| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
1 Department of Chemistry and Physics, We have described a
0.4-Hz rhythm in renal sympathetic nerve activity (SNA) that is tightly
coupled to 0.4-Hz oscillations in blood pressure in the unanesthetized
rat. In previous work, the relationship between SNA and fluctuations in
mean arterial blood pressure (MAP) was described by a set of two
first-order differential equations. We have now modified our earlier
model to test the feasibility that the 0.4-Hz rhythm can be explained by the baroreflex without requiring a neural oscillator. In this baroreflex model, a linear feedback term replaces the sympathetic drive
to the cardiovascular system. The time delay in the feedback loop is
set equal to the time delay on the efferent side, ~0.5 s (as
determined in the initial model), plus a time delay of 0.2 s on the
afferent side for a total time delay of ~0.7 s. A stability analysis
of this new model yields feedback resonant frequencies close to 0.4 Hz.
Because of the time delay in the feedback loop, the proportional gain
may not exceed a value on the order of 10 to maintain stability. The
addition of a derivative feedback term increases the system's
stability for a positive range of derivative gains. We conclude that
the known physiological time delay for the sympathetic portion of the
baroreflex can account for the observed 0.4-Hz rhythm in rat MAP and
that the sensitivity of the baroreceptors to the rate of change in
blood pressure, as well as average blood pressure, would enhance the
natural stability of the baroreflex.
autonomic nervous system; stability; sympathetic drive
WE HAVE PREVIOUSLY DESCRIBED a low-frequency 0.4-Hz
rhythm in renal sympathetic nerve activity (SNA) in the unanesthetized rat (2) that is strongly coupled to low-frequency oscillations in mean
arterial blood pressure (MAP). The source of this periodicity is
unknown because there is no other physiological system that is known to
oscillate at 0.4 Hz. This rhythm could be due either to a central
oscillator or to a resonant oscillation in the baroreflex control
system. For example, the 2- to 6-Hz oscillation and the 10-Hz rhythm
seen in the sympathetic nerve discharge of cats are thought to be
generated by brain stem circuits (1, 18). Although these rhythms are
seen in baroreceptor-denervated cats (1, 18), removal of the baroreflex
eliminates the 0.4-Hz rhythm in rats (10). The goal of this study is to
determine whether a feedback oscillation could explain the 0.4-Hz
rhythm.
Previously, we developed a linear model that relates efferent renal SNA
to fluctuations in MAP (4). This model uses a first-order differential
equation to describe the response of MAP to sympathetic drive. The
present differential equation for the baroreceptor reflex is built on
top of this model. In place of sympathetic drive, we now use the
baroreflex compensation as an input to the same first-order
differential equation. We model the baroreflex compensation as a
linear, proportional-plus-derivative feedback term with a time delay.
We analyze the stability of our baroreflex model to gain insight into
how the stability of the baroreflex depends on its sympathetic limb. In
general, a control system becomes unstable when the gain is too large
or any time delays within the feedback loop become too long. When the
system is perturbed, the perturbation dies away if the system is stable
or grows if the system is unstable. As the perturbation dies away (or
grows), the system oscillates with a resonant frequency determined by
the feedback loop. By definition, a marginally stable system verges on
becoming unstable. When a marginally stable system is perturbed, the
perturbation neither grows nor decays; the system simply oscillates
with its resonant (or marginal) frequency.
Based on time constants derived from earlier work, this new baroreflex
model predicts a resonant feedback oscillation in the 0.4-Hz range. A
Nyquist stability analysis (6) of our model yields stability criteria
for the open-loop gain that depend on a dimensionless ratio of the
model time constants. (A Nyquist analysis is a graphical method
especially suited to problems with a time delay.) The stability
analysis shows that the addition of a derivative feedback term enhances
the stability of the baroreflex within a certain parameter range.
Subjects.
Previously, we simultaneously recorded MAP and renal SNA in conscious
Sprague-Dawley rats during behavioral stress trials. These earlier data
provide certain key values used in the current model. Data were
measured on seven Sprague-Dawley rats (Harlan Industries, Indianapolis,
IN), weighing between 375 and 450 g, whose SNA and MAP response to
stressful and nonstressful tones has been described previously (12,
13).
Surgery.
The rats were anesthetized with pentobarbital sodium (65 mg/kg) in
preparation for implantation of the arterial catheter and renal nerve
electrodes. A Teflon catheter (no. 30 LW, ID 0.012; Small Parts, Miami
Lakes, FL) was inserted into the aorta by way of the caudal artery. A
sympathetic nerve coursing over the aorta toward the kidney was
identified through a flank incision. A small section of this nerve,
usually from the cephalad angle formed by the renal artery and aorta,
was dissected free of connective tissue and placed on fine, closely
spaced (0.4-0.8 mm), bipolar gold electrodes (A-M Systems,
Seattle, WA). The exposed nerve and electrode were encased in silicon
gel (Wacker Chemie, Munich, Germany). The distal ends of the catheter
and wires soldered to the end of the electrodes were tunneled under the
skin, exited at the nape of the neck, and led through a flexible
tether.
Experimental procedures and protocol.
The rats were tested in the conditioning paradigm starting 1 day after
surgery. The catheter was fitted to a pressure transducer (Cobe model
CDX-III), and blood pressure was recorded on a Grass model 7 polygraph.
The electrical signal from the renal sympathetic nerve was amplified
([times] 50) and band-pass filtered between 30 Hz and 3 kHz by a
Grass P511K differential amplifier and displayed on a Tektronix 5111 oscilloscope. Blood pressure and renal SNA were recorded during
presentation of five or more of each tone, after which the rat was
returned to its cage.
Data acquisition and analysis.
The blood pressure and SNA data were digitally sampled at 10,000 Hz.
Sampling began 9 s before presentation of the tone and continued until
6 s after its termination. In subsequent analysis, blood pressure was
reduced to a 1,000 samples/s signal by saving every 10th point. MAP was
then calculated for each pulse. This signal was averaged over 10 points
to produce 100 samples/s files. The initial highly detailed nerve
traffic signal was full-wave rectified and averaged over every 100 points to produce another 100 samples/s signal. This process retains
cumulative information from the initial 10,000 samples/s signal.
Model.
We used the physiological responses to the behavioral tests to develop
a model for sympathetic drive that took the SNA data time series as its
input and produced MAP fluctuations as its output (4). A key equation
of that model is
ABSTRACT
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References
INTRODUCTION
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References
METHODS
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References
where p is the fluctuation in MAP (the output) and
(1)
is the rate of change of p. The constant
T is the characteristic time for the frequency-response
function between the cardiovascular system and the effector elements
controlled by sympathetic nerves, and F(t)
represents the influence of sympathetic drive on MAP. The function
F(t) was obtained from our input SNA at an earlier time 
e, which represents the time delay on the efferent
side of the baroreflex. We lump together the other factors that may affect arterial blood pressure into an unknown function
u(t). For example, u(t)
includes any vagal effects on heart rate and the autoregulatory
function of the blood vessels themselves. Table 1 displays the time constants T and
e for the model of sympathetic drive that were
previously determined (4).
Table 1.
Time constants for the sympathetic drive model parameters
|
(2) |
is the
time delay in the feedback loop. Our analysis indicates how the
baroreflex (as a closed loop) would respond to an external input
v(t). The function v(t)
represents inputs to the cardiovascular system outside of the
baroreflex, such as respiration. Note that v(t)
differs from u(t) (in the earlier model), which
represented inputs to the cardiovascular system outside of sympathetic
drive. We include a derivative feedback term because the baroreceptors
respond not only to the magnitude of arterial pressure but also to the
rate of change of arterial pressure (16).
Again, the left-hand side (LHS) of our baroreflex model (Eq. 2) is identical to the LHS of Eq. 1, which we used to
relate SNA to MAP with the same time constant T (4). The time
delay for the feedback loop is equal to e + a, where e ~0.5 s occurs between
efferent nervous activity and MAP fluctuations (reported in Ref. 4) and
the afferent time delay a occurs on the afferent side of
the baroreflex. The afferent time delay between arterial baroreceptor
stimulation and the onset of the efferent nerve activity is on the
order of ~0.2 s (8), which gives a total time delay of ~0.7 s.
Green and Hefron (8) reported a range of values between 0.15 and
0.30 s for a, which corresponds to a mean value of
a = 0.2 s and a maximum uncertainty 
a = 0.08 s.
To illustrate how our model responds to a perturbation, we ran
simulations that were initialized with a short segment of blood pressure data from our previous tests (12, 13). The initialization period was during the initial 2-3 s of the 9-s control before the
onset of the behavioral tests. We approximated Eq. 2 with a
finite difference equation and solved for
pj at discrete times
tj corresponding to a time step of 0.01 s.
The resulting simulation shows how the system responds to a
"kick." The time constants T and came from Table 2.
The gains G and Gd were chosen to make the system
marginally stable, with Gd set at its optimal value
(defined in RESULTS).
Stability analysis. The stability of our model is determined by the roots of its characteristic equation, namely
|
(3) |
is a complex eigenvalue: = µ + i
.
If the rate at which perturbations decay or grow (µ) > 0, the
system is unstable. The letter i denotes
The complex part
of (i.e., ) is the angular frequency of the system: = 2
f, where f the natural frequency of the system. The
presence of the time delay makes the system infinite-dimensional.
Consequently, the characteristic equation has an infinite number of
roots. In general, the behavior of the system is determined by the root with the most positive real part (the dominant root).
First we consider the solution to Eq. 3 corresponding to
Gd = 0, which has been studied by Hayes (7, 9).
One can find the frequency for marginal stability (the system is
neither stable nor unstable) by setting the real part of (µ) equal
to zero. With Gd equal to zero, the marginal frequency satisfies
|
(4) |
|
(5) |
T > 0 (7). The corresponding marginal
gain is given by
|
(6) |
, we
iteratively solve Eq. 3 for to determine the stability and
natural frequency of the system. To find the dominant root, we start
with G near the marginal gain and Gd = 0. We use the root
from Eq. 4 as the initial guess. After we find the root for the
new G, we slightly modify G and/or Gd again and use
the newly found root as the initial guess for the next root. In this
way, we can find the dominant roots over a range of gains, with the
roots varying in a continuous fashion as a function of the gains.
To obtain a general picture of the stability of our model, we performed
a Nyquist stability analysis in the complex plane (6). Nyquist
stability analysis is a powerful method for determining whether all the
roots of the characteristic equation (Eq. 3) lie to the left
of zero in the complex plane. A Nyquist stability analysis utilizes the
Laplace transform of our baroreflex model. In Fig.
1, we show a block diagram of Eq. 2
where each block represents a transfer function in the complex
s-plane. H is the transfer function for the MAP
response to sympathetic drive, and G is the transfer function
for the sympathetic side of the baroreflex. P(s) is
the Laplace transform of the MAP fluctuations and
V(s) is the Laplace transform of the external input
v(t) to the system.
|
|
(7) |
1, 0) on the negative real axis in the complex
plane. Because of the time delay, a Nyquist plot of our open-loop
transfer function looks like a spiral. So, stability is determined by
the magnitude of GH when the spiral curve first crosses the
real axis at the crossover frequency. If the magnitude of GH is
>1, then the spiral curve encircles (1, 0) and the model is
unstable. By considering the crossover frequency, we determine stability constraints on the gains G and Gd as a function
of the dimensionless ratio /T.
| |
RESULTS |
|---|
|
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References
|
|---|
Figure 2 illustrates how our model responds
to a perturbation. In particular, it shows a simulation for 8 s as
compared with an actual data time series during the control period of a
trial with rat E. After the point indicated by the asterisk,
the system is freely running in response to the initial perturbation.
The response is an oscillation that is not due to any "inertia"
in the system. Rather, the oscillation is the result of the time delay
in the feedback loop. Notice the good agreement between the simulation
and the data time series while the system is freely running. For this
simulation, T = 1.3 s; this value is the mean characteristic
time for rat E (4). The time delay = 0.8 s; this value
equals e reported in Ref. 4 plus 0.2 s for
a (8). The gains were set to G = 3.6, Gd = 0.37 so that the system was marginally stable. For these parameters,
the system's natural frequency is 0.44 Hz.
|
In Table 2, we show the results of
calculations for each of the seven rats. For the characteristic time
T, we used the mean value reported for each rat (4). For the
time delay , we used the mean value of e plus 0.2 s
(8). For these time constants, we calculated the marginal gains and the
marginal frequencies from Eqs. 4 and 6, with the
derivative gain set to zero. The marginal frequencies are close to 0.4 Hz, and the marginal gains are reasonable physiological values for
open-loop gains of the baroreflex (16). The uncertainty in the time
delay is related to the uncertainties in e and
a via
|
e and a were
determined by independent methods. The uncertainties in f and G
were obtained from the range of results produced when T and were set equal to their extreme values.
|
In Table 3 we show the solution of Eq. 3 for rat B over a range of gains for the general case
(Gd 
0). For Table 3, we used the following values for
the time constants: T = 3 s, = 0.8 s. This
value of T for rat B was reported in Ref. 4. The time
delay is equal to the value of e for rat B
plus 0.2 s for a. For these time constants, the
marginal frequency is 0.35 Hz and the marginal gain is G = 6.6. For the
range of gains shown in Table 3, the oscillation frequencies remain
within 10% of the marginal frequency. As µ becomes more negative,
perturbations will decay away more rapidly. Note especially that for a
given G the presence of the derivative feedback term enhances the
stability of the system for small positive values of Gd.
|
A general picture of our model for the baroreflex can be obtained by
doing a Nyquist stability analysis. Figure
3A shows a Nyquist plot for an
unstable system (T = 3 s, = 0.8 s, G = 7, Gd = 0.0). The time constants are for rat B, and
Fig. 3A corresponds to line 7 of Table 3. Figure
3B shows a Nyquist plot for a stable system
(T = 3 s, = 0.8 s, G = 7, Gd = 0.4).
The addition of the derivative feedback term has made the system
stable. For Gd = 0, the radius of the spiral approaches
zero, as seen in Fig. 3A, whereas for nonzero Gd
the radius of the spiral approaches a limit equal to Gd,
as seen in Fig. 3B.
|
In the limit of large , the open-loop transfer function reduces to
GH 
Gdei
, which has a
magnitude equal to Gd. Thus, according to the Nyquist stability criteria, we must have |Gd| < 1 so that the curve GH(i) does not enclose
(1, 0). Further analysis shows that we must also have
|G| < 1, or, to ensure stability for
|G| > 1
|
(8) |
In Figure 4, we show a family of curves for
C(G, Gd) as a function of Gd for G = 5, 6, 7, and 7.34. In this plot, we used the same time constants
(appropriate for rat B ) as we used in Table 3 to give a
dimensionless ratio /T = 0.26. Each entry from Table 3 is
marked by an "x" on the appropriate curve. For a given
proportional gain G, the addition of a derivative feedback term
enhances the stability of the system for Gd > 0 but
decreases the stability of the system for Gd < 0. Figure
3A corresponds to the point on the curve labeled by "G = 7" at Gd = 0.0. Figure 3B corresponds
to the point at Gd = 0.4 on the same curve. Along a curve
for a fixed proportional gain G, there is an optimal derivative gain
Gd at the minimum of the curve, where the system is most stable. The top curve represents the maximum possible gain
(Gmax) the system can have without being unstable at the
optimal value of Gd. The addition of a derivative
feedback term has increased the maximum possible gain G from a marginal
gain of 6.6 to a value of ~7.34.
|
If Gd is set at its optimal value while G is varied, the natural frequency of the system remains about constant (invariant), as shown in Table 4. For the simulation shown in Fig. 2, G = Gmax and Gd is equal to its optimal value so that the simulation frequency is the invariant frequency of rat E. If the system remains near the optimal value of Gd, small changes in the system parameters will leave the frequency of the system unchanged.
To see the full advantage of the derivative feedback term, one should consider dynamic effects such as the rate at which perturbations decay (which is given by the real part of the eigenvalue µ.) For example, from Tables 3 and 4, for a proportional gain of G = 5.0, the rate at which perturbations decay doubles in going from no derivative feedback term to the optimal derivative gain Gd = 0.38. For a proportional gain of G = 6.0, making Gd = 0.40 triples the rate at which perturbations decay compared with only proportional feedback. The faster the cardiovascular system brings blood pressure fluctuations under control, the smaller the stress on cardiovascular organs.
|
From Eq. 8, we obtain an estimate of the maximum possible gain Gmax for stability
|
(9) |
1. The larger the ratio T/, the
larger Gmax can be. The right-hand side of Eq. 9
has a maximum at Gd = 0.44, which is an approximation for
the optimal value of Gd. For this value of
Gd, Gmax is ~1.82 T/. Our
largest value of T/ is 8.8 for rat A, making
Gmax < 16 for the time constants that we measured in
rats.
| |
DISCUSSION |
|---|
|
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References
|
|---|
The major finding of this study is that the 0.4-Hz periodicity in MAP
observed in rats (2) can be explained by the time delay obtained from
the known parameters of the baroreflex. The calculation of the marginal
frequency (for Gd = 0) depends solely on the known time
constants T and . As shown in Table 3, the frequencies of
the system remain near the marginal frequency over a range of gains.
Our analysis also indicates that our measured time constants for the
sympathetic limb of the baroreflex are sufficient to explain the 0.4-Hz
MAP rhythm. Other feedback loops such as vagal effects and the
renin-angiotensin system are not necessary for the generation of the
0.4-Hz rhythm.
We find that the time delay associated with the sympathetic limb of the baroreflex is ~0.7 s and that most of this time delay is associated with the efferent side of the baroreflex. The afferent time delay of ~0.2 s reported by Green and Hefron for cats (8) is consistent with our observations during behavioral conditioning trials for rats, where we reported an ~0.2-s delay between the start of a tone and the onset of a "sudden burst" of SNA (12).
To produce the coherence between MAP and SNA near 0.4 Hz observed by Brown et al. (2), the 0.4-Hz rhythm must be persistent across the data segments that are averaged together during the analysis. A strong coherence was not seen for respiratory interactions at ~1.5 Hz in the conscious rat because of the variability of respiration. Conversely, respiratory interactions were evident after anesthesia, which removed respiratory variability. A conceivable explanation for the relative invariance of the 0.4-Hz rhythm is that the baroreflex feedback loop remained in the "valley" associated with the optimal value of the derivative gain Gd, where the resonant frequency remains constant while the proportional gain G is varied (refer to Table 4). In this situation, the resonant frequency of the baroreflex would remain invariant even if the system's gain G changed as a result of a change in operating conditions.
Limitations. Our model only captures the sympathetic limb of the baroreflex and says nothing about how the numerous other feedback systems (e.g., vagal effects, the renin-angiotensin system, autoregulation) work together to regulate MAP. In addition, our model is linear and does not predict whether nonlinear effects could stabilize the baroreflex for gains that do not satisfy the stability criteria given here.
Though our model only captures the sympathetic side of the baroreflex, we believe for rats that the parasympathetic side is generally not critical in the generation of the 0.4-Hz rhythm. The vagal portion of the baroreflex acts on a faster time scale than the sympathetic portion. Consequently, vagal effects have the wrong time constants to generate the 0.4-Hz rhythm. Moreover, we were able to predict fluctuations in MAP using only sympathetic nerve traffic as our input (and ignoring parasympathetic activity) (4). The idea that a time delay in the baroreflex can be responsible for oscillations in MAP has been proposed for humans (5, 17) and for dogs (11). Our model for the baroreflex for rats is consistent with these earlier works. Our open-loop transfer function consists of a time delay in series with a low-pass filter, which is how the feedback loops are modeled in (17) and in (11). These earlier works are different from our model in that they contain both parasympathetic and sympathetic sides of the baroreflex acting on a beat-to-beat model of the circulation. However, our model is the only one to include a derivative term in the feedback and to show that a derivative term enhances the stability of the baroreflex. The simplicity of our model allows us to analyze the stability in detail and to understand the physiological limitations on the maximum value of the feedback gain G. The frequency response of our model is qualitatively similar to the response function of the deBoer model (5).Perspectives
Because of the difficulty of defining and measuring the open-loop baroreflex gain (14), we do not think there is a good value for the open-loop gain of the baroreflex for the rat reported in the literature. In particular, the sensitivity of the baroreflex control of heart rate is not a good measure of the ability of the baroreflex to control MAP (14). So, we can only speak in terms of the order of magnitude of the baroreflex gain. Roughly speaking, the open-loop gain for animal models seems to be small (i.e.,
10) compared with the
gains in engineering control systems (16). Consistent with these
experimental results, our model also indicates small gains for the
baroreflex on the order of 10. As explained by our stability analysis
in Eq. 9, the baroreflex gain G is small because of the
presence of a considerable time delay in the feedback loop.
In addition, our model gives an explanation of the physiological benefit of including rate information in the baroreflex. The addition of a derivative term in the feedback loop enhances the stability of the baroreflex for a positive range of derivative gains, with an optimal derivative gain Gd of ~0.44. Thus, with the presence of a derivative term, the baroreflex gain G can be larger, which leads to more precise regulation of MAP. The full advantage of the derivative feedback term is seen in its effect on the rate at which perturbations decay. From Tables 3 and 4, the addition of a derivative feedback term can enhance the rate of decay by a factor of two or three.
| |
APPENDIX |
|---|
|
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References
|
|---|
We will give an argument for the validity of Eq. 8 based on the concept of continuity. Consider the stability criterion function
|
First, we know that Eq. 8 is true in the limit as
Gd 0. In this limit, our model Eq. 2 reduces to
the linear delay equation studied by Hayes (9). Setting Gd
equal to zero in Eq. 8 gives
|
Secondly, C = 0 when the system is marginally stable as we now
show. Referring to the Nyquist stability plots in Fig. 3, stability is
determined at the crossover frequency 
when the
contour first crosses the real axis (6). When the system is marginally stable, the contour passes through the point (1, 0) so that
|
(10) |
, we obtain
|
(11) |
We can also write Eq. 10 as
![<FR><NU>G + <IT>i</IT>G <SUB>d</SUB>&ohgr;<SUB>&pgr;</SUB><IT>T</IT></NU><DE>1 + <IT>i</IT>&ohgr;<SUB>&pgr;</SUB><IT>T</IT></DE></FR> [cos (&ohgr;<SUB>&pgr;</SUB>&tgr;) − <IT>i</IT> sin (&ohgr;<SUB>&pgr;</SUB>&tgr;)] = −1](/content/vol273/issue6/fulltext/R1878/img018.gif)
|
(12) |
|
(13) |
and
setting the result equal to Eq. 11, we find
|
(14) |
In summary, Eq. 8 is true for Gd = 0 (when the curves cross the Gd = 0 axis in Fig. 4) and when C = 0 (when the curves cross the zero axis). Because the stability of the system can only change when C = 0, Eq. 8 must be true for all other points in Fig. 4 by continuity.
| |
ACKNOWLEDGEMENTS |
|---|
We thank Eugene Bruce and Bruce Walcott for their extremely helpful discussions and advice.
| |
FOOTNOTES |
|---|
This work was supported by American Heart Association (National) Grant 94012420, National Aeronautics and Space Administration EPSCoR Grant WKU-522611 to the University of Kentucky, National Heart, Lung, and Blood Institute Grant HL-19343, and Kentucky Spinal Cord and Head Injury Research Trust Grant RB-9601-K3 and a grant from the Tobacco and Health Research Institute.
Address reprint requests to D. R. Brown.
Received 19 September 1996; accepted in final form 29 August 1997.
| |
REFERENCES |
|---|
|
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References
|
|---|
1.
Barman, S. M.,
H. S. Orer,
and
G. L. Gebber.
A 10-Hz rhythm reflects the organization of a brainstem network that specifically governs sympathetic nerve discharge.
Brain Res.
671:
345-350,
1995[Medline].
2.
Brown, D. R.,
L. V. Brown,
A. Patwardhan,
and
D. C. Randall.
Sympathetic activity and blood pressure are tightly coupled at 0.4 Hz in conscious rats.
Am. J. Physiol.
267 (Regulatory Integrative Comp. Physiol. 36):
R1378-R1384,
1994
3.
Burgess, D. E.,
J. C. Hundley,
D. R. Brown,
S.-G. Li,
and
D. C. Randall.
A first-order linear differential-delay equation for the regulation of mean arterial blood pressure (Abstract).
FASEB J.
10:
A596,
1996.
4.
Burgess, D. E.,
J. C. Hundley,
S.-G. Li,
D. C. Randall,
and
D. R. Brown.
Multifiber renal SNA recordings predict mean arterial blood pressure in unanesthestized rat.
Am. J. Physiol.
273 (Regulatory Integrative Comp. Physiol. 42):
R851-R857,
1997
5.
DeBoer, R. W.,
J. M. Karemaker,
and
J. Strackee.
Hemodynamic fluctuations and baroreflex sensitivity in humans: a beat-to-beat model.
Am. J. Physiol.
253 (Heart Circ. Physiol. 22):
H680-H689,
1987
6.
Distefano, J. J.,
A. R. Stubberud,
and
I. J. Williams.
Schaum's Outline of Feedback and Control Systems. New York: McGraw-Hill, 1967.
7.
Glass, L.,
and
M. C. Mackey.
From Clocks to Chaos, the Rhythms of Life. Princeton, NJ: Princeton University Press, 1988, p. 72-78.
8.
Green, J. H.,
and
P. F. Hefron.
Studies upon the relationship between baroreceptor and sympathetic activity.
Q. J. Exp. Physiol.
53:
23-32,
1968.
9.
Hayes, N. D.
Roots of the transcendental equation associated with a certain difference-differential equation.
J. Lond. Math. Soc.
25:
226-232,
1950.
10.
Jacob, H. J.,
A. Ramananthan,
S. E. Parz,
M. J. Brody,
and
G. A. Myers.
Spectral analysis of arterial pressure liability in rats with sinoaortic deafferentation.
Am. J. Physiol.
269 (Regulatory Integrative Comp. Physiol. 38):
R1481-R1488,
1995
11.
Madwed, J. B.,
P. Albrecht,
R. G. Mark,
and
R. J. Cohen.
Low-frequency oscillations in arterial pressure and heart rate: a simple computer model.
Am. J. Physiol.
256 (Heart Circ. Physiol. 25):
H1573-H1579,
1989
12.
Randall, D. C.,
D. R. Brown,
L. V. Brown,
and
J. M. Kilgore.
Sympathetic nervous activity and arterial blood pressure control in conscious rat during rest and behavioral stress.
Am. J. Physiol.
267 (Regulatory Integrative Comp. Physiol. 36):
R1241-R1249,
1994
13.
Randall, D. C.,
D. R. Brown,
L. V. Brown,
J. M. Kilgore,
M. M. Behnke,
S. K. Moore,
and
K. R. Powell.
Two-component arterial blood pressure conditional response in rat.
Integrative Physiol. Behav. Sci.
28:
258-269,
1993[Medline].
14.
Sagawa, K.
Baroreflex control of systemic arterial pressure and vascular bed.
In: The Handbook of Physiology. The Cardiovascular System. Peripheral Circulation and Organ Blood Flow. Bethesda, MD: Am. Physiol. Soc., 1983, sect. 2, vol. III, pt. 2, chapt. 14, p. 453-496.
15.
Saul, J. P.,
R. D. Berger,
P. Albrecht,
S. P. Stein,
M. H. Chen,
and
R. J. Cohen.
Transfer function analysis of the circulation: unique insights into cardiovascular regulation.
Am. J. Physiol.
261 (Heart Circ. Physiol. 30):
H1231-H1245,
1991
16.
Scher, A. M.
Control of arterial blood pressure.
In: Physiology and Biophysics, edited by T. C. Ruch,
and H. D. Patton. Philadelphia, PA: Saunders, 1974, vol. II, p. 146-169.
17.
Wesseling, K. H.,
and
J. J. Settels.
Baromodulation explains short term blood-pressure variability.
In: Psychophysiology of Cardiovascular Control, edited by J. F. Orlebeke,
G. Mulder,
and L. J. P. VanDoornen. New York: Plenum, 1985, p. 69-97.
18.
Zhong, S.,
Z. Huang,
G. L. Gebber,
and
S. M. Barman.
Role of the brain stem in generating the 2- to 6-Hz oscillation in sympathetic nerve discharge.
Am. J. Physiol.
265 (Regulatory Integrative Comp. Physiol. 34):
R1026-R1035,
1993
This article has been cited by other articles:
|
|
 |
|
  H. van de Vooren, M. G. J. Gademan, C. A. Swenne, B. J. TenVoorde, M. J. Schalij, and E. E. Van der Wall Baroreflex sensitivity, blood pressure buffering, and resonance: what are the links? Computer simulation of healthy subjects and heart failure patients J Appl Physiol, April 1, 2007; 102(4): 1348 - 1356. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 D. R. Brown, L. A. Cassis, D. L. Silcox, L. V. Brown, and D. C. Randall Empirical and theoretical analysis of the extremely low frequency arterial blood pressure power spectrum in unanesthetized rat Am J Physiol Heart Circ Physiol, December 1, 2006; 291(6): H2816 - H2824. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 P. E. Hammer and J. P. Saul Resonance in a mathematical model of baroreflex control: arterial blood pressure waves accompanying postural stress Am J Physiol Regulatory Integrative Comp Physiol, June 1, 2005; 288(6): R1637 - R1648. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
D. C. Randall, B. R. Baldridge, E. E. Zimmerman, J. J. Carroll, R. O. Speakman, D. R. Brown, R. F. Taylor, A. Patwardhan, and D. E. Burgess Blood pressure power within frequency range ~0.4 Hz in rat conforms to self-similar scaling following spinal cord transection Am J Physiol Regulatory Integrative Comp Physiol, March 1, 2005; 288(3): R737 - R741. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 D. C. Randall, F. H. Wilbur, and T. J. Burkholder Two models for an effective undergraduate research experience in physiology and other natural sciences Advan Physiol Educ, June 1, 2004; 28(2): 68 - 72. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. Ursino and E. Magosso Role of short-term cardiovascular regulation in heart period variability: a modeling study Am J Physiol Heart Circ Physiol, April 1, 2003; 284(4): H1479 - H1493. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
D. E. Burgess, D. C. Randall, R. O. Speakman, and D. R. Brown Coupling of sympathetic nerve traffic and BP at very low frequencies is mediated by large-amplitude events Am J Physiol Regulatory Integrative Comp Physiol, March 1, 2003; 284(3): R802 - R810. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
P. A. Lanfranchi and V. K Somers Arterial baroreflex function and cardiovascular variability: interactions and implications Am J Physiol Regulatory Integrative Comp Physiol, October 1, 2002; 283(4): R815 - R826. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
H.-K. Liu, S.-J. Guild, J. V. Ringwood, C. J. Barrett, B. L. Leonard, S.-K. Nguang, M. A. Navakatikyan, and S. C. Malpas Dynamic baroreflex control of blood pressure: influence of the heart vs. peripheral resistance Am J Physiol Regulatory Integrative Comp Physiol, August 1, 2002; 283(2): R533 - R542. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
S. C. Malpas Neural influences on cardiovascular variability: possibilities and pitfalls Am J Physiol Heart Circ Physiol, January 1, 2002; 282(1): H6 - H20. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
P. van de Borne, M. Rahnama, S. Mezzetti, N. Montano, A. Porta, J. P. Degaute, and V. K. Somers Contrasting effects of phentolamine and nitroprusside on neural and cardiovascular variability Am J Physiol Heart Circ Physiol, August 1, 2001; 281(2): H559 - H565. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
S.-J. Guild, P. C. Austin, M. Navakatikyan, J. V. Ringwood, and S. C. Malpas Dynamic relationship between sympathetic nerve activity and renal blood flow: a frequency domain approach Am J Physiol Regulatory Integrative Comp Physiol, July 1, 2001; 281(1): R206 - R212. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
J. V. Ringwood and S. C. Malpas Slow oscillations in blood pressure via a nonlinear feedback model Am J Physiol Regulatory Integrative Comp Physiol, April 1, 2001; 280(4): R1105 - R1115. [Abstract] [Full Text] [PDF]
|
|
|
|
and 
.
