Gzrdiovasczdar
Research
ELSEVIER
Cardiovascular Research 3 1 ( 1996) 40@-409
Short- and long-term variations in non-linear dynamics
of heart rate variability
JIzrrgen K. Kanters a7by
* , Michael V. HGjgaard a, Erik Agner a, Niels-Henrik
Holstein-Rathlou
b
a Department
of Internal Medicine, Coronary
Care Unit, Elsinore Hospital, DK-3000 Helsing@r, Denmark
b Department
of Medical Physiology,
Uniuersity of Copenhagen,
DK-2200 Copenhagen N. Denmark
Received 13 February 1995; accepted 24 May I995
Abstract
Objectives:
The purpose of the study was to investigate the short- and long-term variations in the non-linear dynamics of heart rate
variability, and to determine the relationships between conventional time and frequency domain methods and the newer non-linear
methods of characterizing heart rate variability. Methods:
Twelve healthy subjects were investigated by 3-h ambulatory ECG recordings
repeated on 3 separate days. Correlation dimension, non-linear predictability, mean heart rate, and heart rate variability in the time and
frequency domains were measured and compared with the results from corresponding surrogate time series. Results: A small significant
amount of non-linear dynamics exists in heart rate variability. Correlation dimensions and non-linear predictability are relatively specific
parameters for each individual examined. The correlation dimension is inversely correlated to the heart rate and describes mainly linear
correlations. Non-linear predictability is correlated with heart rate variability measured as the standard deviation of the R-R intervals and
the respiratory activity expressed as power of the high-frequency band. The dynamics of heart rate variability changes suddenly even
during resting, supine conditions. The abrupt changes are highly reproducible within the individual subjects. Conclusions:
The study
confirms that the correlation dimension of the R-R intervals is mostly due to linear correlations in the R-R intervals. A small but
significant part is due to non-linear correlations between the R-R intervals. The different measures of heart rate variability (correlation
dimension, average prediction error, and the standard deviation of the R-R intervals) characterize different properties of the signal, and
are therefore not redundant measures. Heart rate variability cannot be described as a single chaotic system. Instead heart rate variability
consists of intertwined periods with different non-linear dynamics. It is hypothesized that the heart rate is governed by a system with
multiple “strange” attractors.
Keywords:
Chaos: Non-linear phenomena; Heart rate variability
1. Introduction
The heart is regulated by a feedback system that includes the sinus node, the baro- and chemoreceptors,
humoral factors, and the central nervous system. Such a
complex
system, involving
both positive
and negative
feedback loops, can give rise to very complicated
dynamics. This is evident in the heart rate, which appears to
fluctuate randomly.
The simplest method of quantifying
heart rate variability
(HRV) is the standard deviation
(SD,,) of the intervals between consecutive R-waves in
the electrocardiogram (ECG). SD,, has been shown to be
* Corresponding author. Tel.: + 454829 2305; FAX: + 45 4829 23 17;
e-mail: [email protected].
Elsevier Science B.V.
SSDI OOOS-6363(95)00085-2
a strong
independent
risk
factor
of mortality
in patients
who have suffered an acute myocardial infarction [l].
However, the clinical utility of this observation is limited
becausethe conventional measuresof HRV only permit
the definition of a high-risk group, but lacks the sensitivity
and specificity to identify individual subjects at high risk
for suddencardiac death.
The origin of the fluctuations in the heart rate is not
known in detail. In healthy subjects an important mechanism is variations in sinus node activity mediated by the
autonomousnervous system [2,3]. In various pathological
conditions the heart rhythm can also be influenced by
ectopy and re-entries. One approach that has been used to
Time
for primary
review
23 days.
J.K. Kanrers
et al./ Cardiouascular
identify the sources of heart rate variability is spectral
analysis of time series of consecutive R-R intervals [2].
Spectral analysis allows a distinction of physiological processes based on their characteristic time scales. Studies
using spectral analysis have identified frequency domains
influenced by the autonomic nervous system [2]. A highfrequency domain (above 0.15 Hz) is related to respiratory
activity (respiratory sinus arrhythmia), and it appears to be
caused mainly by variations in parasympathetic activity.
The low-frequency domain (from 0.04 to 0.15 Hz) is
mediated by both the sympathetic and the parasympathetic
nervous system, and reflects to a large extent regulatory
activity due to the baroreflexes [3].
Since simple measures of HRV like the standard deviation and the power spectrum are linear methods, they will
be inadequate for characterizing the dynamics, if HRV has
significant non-linear dynamics. There has been a
widespread effort to apply methods from non-linear dynamical systems theory to heart rate data. In a recent study
[4] of patients with episodes of ventricular tachycardia, it
was found that the pointwise correlation dimension (a
non-linear measure of complexity) could be used to distinguish patients who developed ventricular fibrillation during the monitoring period from those who did not. Using
the standard deviation as a measure of heart rate variability, it was not possible to discriminate between the two
groups [5].
Although there is preliminary evidence indicating that
methods developed for the characterization of non-linear
dynamical systems may be clinically useful in discriminating between patients with a high and a low risk of sudden
cardiac death, several issues need to be resolved before the
methods can be used routinely in clinical practice. One
issue is the stability and reproducibility of the methods.
Therefore one aim of this study was to investigate, in a
group of normal subjects, the day-to-day variation of two
commonly used statistical measures: the correlation dimension calculated by the Grassberger-Procaccia algorithm [6]
and the degree of non-linear predictability [7] in the time
series. A second issue is to determine how the various
measures of heart rate variability are related: i.e., are they
independent measures describing different aspects of the
dynamics, or are they redundant in the sense that they
describe the same phenomena. To test this we looked for
correlations among the different measures of heart rate
variability.
2. Methods
Twelve healthy subjects (6 female and 6 male healthcare workers) between 25 and 38 years of age (3 1.4 f 1.1
years, mean f s.e. (standard error of mean>) were examined on 3 different days, each time in the morning. None
of them received any medication. The study was approved
by the local review board, and the subjects provided
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31 (19%)
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401
informed consent. The subjects were placed in the supine
position in quiet surroundings to minimize external influences. They were instructed to stay awake during the
recording period. The ECG was recorded digitally for
approximately 3 h by a Custo Mega R6 ambulatory ECG
recorder (Gusto Med, Munich, Germany) with a sampling
rate of 500 Hz. The Custo Mega R6 ambulatory ECG
recorder allows on-line determination of the R-R intervals
using an autocorrelation technique for R-wave detection.
After the R-R interval is determined, the segment of the
digitally recorded ECG signal is compressed and stored
together with the R-R interval for later off-line retrieval.
After discarding the first 500 R-R intervals the next 8 192
consecutive intervals were used for further analysis. All
R-R intervals were manually reviewed, and premature
ventricular and supraventricular beats and artifacts were
removed. All subjects had less than 1 premature beat per
hour. The time series of 8192 R-R intervals were analyzed
using spectral analysis, correlation dimension, and non-linear predictability. In addition, the mean length and the
standard deviation of the R-R intervals were calculated
using standard procedures.
2.1. Spectral analysis
Spectral analysis was done using a Fast Fourier Transform (FFT). The power spectrum was calculated as the
square of the absolute values of the corresponding Fourier
transform [8]. No window was applied prior to the analysis. Since the time series consisted of consecutive R-R
intervals, the frequency has units of per beat (beat- ’ 1. The
average heart rate was approximately 60 beats per min,
and the unit beat - ’ is therefore approximately equal to Hz.
For simplicity, we will in the following use Hz as the unit
for frequency instead of beat- ‘. The area in the power
spectrum between 0.15 and 0.40 Hz was defined as the
high-frequency area (I-IF) and the area between 0.04 and
0.15 Hz was defined as the low-frequency area (LF). The
power in these areas was calculated by summing the
corresponding coefficients of the power spectrum.
2.2. Surrogate series
The application of the surrogate data technique [9] to
R-R intervals has previously been described in detail [lo].
The purpose of the surrogate data technique is to test the
significance of the results of the various algorithms (e.g.,
correlation dimension and non-linear predictability) when
applied to the experimental data sets. The main idea is to
construct a set of surrogate time series that have some
specific properties in common with the experimental time
series, and that can be attributed to a specific type of
process [9].
This is necessary since it has been shown [ 11,121 that
several of the algorithms used for the detection of chaos
may suggest the presence of chaos despite the fast that the
402
J.K. Kanters
et al. / Cardiovascular
time series is the result of a simple linear stochastic
process. To avoid this bias in the methods, one can
generate a set of surrogate time series that have the same
linear correlations as the experimental time series. The
algorithm can then be applied to both the experimental and
the surrogate series. Because one can construct as many
surrogate series as one pleases, the statistical significance
of the result obtained from the experimental series can be
calculated. If a significant difference between the experimental and the surrogate series is found, the null hypothesis that the results of the applied algorithm are due to
linear correlations in the data can be rejected.
A surrogate time series was generated by first fitting a
function h so that the original time series had a Gaussian
distribution. The transformed series was then Fouriertransformed using an FFI algorithm. The phase values of
the Fourier-transformed series were then replaced with
random numbers uniformly distributed on the interval 0 to
27r. An inverse Fourier transform was then performed, and
the resulting data series was transformed with the inverse
function, h - I. This process results in a surrogate time
series that has the same mean, standard deviation and
power spectrum as the original process, but where any
existing non-linear correlations in the original time series
(subsequent R-R intervals) have been destroyed. By repeating this process with new random values for the phase
of the Fourier-transformed series, an arbitrary number of
surrogate time series can be generated. In the present study
we generated 10 surrogate time series for each experimental time series.
To compare the results of the applied algorithms between the experimental and surrogate data series we calculated:
CT= ( Esurr- xexp)Psurr
where x,,r is the result (the calculated correlation dimension or the average prediction error) of the applied algorithm on the experimental data, and X,,, and SD,,, is the
mean and the standard deviation of the results (the calculated correlation dimensions or the average prediction errors) on the 10 surrogate data series, respectively. A value
of cr greater than 2 allows one to reject the null hypothesis
that the result is due to linear correlations within the
experimental time series, and is indicative of true non-linear dynamics in the data.
2.3. Correlation dimension
The correlation dimension of a time series from a
chaotic system describes the complexity of the time series
[6,10], and it can be thought of as a measure of the number
of independent variables necessary to describe the system.
The correlation dimensions were calculated using the algorithm described by Grassberger and Procaccia 161. To
calculate the correlation dimension, the state space of the
system was reconstructed using delay coordinates [13]. To
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31 11996)
400-409
reconstruct an n-dimensional state space, a series of n-dimensional vectors (I$> is constructed from the R-R
intervals as
m= (n-
l)r+
1,...,8192
(2)
where r represents a fixed delay (T= 1 in the present
study), and RR, is the length of the i’th R-R interval. The
integer n is the embedding dimension. In the present study
we used a value of 30, which was found optimal under our
experimental conditions [lo]. The correlation integral [ C( 1>]
can then be calculated from the constructed vectors as:
1
Number of vector pairs ( i , j) where i < j and
C(1) = ~
x
whose distance from each other is less than I I
N(N- 1)
(3)
where N is the total number of vectors. For a deterministic
system, we will have that for small values 1, C(l) is
proportional to lD, where D is the correlation dimension.
Therefore, when log[C(1)] is plotted against log(l), D can
be determined as the slope of this curve.
The correlation dimensions of both the experimental
and the surrogate time series were determined using identical procedures.
2.4. Non-linear predictability
In contrast to the correlation dimension, which as described above is a measure of the complexity of the time
series, non-linear predictability provides a direct test for
the presence of determinism in the data [7]. A time series
from a non-linear deterministic system will be predictable
because of the structure when viewed in the state space [7].
This order is destroyed in the surrogate series, and a
finding of a significantly better predictability in the experimental than in the surrogate time series indicates the
presence of non-linear determinism in the experimental
time series.
As described in detail elsewhere [lo], the test for nonlinear predictability is performed by splitting the data set
into two parts, each part 4096 points long. The first part is
the learning set, and it is used to construct the predictor.
The second half constitutes the test set, and it is used to
test the efficiency of the predictor. In a deterministic
system, two states (vectors) that are close to each other in
the state space evolve, at least initially, along nearly
identical paths. After reconstructing the state space using
delay vectors (see “Correlation dimension” above), we
identify the 5 vectors <I&~““, . . . ,I?ky)
in the learning
set that are closest to the vector, RRi
+ Iert, from the test set.
For each of the identified vectors in the learning set, the
first coordinate of the following
vectors (I@yl,.
.. ,
F&E”,“,) are all predictions of the length of the next R-R
J.K. Kanters
et al./ Cardiovascular
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31 (1996)
403
400-409
3. Results
interval. The final prediction of the next R-R interval is
calculated as the weighted average of these five predictions, where the weights are inversely proportional to the
distances between the test vector and the corresponding
vectors from the learning set.
The prediction error is the absolute value of the difference between the actual length of the next R-R interval
and the value predicted from the above procedure. The
average prediction error is calculated by averaging over all
4096 predictions made on the test set. In the present study
we used an embedding dimension of 3 when constructing
the vectors in the learning and the test sets. This has
previously been shown to give the best predictions (the
smallest value for the average prediction error) [lo].
Fig. 1 shows representative recordings of R-R intervals
in 3 subjects on 3 consecutive days. Inspection of the
recordings reveals a high reproducibility of the pattern of
heart rate variability within the same subject. On the other
hand, there are clear differences in the patterns between
subjects. This was a consistent finding in all 12 subjects.
Every subject appeared by simple inspection to have a
unique pattern in heart rate variability, and we could not
detect characteristic patterns that allowed a categorization
of the study population into subgroups having the same
type of dynamics.
Fig. 2 shows a typical return map, constructed by
plotting consecutive R-R intervals against each other. All
subjects had similar “comet-shaped” return maps. Thus,
the differences in the dynamic patterns apparent in the
original time series did not result in differences in the
shape of the return maps. The “comet shape” implies that
the variability between consecutive R-R intervals increases as the instantaneous heart rate decreases. Fig. 2
also shows a return map constructed from a surrogate time
series. The surrogate time series is designed to have the
same amplitude distribution, and the same power spectrum
(linear correlations) as the original time series. Any difference between the return maps will therefore be due to the
2.5. Statistics
The contributions to the total variation of a given
variable from the variation within and between subjects
were estimated by a one-way analysis of variance
(ANOVA). Correlations between variables were estimated
by standard linear correlation analysis. The relationships
between the various measures of heart rate variability were
determined using multiple linear regression with forward
stepwise variable selection [14]. A P-value less than 0.05
was considered significant.
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Fig. 1. Three representative recordings of R-R intervals. Each column represents one subject. Lower, middle, and upper panels are day 1, 2, and 3,
respectively. The x-axis is the number of the beat, and the y-axis is the length of the R-R interval in milliseconds (ms).
J.K. Kanters
404
et al. / Cardiovascular
Research
31 (1996)
400-409
1600
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Surrogatedataseries
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600
400
600
I
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400
600
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400
1600
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Fig. 2. Return map of consecutive R-R intervals of (left) n experimental and (right) a surrogate time series. The x-axis is the length of the first R-R
interval in milliseconds and the y-axis is the length of the following R-R interval, also in milliseconds.
presence of non-linear temporal correlations in the original
series. Evidently the shapes of the two return maps are
different. Indeed, in the surrogate time series, the variability between consecutive R-R intervals decreases as the
length of the R-R intervals increases. This is the opposite
of that observed in the experimental time series. This was
a consistent finding in all subjects, and with all the generated surrogate series.
Table 1 shows the results of the ANOVA on the various
measures of heart rate variability. As expected, the variation within the subjects (SD,,) was significantly lower
than the variation between subjects (SD,) with all measures. The within-subject variability is a measure of the
reproducibility of a given method, and the 95% confidence
interval for the difference between two consecutive measurements is given by kt(u!f)fi
X SD,, where t(df> is
the 97.5% value for the Student t-distribution with df
degrees of freedom [14]. Thus, the difference between two
Table 1
Mean and standard deviation (SD,,) for the entire data set (12 subjects
with 3 observations each)
MeanIt SD,,
Correlation
dimension
PE
9.6*
(ms)
R-R mean
=b
(ms)
HF
LF
(a.u.1
(au.)
(ms)
1.7
33.4* 11.2
922 f103
94.0* 17.9
6.8f4.4
11.8f6.1
SD,,
SD,,
1.2 '
2.5
6.4 *
35 *
17.6
177
27.5
7.0
9.8
11.0 *
2.4 *
2.4 *
SD,, denotes the standard deviation within subjects, and SDb, denotes
standard deviation between subjects, as calculated from the ANOVA.
* P < 0.01 compared to SD,,. PE = average prediction error, R-R,,,,
= mean length of the R-R interval; SD,, = the standard deviation of the
R-R intervals; HF = total power in the high-frequency range of me
power spectrum (0.15-0.40 Hz); LF = total power in the in the lowfrequency area of the power spectmm (0.04-0.15 Hz). ms = milliseconds.
a.u. = arbitrary units.
consecutive measurements has to be outside the 95%
confidence interval before the difference can be considered
statistically significant. Clearly, the narrower the confidence interval, the more sensitive the method will be with
regard to detecting changes in the dynamic characteristics
of the heart rate variability. Using the formula above, the
95% confidence intervals for the various measures can be
calculated to be: correlation dimension + 3.5; average
prediction error k 18.7 ms; SD,, f 32.1 ms.
The results of the surrogate data analysis are shown in
Table 2. The estimate for the correlation dimension in the
original time series was only slightly less than that in the
surrogate data, and the difference expressed in units of (+
was only borderline significant (a = 2 corresponds to P =
0.05 [9]). The small difference shows that the absolute
value of the correlation dimension in HRV is not describing the dimension of an underlying attractor, but is due
rather to the presence of linear correlations. The correlation dimension was correlated to the length of the mean
R-R interval, as shown in Fig. 3. The correlation was
weak, with a correlation coefficient of 0.62 (P < 0.05).
Thus, the correlation accounted for less than 40% of the
total variation in the correlation dimension. No significant
correlations with the other measures of variability were
Table 2
Mean and standard error of the correlation dimension, and the average
prediction error in the experimental and the surrogate time series.
Experimental
Surrogates
lJ
Correlation dimension
Prediction error
(ms)
9.6kO.3
10.3f0.4
2.1f0.8
33.4* 1.9
35.8*2.1
0.85 1.0
*
o (cf. JZq. 1) indicates the significance of the differences between the
experimental and the surrogate series.
* P<O.O5(u>2).
J.K. Kaniers
et al. / Cardiovascular
Research
405
31 (1996) 400-409
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1000
RR-intenral
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1100
1200
(msec)
Fig. 3. Correlation dimension of R-R intervals as a function of the average duration of the R-R intervals. The x-axis is the average length of the R-R
intervals in milliseconds. The y-axis is the correlation dimension.
nal time series than in the surrogates, but the difference did
not reach statistical significance. The prediction error was
correlated with several other measures, but a multiple
detected (mean heart rate, SD,,, LF, HF, correlation
dimension, and PE of the R-R intervals).
The prediction error was also slightly less in the origi2.OE+7
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Fig. 4. Prediction error of R-R intervals as a function of the power in the high-frequency (HF) band (0.15-0.40 Hz). The x-axis is the average prediction
error in milliseconds. The y-axis is the HF power in arbitrary units.
J.K. Kanters
406
et al./Cardiovascular
regression analysis using mean heart rate, SD,, , LF, HF,
and correlation dimension as independent variables revealed that a linear model which included overall HRV
expressed as SD,, and HF, provided an optimal model
with a coefficient of determination (R2> equal to 0.77. No
significant improvements were obtained by including any
of the remaining variables. Fig. 4 shows a scattergram of
corresponding values of the prediction error and the HF.
A major assumption when trying to predict the future
values of the R-R intervals is that the dynamics in the
learning set are similar to that in the test set: in other
words, that the time series is stationary. Inspection of the
data set suggested that the lack of significance in the
prediction error between original and surrogate time series
was due to some subjects actually having a lower prediction error in their surrogate series than in the original
series. As apparent from Fig. 1, there are in some cases
abrupt changes in the character of the dynamics, and it is
clear that a model created on the first half of such a time
series will give poor predictions when applied to the
second half of the time series. In contrast, the surrogate
series are by design stationary. To test this possibility, we
re-evaluated our data by excluding time series where the
standard deviation of the R-R intervals differed more than
25% between the learning set (the first 4096 R-R intervals) and the test set (the last 4096 R-R intervals). This
method is coarse and only excludes the time series having
the most pronounced non-stationaries. When predictability
was re-examined on this subset, the difference in prediction error between the original and the surrogate series was
increased to 2.8 + 0.8 (P < 0.05, n = 25) when expressed
as (+ units (Eq. 1).
To test the character of the observed non-stationarity,
we investigated further the time series shown in the lower
panel of the middle column in Fig. 1. This time series was
chosen because it consisted of 4 well-defined regions with
two types of dynamics, termed A and B in the following.
The A-type behavior was characterized by relatively short
R-R intervals and a low variability. It was presented in the
Research
u
Prediction
error (ms)
Overall
A, vs. A,
B, vs. B,
A,vs.B,
0.6
24.5
3.1
19.6
2.1
31.6
- 13.3
48.4
Overall indicates that the first 4096 R-R intervals are used for the
learning set, and that the last 4096 as the test set. A, vs. A, is the
prediction error when the A, segment (see text for details) is used as the
learning set, and the A, segment as the test set. B, vs. B, : B, isthe
learning set, and B, me test set. A I vs. B, : A, is the learning set, and B,
the test set. ms = milliseconds.
intervals A, (beats l-1024) and A, (beats 4200-5223).
The B-type had longer R-R intervals, and higher variability, and it was found in the intervals B , (beats 2788-3812)
and B, (beats 6488-7512). The return maps (Fig. 5) of the
two types of behavior showed the typical “comet-shaped”
pattern for the A behavior, but a more rounded shape for
the B behavior.
To test whether these intervals represented episodes
with different types of non-linear dynamics, we first calculated the prediction error within each ‘type of dynamics
using A i as the learning set and A, as the corresponding
test set, and similarly using B, as learning set and B, as
the test set. In both cases the prediction errors were
significantly smaller in the original data than in the corresponding surrogates (Table 3). However, when we tried to
predict the B behavior from the A behavior (A, as the
learning set and B, as the test set), it was impossible to
make meaningful predictions. The prediction error in the
original series was considerably larger than in the surrogate series, demonstrating that the two types of dynamics
are different. Note that the differences in the prediction
errors were modest, and it was only when used together
with the surrogate data that it was possible to distinguish
between different types of non-linear dynamics in the data.
1500
fi
2
1000
z
500
400-409
Table 3
tr values and absolute prediction errors for the dataset shown in the
lower panel of the middle column in Fig. 1
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:
31 (1996)
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500
I
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iooo
msec
I
1
1500
1000
I
500
’
I
1000
’
I
1500
msec
Fig. 5. Return map of successiveR-R intervals of (left) type A behavior and (right) type B behavior. The x-axis is the length of the first R-R interval ir
milliseconds and the y-axis is the length of the following R-R interval, also in milliseconds.
J.K. Kanters
et al. /Cardiovascular
4. Discussion
In the present study we tested the reproducibility of
both linear and non-linear measures of heart rate variability in a group of normal, healthy subjects. Although the
variability was significantly less within a given subject
than between subjects, there still was a considerable dayto-day variation within a given subject. Expressing the
day-to-day variability
as a coefficient of variation
(SD,,/mean)
it varied from 12 to 35% for the various
measures. It was lowest for the SD,, and the correlation
dimension, and highest for HF. This variability is surprising, considering that the study was designed to minimize
external influences by placing the subjects in quiet surroundings. In addition, physical activity was minimal,
since all the subjects remained in the supine position for
the duration of the recording. Finally, the recordings were
all made at the same time in the morning. The variabilities
in the numerical measures were also surprising, since
visual inspection of the recordings from an individual
subject revealed a considerable similarity in the overall
pattern of heart rate variability from day to day (see Fig.
1).
Knowledge of the day-to-day variability has direct implications for the clinical utility of the measures of heart
rate variability. For a given test to be useful to the clinician
in making either a diagnosis or a prognosis for a patient, it
should have a high sensitivity and specificity. With the
reproducibility found in the present study, the SD, or the
correlation dimension would have to change by at least
34% between two consecutive measurements before the
change can be considered significant. This is a substantial
change, and it explains why some studies have found that
while the group average of SD, may allow the identification of a high-risk group, it cannot be used to detect
individual subjects at high risk [l]. It therefore seems safe
to conclude that if only one or a few determinations are
made in a single patient, none of the present measures will
be useful to identify individuals at high risk for sudden
cardiac death. The sensitivity could be improved by continuously calculating the measures while the patients are
monitored. In this case, a change in a variable can be
detected with greater certainty due to the large number of
measurements. Since the various measures describe different properties of the signal (see below), it is also possible
that a combination of several measures can improve the
ability to detect patients at high risk for sudden cardiac
death.
The present study confirms our previous observation
that although the correlation dimension is significantly
larger in the surrogate series than in the original series, the
absolute value of the difference is modest [lo]. The correlation dimension found in the present study is similar to
that reported in previous studies [ 10,15,16]. The prediction
error was also greater in the surrogate series than in the
original series, but this difference only reached statistical
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407
significance when we excluded the time series with the
most pronounced non-stationarities. The similarities between the results obtained in the original and the surrogate
time series suggest that the results of the non-linear algorithms to a large extent are determined by the linear
properties of the time series, and does not directly reflect
the properties of a single “strange” attractor. A similar
conclusion was reached in two recent studies also using
surrogate data techniques on HRV in healthy human subjects under normal conditions [ 17,181. Methods like coarse
graining spectral analysis [19] and approximate entropy
[20] are alternative approaches to characterizing non-linear
dynamic systems. Although these methods have been applied to HRV, they have not been used in conjunction with
surrogate data, and it is therefore unknown whether they
are superior to the non-linear measures used in the present
study for capturing the non-linear characteristics of HRV.
However, although the correlation dimension to a large
extent is influenced by linear correlations within the heart
rate, it is not a redundant measure since it apparently
expresses properties of the signal that is not captured by
for example the SD,,. We only found a correlation between the mean heart rate (expressed as the average length
of the R-R intervals) and the correlation dimension. The
correlation was weak, and there was a considerable variation in the correlation dimension that could not be explained by the correlation with the mean heart rate. No
other significant correlations were found and, especially
important, there was no correlation between heart rate
variability measured as the SD, and the correlation dimension. Thus, although the correlation dimension to a
large extent is influenced by linear correlations within the
heart rate, it is not a redundant measure since it apparently
expresses properties of the signal that are not captured by,
for example, the SD,,.
Like the correlation dimension, the average prediction
error also appeared to reflect unique properties of the heart
rate variability. Using a multiple regression analysis, we
found two independent predictors of the average prediction
error; the SD,, and HF. It is not surprising that there
should be a correlation between the average prediction
error and SD,,. This is to be expected, since the larger the
variation of the individual R-R intervals, the larger the
absolute value of the prediction error will be. The independent correlation between the prediction error and HF is
more interesting. The HF power reflects to a large extent
the respiratory activity. This suggests that a large prediction error is associated with a pronounced respiratory sinus
arrhythmia, and indicates that the respiratory sinus arrhythmia is poorly described by the low-order models use in the
present study to predict the lengths of the R-R intervals.
Since breathing is influenced by higher nervous centers, it
appears likely that respiration involves a high-order component that will be difficult or impossible to capture with a
low-order dynamic model.
Although the results of the non-linear algorithms to a
408
J.K. Kanters
et al./Cardiovascular
large extent are determined by the linear properties of the
time series, the significant, albeit small, difference between
original and surrogate data indicates the presence of nonlinear dynamic structure within the original data. This
structure is visible when comparing the return maps of the
original and the surrogate series (Fig. 2). Even in this
2-dimensional plot it is easily seen that the structure is
very different: the original data have the largest variability
within the long R-R intervals, whereas the surrogates have
the variability concentrated in the region with the shorter
R-R intervals. This is direct evidence that a linear model
will not be able to describe adequately the dynamics
observed in heart rate variability.
In several of the investigated subjects, there were abrupt
changes in the character of the dynamics. This is exemplified by the recordings shown in Fig. 1. Within the same
subject, similar changes were observed on all 3 days. This
suggests that the transitions in the dynamics were intrinsic
to the regulatory system, and not the result of some
external perturbation. To test whether the transitions represented distinct dynamic states, we tested the predictability
on visually similar short subsets of data from one subject.
The predictability is especially useful in this situation,
since it requires fewer data points than the correlation
dimension. The non-linear predictability showed clear signs
of non-linear determinism when pairwise similar regions
were used in the analysis. This is an important observation,
since in the investigated subject the domains used to
construct the predictor (the learning sets) were seperated
from the domains (the test sets) where predictions were
made by a domain with a different type of dynamics. This
results indicates that each separate type of dynamics was
stable from period to period.
In a previous study [lo] we concluded that heart rate
variability was not due to a low-dimensional chaotic system. This conclusion was based on the use of surrogate
data. We suggested that an alternative model could be a
system where several “strange” attractors coexisted, and
where the system “jumped”
from one attractor to the
other, the jumps being elicited by small disturbances.
Since the jumps observed in the present study appear to be
between domains with distinct differences in the dynamics,
it is possible that these jumps represent a transition from
dynamics dominated by one type of attractor to dynamics
dominated by another attractor.
The suggestion that heart rate variability is governed by
a set of competing attractors is not incompatible with the
observation that the correlation dimension does not appear
to reflect the properties of an underlying “strange” attractor. The correlation dimension is designed to characterize
systems where the dynamics are dominated by a single
attractor [6]. A possibility could be to subdivide the time
series into subsets of uniform dynamics, and calculate the
correlation dimension on each subset. However, the algorithm requires a fairly large number of data points to give
meaningful results [21], and in many cases, the episodes
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31 (19961400-409
appeared to be of short duration (see Fig. 11, making it
impossible to obtain long time series suited for such an
analysis. Non-linear predictability is better suited for
shorter time series, and was useful in the present study to
identify subsets with similar types of non-linear determinism. But even this method requires time series of sufficient
length so that adequate learning and test sets may be
defined.
Heart failure has been shown to be associated with a
decrease in the correlation dimension [4]. Other studies
have shown that patients with severe heart failure and a
poor prognosis have return plots characterized by a tight
clustering of the data points [22]. The tendency to clustering is correlated with the concentration of norepinephrine
in the blood [23]. We speculate that these findings may be
due to a reduced tendency for the system to jump between
multiple attractors. This hypothesis clearly needs further
experimental investigation.
Although the present study has shown a fairly large
day-to-day variability in the various measures of heart rate
variability, it has also shown that the SD,, the correlation
dimension, and the average prediction error describe different properties of the signal. Future research should
address the question of whether combined use of several
different measures of heart rate variability improves the
identification of high-risk individuals.
In conclusion, the study confirms that the correlation
dimension of HRV is mostly due to linear correlations in
the R-R intervals. A small but significant part is due to
non-linear correlations between the R-R intervals. The
correlation dimension is weakly correlated to the mean
heart rate, but not to any of the other measures of heart
rate variability. It therefore measures properties distinct
from that of the conventional measure of heart rate variability, the SD,,. The prediction error was shown to
correlate with the respiratory activity in the I-IF-band of the
power spectrum. The overall pattern of heart rate variability was highly reproducible within a given subject. The
abrupt changes noted in heart rate variability appear to be
highly reproducible within a given subject, and it is hypothesized that the jumps may represent shifts between
different attractors for the system.
Acknowledgements
The present study was supported by grants from the
Danish Heart Foundation, Danish Medical Research Council (12-3 164-l), and the Novo-Nordisk Foundation. The
authors wish to thank Drs. 0. Skott and P.P. Leyssac for
helpful suggestions in the preparation of the manuscript.
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