Download Chaos-related deterministic regulation of heart rate variability in time

Gzrdiovascular
Research
ELSEVIER
Cardiovascular Research 31 (19%) 410-418
Chaos-related deterministic regulation of heart rate variability in timeand frequency domains: effects of autonomic blockade and exercise
Inger Hagerman a** , Margareta Berglund a, Mikael Lorin b, Jacek Nowak ‘, Christer Sylvh
a
a Departmenr
of Cardiology,
Karolinska
Institute, Huddinge Universily
Hospital, S-141 86 Huddinge,
Sweden
b Department
of Technical Assistance, Karolinska
Institure, Huddinge Hospital, S-141 86 Huddinge, Sweden
’ Department
of Clinical Physiology,
Karolinska
Institute, Huddinge University Hospital. S-141 86 Huddinge, Sweden
Received 27 October 1994; accepted 19 June 1995
Abstract
Objectives:
To study non-linear complexity or chaotic behaviour of heart rate in short time series and its dependence on autonomic
tone. Methods: Ten healthy individuals (5 men, mean age 44 years) were investigated at rest, after intravenous injections of propranolol
(0.15 mg/kg), followed by atropine (0.03 mg/kg). On another occasion, investigation was made during exercise on a bicycle ergometer
at 40% and at 70% of maximal working capacity. Heart rate variability was assessed by: local sensitive dependence on initial conditions
as quantitated by the dominant Lyapunov exponent, coefficient of variation of heart rate, power spectral analysis of high- and
low-frequency bands and the l/f-slope of the very-low-frequency
band and time domain analysis. Results: The approximate dominant
Lyapunov exponent was positive at rest and remained positive during autonomic blockade and during exercise. The exponent decreased
significantly with propranolol + atropine and even more soduringexercisebut did not attainzero. At baselineapproximatepredictability
was lost after about 30 s whereasafter autonomicblockadeor exerciseit was lost after about60 s. The l/f-slope remainedunaltered
around - 1. As expected, power in high- and low-frequency bands as well as time domain index decreased significantly with autonomic
blockade. The low-frequency band and time domain index were affected by exercise. Conclusions: Heart rate variability of sinus rhythm
in healthy individuals has characteristics suggestive of low-dimensional
chaos-like determinism which is modulated but not eliminated by
inhibition of autonomic tone or by exercise. The dominant Lyapunov exponent characterises heart rate variability independent or the other
investigated measures.
Keywords:
Heart rate variability; Exercise; Chaos; Non-linear phenomena; Autonomic nervous system
1. Introduction
Application of chaos theory to complex biological processesanalysis is based on concepts and models derived
from non-linear dynamics. One area of interest in cardiology is to identify and classify electrical instability which
may precede arrhythmias [l]. Heart rate response to a
change in parasympathetic efferent activity is extremely
rapid, occurring usually on a beat-to-beat basis [2]. The
change in heart rate in response to changes in efferent
sympathetic activity is a slower process [3]. These and
other oscillators result in a heart rate variability signal that
* Corresponding
7464120.
author.
Tel.:
(+ 46-8)
tXO3-6363/%/$15.00
0 1996 Elsevier
SSDI 0008-6363(95)00084-4
7464126;
fax:
is made up of componentswith differing time scalesand
corresponding differing frequency components 143. In a
mathematical model of vagally driven sinoatrial node,
Michaels et al. [S] found that irregular dynamics obeying
the rules derived from other chaotic systemswere present
during vagal stimulation of the sinus node. Application of
the same analytical tools to the analysis of simulation of
reflex vagal control of sinus rate suggestedthat chaotic
dynamics could be obtained in the physiologically relevant
case of the baroreceptor loop. As outlined by Denton [6],
the sinusrhythm is constituted by the sinoatrial (SA) node
controlled by multiple non-linear mechanisms such as
parasympathetictone, sympathetic tone, hormones,preload,
(+46-8)
Tie
Science B.V.
All tights
reserved
for primary
review
40 days.
I. Hagerman
et al./ Cardiovascular
afterload, most of which have a long feedback-loop compared with the basic cycle length - “a near perfect
substrate for the generation of chaos”. Fast Fourier transformation of heart rate reveals not only high- and lowfrequency peaks. The major spectral density is located in
the very-low-frequency band without any distinct peaks.
This density may be due to a complexity of oscillators
with overlapping time scales.
Depressed heart rate variability analysed in time and
frequency domains has been shown to predict sudden
death in patients with previous myocardial infarction [8].
Techniques derived from non-linear dynamics and chaos
theory may be of complementary value in identifying
patterns and mechanisms that are not detectable with traditional statistics based on linear models.
Deterministic chaos exhibits a number of characteristics
that distinguish it from periodic and random behaviour, in
particular by sensitive dependence on initial conditions,
which means that small changes in the state variables at
one point will create large differences in the behaviour of
the system at some future point. This manifests itself
graphically as adjacent trajectories that diverge widely
from their initial close positions. The Lyapunov exponent
is a quantitative measure of the average rate of this separation. A positive Lyapunov exponent indicates sensitive
dependence on initial conditions and thus loss of predictability, indicative of deterministic chaos [9].
The aim of this study was to find in healthy volunteers
a design applicable to clinical conditions to test heart rate
on the one hand for non-linear complexity or chaotic
behaviour and on the other hand for its dependence on
autonomic tone. Clearly, this requires an approximation of
an ideal condition since during clinical conditions the time
Research
411
31 11996) 410-418
of recording is limited due to both logistics and the status
of the patients.
2. Methods
Ten healthy individuals, 5 men and 5 women were,
after informed consent, enrolled for investigation. Median
age was 44 years (33-51), they were all healthy without a
history of cardiovascular disease, normotensive, in sinus
rhythm and without any medication. Measurements during
rest and with autonomic blockade were made between 9.00
and 11.OO in the morning and measurements during exercise were made between 1.00 and 3.00 in the afternoon.
2.1. Autonomic blockade
After a rest in the supine position for 10 min, recording
was done of 750 consecutive beats. Recordings were repeated 5 minutes after i.v. administration of propranolol
0.15 mg/kg over 120 s and again after atropine 0.03
mg/kg over 30 s. Thus, the last measurement was performed under dual autonomic inhibition [ 10,111.
2.2. Exercise stress test
Exercise test was performed on two separate occasions.
Using an ergometer bicycle with an automatic device for
continues load increase of 10 W/min (Siemens-Elema,
Sweden), the subjects were encouraged to continue work
until stopped by fatigue or dyspnoea, in order to determine
maximal working capacity. On a second occasion, more
than 24 h later, sampling was made at submaximal work
Table 1
Effects of varying constants on approximate dominant Lyapunov exponents (bits of information * s- ’ ) of heart rate at baseline and following propranolol
and atropine
No.
Grid
Min.
Max.
Evol.
Approximate dominant Lyapunov exponents
ANOVA
Rest
Propranolol
P *
Rromanolol+ atrooine
P *
1
2
3
4
5
6
7
8
9
10
11
12
3
3
3
3
6
6
6
6
10
10
10
10
1
1
3
3
1
1
3
3
1
1
3
3
2
2
2
2
2
2
2
2
2
2
2
2
10
15
10
15
10
15
10
15
10
15
10
15
0.68 f 0.03
0.47 f 0.04
0.56 f 0.02
0.41 f 0.02
0.69 f 0.03
0.46*0.04
0.54 f 0.02
0.40 f 0.02
0.67 f 0.03
0.46 f 0.03
0.53 f 0.02
0.38 f 0.03
0.76 f 0.03
0.52 f 0.03
0.56 f 0.02
0.40*0.02
0.75 f 0.03
0.5 1 f 0.03
0.55 f 0.02
0.40 f 0.03
0.75 f 0.03
0.50 f 0.03
0.54 f 0.02
0.41 *0.02
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
0.39
0.24
0.33
0.23
0.39
0.26
0.33
0.23
0.40
0.25
0.33
0.22
f
rt
f
f
+
f
f
f
f
f
*
*
0.07
0.05
0.05
0.04
0.07
0.05
0.05
0.04
0.07
0.05
0.05
0.04
<
<
<
<
<
<
<
<
<
<
<
<
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
<: 0.01
< 0.01
< 0.01
< 0.01
< 0.01
< 0.01
< 0.01
< 0.01
< 0.01
< 0.01
< 0.01
< 0.01
Nos. 1-12, varying constant combinations. Grid = grid resolution; Evol. = evolution time; Min. = minimum separation of replacement (%); Max. =
maximum separation of replacement (a).
Compared to rest.
P-values from analysis of variance for the 12 combinations are also given.
l
412
I. Hagerman
et al./ Cardiovuscular
Research
31 (1996)
410-418
loads of 40% and at 70% of maximum working capacity.
Prior to sampling of time series of 750 consecutive heart
beats, the subjects worked 6 min on each level in order to
attain steady-state conditions during sampling of the succeeding 750 consecutive heart beats [12,13].
2.3. Lyapunov exponent measurement of heart rate variability
The Lyapunov exponent for time series was calculated
according to Wolf [14,15] for 750 consecutive heart beats
at a sampling frequency of 20 kHz. Interbeat intervals
differing more than 30% from the mean interval of 5
preceding beats were manually edited and excluded from
the calculation. Excluded signals were mainly artefacts due
to missense of the T-wave. The computer program estimates the dominant Lyapunov exponent in a given time
series by averaging the exponential rate of divergence of
short segments of the delay reconstructed orbit (trajectory).
To minimise a systematic artefact, every time series was
evaluated with 12 different combinations of equation constants for grid resolution, evolution time, minimum and
maximum separation of time points. Time delay, embedding dimension, time step and orientation error were held
constant at 1, 3, 1 and 30, respectively (Table 1). Time
delay is the delay with which the return map is constructed
and time step the step taken between succeeding return
maps. The Lyapunov exponent is positive for chaotic
behaviour, zero or negative for periodic behaviour and
around zero when noise prevails.
2.4. Frequency domain measurements of heart rate variability
Frequency domain analysis was performed on 750 consecutive beats using a Holter ECG device (Daltec Biomedical). If extrasystoles appear, the preceding and succeeding
intervals are substituted through linear interpolation with
the following normal R-R. Two or more extrasystoles
generate a new calculation period. From the time series of
equidistant R-R values the power spectrum is calculated
through Fast Fourier transformation [16] within 2 frequency bands: high-frequency power (HF) 0.40-0.15 Hz
0
250
500
750
Heart Beat
Fig. 2. Example of approximate dominant Lyapunov exponent calculation
(bits of information * s- ‘1 derived from 750 heart beats.
and low-frequency power (LF) 0.15-0.04 Hz. As HF is
suggested as an index of cardiac parasympathetic activity
while LF reflects sympathetic activity with vagal modulation [4,17], these parameters were chosen to describe the
influence of autonomic blockade on heart rate variability.
Spectral data are expressed as a percentage of total power
and as the LF/HF ratio, which has been shown to be a
marker of changes in the sympatho-vagal balance [17].
2.5. I /f measurements of heart rate variability
For all time series a regression line relating the log of
spectral amplitude to the log of frequency was constructed.
This regression line is called a l/f X plot because of the
inverse relation between amplitude (or power) and frequency that character& spectra for heart rate variability
and other fractal processes [9,27]. The slope of the regression line equals the exponent x in the l/f”
plot. This
exponent can be used to characterise the overall distribution of frequencies and their amplitudes and has been
suggested to indicate the complexity of a process [ 18-201.
2.6. Time domain measurement of heart rate variability
Time domain analysis was performed on 750 consecutive beats using a Holter ECG device (Daltec Biomedical).
The heart rate variability is displayed as an interval histogram, with the interval on the horizontal axis and the
number of beats in each time increment on the vertical
axis. A triangular index is calculated as described by
Cripps and colleagues [21]. An index < 25 ms has been
associated with increased risk for sudden cardiac death in
patients with previous myocardial infarction [22].
2.7. Statistics and ethics
Fig. 1. ‘Three-dimensional representation of 750 R-R intervals (ms) from
one subject at baseline. The x-axis shows the R-R interval X, tbe y-axis
and z-axis the R-R intervals x - 1 and x - 2, respectively.
The study was approved by the local Ethics Committee.
All data are presented as mean f s.e.m. The data were
subjected to one-way analysis of variance (ANOVA) for
I. Hagerman
ef ul./Cardiovascular
Research
31 (1996)
413
410-418
Table 2
Effects of propranolol and atropine on heart rate variability
Propranolol
Rest
P’
Rropranolol +
atropine
P’
ANOVA
< 0.01
NS
86
+4
2.4 to.7
< 0.001
< 0.001
< 0.01
NS
NS
< 0.05
0.30*0.05
0.94 + 0.04
2.0 rt0.6
3
rto.7
2.0 *0.07
70
+7
< 0.01
NS
< 0.001
HR
(beats/min)
HR (var.%)
Approximate
Lyapunov
l/f
I-F (o/o)
LF (%)
LF/HF
Tl h-d
60
+3
2.4 f 1.8
71
+3
1.8 f 0.2
0.52f
0.03
-0.99f
0.04
7.0 rt 1.0
23
2 4.0
3.06*
0.06
200
*19
OSf
-0.73f
4.0
12
3.0
218
0.03
0.19
f 1.0
+ 2.0
* 0.06
f26
< 0.001
NS
NS
< 0.01
< 0.001
< 0.01
< 0.001
< 0.001
NS
< 0.001
< 0.0001
< 0.01
< 0.001
* Compared to rest.
HR = heart rate; Lyapunov exponent (bits of information * s-l), mean from 12 combinations of constants for 10 subjects; l/f=
distribution; HF = high frequency; LF = low frequency; TI = time domain index.
Values are given as mean f s.e.m. P-value or non-significance (NS) for ANOVA is given.
repeated measurements with identification of group differences by Fisher’s protected least significance test. Differences from zero was tested according to the normality
distribution. Statistical significance was accepted at P-values below 0.05.
similar discriminative capacity for changes in Lyapunov
exponents during autonomic blockade (Table 1). Small
variations in the magnitude of information due to the
selection of constants as made in this study were therefore
considered to be of minor importance. Consequently, all
12 constat combinations were used when computing mean
approximate Lyapunov exponents. Further, for each constant combination, the mean Lyapunov exponent was divided by the standard deviation. All ratios were statistically different from zero with mean values of 5.7 f 0.7,
9.6 + 2.8 and 3.9 f 0.7 for rest, propranolol and atropine.
3. Results
3.1. Lyapunov exponent -
role of constant selections
Fig. 1 shows a representative 3-dimensional representation of the cigar-shaped attractor generated by 750 beats of
one subject at baseline. The quantity of 750 beats was
found to generate a steady-state level of the dominant
Lyapunov exponent (Fig. 2). This was further evaluated by
calculating the Lyapunov exponent at baseline after 500,
625 and 750 beats, n = 10. ANOVA did not show any
non-homogeneity, thus further suggesting that a steady
state was obtained. ‘All 12 combinations of constants had
3.2. Autonomic blockade (Table 2)
Following propranolol and atropine heart rate, as expected, decreased and increased. The variability coefficient, however did not change for propranolol but for
atropine. The mean Lyapunov exponent that was about
+ 0.5 bits of information * s- ’ at baseline, increased after
Table 3
Effects of submaximal exercise at 40 and 70% of symptom-limited work load on heart rate variability
Rest
Exercise 40%
P’
Exercise 70%
HR (beats/min)
HR (WI%)
Lyapunov
l/f
HF (%)
LF (o/o)
LF/HF
TI (ms)
67
109
k4
1.8 f 0.2
0.56f
-0.90*
4.0 f
0.04
0.04
0.7
17.4 f 3
4.9 f 0.7
204
*I9
*4
2.3 dcO.4
0.45 f 0.05
-0.95*0.01
4.4
3
2
93
slope of l/f
< 0.001
146
NS
NS
3.2 f0.7
0.24kO.l
- 1.02 f 0.05
5.7 *3
3
+2
NS
f 1.5
NS
*1
l 0.01
< 0.001
< 0.001
+7
< 0.001
*4
I.1 f0.1
74
k6
P’
ANOVA
< 0.001
< 0.001
NS
NS
< 0.01
< 0.001
NS
NS
NS
NS
< 0.001
< 0.001
< 0.001
< 0.001
< 0.001
< 0.001
* Compared to rest.
HR = heart rate; Lyapunov exponent (bits of information * s- ’ ), mean from 12 combinations of constants for 10 subjects; 1/f = slope of l/f
distribution; HF = high frequency; LF = low frequency; Tl = time domain index.
Values are given as mean f s.e.m. P-value or non-significance (NS) for ANOVA is given.
414
I. Hagerman
et al./Cardiovascular
Research
31 (1996)
410-418
Table 4
Heart rate (beats/min) calculated for the first, middle and last 30 beats of
the 750 beats for the different interventions
05 0,5 -
Baseline
Propranolol
Propranolol
+ atropine
O-40,3 -
Baseline
40% exercise
70% exercise
0
20
Workload,
40
60
80
100
Begin
71*3.5
59 f 2.4
88 f 3.6
Middle
70*3.2
59rt2.3
87f-3.6
End
70&-3.4
59k2.3
86k3.5
ANOVA
NS
NS
0.0005
72 i- 2.9
llOf3.3
149f4.2
7ljI3.07
113f3.3
152 f 4.2
70+3.1
114*3.1
155rt4.4
NS
0.002
O.OGQl
Values are given as mean f s.e.m. f-value or non-significance (NS) for
ANOVA is given.
Watts
Fig. 3. Approximate dominant Lyapunov exponent (bits of information* SC’) at rest and at 40 and 70% of symptom-limited ergometer
exercise load. Values are given as mean f s.e.m.
propranolol whereas following propranolol + atropine it
decreased to almost half the value at baseline. With a
registration accuracy of 20 kHz corresponding to 14 bits,
this means that predictability is lost after about 30 and 60 s
at baseline and after propranolol + atropine, respectively.
I-IF% and LF% decreased similarly following propranolol
and with a further decrease following addition of atropine.
This means that the major spectral power after autonomic
blockade resides in the very-low-frequency band. The l/f
exponent, however, did not change and was around - 1 for
the 3 measurements. The TI index did not change with
propranolol whereas it decreased substantially after atropine.
3.3. Submaximal exercise stress test (Table 3)
Mean work loads at 40 and 70% submaximal exercise
were 74 f 7 and 132 Ifr 12 W/min, respectively. As expected, mean heart rate increased with increasing work
load. The variability coefficient was unchanged with increasing work load. The Lyapunov exponent decreased
curvilinearily (Fig. 3) but remained at 70% work load
different from zero (P < 0.01). The decrease was compa-
rable to that following autonomic blockade with propranolo1 + atropine. Contrary to following autonomic blockade the spectral HF% did not change from baseline whereas
the LF% decreased to a similar degree as after autonomic
blockade. The l/f exponent did not change with exercise.
Incidentally, the Lyapunov exponent and TI index during
both autonomic blockade and exercise showed similar
patterns of relative changes as opposed to the other indices
of heart rate variability.
3.4. Stationary of time series
For calculation of the Lyapunov exponent it is essential
that non-stationarity of the heart rate is kept to a minimum.
Table 4 shows that while at baseline no differences in
heart rate were observed, during the interventions heart
rate for the first 30, middle 30 and last 30 beats of the 750
beats showed minimal differences. The dominant Lyapunov exponent based on the first 375, the last 375 and the
total 750 beats for the different interventions did not differ
(Table 5). Also no difference was seen in the behaviour at
baseline and during the interventions. From Table 5 it is
also evident that the relative change of the Lyapunov
exponent following autonomic blockade and with exercise
was the same when calculations were based on only the
first 375, on only the last 375 or on the total 750 beats.
Table 5
Approximate dominant Lyapunov exponents (bits of information * s- ’ ) based on the first and last 375 beats and the total of 750 beats for the different
interventions
Begin-Middle
Middle-End
Total
ANOVA
Baseline
Propranolol
Propranolol
+ a&opine
0.574 f 0.04
0.564 f 0.03
0.419 f 0.06
0.535 f 0.04
0.626 f 0.04
0.410 It 0.05
0.521 f 0.03
0.552 f 0.03
0.299 f 0.05
NS
NS
NS
Baseline
40% exercise
70% exercise
0.614kO.05
0.500*0.06
0.329 f 0.05
0.585 f 0.04
0.513fO.08
0.336 f 0.07
0.555 f 0.04
0.445 f 0.07
0.243 f 0.05
NS
NS
NS
Values are given as mean f s.e.m. P-value or non-significance (NS) for ANOVA is given.
I. Hagerman
et al./Cardiovascular
4. Discussion
In healthy individuals, heart rate variability of sinus
rhythm has characteristics of chaos-like determinism, with
a positive Lyapunov exponent and l/f-like
broad-band
spectrum with an exponent of approximately - 1. This
conclusion is at variance with Kanters et al. [23] who
suggested a lack of evidence for low-dimensional chaos in
heart rate variability. They, however, did not calculate the
Lyapunov exponent. Lyapunov exponents have proven to
be the most useful dynamic diagnostic for chaos-like systems [14] and any system containing at least one positive
exponent is defined as being chaotic, with the magnitude
of the exponent reflecting the time scale on which system
dynamics being deterministic yet become unpredictable. A
non-linear deterministic
low-dimensional
chaotic behaviour describes that the process is deterministically and
homeostatically constrained within a certain range whereby
it cannot wander off into infinity like a random process [6].
These results are in keeping with Osaka et al. [24] who
calculated the correlation dimension on heart rate registered for 150 s before and after autonomic blockade.
Deterministic non-linear estimates of heart rate variability
can thus be determined with sufficient and meaningful
accuracy under clinical conditions. This is of considerable
interest as Skinner et al. [25] reported that a reduction in
correlation dimension precedes lethal arrhythmias by hours.
Development of diagnostic deterministic tools for the prediction of electrical instability and lethal arrhythmias may
therefore become a new field of exploration.
Do the Lyapunov exponent and the correlation dimension give different information as regards the complexity
of a process? The correlation dimension describes the
complex structure of the attractor approximating the fractal
dimension and thereby indicating the number of oscillators
on which the attractor is dependent. The Lyapunov exponent is a dynamic measure and describes the sensitive
dependence on initial conditions - the hallmark of a
deterministic chaotic process. Thus with the dominant
Lyapunov exponent as the analytical tool, information on
the importance of underlying oscillators has to be obtained
through experimental interventions. For example, in the
present experiments vagal withdrawal was shown to be a
more important determinant of the dominant Lyapunov
exponent than sympathetic tone.
The method for calculating dominant Lyapunov exponents in this study is developed for experimental data,
typically consisting of discrete measurements of a single
variable [ 141. The computer program creates a database of
delay-reconstructed data in a grid of a dimension defined
by the parameter “embedding dimension”, which was
held constant at 3, considered sufficient with regard to the
amount of data available [14,15]. If the embedding dimension chosen is too large, one can expect that noise in the
data will tend to decrease the density of points defining the
attractor. Noise is an infinite dimensional process which,
Research
31 (1996)
410-418
415
unlike the deterministic component of the data, fills each
available phase space dimension in a reconstructed attractor. By varying “grid resolution”, the data are stored in
different numbers of boxes. Typical values of grid resolution are in the range of 6 to 12 and, within reasonable
bounds, the choice of grid resolution affects the efficiency,
but not the accuracy of the exponent. In systems where the
mechanism for chaos is unknown, exponent stability should
be checked over different evolution times, affecting propagation times and replacement steps in the calculations. The
value should be kept small enough that orbital divergence
is monitored at least a few times per orbit, and sensitive
dependence does not pull the measuring points too far
apart. We chose evolution time to be 1 or 3 and obtained
exponent stability. Minimum separation at replacement is
the smallest length scale on which orbital divergence is
monitored. To reduce the effect of error bar at the location
of each phase space point, this parameter was held relatively small, not exceeding 2%. Minimum separation of
time points, if chosen incorrectly, could affect the exponent estimate, as could maximum separation of time points,
which defines the largest length scale on which orbital
divergence is being monitored. Wolf [14], has for 3- and
4-dimensional reconstructions found a value of lo-15% of
the range of time series values to be optimal. We tried
both, with some variation in exponent stability but with
maintained magnitude of positivity.
The effects of noise in the algorithm of Wolf, as in any
physical system, could partly be measurement noise. Samples obtained from different time series recorded at 20 kHz
were therefore analyzed with a resolution of 5 digits
(10-5), 3 digits (10m3> or 2 digits (lo-*), respectively,
giving a positive Lyapunov exponent of comparable size.
The algorithm was also tested with periodic data, derived
with both high and low resolution from the Verhulst
equation [6], giving a negative and stable Lyapunov exponent. Thus it appears that with the conditions used measurement noise is kept to a minimum and that the algorithm calculates both negative and positive values.
A matter of concern is whether the number of data
points is sufficient for this type of analysis. Correlation
dimension analysis suggests that heart rate in the basal
state has a fractal dimension between 2 and 3 [25]. This
and the use of an embedding dimension of 3 suggest the
use of least 1000 beats. Convergence of the dominant
Lyapunov exponent, however, was demonstrated within
750 data points and was still obtained, and to the same
numerical value, when changing embedding dimension and
time delay to 4 and 3, respectively. Goldberger [19]
achieved a broad-band l/f-like spectrum when sampling
heart rate for 600 beats. Osaka et al. [24] used 140 to 240
beats when calculating the correlation dimension. Thus,
from an empirical point of view 750 beats appear to be
sufficient to calculate an approximate but informative
dominant Lyapunov exponent. A minimum for a reliable
estimate should be 50 to 100 orbits of the attractor with at
416
I. Hagerman
et al./Cardiouascular
least a few points per orbit. By eye-balling we found at
baseline that the average orbit was about 4 beats, roughly
corresponding to the respiratory cycle. Thus we have
studied about 180 orbits with an approximate loss of about
2 bits of information*s-‘.
Another concern is the degree on non-stationarity of the
data [26]. Our registrations were made during stationarity
or near stationarity. Although ANOVA shows that during
the interventions there were small degrees of non-stationarity, it was minimal. It is difficult to see how such nonstationarity could have been avoided. For example, following propranolol + atropine, the average heart rate was 2
beats/min lower at the end of the registration period than
at the start of the registration period. Thus the propranolol
+ atropine effect appears to be slowly eliminated towards
the end of the registration period. To make a more accurate
design would involve infusion of propranolol + atropine
over longer periods of time. Such an approach would
certainly become a major development of the method as
the amount of propranolol + atropine needs to be individualised. Probably the whole experiment would have been
difficult to pursue even in healthy subjects. From this
observation it is also apparent that a longer period of
registration after administration of propranolol + atropine
would involve a larger degree of artefact as the propranolo1 + atropine effect becomes less apparent with time.
The same kind of observation and reasoning holds for the
approximate stationarity observed during submaximal exercise. We thus conclude that we have studied the approximate steady state with a degree of accuracy that is attainable during clinical conditions.
The next question is whether our results are not the
result of non-stationarities. We certainly agree that this is a
significant problem. Especially when the degree of complexity or chaos is estimated as a static property like the
fractal dimension or correlation dimension. With such a
method a changing time series is likely to alter the clustering of data. A dynamic estimate such as the Lyapunov
exponent that with frequent resettings estimates the average rate of orbital divergence should be less sensitive to
non-stationarity and thus perhaps be more suitable for
clinical time series. In any case the positive dominant
Lyapunov exponents that we obtained could not have been
due to the slight non-stationarity we had in our time series.
As the non-stationarity has a trend, such a trend should
decrease the Lyapunov exponent or result in a Lyapunov
exponent not different from zero or a negative Lyapunov
exponent. Following propranolol + atropine, however, the
non-stationarity was a fading propranolol + atropine effect
which if anything should have increased the Lyapunov
exponent.
In the heart, the self-similar fractal branching pattern
has been observed in the right atria1 musculature, in the
cordae tendinae and in the His-Purkinje conduction network [27]. The fractal nature of the His-Purkinje cells
results in a frequency spectrum of the QRS complex,
Research
31 (1996)
410-418
which takes on the form of an inverse power law. This
predicted inverse power law relationship is borne out by
Fourier analysis of the normal QRS waveform. Goldberger
also examined in healthy subjects the variations in heart
rate intervals [27,28] which were found to be substantial.
This variability exhibits an inverse power-law relationship
qualitatively similar to that in the QRS waveform. In a
number of physiological parameters the slope of the line is
believed to be a useful index of spectral reserve, and
therefore healthiness [20]. The exponent in the case of
heart rate variability is reported to be approximately - 1
[27], and this was also obtained in our study. The dominant
Lyapunov exponent decreased with atropine. During exercise the dominant Lyapunov exponent approached but did
not attain zero or a negative value. Also the l/f exponent
remained unaltered around - 1, describing unaltered high
spectral power in the very-low-frequency band. If random
noise had influenced the dominant Lyapunov exponent to
be close to zero, the l/f dimension would probably not
have been stable around - 1. Although we did not observe
any reactivity in the l/f slope, it can be modulated.
Wagner and Persson reported that in a canine model the
l/f
slope of arterial blood pressure sure time series
changed after baroreceptor denervation and blockade of
the autonomic nervous system [29]. Thus even during
physical stress with twice as high a heart rate as during
rest, heart rate is determined by deterministic chaos and a
strange attractor. From rest the dominant Lyapunov exponent decreased curvilinearily and by extrapolation it was
hypothesised to attain zero and thus a stable periodic
behaviour around 90% of symptom-limited exercise capacity. Therefore, in three of the volunteers an attempt was
made to reach steady-state conditions at this level of
exercise capacity but appeared not to be attainable. These
results are in keeping with Nakamura [30], who in healthy
individuals performed prolonged exercise under quasisteady-state conditions until exhaustion and showed
changes in l/f”
slope and derived fractal dimensions
indicative of less complexity in the system, a change
towards stable, periodic behaviour although such behaviour was not attained.
The dimension of the dominant Lyapunov exponent was
affected by pharmacological autonomic blockade, and during exercise probably due to release of vagal withdrawal.
As described by Goldberger [31], the fluctuations in the
normal heart beat on short and longer time scales are
related to the competing influence of the two branches of
the autonomic nervous system, which appear to interact in
a non-linear manner, under healthy conditions [32]. The
decrease in spectral power of both HF and LF bands with
autonomic blockade, and particularly when cardiac vagal
activity was reduced, is consistent with studies by Akselrod and others [4,17,33] as is the decrease in the time
domain index [33]. Montano et al. [34] have recently
evaluated LF and I-IF components in absolute units and in
normalized units, indicating that normalized values of both
I. Hagerman
et al./Cardiovascular
LF and HF components may have the highest degree of
correlation to changes in sympathovagal balance, although
LF and I-IF expressed as percent values of total power also
did correlate significantly. The effect of propranolol was
more pronounced in the LF band, which reflects that both
the sympathetic and the parasympathetic systems mediate
low-frequency fluctuations, while parasympathetic activity
dominates at higher frequencies. The parallel shift in frequency and time domain indexes and Lyapunov exponent
supports a major autonomic influence over non-linear behaviour of heart rate variability, as previously suggested
both from a theoretical [S] and an empirical standpoint
[6,31]. Parasympathetic activity, however, seems to be a
more important determinant than sympathetic activity.
During moderate exercise the power in the LF band
decreased and remained at the same level during submaximal exercise, despite increased sympathetic drive, reflected
by the marked increase in heart rate. This might be due to
the fact that spectral power in the LF band depends not
only on the activity of the sympathetic nervous system but
also on the baroreceptor cardiac reflex. It has been reported that in moderate to severe exercise, the sympathetic
activity is elevated [35] but that the baroreceptor control of
the heart is inhibited [36] and the LF oscillations decrease
[37,38]. Somewhat conflicting, the power in the HF band
did not change significantly during exercise as one might
expect, at least during moderate exercise with reduced
vagal activity. However, similar results were reported by
Perini et al. [38] when expressing the power of each
component as a percentage of the total spectral power.
We have demonstrated that autonomic blockade and
exercise modulates chaos-like determinism of heart rate
regulation and variability, but does not, at least under the
present experimental conditions, transfer heart rate regulation to a stable periodic attractor. Two major determinants
appear to be parasympathetic activity and to a lesser
degree sympathetic activity. The reactivity of the Lyapunov exponent to the two investigated interventions was
similar to that of the TI index, but different from the other
established measures of heart rate variability such as the
coefficient of variation, high- and low-frequency spectral
power, the low-frequency to high-frequency ratio and the
I/f exponent. Therefore the Lyapunov exponent might be
an independent and deterministic characteristic of heart
rate regulation and its integrity.
Acknowledgements
This study was supported by grants from the Karolinska
Institute, ELFA Research Foundation and the Swedish
Heart and Lung Foundation.
References
[l] Chialvo DR. Low dimensional chaos in cardiac tissue. Nature
1990;343:653.
Research
31 (19%)
410-418
417
[2] Rosenblueth A, Simeone FA. The interrelations of vagal and accelerator effects on the cardiac rate. Am J Phvsiol 1934;l l&42-55.
[31 Appel ML, Berger RD. Saul JP, Smith SM, Cohen R. Beat to beat
variability in cardiovascular variables: noise or music? J Am Coil
Cardiol 1989;14:1139-1148.
t41 Akselrod S, Gordon D, Ubel FA, Shannon DC, Barger AC, Cohen
RJ. Power spectrum analysis of heart rate fluctuations: a quantitative
probe of beat-to-beat cardiovascular control. Science 1981;213:220222.
El Michaels DC, Chialvo DR. Matyas EP, Jahfe J. Chaotic activity in a
mathematical model of the vagally driven sinoatrial node. Circ Res
1989;65:1350-1360.
b1 Denton TA, Diamond GA, Helfant RH, Khan S, Karagueuzian H.
Fascinating rhythm: a primer on chaos theory and its application to
cardiology. Am Heart J 1990,120:1419-1440.
[71 Kobayashi M, Musha T. l/f fluctuations of heart beat period. IEEE
Trans Biomed Eng 1982;29:456.
181Bigger JT, Fleiss JL, Steinman R, Rolnitzky LM, Kleiger RE,
Rottman JN. Frequency domain measures of heart period variability
and mortality after myocardial infarction. Circulation 1992;85: 164171.
I91 Ruelle D. Sensitive dependence on initial condition and turbulent
behavior of dynamical systems. Ann NY Acad Sci 1979;316:408416.
[lOI Jose AD. Effect of combined sympathetic and pamsympathetic
blockade on heart rate and cardiac function in man. Am J Cardiol
1%6:18:476.
[III Jose AD, Collision D. Autonomic blockade by propranolol and
atropine to study intrinsic myocardial function in man. J Clin Invest
1%9;48:219.
t121 Jose AD, Stitt F, Collison D. The effects of exercise and changes in
body temperature on the intrinsic heart rate in man. Am Heart J
1970;79:488.
[I31 &.trom H, Jonsson B. Design of exercise test, with special reference
to heart patients. Br Heart J 1976.38289.
[141 Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov
exponent from a time series. Physica 1985;16D:285-317.
iI51 Wolf A. Quantifying chaos with Lyapunov exponents. In: Holden
AV, ed. Chaos. Princeton: Princeton University Press, 1986;273290.
[I61 Cain ME, Ambros D, Witkowski FX, Sobel BE. Fast-Fourier transform analysis of signal-averaged electrocardiograms for the identiflcation of patients prone to sustained ventricular tachycardia. Circulation 1984;69:71 l-720.
[I71 Pagani M, Lombardi F, Guzzetti S, et al. Power spectral analysis of
heart rate and arterial pressure variabilities as a marker of
sympatho-vagal interaction in man and conscious dog. Circ Res
1986;59:178-193.
b31 Lipsitz LA, Mietus J, Moody GB, Goldberger AL. Spectral characteristics of heart rate variability before and during postural tilt.
Circulation 1990,81(6):1803-1810.
t191 Goldberger AL, Rigney DR. Nonlinear dynamics at the bedside. In:
Glass L, Hunter P, McCulloch A, eds. Theory of Heart. Biomechanits, Biophysics and Nonlinear Dynamics of Cardiac Function. New
York Springer Verlag, 1991;583-605.
DO1Eberhart RC. Chaos theory for the biomedical engineer. IEEE Eng
Med Biol Mag 1989;8:41-45.
Dll Cripps TR. Malik M. Fare11 TG, Camm AJ. Prognostic value of
reduced heart rate variability after myocardial infarction: clinical
evaluation of a new analysis method. Br Heart J 1991;65:14-19.
[221 Kleiger RE, Miller JP, Bigger JT, Jr, Moss A. Decreased heart rate
variability and its association with increased mortality after acute
myocardial infarction. Am J Cardiol 1987;59:256-262.
t231Kanters JK, Holstein-Rathlou N-H, Agner E. Lack of evidence for
low-dimensional chaos in heart rate variability. J Cardiovasc Electrophysiol 1994;5:591-601.
[%I Osaka M, Saitoh H, Atarashi H, Hayakawa H. Correlation dimen-
418
[25]
[26]
[27]
[28]
[29]
[30]
[31]
I. Hagennan
et al./Cardiovascular
sion of heart rate variability: a new index of human autonomic
function. Front Med Biol Eng 1991;5:289-300.
Skinner JE, Pratt CM, Vybiral T. A reduction in the correlation
dimension of heartbeat intervals precedes imminent ventricular tibrillation in human subjects. Am Heart J 1992.125731-743.
Elbert T, Ray WJ, Kowalik ZJ, Skinner JE, Graf KE, Birbaumer N.
Chaos and physiology: deterministic chaos in excitable cell assemblies. Am Physiol Sot 199474: l-47.
Goldberger AL, West BJ. Fractals in physiology and medicine. Yale
J Biol Med 1987;60:421-43.5.
Goldberger AL, Fmdley LJ, Blackbum MR. Mandell AI. Nonlinear
dynamics in heart failure: implications of long-wavelength cardiopulmonary oscillations. Am Heart J 1984; 107:612-615.
Wagner CD, Persson PB. Two ranges in blood pressure power
spectrum with different l/f characteristics. Am J Physiol
1994;267:H449-H454.
Nakamura Y, Yamamoto Y, Muraoka I. Autonomic control of heart
rate during physical exercise and fractal dimension of heart rate
variability. J Appl Physiol 1993;74:875-881.
Goldberger AL. Fractal electrodynamics of the heartbeat. Ann NY
Acad Sci 1990,591:402-409.
Research
31 (19%)
410-418
[32] Goldberger AL. Is the normal heart beat chaotic or homeostatic?
News Physiol Sci Am Physiol Sot 1991;6:87-91.
[33] Hayano J, Sakakibara Y, Yamada A, et al. Accuracy of assessment
of cardiac vagal tone by heart rate variability in normal subjects. Am
J Cardiol 1991;67:199-204.
[34] Montana N, Ruscone T, Porta A, Lombardi F, Pagani M, Malliani
A. Power spectrum analysis of heart rate variability to assess the
changes in sympathovagal balance during graded orthostatic tilt.
Circulation 199490: 1826- 183 1.
1351 Ekblom B, Goldbarg A, Kilbom A, ,&strand P-O. Effect of atropine
and propranolol on the oxygen transport system during exercise in
man. Sand J Clin Lab Invest 1972;30:35-42.
[361 Bristow JD, Brown EB, Cunningham DJC, et al. Effect of bicycling
on the baroreflex regulation of pulse interval. Circ Res
1971;28:582-591.
[371 Arai Y, Saul JP, Albrecht P, et al. Modulation of cardiac autonomic
activity during and immediately after exercise. Am J Physiol
1989;256:H13233141.
[38] Perini R, Grixio C, Baselli G, Cerutti S, Veicsteinas A. The influence of exercise intensity on the power spectrum of heart rate
variability. Eur J Appl Physiol 1990,61:143-148.