
original data contain nonlinear properties, the estimation of BinApEn
and BinShan should reveal differences between the original and
surrogate data.
Statistics.
Dependencies between two variables were quantified by Pearson's
correlation coefficient (referred to as R to distinguish it from the parameter r). The dependence between mean BinApEn
versus mean R-R and Shannon entropy versus mean R-R was quantified by the linear regression y = a · x + b. To test the hypothesis that nonlinear
components are still observable in the binary sequences, the
distribution of differences between original and surrogate slopes and
correlation coefficients was used. The probability of rejecting the
null hypothesis that no difference is observable was calculated with
Student's t-test, and P < 0.05 was
considered statistically significant.
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RESULTS |
Approximate entropy.
The results for BinApEn of all 236 24-h ECGs were examined visually by
plotting mean BinApEn against mean R-R interval of each
10-min interval. Figure 2A
shows an example. A linear dependency between mean BinApEn and mean R-R
interval is observable: the longer the R-R interval, the higher the
mean BinApEn and, hence, the more irregular the binary patterns. The
correlation between mean BinApEn and mean R-R interval yielded
R = 0.84. Generally, we found this dependence in all
24-h ECGs. In Figs. 3A and
4A the distributions of slopes
and correlation coefficients of all ECGs are shown. The
distribution of correlation coefficients has a mean of
R = 0.78, showing strong correlation between mean
BinApEn and mean R-R interval in all ECGs. Thus a proper evaluation of linear regression was guaranteed. The distribution of the slopes yielded a mean slope of a = 4.22 × 10
1 s1.
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Fig. 2.
Example of mean binary approximate entropy (BinApEn) of
10-min sequences vs. mean R-R interval of original (A) and
surrogate data (B). Solid line indicates linear regression
with slope a. R, Pearson's correlation
coefficient.
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Fig. 3.
Distribution of slopes of linear regression evaluated
from mean BinApEn vs. mean R-R interval of all ECGs for original
(A) and surrogate data (B).
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Fig. 4.
Distribution of Pearson's correlation coefficients of
mean BinApEn vs. mean R-R interval of all ECGs evaluated for original
(A) and surrogate data (B).
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Next, we evaluated BinApEn for the surrogate data in a similar fashion.
At first glance, the slope of the linear dependence in Fig.
2B is less steep; a and R are smaller
than those of the original data. However, the distribution of
correlation coefficients as depicted in Fig. 4B shows that
the mean coefficient (R = 0.73) of the surrogate data
is only slightly lower than that of the original data. The distribution
of paired differences of correlation coefficients between original and
surrogate data has a mean of 0.05 (P < 0.0001). Thus
surrogate data showed a linear dependence to a slightly lesser extent,
but it is still feasible to evaluate linear regression slopes. On the
other hand, the distribution of slopes of all surrogate data as shown
in Fig. 3B revealed a clear reduction of the mean slope
(a = 2.88 × 101 s1).
The distribution of paired differences of slopes between the original
and surrogate data has its mean at 1.35 × 101
s1, showing a clear deviation from zero mean
(P < 0.0001).
We point out that the evaluation of the linear regression depends on
the correlation between mean BinApEn and mean R-R interval. Consequently, the decrease of the slope of the linear regression for
the surrogate data is partly due to a decrease in the correlation between mean BinApEn and mean R-R interval.
Shannon entropy.
An example of BinShan of 10-min intervals plotted against mean R-R
interval is depicted in Fig. 5A
(data are from same subject as shown in Fig. 2). Overall, in all
ECGs, as mean R-R interval increased, BinShan decreased. This implies
that a shorter mean R-R interval could be associated with more equally
distributed binary patterns. The distribution of slopes yielded
a mean of a = 2.32 × 101
s1 (Fig. 6A).
The mean value of R (Fig.
7A, R = 0.56) guaranteed a proper evaluation of linear regression.
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Fig. 5.
Example (from same subject as in Fig. 2) of binary
Shannon entropy (BinShan) of 10-min sequences vs. mean R-R interval of
original (A) and surrogate data (B). Solid line
indicates linear regression with slope a.
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Fig. 6.
Distribution of slopes of linear regression evaluated
from BinShan vs. mean R-R interval of all ECGs for original
(A) and surrogate data (B).
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Fig. 7.
Distribution of Pearson's correlation coefficients of
BinShan vs. mean R-R interval of ECGs evaluated for original
(A) and surrogate data (B).
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For the surrogate data, values of BinShan are generally increased as
shown in Fig. 5B. Thus a less marked difference between short and long mean R-R interval was observable, and hence,
R is reduced (Fig. 7B, mean R = 0.42). The distribution of slopes was shifted to higher values (Fig.
6B, mean a = 1.04 × 101 s1). The distribution of paired
differences of slopes showed a clear deviation from zero mean (mean
a = 1.28 × 101 s1,
P < 0.0001).
Reproducibility of BinApEn and BinShan.
Two consecutive 24-h ECGs were available for each subject. The slopes
of linear regression of each subject were used to estimate the
reproducibility. The slopes of ECG A were plotted against those of ECG B (Fig. 8). Both
entropies yielded strong correlation between the slopes of both days
(BinApEn: R = 0.78; BinShan: R = 0.85).
This implies a good intraindividual reproducibility of BinApEn and
BinShan. Because the slopes showed a broad distribution, this result
may also imply that each subject has its specific slope of linear
regression.
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Fig. 8.
A: slopes of linear regression of BinApEn of
ECG A (BinApEn A) vs. slopes of linear regression of BinApEn
of ECG B (BinApEn B). B: slopes of linear
regression of BinShan of ECG A (BinShan A) vs. slopes of
linear regression of BinShan of ECG B (BinShan B). Dotted
line indicates optimal reproducibility.
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DISCUSSION |
We used binary sequences derived from R-R tachograms of 24-h ECG
recordings that retain only basic dynamic aspects of the R-R tachogram,
i.e., the acceleration (set to 0) and deceleration (set to 1) of
heartbeat, to estimate approximate and Shannon entropy. This kind of
dynamic symbolization allowed the examination of stationary as well as
many nonstationary segments because the symbolization of differences
between R-R intervals eliminates nonstationarities resulting from a
minor bias underlying the R-R tachogram. We did not calculate entropy
estimations using a static symbolization (e.g., all R-R intervals above
the level of the mean R-R interval were set to 1, and the others were
set to 0). In the literature this kind of transformation is used to
detect so-called "forbidden words," i.e., patterns in successive
R-R intervals, that might be of interest in certain cardiac diseases (11, 32-34). In the context of entropy estimations established in
this study, the latter transformation is not useful because it often
yields long chains of 1's or 0's in nonstationary sequences, resulting in minimal entropy estimations for BinApEn and BinShan that
might be interpreted spuriously.
The evaluation of mean BinApEn of each 10-min interval exhibited two
properties: mean BinApEn strongly correlated with mean R-R interval and
was very reproducible for each subject. Mean BinApEn demonstrated that
short binary patterns were most regular at short R-R intervals and
displayed more irregularity with increasing R-R intervals. BinShan was
maximal for shorter R-R intervals, indicating that all binary patterns
occur with almost the same probability, and was minimal for longer R-R
intervals, exhibiting predominance of certain binary patterns that may
result from phase locking with the respiratory rhythm (see below).
We point out that BinApEn and BinShan deal with two different notions
of regularity. BinApEn quantifies the regularity of short binary
patterns, whereas BinShan quantifies the regularity of the
occurrence of the binary patterns. Thus the two notions complement each other.
Considering only Shannon entropy would lead to the conclusion that the
general behavior of heart period dynamics seems to be more regular at
longer R-R intervals in the sense that certain binary patterns
predominantly occur, whereas other patterns tend to disappear. On the
other hand, the results of BinApEn indicate that for long R-R intervals
the binary patterns in heart period dynamics were those with highest
irregularity. Combining these findings, we can conclude that although
fewer distinct patterns occurred at longer R-R intervals, these
patterns were precisely those reflecting greater irregularity. In other
words, at longer R-R intervals irregular patterns of heart period
dynamics appeared more regularly.
Although we did not differentiate between daytime and nighttime (or
sleep stages), we noted that long R-R intervals are likely to appear at
night, whereas short ones appear during the day. This is shown in Fig.
2, in which two distinct regions are separated at a mean R-R interval
of ~0.85 s. This leads to the conclusion that at night, fewer
distinct dynamic patterns of the R-R intervals occur more regularly,
but the dynamics of these patterns are more irregular than during the day.
This finding fills the gap between the findings of two former studies
conducted in our laboratory. Using the full information of R-R interval
lengths for the evaluation of ApEn, we were able to demonstrate that
heart period dynamics are more irregular at night than during the day
and that the change from day to night or vice versa is probably
accompanied by a phase transition in the notion of synergetics (i.e.,
no linear dependence on mean R-R interval length) (2). In a recent
study, we emphasized that at night, cardiac dynamics reveal a
predominance of binary patterns that can be assigned to distinct
frequency ratios or even phase locking with the respiratory rhythm
(e.g., 4:1, 7:2, 5:1) (1). For example, if 5:1 phase locking is
present, the binary pattern 11001 must occur predominantly and
cyclically recurrent. This predominance was interpreted as an increase
of heart period regularity and an augmentation of musical rhythmicity
in cardiac dynamics. In the present analysis, this pattern was
identified as one of the most irregular patterns, i.e., with the
highest value of BinApEn (20), leading to high values of mean BinApEn. Thus the predominance of binary patterns that results from frequency or
phase locking ratios may still lead to strong irregularities within the
binary patterns. We point out that synchronization in physiological
systems is most often an intermittent phenomenon, detectable during
short periods of time with changing locking ratios (26, 29). A further
distinction of irregularities between synchronized and nonsynchronized
sequences has yet to be established.
The use of surrogate data resulted in a reduction of the slopes of the
linear regression between mean BinApEn and mean R-R intervals. For
short R-R intervals mean BinApEn slightly increased, and for long R-R
intervals mean BinApEn slightly decreased. The values of mean BinApEn
of binary sequences generated from completely random sequences
(independent identical distribution) tend toward a value of ~0.37.
(Note that by construction, purely random sequences are not maximally
irregular in the sense of BinApEn; see e.g., Ref 19.) This indicates
that the randomization procedure destroyed some inherent nonlinear
properties because the values of mean BinApEn tended toward the stated
value even though almost all linear properties were kept constant. The
results for BinShan of the surrogate data can be interpreted in a
similar fashion. In conclusion, the dynamic properties under
consideration cannot solely be described with linear methods but also
show evidence of nonlinearities. Moreover, even binary sequences
contain nonlinear properties that cannot be described with measures of
HRV derived from linear time series analysis.
By focusing on the beat-to-beat acceleration and deceleration of heart
periods, only fast-modulating rhythms in heart period dynamics are
captured, i.e., changes in heart periods due to respiratory sinus
arrhythmia (RSA) and other parasympathetic activity. The effects of
slower rhythms that influence the heart periods, e.g., the blood
pressure or slower variations, can be neglected because they only give
rise to a bias underlying the fastest modulation. These modulations
only affect the symbolization scheme if the bias exceeds the
modulations of the RSA. Hence, our results are primarily attributed to
the vagal activity on the cardiac system. It is well known that the
vagal influence shows a circadian pattern with an increasing strength
at night (4). This is in accordance with the aforementioned binary
pattern types that occur predominantly at longer R-R intervals and may
indicate frequency or phase locking between heartbeat and respiration
but that reveal at least certain frequency ratios between these two
interacting systems. Keeping our results in mind, the interpretation of
an HRV power spectrum can be extended. On one hand, a pronounced
modulation of heart periods by RSA causes high power in the respiratory
frequency band. This implies that the heart periods are modulated more
regularly. On the other hand, the same modulation may result in more
irregular patterns of heart period dynamics, attributing to an increase of complexity.
Moreover, the entropies of binary heart period dynamics turned out to
be highly reproducible for each subject. This fact supports the
findings that each healthy individual maintains the dynamic properties
of the heart periods over at least two days (1). Further investigations
may show how these properties depend on age and are affected by
cardiovascular and autonomic diseases.
In conclusion, the findings of this study have demonstrated that the
binary symbolization of R-R interval dynamics, which at first glance
seems to be an enormous waste of information, gives an important key to
a better understanding of normal heart period regularity. Furthermore,
differential binary symbolization still enables the identification of
nonlinear dynamical properties.
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ACKNOWLEDGEMENTS |
We acknowledge financial support from Weleda, Schwäbisch
Gmünd, Germany (to H. Bettermann and D. Cysarz).
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FOOTNOTES |
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: D. Cysarz, Dept.
of Clinical Research, Gemeinschaftskrankenhaus Herdecke,
Gerhard-Kienle-Weg 4, D-58313 Herdecke, Germany (E-mail:
d.cysarz{at}rhythmen.de).
Received 23 August 1999; accepted in final form 29 December
1999.
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Am J Physiol Heart Circ Physiol 278(6):H2163-H2172
TOP
ABSTRACT
INTRODUCTION
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