Vol. 275, Issue 4, H1419-H1433, October 1998
Mutual information discloses relationship between hemodynamic
variables in artificial heart-implanted dogs
Motohisa
Osaka,
Tomoyuki
Yambe,
Hirokazu
Saitoh,
Makoto
Yoshizawa,
Takashi
Itoh,
Shin-Ichi
Nitta,
Hiroshi
Kishida, and
Hirokazu
Hayakawa
First Department of Internal Medicine and Center of Informatics
Science, Nippon Medical School, Tokyo 113-8603; Department of
Medical Engineering and Cardiology, Division of Organ Pathology,
Institute of Development, Aging, and Cancer, and Department of
Electrical Engineering, Faculty of Engineering, Tohoku University,
Sendai 980-8575, Japan
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ABSTRACT |
A mutual information (MI) method for
assessment of the relationship between hemodynamic variables was
proposed and applied to the analysis of heart rate (HR), arterial blood
pressure (BP), and renal sympathetic nerve activity (RSNA) in
artificial heart-implanted dogs to quantify correlation between these
parameters. MI measures the nonlinear as well as linear dependence of
two variables. Simulation studies revealed that this MI technique
furnishes mathematical features well suited to the investigation of
nonlinear dynamics such as the cardiovascular system and can quantify a
relationship between two parameters. To constitute a model free of the
natural heart, two pneumatically actuated ventricular assist devices
were implanted as biventricular bypasses in acute canine experiments. RSNA was detected with the use of bipolar electrodes attached to the
renal sympathetic nerve. Analysis of data during control revealed that
correlation between HR and RSNA was higher than that between HR and BP
and that between RSNA and BP (P < 0.05). Although RSNA seemed to fluctuate noncorrelatedly with BP in
higher pacing rates, the MI values between them disclosed their strong correlation. Surprisingly, correlation between RSNA and BP was stronger
during a pacing rate of 60 beats/min than during higher pacing rates
and control (P < 0.05). It is
suggested that the baroreflex system may be susceptible to pacing rates
during the total artificial heart state. We calculated the time delay
between HR and RSNA, between RSNA and BP, and between HR and BP by
regarding a time delay at which the maximum MI value between each pair
of parameters was given as a physiological delay. Our results indicate that RSNA leads BP, BP leads HR, and RSNA leads HR during control (P < 0.05). We conclude that this
method could provide a powerful means for measuring correlation of
physiological variables.
cardiovascular regulation; autonomic nervous system; nonlinear
dynamics; spectral analysis; computer simulation
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INTRODUCTION |
CONVENTIONAL ANALYSES of the correlation of two time
series, such as cross-correlation function in the time domain and
coherency in the frequency domain, have been widely used (2, 9, 10, 28,
30, 31). For linear systems, the coherency function C(
) between
x(t)
and
y(t)
can be interpreted as the fractional portion of the mean-square value
at
y(t)
that is contributed by x(t)
at a frequency . Conversely, the quantity 1 
C() is a measure of the mean-square
value of
y(t)
when
x(t)
is not accounted for at the frequency (Ref. 7, p. 172). However, it
does not provide any information about whether they are nonlinearly
correlated. Hence, these techniques are inappropriate to assess
nonlinear dynamics. On the other hand, it has been demonstrated that
the cardiovascular system comprises nonlinear dynamics from micro to
macro in scope (5, 17, 19, 25). In particular, beat-to-beat heart rate
variability has been proven to contain an appreciable amount of fractal
components in the time series (11, 36). Hence, a new measure is needed
to quantify the correlation between various hemodynamic parameters.
Mutual information (MI), to be defined precisely later, can measure the
nonlinear as well as linear dependence of two variables. This method
has been proposed to provide a good criterion for the choice of time
delay in phase-portrait reconstruction from time series by gauging
correlation between an original series and its delayed series (15). MI
can provide a quantitative characterization of chaotic spatial
patterns. Moreover, MI analysis is also applicable to the detection of
nonlinearity of time series (29). We previously applied MI to the
assessment of patterns of occurrence of ventricular premature beats and
reported that the method can quantify randomness of their occurrence
(26). Thus we think this method could be applied to quantify the
relationship between various hemodynamic parameters.
The power spectrum analysis of heart rate (HR) or arterial blood
pressure (BP) variability has been widely recognized as a useful means
for evaluating the autonomic balance noninvasively (2, 3, 6, 9, 10, 23,
25, 27, 28, 30-32). The low-frequency spectra have been generally
thought to reflect the sympathetic activity with the parasympathetic
modulation and the high-frequency spectra to reflect the
parasympathetic activity modulated by respiration. However, a direct
relationship between sympathetic activity and either HR or BP in the
low-frequency spectra has been controversial recently (1, 6, 10, 28). One of the reasons for this is that determination of the origin of the
rhythmic fluctuations in the circulatory system is very difficult
because there are some interactions between hemodynamic parameters. For
example, BP influences HR through the baroreflex, which combines the
effects of the afferent signal from BP to the vasomotor center and the
efferent signals of the sympathetic and parasympathetic nerves to the
sinus node. HR affects BP through the mechanical coupling between the
left ventricle and the vasculature. Thus we thought that an open-loop
identification analysis might be a potential choice for investigating
the origin of these fluctuations. The use of a totally artificial heart
should allow an open-loop analysis of the cardiovascular system with
respect to the heart. Moreover, the artificial heart, including
ventricular assist devices, which can return patients with congestive
heart failure to society, will be in great demand. Although a number of
control algorithms and strategies for driving the artificial heart have
been described during the last two decades, current control methods are
based on information such as hemodynamics, oxygen utilization, and
hormonal factors (4, 21, 22). However, reports on evaluation of the
autonomic nervous system in the artificial heart are still sparse. It
is obvious that the circulatory control system including the baroreflex
system must be carefully investigated to determine the optimal driving
conditions for the artificial heart from a neurophysiological point of
view.
We sought to refine the MI analysis to assess the relationship among
HR, renal sympathetic nerve activity (RSNA), and BP. The purpose of
this study was to propose and test a new method based on MI that could
be applied to hemodynamic data and that could assess alterations in
neurocardiovascular interactions in a variety of physiological
conditions. To test our new technique, we first examined its validity
with simulated data and then evaluated its reliability with data from
dogs during control and during the driven complete artificial
circulation.
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METHODS |
General Procedure
The experimental protocol was approved by the Tohoku University
Committee on Animal Care. Seven adult mongrel dogs of both sexes,
weighing 15-35 kg, were anesthetized by intravenous thiopental sodium (2.5 mg/kg) and ketamine sodium (5.0 mg/kg) administration and
nitrous oxide inhalation under mechanical ventilation. Experimental preparations are shown in Fig. 1. After
tracheal tube intubation, the animals were placed on a volume-limited
respirator (ARF-850, Acoma, Tokyo, Japan). Electrodes for
electrocardiograms were implanted in the left foreleg and both hind
legs.
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Fig. 1.
Schematic drawing of the experimental system. Arrow labeled
"clamp" indicates site where descending aorta was clamped to
identify the observed signal as sympathetic nerve discharge. A/D,
analog to digital; AHF, artificial heart flow; AoP, aortic pressure;
LAP, left atrial pressure; main-Amp, main amplifier; pre-Amp,
preamplifier; PAP, pulmonary artery pressure; RAP, right atrial
pressure; R-C, resistance-capacitance.
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After the left pleural cavity was opened through the fifth intercostal
space, the pericardium was partially incised to expose the left atrium.
Aortic pressure and left atrial pressure (LAP) were monitored
continuously by catheters inserted into the aorta and left atrium
through the left femoral artery and the left appendage, respectively.
For left artificial heart implantation, the intercostal arteries were
dissected to free the descending aorta. A polyvinyl chloride outflow
cannula with a T-shaped pipe was inserted into the descending aorta and
secured with a ligature. An inflow cannula was inserted into the left
atrium through the left atrial appendage. Both cannulas were connected
to the authors' TH-7B pneumatically driven sac-type blood pump via the
built-in valve connectors (37, 38). An outflow cannula was inserted
into the pulmonary artery, and an inflow cannula was inserted into the
right atrium through the right atrial appendage. Both cannulas were
connected to the right pump. After the driving of both pumps was
initiated, ventricular fibrillation was electrically induced. After
ventricular fibrillation was induced, we observed that the
pulse-synchronous discharges in the RSNA did not change their
periodicity and quantity significantly. With respect to the influence
of the fibrillated heart on RSNA, Toda et al. (34) reported that the
fibrillated heart had little influence on the RSNA and was presumably
not essential for maintaining nervous control of circulation as long as
the circulation was maintained by an artificial heart.
The pumps were driven by a pneumatic drive console that we developed
(24). The positive and negative pressure and systolic duration of both blood pumps were selected to maintain hemodynamic derivatives within the control range: mean aortic pressure was maintained within 60-140 mmHg, mean LAP within 5-12 mmHg,
mean pulmonary artery pressure <30 mmHg, and right atrial pressure <5 mmHg (Fig. 1). Pump output was controlled to maintain a cardiac output similar to that during the control state.
Recording of Nerve Activity
The left flank was opened between the iliac crest and the
costovertebral angle, and the left renal artery was then exposed. The
left renal sympathetic nerve bundle was separated from the left renal
artery and surrounding connective tissues. After the nerve sheath was
removed, the nerve was placed on bipolar stainless steel electrodes for
recording of RSNA. The renal sympathetic nerve discharges were led into
a preamplifier with a gain of 1,000 and an amplifier with a gain of 50 (DPA-21 and DPA-11E, DIA Medical System, Tokyo, Japan). Amplified
signals were routed through a band-pass filter with a bandwidth of
30-3,000 Hz and through an amplitude discriminator to a storage
oscilloscope. For analysis, the filtered neurogram was integrated by a
resistance-capacitance circuit with a time constant of 0.1 s.
To identify the observed signal as the sympathetic nerve discharge, we
transiently clamped the descending aorta above the level of the renal
arteries, as indicated in Fig. 1. During the clamp RSNA decreased
rapidly because BP increased at the aortic arch (Fig.
2). However, the reason why BP decreased
abruptly in Fig. 2 is that a catheter to measure BP was left below the
clamping site. We confirmed that the respiratory rhythm of RSNA was
changed significantly in accordance with the changes in the respiratory rhythm of the mechanical ventilator (14). Figure
3 shows the recording from the natural
heart state to a stationary state of biventricular bypass circulation
after electrical induction of ventricular fibrillation. Careful
inspection of Fig. 3 suggests a reciprocal relationship between RSNA
and BP; that is, RSNA bursts occurred primarily when BP was falling,
whereas there was little RSNA when BP was increasing. When the nerve
discharge signal was significantly contaminated by environmental
electromagnetic noise, the animal was covered with wire netting to
avoid production of the noise.
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Fig. 2.
Identification of sympathetic nerve discharge by clamping descending
aorta (des Ao) transiently above level of renal arteries, as indicated
in Fig. 1. During clamp, renal sympathetic nerve activity (RSNA)
decreased because arterial blood pressure (BP) increased at aortic
arch. The reason for the abrupt decrease in BP is that a catheter for
measuring BP was left below the clamping site.
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Fig. 3.
Recording of electrocardiogram (ECG), BP, RSNA, and integrated RSNA.
After electrical induction of ventricular fibrillation, driving of both
artificial pumps was initiated. It took ~1 min to adjust driving
condition until cardiac output was maintained.
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Data Analysis
The signals of hemodynamic parameters and sympathetic nerve activity
were recorded with an ink-jet recorder and on magnetic tape after the
stabilization of all hemodynamic derivatives (almost 20-30 min
after the preparation). After the control data were recorded,
ventricular fibrillation was induced electrically. Time series were
recorded in a stationary state of the hemodynamics during the different
pacing rates from 60 to 160 beats/min. These data were played back from
the magnetic tape.
The occurrence of R waves was detected in each record, and a smoothed
instantaneous HR time series was constructed and sampled at 8 Hz using
the algorithm proposed by Berger et al. (8). All R-R intervals during
control were measured at an accuracy of 1 ms. A time series of R-R
intervals that comprised only beats in normal sinus rhythms and in a
stationary state was selected for the final analysis.
A time series of BP was constructed as that of arterial blood pressure
averaged from beat to beat. Time series of BP and RSNA were splined and
sampled at 8 Hz so that values of all the constructed time series
occurred simultaneously.
MI Analysis
We calculated MI values according to an algorithm proposed by Fraser
and Swinney (15). For a couple of time series,
x(t) and
y(t),
we measured the dependency of the values of
y(t + T) on the values of
x(t),
where T was a time delay and data
length was the power of two. We made the assignment
(s,
q) = [x(t),
y(t + T)] to consider a general
coupled system (S,
Q). MI of this system
[I(S,
Q)] is defined as the answer
to the question, "Given a measurement of
s, how many bits on the average can be
predicted about q?"
where
S and
Q denote the systems;
Ps(s)
and
Pq(q)
are the probability densities at s and
q, respectively; and
Psq(s,
q) is the joint probability density
at s and
q. The larger the value of MI is for
(S,
Q), the stronger the mutual
dependence is between S and
Q. The algorithm of MI will be
described in APPENDIX A.
We denote time series of HR, RSNA, and BP as
HR(t),
RSNA(t), and
BP(t), respectively. The data length
was between 29 and
211 samples, that is, between 64 and 256 s, because the data were sampled at 8 Hz. If
S = Q, the correlation between them
should be perfect, and then
I(S,
Q) = n, where the data length is
2n, because the algorithm is
developed to the discrete case. This point will be discussed in
APPENDIX A. The MI value between the
same two time series is n. Hence, MI
values were normalized by n; in other
words, these values were divided by n.
We calculated mutual information
I(T)
of (S,
Q), where
S is a time series of HR(t),
RSNA(t), or
BP(t) and
Q is another time series with a time delay T; for example,
S = HR(t) and
Q = RSNA(t + T).
T was then between 5 and 5 s in
a step of 0.125 s. The maximum value of I(T)
between S and
Q, where
T was from 5 to 5 s, is denoted
by Imax(S,
Q). We considered a time delay
Tmax(S,
Q), at which the maximum value of
I(T)
of (S,
Q) was given, as a physiological delay between these parameters. The definition of
Tmax(S,
Q) will be discussed later (see
DISCUSSION). If
Tmax(S,
Q) is negative, then
S leads
Q, and if
Tmax(S,
Q) is positive, vice versa.
Coherency Analysis
To compare MI analyses, we calculated coherency among HR, RSNA, and BP.
Ordinary coherence function estimates were computed as the ratio of the
magnitude square of the cross spectra divided by the product of the
autospectra. Coherence is a measure of the statistical link between two
variability series at any given frequency and is expressed as a number
between zero and one. The significant value of coherency will be
discussed in Simulation 4. The
confidence in spectral estimates can be enhanced by dividing data into
multiple epochs and ensemble averaging. Although this decreases
variance and increases confidence, it reduces resolution because a
reciprocal of the length of the data epoch determines the limits of the
low-frequency resolution. Auto- and cross-spectra estimates were
optimized by ensemble averaging between 9 and 31 data epochs of 128 points (16 s) using Welch's method (7). To reduce the loss of
stability, the original data were sectioned using a 50% overlap.
Before a fast Fourier algorithm was used, linear trends were removed
from the data, and data were tapered with the use of a Hanning window.
Statistical Procedures
Values of continuous variables are presented as means ± SE. We used
nonparametric two-way ANOVA (Friedman's test) to test for statistical
significance between the various stages of the experimental study. A
two-tailed P < 0.05 was considered
significant.
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RESULTS |
Simulation Studies
Simulation 1.
To demonstrate an effect of noise added to original simulated data on
MI values between the original time series and its noisy series, a
couple of time series were generated with noise according to the
following formulas (29)
where
R1:R2 = 5:4,
f1:f2 = 10:9, 
= 1.3
, and 
are random numbers uniformly distributed
between 
and . These time series were sampled
at 10 Hz. The term "50% noise" means that
R1:R2: = 5:4:9. The time series
x(t)
represents the torus series without noise. A torus series such as
x(t)
is a simple example of a nonlinear and yet periodic series. We
considered it appropriate to compare the reliability of MI analysis
with that of coherency analysis. With the percentage of noise
increasing, peaks at several main frequencies will be obscured in the
power spectrum of the noise-contaminated torus series. Hence, there is
considered to be no significant correlation between the original torus
series and noisy series that do not show any characteristic spectra due
to the two fundamental frequencies
(f1 and
f2). This is
why it was chosen to determine a threshold value of MI to detect a
significant correlation.
I[x(t),
y(t)]
were computed from an original torus series and noisy torus series that
were 1,024 samples long and contained 10-90% noise. For each
percentage of noise, 10 sets of
I[x(t), y(t)]
were computed by generating different seeds of random numbers 10 times.
I[x(t),
y(t)]
at 10, 20, ..., 90% noise were compared by one-way factorial ANOVA.
Figure 4 shows that the MI value decreased with an increasing percentage of noise
(P < 0.0001). For frequency analysis, these data were sectioned using a 50% overlap, and auto- and
cross-spectra estimates were optimized by ensemble averaging 15 data
epochs of 128 points. Because the ratio
f1/f2
of the two fundamental frequencies
(f1 = 1.59 Hz and
f2 = 1.43 Hz) is rational, all the peaks are harmonics of the frequency
f = f1 f2 (0.16 Hz) in
the power spectrum of
x(t).
Because the three main peaks in the spectrum of
x(t)
became obscure in the spectrum of noisy torus
y(t)
containing >80% noise, the average of
I[x(t),
y(t)] in which
y(t)
contained 70% noise (0.047 ± 0.002) was taken as the threshold
value to discriminate correlated from noncorrelated data. In
particular, an MI value >0.088, which was the average of
I[x(t),
y(t)]
when
y(t)
contained 50% noise, indicated the strong correlation between them.
This result would provide a measure to assess the effect of noise on MI
between an original series and the noisy series. With an increasing
percentage of noise, the main peaks in the spectrum of the noisy torus
y(t)
that were consistent with those in the spectrum of
x(t)
became obscure and the MI value between them decreased. This suggests
that the high MI values corresponded to the fluctuations pertaining to
these main peaks. Hence, in cases in which two time series show several consistent peaks in their spectra and have a high MI value, the respective fluctuations pertaining to these main peaks would be strongly correlated with each other between these time series.
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Fig. 4.
Top: mutual information (MI) between a
simulated torus time series (left)
and its noisy series with increasing percentage of noise. In
middle rows, power spectral density
(PSD, measured in arbitrary units) functions of these series and
coherency between original torus times series and each noisy series are
shown. The units for y-axis of tachograms are arbitrary. MI
between original time series and each noisy series is shown in
bottom panel (SE of each
mean is trivial and bars are not shown).
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In APPENDIX B, we describe two
simulations. One simulation shows that an MI value of zero means that
two time series are neither linearly nor nonlinearly correlated at all.
The other simulation shows the reliability of MI analysis for detecting the time delay between two torus series.
Simulation 2.
Suppose that
x(t)
and
y(t)
are both formed from the same purely random process
z(t),
which has a mean of zero and a variance of
2z, by
The
data length of these series was 16,384 (214) samples. The
sampling rate was 1 Hz. For coherency analysis, these data were sectioned using a 50% overlap, and auto- and cross-spectra estimates were optimized by ensemble averaging 127 data epochs of 256 points. Figure 5 shows the randomness of
x(t)
and
y(t)
and indicates that the coherency spectrum
C() is almost constant and equal to
one. In fact, the coherency spectrum is given by
C() = 1 for all in (0, 0.5),
after the application of some algebra using the definition of the
coherency spectrum. A value of unity indicates a perfectly linear
relationship between
x(t)
and
y(t)
at any frequency. The time series
y(t)
was linearly correlated with
x(t),
and this explains why there was perfect correlation between the
components of the two processes at any given frequency. The coherency
spectrum faithfully reflected the perfect correlation between
x(t)
and
y(t).
On the other hand, the MI value between
x(t)
and
y(t)
equaled unity and likely reflected the perfect correlation between
them. This example indicates that even if
x(t)
and
y(t)
are noise processes, both coherency and MI can quantify the perfect
correlation between them faithfully if they are linearly correlated.
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Fig. 5.
Frequency domain analysis of 2 simulated random time series,
X and
Y, which are perfectly linearly
correlated. PSD is measured in arbitrary units. The units
for x-axis of tachograms are in seconds and the units for
y-axis are arbitrary.
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Simulation 3.
We examined another case in which
x(t)
and
y(t)
were not linearly correlated at all. The time series
x(t)
was a noise process as shown in Fig. 6,
where 1 
x(t) 16,384; if t1 
t2, then x(t1) x(t2).
The time series
y(t)
was generated by one-to-one relation from the
X-Y
plot of Fig. 6. The sampling rate was 1 Hz. The region of the
X-Y
plot is divided into 16 subregions. In each of the four
subregions denoted as A,
B, C,
and D (Fig. 6),
x(t)
and
y(t)
were perfectly linearly correlated. The data length of these series
was 16,384 samples. The coherency spectrum was computed in the same
manner as in Fig. 5. Coherency of
x(t) and
y(t)
at any frequency was <0.05. Although this indicates that x(t)
and
y(t)
were not linearly correlated at all, it provides no information as to
whether they were nonlinearly correlated. On the other hand,
the MI value between the series was just unity, indicating that
x(t)
and
y(t)
had a perfect correlation. This case demonstrates that the
coherency method fails to detect correlation between two time series
that are nonlinearly correlated.
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Fig. 6.
Frequency domain analysis of 2 simulated random time series,
X and
Y, which are nonlinearly correlated.
The time series
x(t)
is a noise process shown in tachogram of
X, where 1 x(t) 16,384; if t1 t2, then
x(t1) x(t2).
The units for x-axis of tachograms are in seconds and the units
for y-axis are arbitrary. The time series
y(t)
is generated by one-to-one relation from the
X-Y
plot (bottom panel). The region of
the
X-Y
plot is divided into 16 subregions. In each of 4 subregions denoted by
A, B, C, and D,
x(t)
and
y(t)
are perfectly linearly correlated. PSD is measured in arbitrary
units. The units for both axes of the
X-Y plot (bottom) are arbitrary.
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Simulation 4.
Using simulated data, we determined that the confidence of coherence
depends on the number of data epochs. In the same manner as that shown
in Fig. 6,
x(t)
and
y(t)
were not linearly correlated at all. The time series
x(t) was a noise process
for which 1 x(t) 1,024; if
t1 t2, then
x(t1) x(t2).
The time series y(t)
was generated by one-to-one relation from the
X-Y
plot of Fig. 7. The data length of these
series was 1,024 samples. The sampling rate was 1 Hz. For
coherency analysis, these data were sectioned without any overlap, and
auto- and cross-spectra estimates were optimized by ensemble averaging
four data epochs of 256 points. For example, assume that a value of
C() = 0.5 is estimated at a
frequency of interest. The normalized random error of the coherence function estimate is then approximated by using the estimate
C() in place of the unknown true
coherence as follows (Ref. 7, p. 311)
where
C() = 0.5 and
n = 4 data epochs. Hence, an
approximate 95% confidence interval for the true value of coherence at this frequency is between zero and one. This formula indicates that the
normalized random error of the coherence function estimate approaches
zero as either n 
or C() 1. Therefore, we
took the data length in simulation 3 to be 16 times
as large as that in this simulation. In
simulation 3 the data were sectioned
with the use of a 50% overlap, and the number of data epochs
was 127. Eighteen frequencies can be counted between 0 and 0.5 Hz at
which the coherence value exceeds 0.5 in Fig. 7, whereas no such
frequencies are present in Fig. 6. Hence, these results indicate that
the confidence of coherence computed by coherency analysis depends on
the number of data epochs. For coherency analysis of the experimental data, the number of data epochs was at least nine. If a coherency value
is then estimated to be 0.5 at a frequency of interest, an ~95%
confidence interval for the true value of coherence is between 0.2 and
0.8. Therefore, coherency values >0.5 were considered significant.
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Fig. 7.
Coherency analysis of 2 simulated random time series,
X and
Y, which are nonlinearly correlated in
the same manner as in Fig. 6. Data length of 1,024 was
one-sixteenth of that in Fig. 6. The time series
x(t)
is a noise process, where 1 x(t) 1,024; if t1 t2, then
x(t1) x(t2).
The time series
y(t)
is generated by one-to-one relation from the
X-Y
plot. PSD is measured in arbitrary units. The units for
both axes of the X-Y plot are arbitrary.
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Experimental Data
Figure 8 shows the representative
tachograms and power spectra of HR, RSNA, and BP during the natural
heart state in a dog. The coherency spectra and phase-angle spectra
between each two variables are also shown in Fig. 8. The frequency of
the maximum peak power in these power spectra is almost the same (0.12 Hz). The coherency values between HR and BP and between RSNA and BP were >0.5 at 0.12 Hz, indicating that they are significantly linearly correlated with each other, whereas coherency between HR and RSNA was
not significant. In the time-domain analysis, the correlation coefficient between HR and BP, that between RSNA and BP, and that between HR and RSNA were 0.17, 0.25, and 0.19,
respectively. Although these correlation coefficients were small, the
hypothesis of zero correlation for each pair was rejected (Ref. 7, p. 101). On the other hand, the maximum MI values between these variables were Imax(HR,
RSNA) = 0.25, Imax(HR, BP) = 0.17, and
Imax(RSNA, BP) = 0.13. These values indicate that these variables have mutual dependence
on one another and, especially, that HR and RSNA have a very strong
correlation. From the cross-spectrum phase-angle estimates at 0.12 Hz,
RSNA led HR by 2.9 ± 0.8 s, BP led HR by 2.1 ± 0.3 s, and RSNA
led BP by 0.9 ± 0.6 s. These standard deviations of time delays at
the frequency were estimated with the use of a formula to compute those
of the phase angles (Ref. 7, p. 301). From MI analysis, RSNA led HR by
2.1 s, BP led HR by 2.0 s, and RSNA led BP by 0.5 s. The time delay
between each pair of these variables as estimated by MI analysis was
within its range as estimated by phase-angle analysis, and the
relationship between them with respect to leading was the same.
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Fig. 8.
Typical example of results of frequency analysis of heart rate (HR),
RSNA, and BP during natural heart state in a dog. Units of measure for
RSNA are arbitrary; units for PSD of HR and BP are
(beats/min)2/Hz and
mmHg2/Hz, respectively. HR:RSNA,
coherency spectrum between HR and RSNA; HR:BP, coherency spectrum
between HR and BP; RSNA:BP, coherency spectrum between RSNA and BP.
|
|
Figures 9 and
10 show tachograms, power spectra of RSNA
and BP, and coherency spectra between RSNA and BP during the artificial heart states with pacing rates of 60 and 160 beats/min, respectively, in the same dog as in Fig. 8. With a pacing rate of 60 beats/min, the
low-frequency peak in the RSNA spectrum was consistent with that in the
BP spectrum. The considerable components appeared in a broader
frequency band of spectra of RSNA and BP for a pacing rate of 160 beats/min compared with those for a pacing rate of 60 beats/min. The
coherency value between RSNA and BP at 0.12 Hz was higher
with a pacing rate of 60 beats/min than with a pacing rate of 160 beats/min. The correlation coefficient between RSNA and BP was higher
with a pacing rate of 60 beats/min than with a pacing rate of 160 beats/min (0.58 and 0.32, respectively). These findings indicate
that the linear correlation between RSNA and BP decreased at a higher
pacing rate. Because HR was constant during the artificial
heart state, only I(RSNA, BP) was
calculated. Imax(RSNA,
BP) at pacing rates of 60 beats/min and 160 beats/min were 0.16 and
0.08, respectively. This indicates that the correlation between RSNA
and BP decreased with an increasing pacing rate. RSNA and BP, however,
were significantly correlated even with a pacing rate of 160 beats/min.
At a pacing rate of 60 beats/min, RSNA led BP by 3.6 ± 0.2 s
according to the cross-spectrum phase-angle estimates at 0.12 Hz,
whereas RSNA led BP by 0.6 s according to MI analysis. At a pacing rate
of 160 beats/min, the coherencies between RSNA and BP at some
frequencies >0.25 Hz were significant and the phase angles at the
respective frequencies were roughly zero radian. These results were
compatible with those from MI analysis in that they were strongly
correlated and the time delays between them were <0.25 s.
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Fig. 9.
Typical example of results of frequency analysis of RSNA and BP with a
pacing rate of 60 beats/min in the same dog as in Fig. 8. Units of
measure for RSNA are arbitrary; units for PSD of BP are
mmHg2/Hz.
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Fig. 10.
Typical example of results of frequency analysis of RSNA and BP with a
pacing rate of 160 beats/min in the same dog as in Fig. 8. Units of
measure for RSNA are arbitrary; units for PSD of BP are
mmHg2/Hz.
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|
During the control state,
Imax(HR, RSNA) > Imax(HR, BP) > Imax(RSNA,
BP) as shown by ANOVA (P < 0.05)
(Fig. 11). This indicates that the correlation between HR and RSNA is stronger than that between
HR and BP and that between RSNA and BP.
Imax(RSNA, BP) was significantly larger at a pacing rate of 60 beats/min than at the
other pacing rates (80-160 beats/min) or in control
(P < 0.05) (Fig.
12).
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Fig. 11.
Comparison of maximum values of mutual information
(Imax) during
natural heart state by ANOVA (n = 7).
Maximum value of
I(T)
between HR(t) and
RSNA(t + T), where
T is time delay from 5 to 5 s,
is denoted by
Imax(HR, RSNA).
P < 0.05.
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Fig. 12.
Comparison of
Imax(RSNA, BP)
between control and total artificial heart states with pacing rates
from 60 to 160 beats/min by ANOVA
(n = 7).
P < 0.05.
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|
The time delay of HR and RSNA
[Tmax(HR,
RSNA)] was significantly different from
Tmax(HR, BP) and
Tmax(RSNA, BP)
during the control state (P < 0.05)
(Fig. 13). These results show that RSNA leads BP, BP leads HR, and RSNA leads HR during the control state. The
values of
Tmax(RSNA, BP)
were significantly different at the pacing rates (60-160
beats/min) or in control (P < 0.05)
(Fig. 14).
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Fig. 13.
Comparison of time delay
Tmax among HR,
RSNA, and BP during control state by ANOVA
(n = 7).
Tmax(S,
Q) is defined as time delay at which
maximum mutual information value of
I(T)
between S and
Q (5 T 5 s) is given. If
Tmax(S,
Q) is negative, then
S leads
Q. If
Tmax(S,
Q) is positive, then vice versa.
P < 0.05.
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Fig. 14.
Comparison of
Tmax(RSNA, BP)
between control and total artificial heart states with pacing rates
from 60 to 160 beats/min by ANOVA (n = 7). P < 0.05.
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| |
DISCUSSION |
In this study we proposed and tested the MI technique to assess
correlation between two time series. In the case in which the MI value
between a pair of variables is >0.088, which is the average of
I[x(t),
y(t)]
where
y(t)
contains 50% noise, it is indicated that these variables are
correlated even if they are contaminated by noise. On the other hand,
some researchers took a coherency value of 0.5 as the threshold value
above which there is a significant correlation between oscillations in
different variables (28, 31). Although the confidence of coherence can be enhanced by dividing data into multiple epochs and ensemble averaging, it might be practically difficult to achieve the normalized random error, which equals 0.1. For example, when a value of coherence equals 0.5, >100 epochs are necessary to achieve ~95% confidence intervals (7). This reduces resolution because the length of the data
epoch determines the limits of the low-frequency resolution. However,
the coherency analysis might have the advantage of gauging the linear
correlation between two time series at any frequency. Hence, to
quantify the relationship between two time series
x(t) and
y(t),
coherency analysis as well as MI analysis should be used. Although the
MI value does not quantify the relationship between the time series at
a frequency of interest, it can quantify the nonlinear as well as
linear dependence between them. In particular, the findings in
Simulation 1 indicate that, in the
case of two time series showing several consistent peaks in their
spectra and a high MI value, the respective fluctuations pertaining to these main peaks would be strongly correlated with each other between
these time series.
During the control state, the frequency of the maximum power in the HR
spectrum was the same as that of the maximum power in the RSNA
spectrum. Moreover, the MI value between them was 0.25, indicating that
they had a strong correlation. On the other hand, the peak power at
0.12 Hz in the RSNA spectrum was in the frequency range from 0.04 to
0.15 Hz, which has been defined as the low-frequency (LF) component in
the power spectra of heart rate variability in humans and dogs (2, 27).
The high-frequency component, from 0.15 to 0.4 Hz, has been considered
to reflect the parasympathetic activity modulated by respiration.
Recently, there have been some debates about whether the LF component
reflects the fluctuation of sympathetic activity (28) or not (1, 6, 10). However, the frequency of the peak power was 0.12 Hz in both the
HR and the RSNA spectra and
Imax(HR, RSNA)
was high (0.273 ± 0.022). According to the analysis in
Simulation 1, these findings suggest
that the LF component in the HR spectrum is modulated at least by the
sympathetic activity.
Gebber and colleagues (16, 33) reported that a rhythmic fluctuation in
the LF range persisted after denervation of all cardiovascular nerves.
This suggests that the rhythmic discharge of the peripheral sympathetic
nerve might result from a sympathetic rhythm of central origin that is
normally entrained to the cardiac cycle by the pressor receptor input
(35). To examine this hypothesis, we thought that the artificial heart
model might be useful. If the cardiovascular regulatory system causes
the rhythmic fluctuations of hemodynamic parameters through the natural
heart, sympathetic nerve activity should not cause rhythmic
fluctuations with the total artificial heart. The rhythmic fluctuations
(0.12 Hz) of RSNA and BP were observed in a pacing rate of 60 beats/min
(1 Hz). It is considered that these fluctuations are spontaneous because the rate of the fluctuations was different from the pacing rate. With this finding, the high values of
Imax(RSNA, BP)
suggest that these cyclic fluctuations are correlated strongly and are not induced by driving the artificial heart.
Using coherency analysis, Brown et al. (10) have reported that
sympathetic interactions are more reliably reflected in the BP power
spectrum than in the HR spectrum in the conscious rat (10). Indeed,
Fig. 8 shows that the coherence between RSNA and BP was stronger than
that between RSNA and HR in the LF band. We consider that these
coherencies might reflect only the linear correlations between the
series. In the present study the MI values among HR, RSNA, and BP
during control indicate that the correlation between HR and RSNA is
stronger than that between HR and BP and that between RSNA and BP. This
might be because the sinus node is affected by efferent sympathetic
nerve directly, although it responds to both sympathetic and vagal
inputs.
We calculated the time delay of each pair of signals derived from the
HR, RSNA, and BP signals by regarding a time delay at which the maximum
MI value between the signals was given as a physiological delay. Such a
time delay T does not indicate a time delay between two time series at a certain frequency of fluctuations originating from respective physiological inputs, for instance, sympathetic or vagal input, but instead means that one of these time
series is most strongly correlated with the time series that is delayed
for the time delay T from the other
time series. The results on detection of the time delay between two
torus series showed the superiority of MI analysis to coherency
analysis. However, we must be very careful in interpreting the delay
relations between these parameters because, during closed-loop
operation of the control state, the relationship between HR and BP
reflects both the feedback from BP to HR through the baroreflex and the
moderating effect of HR on this response by changing BP through changes
in cardiac output. Moreover, efferent sympathetic and vagal activities directed to the sinus node are characterized by discharge largely synchronous with each cardiac cycle that can be modulated by central (vasomotor and respiratory centers) and peripheral (oscillation in
arterial pressure and respiratory movements) oscillators (23). These
oscillators generate rhythmic fluctuation in efferent neural discharge
and construct regulatory feedback loops. Although our results are
difficult to account for precisely, they suggest that RSNA leads BP, BP
leads HR, and RSNA leads HR during the control state in anesthetized
dogs. Our results are compatible with the findings reported by Pagani
et al. (27); however, there have been only a few reports available
regarding the physiological delay between these parameters
(31).
Perhaps the most difficult finding to explain is the small lags between
RSNA and BP during the artificial heart state with pacing rates >60
beats/min. The demonstration in other studies (10, 12) that sympathetic
activity can follow respiration has suggested that sympathetic nerve
activity (SNA) can respond to rapid disturbances. Persson et al. (30)
have reported that the phase lag of SNA and BP was roughly 0.2 s.
Similarly, Fig. 2 shows that SNA can respond to the rapid increase of
BP. On the other hand, there were considerable variabilities of RSNA
and BP at a broader frequency band with a pacing rate of 160 beats/min than with a pacing rate of 60 beats/min (Figs. 9 and 10). With a pacing
rate of 160 beats/min, the coherencies between RSNA and BP at some
frequencies >0.25 Hz were significant, and the phase angles at the
respective frequencies were roughly zero radian. These results were
compatible with those from MI analysis in that they were strongly
correlated and the time delays between them were <0.25 s. Although we
should take the sampling frequency (8 Hz) into consideration to
interpret "roughly zero" lags, these observations raise the
possibility of a rapid response of RSNA to BP with an increasing pacing
rate.
During the artificial heart state,
Imax(RSNA, BP)
was significantly larger with a pacing rate of 60 beats/min than with
other pacing rates. This finding indicates that the fluctuation of
sympathetic nerve is strongly correlated with that of BP with a pacing
rate of 60 beats/min.
Imax(RSNA, BP)
decreased with increasing pacing rate. This might suggest that a
response of the baroreflex system to BP depends on pacing rates of the
artificial heart and that the response can hardly keep up with the
increasing pacing rate. Thus we believe that the neurological effect of
the total artificial heart must be considered when its driving
conditions are selected.
Limitations
We should take effects of anesthesia on the cardiovascular control
system into consideration. Most of the gaseous anesthetic agents,
including the nitrous oxide used in the present study, depress
sympathetic activities in different regions of the sympathetic nervous
system (13, 20). However, Keyl et al. (18) demonstrated, by
means of spectral analysis of heart rate variability, that general
anesthesia had no disadvantageous effects compared with local
anesthesia on the perioperative cardiac autonomic tone during ophthalmic surgical procedures in otherwise healthy patients. Even if
the effects of anesthesia on cardiac autonomic tone must be considered,
we think that the correlation among HR, RSNA, and BP could be assessed
in this study because the frequencies of peak power were consistent in
the spectra of these parameters (Figs. 8 and 9). This is supported by
Brown et al. (10), who found that anesthesia with pentobarbital lowered
spectral power for HR, SNA, and BP but essentially did not change the
coherence between SNA and BP.
In conclusion, MI analysis has many features that are advantageous in
assessing the relationship between various cardiovascular variables.
1) The method consists of
probabilistic processes that do not require linearity of data and thus
is applicable to the assessment of data generated not only by a linear
system but also by a nonlinear one.
2) MI analysis can sensitively
quantify the relationship between two time series and assess a time
delay of the interaction between them.
3) The MI algorithm is
simple and quick to compute. We applied MI analysis to quantify the
correlation among HR, RSNA, and BP in dogs during the natural heart
state and during the total artificial heart state. We conclude, from both theoretical and practical perspectives, that MI analysis has a
potentially wide range of applications in studies of cardiovascular variation.
| |
APPENDIX A |
Mutual information is a measure of information theory.
S denotes the whole system
that consists of possible messages.
Ps(s) is the probability density at a message
s. The average amount of information
gained from a measurement that specifies
s is the entropy
H of a system
where
H(S)
is the quantity of surprise one should feel when reading the result of
a measurement. If the logarithm is taken to the base two,
H is in units of bits. We considered a
general coupled system (S,
Q) and asked, "Given that
s has been measured and found to be
si, what
uncertainty is there in a measurement of
q?" The answer is
where
Pq|s(q|si)
is the probability of a measurement of
q given the measured value of
si and
Psq(s,
q) is the joint probability density
at s and
q. By averaging
H(Q|si)
over si, the
average uncertainty was obtained
where
Hence,
the amount that a measurement of s
reduces the uncertainty of q is
This
is the mutual information. Individual entropies of continuous systems
depend on coordinates, but this algorithm was developed to measure a
coordinate-independent difference of entropies for the discrete case.
We calculated values of mutual information between time series
x(ti)
and
y(ti)
according to an algorithm proposed by Fraser and Swinney (15). The data
length is the power of two, that is,
2n.
x(ti)
and
y(ti)
were changed into
s(ti)
and
q(ti)
in a fashion that preserves orderings, with the constraints that if
x(t1) < x(t2),
then
s(t1) < s(t2)
and 1 s(ti) 2n and with the same
constraints for
q(ti).
We measured how dependent the values of
s(t)
were on the values of
q(t).
We defined a sequence of partitions of the
(s,
q) plane
(G0,
G1,
G2, ..., Gm, ...) such that each partition is a rectangular grid of
4m elements generated by dividing
each axis into 2m equiprobable
segments.
Rm(Km)
denoted an element of
Gm, where Km is an index
that takes one of 4m possible
values. Associated with the sequence of partitions
Gm is a sequence
im, which
converges to
I(S,
Q), where
Psq[Rm(Km)]
is estimated by
N[Rm(Km)]/N0,
where
N[Rm(Km)]
is the number of events observed in partition element
Rm(Km)
and N0 is the
total number of events observed. One representation of
Km is the
ordered pair (i,
j), where
i indicates a range of s values and
j indicates a range of
q values. With the definition of
Rm(Km)
and
The algorithm requires a recursive approach to the definition
of
I(S,
Q). These steps are iterated in
areas in which
Psq has finer
structure, yielding smaller partition elements where they are needed.
When the number 2m of
equiprobable segments of each axis approaches the data length, im converges to
I(S,
Q). Therefore, if
S = Q, then
I(S,
Q) = n, where the data length is
2n. The detailed algorithm is
referred to in Ref. 15.
| |
APPENDIX B |
Simulation 5.
To compare the results of our experimental data, we took the scalar
data consisting of 1,024 random numbers as a time series [r(t)]
and measured how dependent the values of
r(t + T) are on the values of
r(t).
The MI values of
[r(t),
r(t + T)], where
T was a time delay such that 0 T 10, in a step of one, were all zero. This result faithfully reflected
the perfectly random generation of these random numbers. Hence, the MI
value zero indeed indicates that two time series are neither linearly
nor nonlinearly correlated at all.
Simulation 6.
Using the torus series in simulation
1, we examined the reliability of MI analysis to detect
the time delay between the original data
x(t)
and the 0.5-s-delayed noisy data
z(t)
as
I(T)
I[x(t),
z(t + T)] were computed from an
original torus series and a delayed noisy torus series 1,024 samples
long containing 10-50% noise at
T, which was between 1 and 1 s
in a step of 0.1 s. For each percentage of noise, the coherency
spectrum between
x(t)
and
z(t)
was computed in the same manner as that between
x(t)
and
y(t).
Figure 15 shows that the maximum value of I(T)
was at T = 0.5 s in all cases.
This means that
x(t)
leads z(t)
by 0.5 s. In frequency analysis, the time delay at 1.6 Hz, at which the
maximum power appeared with the maximum coherency, was 0.25 s with
x(t)
leading. Although it assumed that MI values relate primarily to the
high power components of the signal, this result suggests that MI
analysis can detect the time delay between two nonlinearly correlated
processes in comparison with phase-angle analysis.
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Fig. 15.
Detection of a time delay between original torus series and
0.5-s-delayed noisy torus series by frequency analysis and MI
analysis.
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|
| |
ACKNOWLEDGEMENTS |
This work is partially supported by a Grant-in-Aid for Medical
Research from the Alumni Association of Nippon Medical School and by a
Grant-in-Aid from the Fukuda Foundation for Medical Technology.
| |
FOOTNOTES |
This study was initiated at the 3rd Workshop, "Various Approaches to
Complex Systems," held at the International Institute for Advanced
Studies in Kyoto, Japan, 1996.
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: M. Osaka, First Dept. of Internal
Medicine, Nippon Medical School, Sendagi 1-1-5, Bunkyo-ku,
Tokyo 113-8603, Japan.
Received 29 April 1998; accepted in final form 29 June 1998.
| |
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Am J Physiol Heart Circ Physiol 275(4):H1419-H1433
0002-9513/98 $5.00
Copyright © 1998 the American Physiological Society
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Abstract
Introduction
![<IT>I</IT>(<IT>S</IT>, <IT>Q</IT>) = ∫ <IT>P</IT><SUB>sq</SUB>(<IT>s</IT>, <IT>q</IT>) log {<IT>P</IT><SUB>sq</SUB>(<IT>s</IT>, <IT>q</IT>)/[<IT>P</IT><SUB>s</SUB>(<IT>s</IT>)<IT>P</IT><SUB>q</SUB>(<IT>q</IT>)]}d<IT>s</IT>d<IT>q</IT>](/content/vol275/issue4/fulltext/H1419/img001.gif)
![<IT>x</IT>(<IT>t</IT>) = [<IT>R</IT><SUB>1</SUB> + <IT>R</IT><SUB>2</SUB> sin (2&pgr; <IT>f</IT><SUB>2</SUB><IT>t</IT> + &phgr;)] sin (2&pgr; <IT>f</IT><SUB>1</SUB><IT>t</IT>)](/content/vol275/issue4/fulltext/H1419/img002.gif)
![<IT>y</IT>(<IT>t</IT>) = [<IT>R</IT><SUB>1</SUB> + <IT>R</IT><SUB>2</SUB> sin (2&pgr; <IT>f</IT><SUB>2</SUB><IT>t</IT> + &phgr;)] sin (2&pgr; <IT>f</IT><SUB>1</SUB><IT>t</IT>) + &xgr;](/content/vol275/issue4/fulltext/H1419/img003.gif)
![<FR><NU><RAD><RCD>2</RCD></RAD>[1 − <IT>C</IT>(ω)]</NU><DE><RAD><RCD><IT>C</IT>(ω)</RCD></RAD><RAD><RCD><IT>n</IT></RCD></RAD></DE></FR> = 0.5](/content/vol275/issue4/fulltext/H1419/img006.gif)
![= −∫[<IT>P</IT><SUB>sq</SUB>(<IT>s</IT><SUB><IT>i</IT></SUB>, <IT>q</IT>)/<IT>P</IT><SUB>s</SUB>(<IT>s</IT><SUB><IT>i</IT></SUB>)] log [<IT>P</IT><SUB>sq</SUB>(<IT>s</IT><SUB><IT>i</IT></SUB>, <IT>q</IT>)/<IT>P</IT><SUB>s</SUB>(<IT>s</IT><SUB><IT>i</IT></SUB>)]d<IT>q</IT>](/content/vol275/issue4/fulltext/H1419/img009.gif)
![= −∫∫<IT>P</IT><SUB>sq</SUB>(<IT>s</IT>, <IT>q</IT>) log [<IT>P</IT><SUB>sq</SUB>(<IT>s</IT>, <IT>q</IT>)/<IT>P</IT><SUB>s</SUB>(<IT>s</IT>)]d<IT>s</IT>d<IT>q</IT>](/content/vol275/issue4/fulltext/H1419/img011.gif)
![<IT>H</IT>(<IT>S</IT>, <IT>Q</IT>) = − ∫∫<IT>P</IT><SUB>sq</SUB>(<IT>s</IT>, <IT>q</IT>) log [<IT>P</IT><SUB>sq</SUB>(<IT>s</IT>, <IT>q</IT>)]d<IT>s</IT>d<IT>q</IT>](/content/vol275/issue4/fulltext/H1419/img013.gif)
![<IT>i</IT><SUB><IT>m</IT></SUB> = <LIM><OP>∑</OP><LL><IT>K</IT><SUB><IT>m</IT></SUB></LL></LIM> <IT>P</IT><SUB>sq</SUB>[<IT>R</IT><SUB><IT>m</IT></SUB>(<IT>K</IT><SUB><IT>m</IT></SUB>)] log (<IT>P</IT><SUB>sq</SUB>[<IT>R</IT><SUB><IT>m</IT></SUB>(<IT>K</IT><SUB><IT>m</IT></SUB>)]/{<IT>P</IT><SUB>s</SUB>[<IT>R</IT><SUB><IT>m</IT></SUB>(<IT>K</IT><SUB><IT>m</IT></SUB>)]<IT>P</IT><SUB>q</SUB>[<IT>R</IT><SUB><IT>m</IT></SUB>(<IT>K</IT><SUB><IT>m</IT></SUB>)]})](/content/vol275/issue4/fulltext/H1419/img016.gif)
![<IT>P</IT><SUB>s</SUB>[<IT>R</IT><SUB>m</SUB>(<IT>K</IT><SUB>m</SUB>)] = <IT>P</IT><SUB>q</SUB>[<IT>R</IT><SUB><IT>m</IT></SUB>(<IT>K</IT><SUB><IT>m</IT></SUB>)] = (1/2)<SUP><IT>m</IT></SUP>](/content/vol275/issue4/fulltext/H1419/img017.gif)
![<IT>i</IT><SUB>m</SUB> = <IT>m</IT> log (4) + <LIM><OP>∑</OP><LL><IT>i</IT>,<IT>j</IT>=0</LL></LIM> <IT>P</IT><SUB>sq</SUB>[<IT>R</IT><SUB><IT>m</IT></SUB>(<IT>i</IT>, <IT>j</IT>)] log {<IT>P</IT><SUB>sq</SUB>[<IT>R</IT><SUB><IT>m</IT></SUB>(<IT>i</IT>, <IT>j</IT>)]}](/content/vol275/issue4/fulltext/H1419/img018.gif)
![= {<IT>R</IT><SUB>1</SUB> + <IT>R</IT><SUB>2</SUB> sin [2&pgr; <IT>f</IT><SUB>2</SUB>(<IT>t</IT> + 0.5) + &phgr;]} sin [2&pgr; <IT>f</IT><SUB>1</SUB>(<IT>t</IT> + 0.5)] + &xgr;](/content/vol275/issue4/fulltext/H1419/img020.gif)
