Department of Pharmacology and Toxicology, Michigan State
University, East Lansing, Michigan 48824-1317
 |
INTRODUCTION |
Sympathetic nerve
discharge (SND) contains a strong cardiac-related rhythm in
baroreceptor-innervated cats. The cardiac-related discharges of
different sympathetic nerves are highly correlated as indicated by
ordinary coherence values approaching one (Barman et al.
1992
; Gebber et al. 1990
; Kocsis
1995
). The tight linkage is manifested by a sharp peak at the
cardiac frequency in the nerve to nerve coherence function
(CohSND-SND) that rises above background. This
peak is eliminated by surgical baroreceptor denervation (Gebber
et al. 1994
). Two studies in the cat (Gebber et al.
1994
; Kocsis 1995
) and one in man (Kocsis
et al. 1999
) have used a computational technique called partial
coherence analysis to examine the mechanisms responsible for linkage of
the cardiac-related discharges of pairs of sympathetic nerves. This
method mathematically removes the portions of the cardiac-related SND
that are coherent to the arterial pulse (AP). We assume that AP
represents pulse-synchronous baroreceptor afferent nerve activity
(Gebber et al. 1994
; Kocsis 1995
).
Theoretically, partialization of the nerve to nerve coherence function
using AP (CohSND-SND/AP) will eliminate the peak
at the cardiac frequency if the central circuits governing the
discharges of the two nerves share baroreceptor inputs but are not
otherwise connected and the relationship between AP and SND is strictly linear (Bendat and Piersol 1966
; Rosenberg et al.
1998
). Contrary to this model of baroreceptor-sympathetic
interactions, there were numerous cases reported by Gebber et
al. (1994)
, Kocsis (1995)
, and Kocsis et
al. (1999)
in which partialization reduced but did not
eliminate the cardiac-related peak. This observation indicated that the
cardiac-related discharges of sympathetic nerves pairs were not solely
linearly correlated to pulse- synchronous baroreceptor inputs.
Additional factors contributing to residual coherence may include
coordination arising from physical interconnections of the central
circuits generating the activities of different nerves (Gebber
et al. 1994
) and nonlinear relationships between pulse
synchronous baroreceptor nerve activity and SND that are common to
different sympathetic nerves (Kocsis 1995
). There is evidence for some form of coupling between the central circuits generating SND in that after baroreceptor denervation, statistically significant CohSND-SND at frequencies near the
heart rate is still observed (Gebber et al. 1994
;
Kocsis et al. 1990
).
In the accompanying paper (Lewis et al. 2000
), we
observed two modes of coordination between AP and SND in the
cardiac-related band, phase walk and phase-locking. The phase walk was
characterized by a progressive cycle-by-cycle change in phase angle
between peak systole and SND. This relationship may be viewed as
nonlinear. In contrast, during phase-locking the range of phase angles
is narrow, but the variation is apparently random. This relationship may be viewed as linear. On the basis of these observations, the current study was designed to answer two questions. First, what proportion of cardiac-related SND is the consequence of the nonlinear phase walk? Second, does the nonlinear phase-walk account for instances
in which nerve-to-nerve coherence function partialized by AP
(CohSND-SND/AP) contains a peak exceeding
background at frequencies near the heart beat? To address the first
question, we used partial spectral analysis, comparing the power in the autospectrum of SND before (ASSND) and after
partialization using AP (ASSND/AP). To address
the second question, we used partial coherence analysis as outlined in
the preceding text.
 |
METHODS |
Experimental subjects and anesthesia
Two groups of cats were used in this study. The first group of
24 cats of either sex (weight range 2.3-4.1 kg) were initially anesthetized with 2.5-3.5% isoflurane in oxygen. Following
cannulation of a femoral vein, urethan (1.2-1.4 g/kg iv) was
administered, and isoflurane inhalation was terminated. This dose range
of urethan has been reported to maintain a surgical level of anesthesia
for a period of 8-10 h (Flecknell 1987
), which exceeded
the duration of our experiments. Blood pressure was measured from a
catheter inserted into the femoral artery. The cats were paralyzed and artificially ventilated with room air, and a bilateral
pneumothoracotomy was performed. End tidal CO2
was maintained between 3.5 and 4.0% by adjusting the parameters of
ventilation. Rectal temperature was maintained near 38°C using a heat
lamp. As previously described (Barman et al. 1992
),
potentials were monophasically recorded with bipolar platinum
electrodes from the central ends of the cut postganglionic sympathetic
inferior cardiac nerve (CN) and vertebral nerve (VN) near their exits
from the left stellate ganglion and the left postganglionic sympathetic
renal nerve (RN). Nerve recordings were made with a band-pass filter
set at 1-1000 Hz (Grass Instruments 7P3 preamplifier), so that
envelopes of multiunit spikes appeared as slow waves (Barman et
al. 1992
; Cohen and Gootman 1970
). We recorded
an 80-s epoch of data and then performed baroreceptor denervation by
bilateral sectioning of the carotid sinus, aortic depressor and vagus
nerves (Barman et al. 1992
). Sectioning of these nerves
eliminated the cardiac-related rhythm in SND and the inhibition of SND
produced by raising blood pressure with a bolus intravenous injection
of norepinephrine. Following baroreceptor denervation, we recorded a
further 80-s epoch of data.
The second group of 10 cats was used to compare AP-SND phase angles at
different steady-state levels of blood pressure described in the
accompanying paper (Lewis et al. 2000
). These cats were anesthetized
and ventilated as described in the preceding text, and the baroreceptor
nerves were intact. Blood pressure was measured from the brachial
artery, and the CN activity was recorded. Blood pressure was controlled
at various steady-state levels by altering the parameters of a
phenylephrine infusion.
The All-University Committee on Animal Use and Care of Michigan State
University approved all protocols used in these experiments.
Data analysis
FREQUENCY DOMAIN ANALYSIS.
Spectral analysis was performed using the fast Fourier transform (FFT)
after SND had been low-pass filtered at 100 Hz. The sampling rate of
200 Hz gave a resolution of 0.2 Hz/bin. The power density spectra
(autospectra) and ordinary coherence functions (normalized cross
spectra) were averages of 32 5-s data windows, with a 50% overlap. The
autospectrum of a signal shows how much power is present at each
frequency and the coherence function measures the strength of
correlation (scale 0-1.0) of two signals. The spectra are displayed
over 0-10 Hz.
Partial autospectral and coherence analyses were performed using the
algorithms of Jenkins and Watts (1968)
. Partial
autospectral analysis involves the mathematical elimination of the
portion of a given signal (S1) that is determined or predictable on the basis of a second signal (S2). The partial autospectra at a given frequency (f) is defined as
where ASS1/S2 is the autospectrum of S1
partialized by S2, ASS1 is the ordinary
autospectrum of S1, and CohS1-S2 is the ordinary coherence between signals S1and S2.
If the power in ASS1 at f is entirely
predicted by S2, then partilization with S2 will remove all the power
in S1 at that frequency. If, however, power in
ASS1 at f is not fully predicted by
S2, then residual power will be present in
ASS1/S2(f)
(Jenkins and Watts 1968
; Rosenberg et al.
1998
).
A macro written in Microsoft Excel 7.0 was used to measure the power
above background activity in the cardiac-related band of
ASSND before and after partialization with AP. A
line was fitted to connect the left and right limits of the sharp peak
surrounding the cardiac frequency in the autospectrum of SND, and power
in this band was calculated as the area above this line.
Partial coherence analysis is the computation of the coherence between
two signals, S1 and S2, after the removal of the components from each
signal that are predictable on the basis of a third signal, S3. The
partial coherence function
[CohS1-S2/S3(f)]
measuring the relationship between S1 and S2 at frequency f
after removal of S3 is defined as
where ASS1/S3 and
ASS2/S3 are as defined in the preceding text and
|CSS1-S2/S3| is the amplitude of the residual
cross spectrum. The residual cross spectrum is defined as
where CSS1-S2,
CSS1-S3, and CSS2-S3 are
the ordinary cross spectra.
If the relationship between S1 and S2 at f is solely
dependent on S3, then partialization of the coherence between S1 and S2
using S3 will approach 0. If, however, the relationship between S1 and
S2 at f is not solely dependent on S3, then these two
signals will remain significantly correlated after partilization using S3 (Jenkins and Watts 1968
; Lopes da Silva et al.
1980
). A coherence value
0.1 on a scale of 0 to 1 is
statistically significant when 32 windows are averaged (Benignus
1970
).
TIME SERIES ANALYSIS.
Sympathetic nerve activity was digitally filtered (symmetric,
nonrecursive band-pass filter with a Lanczos smoothing function, RC
Electronics, Santa Barbara, CA), with a band-pass width of 4 Hz, and
the center frequency was matched to that of the sharp peak at the
frequency of the heart beat in the autospectrum of SND (Lewis et
al. 2000
). Following filtering we made cycle-by-cycle measurements of the interval (ms) between the peak of the AP and the
next peak of sympathetic nerve activity as described in the accompanying paper (Lewis et al. 2000
). The interval
between the peak of the AP and the next peak of nerve activity was
converted to a phase angle (
) in degrees using the formula
where t is the AP-SND interval (ms) and T
is the interval (ms) between the peaks of the APs that immediately
preceded and followed the nerve slow wave. The resolution of
measurement of the phase angle was 5.4°/bin (sampling period was 5 ms) when the period of the cardiac cycle was 333 ms (heart rate, 3 Hz).
STATISTICAL ANALYSIS.
Coherence values before and after partialization with AP and
baroreceptor denervation and autospectral power in the cardiac-related band before and after partialization with AP were compared using a
Student's paired t-test. Comparisons of values between
groups were performed using unpaired t-test. Statistical
tests were performed using Statview 5.0 (Abacus Concepts). Coherence
values were Z-transformed before the statistical comparisons
were made.
 |
RESULTS |
Figure 1 shows the relationships
between CN, VN, and RN discharges and AP in a baroreceptor-innervated
cat. The original recordings in Fig. 1 (top right)
demonstrate that the discharges of CN, VN, and RN were related in a 1:1
fashion to AP. Consequently, most of the power in
ASSND was concentrated in a narrowband around the
cardiac frequency, which is referred to as the cardiac-related band
(see ASSND Fig. 1, 2nd row). In
ASAP (Fig. 1 top left), and to a
lesser extent in ASSND (Fig. 1, 2nd
row), in addition to the peak at the cardiac frequency, there are
peaks at harmonics of the heart rate. The ordinary coherence functions
demonstrate that there was significant coherence between SND and AP
(Fig. 1, 3rd row) and between nerve pairs (Fig. 1,
bottom row) at the frequency of the heart rate and at higher
harmonics of the heart rate.

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Fig. 1.
Cardiac-related rhythm in sympathetic nerve discharge (SND) in a
baroreceptor innervated cat. Top right: oscilloscopic
records of arterial pulse (AP), and discharges (low-pass filtered at
100 Hz) of cardiac nerve (CN), vertebral nerve (VN), and renal nerve
(RN). Top left: autospectrum (AS) of AP. Second
row, left to right:
ASCN, ASVN, and
ASRN. Third row: ordinary coherence
functions relating AP to CN, VN, and RN activities. Bottom:
ordinary coherence functions relating CN and VN, CN and RN, and VN and
RN discharges. Spectral analysis was performed on 32 5-s windows with
50% overlap, and have a frequency resolution of 0.2 Hz in this and
subsequent figures.
|
|
Partialization of the autospectra of SND with AP removes the component
of the ASSND that is coherent with
ASAP. Two patterns of results were noted. The
first of these is illustrated in Fig. 2.
Here the ASSND (top row) shows a major
peak at the cardiac frequency. After partialization with AP (2nd
row), a large portion, but not all, of the cardiac-related peak is
removed for all three nerves. The residual peak in the
ASSND/AP indicates that there is activity in all
three nerves in the cardiac-related band that was not coherent to
activity in the AP. The residual power peaked at a lower frequency than
in the original ASSND (3.4 vs. 3.8 Hz). Power in
the cardiac-related band of the ASSND/AP
represented 34, 37, and 40% of the original power in the
cardiac-related band of ASCN,
ASVN, and ASRN,
respectively. A residual peak containing >1% of the original
cardiac-related power was observed in ASSND/AP in
14 of 24 experiments for all three nerves.

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Fig. 2.
Partial spectral and coherence analysis of cardiac-related rhythm in
discharges of CN, VN and RN. Top row, left to
right: ASCN,
ASVN, and ASRN.
Second row: AS have been partialized using the AP, from
left to right, ASCN/AP,
ASVN/AP, and ASRN/AP.
Third row: ordinary coherence functions are given, from
left to right, CohCN-VN,
CohCN-RN, and CohVN-RN.
Bottom: partialized coherence functions are, from left
to right, CohCN-VN/AP,
CohCN-RN/AP, and
CohVN-RN/AP.
|
|
In the example shown in Fig. 2, partialization of the nerve-nerve
coherence functions with AP produced a small reduction in peak
coherence within the cardiac-related band (compare 3rd and 4th rows) and moved the peak to the same frequency as peak
residual power in the ASSND/AP. The residual peak
in CohSND-SND/AP exceeded background levels of
coherence at frequencies >2 Hz.
The second type of outcome observed by partializing
ASSND with AP is shown in Fig.
3. In this case, cardiac-related power was almost completely removed in ASSND/AP
(residual cardiac-related power in ASCN/AP,
0.1%; ASVN/AP, 0.5%; and
ASRN/AP, 0.3%). In 10 of 24 experiments,
partialization of ASSND with AP removed >99% of
the cardiac-related power for all three nerves.

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Fig. 3.
Partial spectral and coherence analysis of cardiac-related rhythm in
discharges of CN, VN, and RN in a 2nd experiment. Top, left
to right, ASCN,
ASVN, and ASRN.
Second row, left to right:
ASCN/AP, ASVN/AP, and
ASRN/AP. Third row, left to
right: CohCN-VN,
CohCN-RN, and CohVN-RN.
Bottom row, left to right:
CohCN-VN/AP, CohCN-RN/AP,
and CohVN-RN/AP.
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|
In the case shown in Fig. 3, there was no clear peak in
CohCN-VN at the cardiac frequency (3rd row,
left), due to the high level of coherence at all frequencies >2
Hz. It is important to note that since the coherence function is a
normalized value it measures the correlation between the activity of
two signals at a given frequency, independent of the magnitude of that
activity (Bendat and Piersol 1966
). Partialization with
AP in this instance did not reduce coherence. For
CohCN-RN and
CohVN-RN, however, there was a peak in
coherence at the frequency of the heart beat (3rd row), and
this peak was substantially reduced by partialization with AP
(bottom).
The data from 14 experiments in which the
ASSND/AP contained >1% residual cardiac-related
power (group 1) and from the 10 experiments in which the
ASSND/AP contained <1% residual cardiac-related power (group 2) are summarized in Table
1. There was no significant difference
between these two groups with respect to mean arterial pressure,
cardiac frequency, CohAP-SND, or peak
CohSND-SND within the cardiac-related band.
CohAP-SND values were not significantly different
for the three nerves, and there was no difference between the
CohSND-SND values for the three nerve pairs. For each of
the three nerve pairs in both groups, CohSND-SND in the
cardiac-related band was significantly reduced by partialization with
AP. However, the reduction was significantly less for group
1 than for group 2 for each of the nerve pairs.
Figure 4 shows nerve to nerve
coherence functions for the three nerves following baroreceptor
denervation (CohSND-SND(Den)) superposed on to the same scale as
CohSND-SND/AP. In Fig. 4A, which is from the
same experiment as Fig. 2, CohSND-SND(Den) is lower than the CohSND-SND/AP at most frequencies
and particularly within the cardiac-related band due to the elimination
of the peak above background following sectioning of the baroreceptor nerves. CohSND-SND(Den) at the cardiac frequency
was significantly lower than peak CohSND-SND/AP
within the cardiac-related band in group 1 for each of the
nerve pairs (Table 1). In Fig. 4B, which is from the same
experiment as Fig. 3, there was little difference between
CohSND-SND(Den) and
CohSND-SND/AP. In group 2, statistically significant differences were not detected between CohSND-SND(Den) and
CohSND-SND/AP at the cardiac frequency
(Table 1).

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Fig. 4.
Comparison of effects of partialization using AP and baroreceptor
denervation on coherence functions relating the discharges of
sympathetic nerve pairs. Data in A and B
are from the experiments illustrated in Figs. 2 and 3, respectively.
Left to right: the partialized (dark
line) and baroreceptor denervated (light line) coherence functions are
superposed on the same scale for CohCN-VN,
CohCN-RN, and CohVN-RN pairs.
|
|
We used time series analyses to investigate the mode of relationship
between AP and SND in all 24 baroreceptor-innervated cats. In the 14 cases in which we found residual cardiac-related power in the
ASSND/AP (group 1), phase walk was the
predominant mode of coordination between AP and SND. In the 10 cases in
which residual cardiac-related power in ASSND/AP
was <1% of control (group 2), phase-locking was the
predominant mode of AP-SND relationship. Figure
5, A and B,
illustrates a time series for 20-s data epochs taken from the
experiments shown in Figs. 2 and 3, respectively. Positive and negative
values of phase angle refer to lags and leads of activity in the second
signal relative to the first. From top to bottom,
the time series in Fig. 5, A and B, show the phase angle between peak systole and CN, VN, and RN activities. In Fig.
5A a phase walk was observed for all three nerves, occurring over a portion of the cardiac cycle, with a period of ~3.3 s. The
range of the walk was 170, 160, and 220° for AP-CN, AP-VN, and AP-RN,
respectively. For all three nerves, the timing of the cardiac-related
slow wave was progressively delayed relative to peak systole from heart
beat to heart beat until reaching maximal values (AP-CN, 110°; AP-VN,
95°; and AP-RN, 175°) at which point a transition back to the
starting values (AP-CN,
50°; AP-VN,
55°; and AP-RN,
45°)
occurred more rapidly, generally within a few heart beats. In this
experiment, the walk was less regular for RN than for the other two
nerves. Heart rate was constant over this data epoch (240 beats/min),
and the blood pressure was stable (130/100 mmHg). In Fig.
5B, strong phase-locking was observed between each nerve and
the AP. The bands of phase angles were restricted to 55° for AP-CN,
50° for AP-VN, and 55° for AP-RN. Heart rate (210 beats/min) and
blood pressure (190/145 mmHg) were stable over the recorded epoch.

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Fig. 5.
Time series showing cycle-by-cycle measurements of phase angles between
peak systole (AP) and peak of cardiac related slow waves in CN
(top), VN (middle), and RN
(bottom). Data in A are from the
experiment illustrated in Figs. 2 and 4A. Data in
B are from the experiment illustrated in Figs. 3 and
4B. Positive and negative phase angles denote,
respectively, lags and leads of the slow wave relative to AP.
|
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Figure 6, A and B,
illustrates the nerve to nerve relationships for the same 20-s epochs
that are shown in Fig. 5, A and B, respectively.
From top to bottom, time series in Fig. 6,
A and B, show the CN-VN, CN-RN, and VN-RN phase
relationships. Phase-locking was the predominant mode of coordination
between the cardiac-related slow waves for the three nerve pairs in
both experiments. The CN-VN phase angles were largely restricted to a
band from
20 to 20° in Fig. 6A and from
5 to 15° in
Fig. 6B. In both experiments, phase-locking of the
cardiac-related slow waves of CN and VN was tighter than that observed
for the other two nerve pairs as might be expected from the higher
CohCN-VN within the cardiac-related band and at
all frequencies >2 Hz (Figs. 2 and 3).

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Fig. 6.
Time series showing cycle-by-cycle measurements of phase
angles between peaks of cardiac related slow waves in activities of,
top to bottom, CN-VN, CN-RN, and VN-RN
pairs. Data in A and B are from the
experiments illustrated in Figs. 2 and 3, respectively.
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For the experiments in group 1, on average, there were no
significant differences in the percentages of residual cardiac-related power for the three nerves (ASCN/AP, 25 ± 13% of control; ASVN/AP, 23 ± 11% of
control; and ASRN/AP, 22 ± 11% of
control). Nevertheless, there were individual cases in which
partialization differentially reduced cardiac-related power. An example
of this is shown in Fig. 7. On the
left, from top to bottom, the time
series show the phase angle between peak systole and cardiac-related
slow waves of CN, VN, and RN discharges. The AP-CN and AP-VN time
series show phase walk starting at approximately
10°, with
progressive phase delays of the slow waves in SND relative to systole,
up to a maximum phase angle of ~100° before rapidly returning to
10° (range 110°). The AP-RN phase walk started at around
30° and consisted of progressive delays of RN to a maximal value of 40°
(range 70°). The walk was less regular for AP-RN than for the other
two nerves. The ASSND and the
ASSND/AP for the three nerves are given on the
right of the corresponding time series. Residual cardiac-related power
in ASCN/AP and ASVN/AP was
31 and 28% of control respectively, but the
ASRN/AP contained only 15% residual
cardiac-related power. This suggests that the range of the phase walk
may be related to the amount of residual cardiac-related power in
ASSND/AP.

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Fig. 7.
Relationship between magnitude of AP-SND phase walk and reduction in
cardiac related power in ASSND produced by partialization
using AP. Left: time series showing cycle-by-cycle
measurements of, top to bottom, AP-CN,
AP-VN, and AP-RN phase angles. Middle: ASCN,
ASVN, and ASRN. Right:
ASCN/AP, ASVN/AP, and ASRN/AP.
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For the 14 experiments in which ASSND/AP
contained residual cardiac-related power (group 1), there
was a statistically significant correlation between the range of the
phase walk, measured as the standard deviation of the phase angles
between AP and SND, and the residual cardiac-related power in
ASSND/AP (percent of control) for each of the
three nerves (CN, r = 0.72, P = 0.002;
VN r = 0.70, P = 0.003; RN
r = 0.70, P = 0.004). In addition,
there was a significant inverse correlation between
CohAP-SND at the cardiac frequency and the
residual cardiac-related power in ASSND/AP (CN r =
0.72, P = 0.002; VN
r =
0.69, P = 0.004; RN
r =
0.83, P = 0.0001). Neither the
range of the phase walk nor the CohAP-SND at the
cardiac frequency was correlated significantly with MAP (range 110-195
mmHg). The relationships between residual power and
CohAP-SND and range of the phase walk suggest
that CohAP-SND and range of phase angles may be
negatively correlated. In fact, Fig. 8
illustrates that the inverse relationship between
CohAP-SND and range of phase angles was very
similar for both the 14 cases in which the predominant mode of
coordination was phase walk (group 1) and the 10 cases in
which phase-locking (group 2) was observed. This observation
that CohAP-SND was dependent on the range of AP-SND phase angles independent of the mode of coordination is explained in the APPENDIX.

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Fig. 8.
Relationship between CohAP-SND and the standard deviation
of the phase angles between AP and SND during phase walk
( ) and phase-locking ( ). The slopes of
the fitted lines for the 2 groups were not significantly different, and
for both modes of coordination the relationship was highly significant
(P = 0.0001).
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In 12 of the 14 experiments in group 1, residual power in
ASSND/AP peaked at a frequency 0.4 Hz below that
of the original cardiac-related peak in the
ASSND. This observation may be related to the
fact that in most cases, the phase walks were asymmetric, showing slow
progressive delays of nerve slow waves relative to timing of systole
(Figs. 5A and 7). In such cases, the slow-wave-to-slow-wave intervals during most of the phase walk were of longer duration (and
thus at a lower frequency) than the systole-systole intervals. There
were two instances in which residual power in
ASSND/AP consisted of two distinct peaks, one
below and one above the cardiac frequency. One of these cases is
illustrated in Fig. 9. In this
experiment, the initial peak in ASSND was at a
frequency of 3.0 Hz, and the ASSND/AP show two
clear peaks, one at 2.6 Hz, the other at 3.4 Hz. The larger of the two
peaks was at the lower frequency. Figure 10 shows a 20-s time-series plot of
AP-SND for the three nerves from this experiment. In this case, the
phase walk involves progressive delays to a maximal phase angle
followed by progressive shortening to a minimal phase angle for each of
the three nerves and is therefore more symmetrical than the examples
shown in Figs. 5A and 7. As a consequence, the intervals
between successive peaks of cardiac-related slow waves in SND were
longer than the AP-AP intervals in approximately the first half of the
walk (as phase angle is increasing) and shorter in the second half (as
phase angle is decreasing), resulting in residual power in
ASSND/AP at frequencies both lower and
higher than the heart rate. In this case,
CohSND-SND/AP contained two peaks within the
cardiac-related band at the frequencies of the two peaks in
ASSND/AP (Fig. 9).

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Fig. 9.
Partial spectral and coherence analysis of cardiac-related rhythm in
CN, VN, and RN discharges. Top row, left to
right: ASCN, ASVN, and
ASRN. Second row, left to
right: ASCN/AP, ASVN/AP, and
ASRN/AP. Third row, left to
right: CohCN-VN, CohCN-RN, and
CohVN-RN. Bottom row, left to
right: CohCN-VN/AP,
CohCN-RN/AP, and CohVN-RN/AP.
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Fig. 10.
Time series showing cycle-by-cycle measurements of, top
to bottom, AP-CN, AP-VN, and AP-RN phase angles. Data
are from the experiment illustrated in Fig. 9.
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Steady-state blood-pressure changes
From the steady-state blood-pressure experiments in the
accompanying paper (Lewis et al. 2000
), we examined 18 epochs from 7 animals in which phase walk was the predominant mode of
AP-SND coordination, and 18 epochs from 10 animals in which strong
phase-locking was the predominant mode. In all 18 cases of phase walk,
ASCN/AP contained a residual cardiac-related
peak, with residual power ranging from 7 to 69% (mean 33%) of that in
the control autospectrum, ASCN. In all of the 18 epochs of strong phase-locking, ASCN/AP contained
<1% residual cardiac-related power. Figure
11 shows the data from one experiment
at three steady-state blood pressures. These data are from the example
shown in Fig. 5 of the accompanying paper. In Fig. 11, A-C,
the plots are, from left to right, time series of
cycle-by-cycle measurements of the phase angle of CN relative
to peak systole, ASCN, and
ASCN/AP. In Fig. 11A, recorded with a
systolic blood pressure near 114 mmHg, values of AP-CN phase angles
were scattered over 360°, and there is no evidence of a peak in
ASCN at the cardiac frequency, although a clear
peak is seen at 10 Hz. The emergence of a 10-Hz rhythm in SND at low blood pressures has been well characterized (Barman et al.
1992
). Because there was no discernable cardiac-related power,
partialization of ASCN with AP had no effect. In
Fig. 11B, the systolic blood pressure was raised to near 200 mmHg, and at this level of blood pressure, phase-locking was
predominant. The values of AP-CN phase angles were restricted to a band
from 0 to 90°. Under these conditions, there was a sharp
cardiac-related peak in ASCN, which was almost completely removed by partialization with AP. In Fig. 11C,
the systolic blood pressure was raised to near 280 mmHg, and under these conditions, a phase walk between 0 and 210° with a period of
~3.8 s was observed. Note that in this case the phase walk consists
of progressive delays in AP-CN phase angle followed by progressive
advances. There is a sharp peak at the cardiac frequency (3.4 Hz) in
ASCN and considerable residual cardiac-related
power is apparent in ASCN/AP (59% of control).
Note also that the residual power consists of two peaks, one at 3.0 Hz,
the other at 3.4 Hz, consistent with a phase walk involving both
progressive delays and progressive advances in AP-CN phase angle.

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Fig. 11.
Time series and spectral analysis from 1 baroreceptor-innervated cat at
3 steady-state levels of blood pressure. A-C,
left to right: time series showing
cycle-by-cycle measurements of AP-CN phase angles, ASCN,
and ASCN/AP. A: systolic blood pressure was
114 mmHg; there is no coordination between AP and CN and no
cardiac-related power in ASCN. B: systolic
blood pressure was 200 mmHg and phase-locking was observed.
Cardiac-related power in ASCN was almost completely removed
by partialization with AP (ASCN/AP). C: at a
systolic pressure of 280 mmHg, phase walk was observed, and
cardiac-related power in ASCN/AP was only slightly reduced
from control (ASCN). The data shown here are from the
experiment shown in Fig. 5 in the accompanying paper (Lewis et
al. 2000 ).
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DISCUSSION |
In the accompanying paper, we have described two modes of
baroreceptor-sympathetic nerve coordination, phase walk and phase-lock, as reflected by the temporal relationship between SND and AP. The
former mode represents a nonlinear interaction of AP and SND, while the
latter represents a linear interaction (Lewis et al. 2000
). In this study, we have demonstrated that partialization of the autospectrum of SND with AP leaves considerable residual cardiac-related power when the predominant mode of coordination is
phase walk but removes virtually all cardiac-related power during
phase-locking. Residual cardiac-related power in
ASSND/AP results from activity in the
cardiac-related band of SND that is not coherent with the AP
(Jenkins and Watts 1968
; Rosenberg et al.
1998
). Thus while virtually all the cardiac-related SND is
coherent to AP during phase-locking, this is not the case during phase walk.
In the accompanying paper, the phase walk has been described in terms
of relative coordination, which is a characteristic feature of forced
nonlinear oscillators. We have suggested that the phase walk is caused
by the influence of a third factor on the AP-SND relationship, such as
blood pressure oscillations or an interaction with the respiratory
system (Lewis et al. 2000
), although such interactions
remain to be fully characterized and were not investigated in this
study. Whatever the mechanism generating the phase walk, the result is
a tendency for a significant portion (as much as 69%) of
cardiac-related SND to occur at a frequency distinct from that of the
heart rate. In most cases in the present study, cardiac-related
activity was attracted to a lower frequency, evidenced by a residual
peak in ASSND/AP at a frequency lower than that
of the heart rate. Such peaks are consistent with phase walk
characterized by progressive delays of cardiac-related sympathetic slow
waves relative to the timing of systole with abrupt snap backs from a
point of maximal delay.
Theoretically, relative coordination between two signals does not
result in any linear coherence (Hoyer et al. 1997
)
because one signal is attracted to, but never quite achieves, the
frequency of the other. Under these circumstances the phase walk is
through 360° (Kelso 1995
). In this sense, the AP-SND
phase walks that we have described do not fit the classical definition
of relative coordination since during phase walk there was still high
CohAP-SND at frequencies within the
cardiac-related band, and the phase walk occurred over <360°. Thus
it is probable that we never saw pure phase walk but rather a mixture
of phase walk and phase-locking. This may reflect the fact that our
system was never truly at a "steady state" in that the excitability
of the central sympathetic oscillator(s), and/or the strength of the
pulse synchronous baroreceptor input was continually changing in a
cyclic fashion.
The range of phase angles during phase walk was found to be predictive
of the amount of residual cardiac-related power observed in
ASSND/AP. When the phase angle between AP and SND
changes progressively from one cycle to the next, then the frequency of
SND is different from the heart rate (which in our experiments was
essentially constant due to the vagolytic actions of gallamine and the
sectioning of one cardiac sympathetic nerve). Thus the range of the
phase angles in the phase walk is proportional to the time that the frequency of SND is divergent from the frequency of AP and therefore should be proportional to the amount of residual power in
ASSND/AP in the cardiac-related band.
CohAP-SND was inversely related to the range of
AP-SND phase angles irrespective of whether phase walk or phase-locking
was the predominant mode of coordination between the two signals. In
fact, the slopes describing these relationships were the same for both
modes of coordination (see Fig. 8). The range of phase angles between
AP and SND observed during phase-locking is proportional to the amount
of random perturbation of the 1:1 relationship between the two signals.
As mathematically derived in the APPENDIX, such randomness
is as effective in reducing coherence between AP and SND as a
nonlinearity producing a phase walk with a similar range of phase angles.
Given that phase walk and phase-locking with equivalent ranges of phase
angles result in the same CohAP-SND and that the
CohAP-SND determines the portion of
ASSND that is removed by partialization using
AP, then why is it that a phase walk resulted in considerable residual cardiac-related power, while the random fluctuations that
occur in phase-locking did not? The phase walk alters the frequency of
SND slightly from that of the heart rate but in a structured manner,
and this activity therefore rises above background to form part of the
cardiac-related band that is seen in ASSND/AP. Random perturbations of the AP and SND relationship should be as
effective in producing residual power as a nonlinearity (Bendat and Piersol 1966
). However, due to the apparently random
fluctuations in AP-SND phase angles during phase-locking the power in
ASSND/AP at any given frequency within the
cardiac-related band would be indistinguishable from background activity.
Gebber et al. (1994)
and Kocsis (1995)
have previously reported that CohSND-SND/AP
remains high within the cardiac-related band. Kocsis
(1995)
commented on the possibility of a nonlinearity contributing to residual coherence but did not elaborate on the nature
of the nonlinearity. Gebber et al. (1994)
proposed that physical interconnection (coupling) of the central circuits governing discharges of different sympathetic nerves might also lead to residual
CohSND-SND/AP in the cardiac-related band. We
have demonstrated that a nonlinearity is present in the relationship
between AP and SND and that this nonlinearity results in
residual cardiac-related power in ASSND/AP. The
interpretation of CohSND-SND/AP in the cardiac-related band needs to be reexamined in this light.
The phase walk that produces residual cardiac-related power in
ASSND/AP is also responsible for the peak in the
residual CohSND-SND/AP within the cardiac-related
band in group 1. It would seem logical to assume that the
nonlinear AP-SND relationship is the result of central circuits for all
three nerves responding similarly to their baroreceptor inputs.
However, commonality of response to shared baroreceptor inputs
cannot be the only factor involved in generating residual
CohSND-SND/AP because significant coherence remains following baroreceptor denervation (Gebber et al.
1994
; Kocsis et al. 1990
; also Table 1). This
would require either some other form of shared input (noise) or
physical interconnections to be present generating some coherence
between nerve pairs.
There is evidence for some noise affecting the relationship between SND
and AP, particularly during phase-locking, in that there are apparent
random variations in phase angles. However, nerve-to-nerve phase angles
appear less disrupted by these random variations between SND and AP.
This suggests that either the source of noise is common or that there
are physical interconnections between the nerves leading to less
apparent noise in the nerve-to-nerve relationship, either of which
would produce residual CohSND-SND/AP. The
relationship between these three factors may be very complex. For
example, any nonlinearity in the AP-SND relationship may be shared not
only because of common responses to similar inputs to the central
circuits for different nerves but also because of physical
interconnections between these circuits, therefore the extent to which
each mechanism contributes to residual
CohSND-SND/AP in the cardiac-related band cannot
be stated.
In conclusion, there are two predominant modes of coordination between
AP and SND, phase-locking and phase walk. To an extent, these two modes
may always be coexistent. The effect of phase walk is to move a
considerable portion of the cardiac-related activity to frequencies
other than that of the heart rate, such that residual peaks are seen in
ASSND/AP and in
CohSND-SND/AP. However, during both modes of
coordination, other factors must be involved in generating
CohSND-SND/AP because there is still significant
coherence at frequencies near the heart rate after sectioning of the
baroreceptor nerves. Such factors may include physical interconnections
between central circuits generating SND and common inputs of
nonbaroreceptor origin.
The cross spectrum is defined as the Fourier transform of the cross
covariance function
Cross covariance is the second moment of the joint probability density
function relating xS1and
xS2. Coherence, therefore is directly
dependent on the joint probability density function of
xS1and
xS2.
This study was supported by National Heart, Lung, and Blood
Institute Grant HL-13187.
Address for reprint requests: G. L. Gebber (E-mail:
gebber{at}msu.edu).
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. Section 1734 solely to indicate
this fact.