INTRODUCTION

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IT IS WELL ESTABLISHED that blood pressure in humans can contain a distinct oscillation at 0.1 Hz, often referred to as the Mayer wave (26, 38). Experiments in a variety of animal models have shown that this oscillation is due to the action of the sympathetic nervous system on the vasculature. Although the oscillation in blood pressure is shifted to 0.4 Hz in the rat (7) and to 0.3 Hz in the rabbit (22), changes in the strength of this oscillation have been proposed to reflect changes in the mean level of sympathetic nerve activity (SNA) and/or baroreflex gain (6), raising the possibility that measurement of the strength of this oscillation may be used as a diagnostic measure of neural control of the cardiovascular system in humans (1, 10, 26).

Current evidence favors the concept of feedback in the baroreflex loop as the origin for the 0.1-Hz oscillation in blood pressure (5, 6, 13, 25, 41). In this model, a change in blood pressure is sensed by the arterial baroreceptors altering the afferent signal to the central nervous system (CNS) and subsequently the mean SNA level, which, in turn, alters vascular tone in the target organ. In the feedback representation, changes in blood pressure at a certain frequency undergo a phase shift of 180 degrees, which, combined with the negative feedback sign (1), results in positive feedback, which sustains the oscillation at that frequency.

Burgess et al. (8) proposed a linear feedback model to account for this oscillation, adopting a particular form of linear controller to represent the neural controlling mechanism, employing both mean arterial pressure (MAP) and rate-of-change of MAP (proportional-derivative, or PD) to determine the SNA signal. However, this structure requires a very strict relationship between the vasculature and controller parameters to exist to maintain sustained (and stable) oscillations. Thus stimuli that alter gain along the feedback loop (e.g., altered baroreflex gain) would predispose the oscillation toward either instability (gain increase) or asymptotic stability (gain decrease) and it would increase without bound or cease altogether. For a stable oscillation to be maintained during such changes, a linear model implies continuous adaptation. Such a possibility would suggest that the oscillation is deliberate and has a useful function, as yet unknown, but is not simply a by-product of time delays in the baroreflex loop.

In the present study, we explore the hypothesis that a nonlinear feedback model is better able to explain the low-frequency oscillation, without the requirement for strict parametric relationships. This nonlinear model is also able to explain the frequency variability of the oscillation across different species and goes some way toward explaining the amplitude variation and presence/absence of the oscillation under certain physiological conditions. In addition, we test whether the nonlinearity is confined to the CNS or also extends to the vasculature response to SNA by constructing models from blood pressure and SNA data collected in conscious rabbits.

Glossary

SNA	Sympathetic nerve activity
MAP	Mean arterial pressure
CNS	Central nervous system
g( )	Nonlinearity (static) in central nervous system
e1	Preganglionic (efferent) delay
e2	Postganglionic (efferent) delay
v( )	Nonlinearity (static) in vasculature
Gv(s)	Linear vasculature dynamics
Gb(s)	Linear baroreceptor dynamics
a	Afferent delay
e	Total efferent delay
K(s)	Burgess' CNS controller
G(s)	Burgess' vasculature dynamics
H(s)	Burgess' feedback dynamics (afferent delay)
v	Lag in vasculature dynamics
kp	Proportional control gain
kd	Derivative control gain
k*d	Derivative control gain as a multiplier on v
N(a)	Combination nonlinearity in baroreflex loop
g(x)	Generalized sigmoidal characteristic
(x*, y*)	Center of symmetry of sigmoid characteristic
r*	Vertical range of sigmoid characteristic
,	Sigmoid shape parameters
wo	Frequency of oscillation, rads/s
G(z)	Discrete-time vasculature dynamics
n	Degree of denominator of G(z)
m	Degree of numerator of G(z)
d	Number of steps delay in G(z)
ai	Denominator coefficients in G(z)
ARX	AutoRegressive with eXogenous input model
MSEMAP	Mean squared error in MAP model prediction
MAPa	Actual MAP value
MAPm	Modeled MAP value
N	Number of data points

	METHODS

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For comparative purposes, model parameters previously published by Burgess et al. (8) for rats were used to compare the efficacy of a linear vs. nonlinear model in the synthesis of an oscillation at 0.4 Hz in rats.

Subsequently, to validate the model further and explore the location of the nonlinearity, resting levels of SNA and MAP were recorded in conscious rabbits (weight 2.5-3.0, mean 2.6 kg, 5 rabbits in total) for a 35-min period. Animals underwent surgery, at least 7 days before the recording, to implant a recording electrode around the left renal sympathetic nerve as previously described (28). Experiments were previously approved by the University of Auckland Animal Ethics Committee. Arterial pressure was measured from a catheter inserted in a central ear artery. SNA was amplified, filtered between 50 and 5,000 Hz, full-wave rectified, and integrated using a low-pass filter with a time constant of 20 ms. This integrated SNA signal and arterial pressure were continuously recorded throughout the experiment and were sampled at 500 Hz using an analog-to-digital data-acquisition card (National Instruments). Calibrated signals were displayed on a computer screen and saved to disk using a program written in the LabVIEW graphical programming language (National Instruments). For model development, data were subsequently filtered using a seventh-order Butterworth filter at 1 Hz and resampled at 2 Hz.

	RESULTS

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A Mathematical Vasculature Model

View larger version (18K): [in this window] [in a new window]   Fig. 1.   Block diagram of the central nervous system and vasculature that give rise to the components of the feedback model (see Glossary for definitions).

The pure time delays _a and _e = _e1 + _e2 are due to conduction time along the efferent and afferent nerves and neurotransmission.

A Linear Feedback Model

(1)

(2)

(3)

View larger version (13K): [in this window] [in a new window]   Fig. 2.   Components of the linear feedback system developed by Burgess et al. (8).

View larger version (17K): [in this window] [in a new window]   Fig. 3.   Variations in mean arterial pressure (MAP) (mmHg) around the mean level derived using a proportional-derivative controller, which indicates that the oscillation is difficult to sustain without the oscillation either growing (Kp = 4, dotted line) or decaying (Kp = 3, dashed line).

However, using a linear feedback model, these oscillations are difficult to sustain, without the oscillation amplitude either growing or decaying. This is illustrated in Fig. 3 for small variations in k_p. Increasing k_p gives an unstable response, while decreasing k_p results in asymptotic stability and the oscillation dies out. Burgess et al. determined the required condition for marginal stability as

(4)

A corollary of this requirement is that any variation in the parameters of the vasculature, due to hormonal effects, etc., must be accompanied by a corresponding adjustment in the other (e.g., CNS) parameters, to sustain the marginal stability condition (4). This implies that the feedback loop is continuously self-adaptive.

A Nonlinear Feedback Model

Our proposed nonlinear feedback model is shown in Fig. 4, which is similar to that in Fig. 2, with the following exceptions: 1) the PD controller has been replaced with a proportional controller, and 2) there is an amplitude-limiting sigmoidal nonlinearity in the forward path. This could belong to either the neural controller or the vasculature itself.

View larger version (12K): [in this window] [in a new window]   Fig. 4.   Components of the proposed nonlinear feedback loop (see Glossary for definitions).

Note that a nonlinear oscillation model does not preclude the use of a PD (or other) form of "controller" representation in the CNS. The choice of a gain is used purely for simplicity. Assuming a globally uniform relationship between blood pressure and SNA,¹ a sigmoidal characteristic is proposed in Fig. 5 where

(5)

Note that this representation is separate from the negative feedback (1) effect, accounting for the "forward" S shape, as opposed to the more familiar reverse S seen more commonly in the baroreflex literature. The parameters r*, , , x*, and y* specify the shape and position of the characteristic. It is known (4, 37) that a sigmoidlike characteristic represents the steady-state relationship between MAP and SNA and that the parameters describing the function vary with physiological condition. Figure 6 shows some of the possibilities, with the vasculature parameters specified in A Linear Feedback Model. Note also that the sigmoid as shown includes the effect of negative feedback (as shown in Fig. 4), accounting for the alternative orientation of the sigmoid curve compared to that normally presented for the arterial baroreflex.

View larger version (11K): [in this window] [in a new window]   Fig. 5.   Amplitude-limiting sigmoid characteristic between MAP (x*) and sympathetic nerve activity (SNA; y*). Note that this assumes a globally uniform relationship between blood pressure and SNA, which is not the case in all situations (see Limitations).

View larger version (24K): [in this window] [in a new window]   Fig. 6.   Variations in MAP (mmHg) around the mean level derived using a sigmoid nonlinearity in the forward path. The four cases, a, b, c, and d, are examples of changes in central nervous system parameters listed in Table 2 and indicate that under wide variations in the parameters of the sigmoidal curve (Fig. 5), the oscillation is stable.

The MAP responses in Fig. 6 correspond to the sigmoid parameters shown in Table 2, with  set equal to unity.

Stability Analysis

View larger version (22K): [in this window] [in a new window]   Fig. 7.   Stability analysis for the nonlinear feedback loop indicating the limit cycle is stable because increases in the oscillation amplitude cause movement back toward a region of stability.

Note that a stable limit cycle occurs where the [1/N(a)] curve intersects the GH(j) curve. The limit cycle is stable (i.e., operation tends toward this point) because increases in the oscillation amplitude, a, causes movement along the [1/N(a)] toward a region of stability, resulting in a decrease in a and movement back to point P. Note also that the point Q is given by

(6)

(7)

Depending on the sigmoid parameters, the point Q may lie to the left of point P (see example responses in Fig. 6), in which case no limit cycle occurs. On the basis of the available analysis, movement of Q to the left can be caused by an increase in r*. Movement of P to the right can be caused by a decrease in k_p (CNS gain). The value of  (curvature parameter) does not affect the rightmost limit of the [1/N(a)] curve. Because different points on the [1/N(a)] line correspond to different values of a, the specific intersection point of the [1/N(a)] line and the GH(j) curve determines the oscillation amplitude.

An analytic approximation for the describing function, in terms of the parameters of the sigmoid, has been evaluated, which allows some insight into the change in oscillation amplitude for variations in the baroreflex curve. However, this analysis is excluded for brevity, but the interested reader is referred to Ref. 17.

Interspecies Frequency Variability

It is of particular interest to examine the variation in oscillation frequency with the model time constant and delays, because this relates particularly to interspecies variations. The oscillation frequency may be determined as the frequency (in rads/s), which causes the phase of GH(j) to equal radians exactly. This occurs at a frequency _o, where

(8)

Unfortunately, Eq. 8 does not have an analytic solution, but can be solved by numerical optimization. A combined Gauss-Newton/quasi-Newton method is employed, which uses an analytic derivative calculation (easily calculated from Ref. 8) in the gradient minimization. This allows values of _o to be determined for different values of _v and _e + _a. Note that the efferent and afferent time delays are combined, because _o depends only on the total delay value. Figure 8 shows the variation in oscillation frequency (in Hz) with variations in lag and delay terms. In particular, the 0.1-, 0.3-, and 0.4-Hz frequencies have been highlighted, showing the required relationship between _v and _e + _a for humans, rabbits, and rats.

View larger version (31K): [in this window] [in a new window]   Fig. 8.   A and B: variation in the oscillation frequency (Osc. Freq.) with changes in the total delay and vasculature lag. The solid lines drawn at 0.4, 0.3, and 0.1 Hz reflect the known oscillations in blood pressure for rats, rabbits, and humans, respectively (see text for explanation).

Note that the oscillation frequency is much more sensitive to variations in delay than in time constant. Given that the smooth muscle characteristics are similar across species, the frequency variation across species is therefore explained by the difference in conduction times due to variation in size of species. Note also that this will (asymptotically) approach a limit, when the neurotransmission delay (relatively constant between species) dominates over the nerve conduction time.

A final comment relates to the use of a PD model for the baroreflex, instead of the pure gain term. The addition of such a term would add positive phase to the GH(j) plot in Fig. 7, causing it to rotate in an anticlockwise direction. The net result of this is that the intersection point of the GH(j) and [1/N(a)] curves changes, with a slight increase in the oscillation frequency.

Location of Nonlinearity Within Feedback Loop

With the data recorded for a group of rabbits, a model may be constructed for the vasculature section, as in Fig. 1. The synthesis and comparison of linear and nonlinear models allow a conclusion to be made regarding the presence (or absence) of nonlinearity in the vasculature section. Note that the data employed in this modeling exercise are taken from animals under resting levels of MAP and SNA, and thus lie well within the limiting values of MAP and SNA that are known to introduce nonlinear effects (4, 29). The data also contain an oscillation at ~0.3 Hz.

To focus on the MAP/SNA system, models were constructed for MAP, using only SNA as an input. Although these models are not likely to explain the complete variation in MAP [which can be attributed to many factors, such as heart-rhythm, respiration, etc. (18, 33)], the focus here is on the comparative performance of linear and nonlinear models. Note that the data are collected from baroreceptor-innervated animals, for the following reasons: It is desired to measure the characteristics under normal feedback (oscillatory) conditions, and although in closed-loop mode, the relation between SNA and MAP contains both the feed-forward effects of SNA on MAP as well as the feedback impact of MAP on SNA, the identification of the vasculature dynamics is nevertheless mathematically justified, because the CNS has a significant nonlinear component and prediction error methods are used for parameter identification (24).

Furthermore, it is important that a vasculature model be generated from data containing a low-frequency oscillation, so that any possible contributing nonlinear effects can be identified. Over the five animals considered, one rabbit had a strong low-frequency oscillation, two had less significant low-frequency oscillations, and the remaining two had no perceptible low-frequency component. Given the desired experimental conditions listed above, the rabbit with the significant oscillation was selected for modeling.

The original data were recorded at a sampling frequency of 500 Hz, but the data are then filtered and resampled at a frequency of 2 Hz. Use of a sampling rate of 2 Hz, with a seventh-order anti-aliasing filter cut-off of 1 Hz, eliminated the variations in the MAP data due to the heart rhythm. The spectrum of the resulting MAP signal is shown in Fig. 9, where the oscillation at ~0.25 Hz is manifested as a resonant peak.

View larger version (51K): [in this window] [in a new window]   Fig. 9.   Smoothed spectrum of resampled and filtered MAP against frequency (freq) from a single rabbit indicating an oscillation between 0.2 and 0.3 Hz.

In the (resampled) data set from the selected animal, a total of 942 points is available. The first 500 points were used for training the model, with the next 442 points used for model validation.

The complete data set was detrended (the mean values of SNA and MAP were removed, respectively), resulting in a variational model. Although such a data transformation is not required in a nonlinear model, linear modeling requires the removal of the dc (zero frequency) component to avoid biased parameter estimates. To facilitate a meaningful comparison, the transformation was applied to both model types.

A linear model for the vasculature section. A linear discrete-time AutoRegressive with an eXogenous input (ARX) model of the form

(9)

Initially, the values for n, m, and d must be determined. This was performed using a loss function analysis, which compares the performance of different model structures. Figure 10 shows the loss function plotted for variations in n, m, and d, respectively. The chosen order and delay values, based on simultaneous optimization, are

(10)

Note that the choice of d = 2 gives two-steps delay with a sampling period of 0.5 s, resulting in a total delay of 1 s, which accords well with measurements made via other routes (29). Although this may appear to imply that SNA would not affect MAP within this time period, the use of d = 2 for the identification of the vasculature component is justified mathematically.

View larger version (13K): [in this window] [in a new window]   Fig. 10.   Loss function variations for changes in model structure parameters n (A), m (B), and d (C).

View larger version (21K): [in this window] [in a new window]   Fig. 11.   Performance of the linear autoregressive with exogenous input (ARX) model for the training data (A) and the validation data (B). The solid line indicates the actual detrended MAP (mmHg), dotted line the predicted MAP.

A nonlinear model for the vasculature section. A nonlinear model was constructed using an artificial neural network (ANN). ANN models are data based and can use exactly the same input structure as the previous linear model, providing a solid base for comparison. The corresponding nonlinear model, using the same model orders and delays as in Eqs. 9 and 10, is given as

(11)

where the nonlinear function, v, is synthesized using an ANN.

View larger version (22K): [in this window] [in a new window]   Fig. 12.   Performance of the nonlinear ARX model for the training data (A) and the validation data (B). The solid line indicates the actual detrended MAP (mmHg), dotted line the predicted MAP.

Model comparisons. Figures 11 and 12 show little difference between results for the linear and nonlinear models. This difference is quantified in terms of the mean squared error (MSE) for both sets of results in Table 5, both for the single-step and multi-step prediction errors. The MSE is defined as

(12)

where MAP_a is the actual MAP value; MAP_m is the modeled MAP value, and N is the number of data points considered.

View larger version (14K): [in this window] [in a new window]   Fig. 13.   DC gain characteristic of artificial neural network model showing MAP (steady-state MAP in mmHg) against SNA (steady-state SNA in µV).

	DISCUSSION

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The main finding of this study is that a nonlinear feedback model can account for the oscillation in blood pressure at 0.3 Hz in rabbits. Models constructed from conscious rabbits under resting conditions deduce that no nonlinearity is contained within the vasculature, but rather is contained within the CNS. The advantage of the nonlinear feedback model is that it is stable under a wide variety of situations where gain at various points along the feedback loop may be altered. Our model indicates that such changes in gain would not produce changes in the frequency of the oscillation but rather change its strength. Conversely, a linear model as previously proposed by Burgess et al. (8) would be unable to sustain an oscillation with changes in gain unless the loop had a system for continuous adaptation. In other words, the CNS would have to go to great lengths to sustain the oscillation. Thus, if the feedback loop giving rise to the oscillation is truely linear, and not nonlinear as we propose, then the oscillation is deliberate and not simply a by-product of various time delays in the circuit. The conclusion to draw from such a linear process is that the oscillation has a functional purpose and that the CNS works hard to maintain its existance.

It is well established that the oscillation at 0.3 Hz in the rabbit is analogous to 0.4 Hz in the rat and 0.1 Hz in the human. Our nonlinear model is able to account for these species differences through changes in conduction time and indicates that changes in vasculature lag have little impact [generally considered relatively similar across species (40)]. The asymptotic behavior of oscillation frequency vs. species size is also relatively easily explained by observing (in Fig. 8) that oscillation frequency reduces asymptotically with decreasing conduction time. Recent research indicates that the oscillation is ~0.4 Hz in mice (23). It is likely that this frequency is close to the maximal rate achievable, given that the conduction time in such species has approached its asymptotic limit.

There is now evidence from several animal models that sympathetic overactivity can initiate and/or subsequently maintain a blood pressure increase. In humans, essential hypertension is associated with elevated plasma norepinephrine levels, whereas muscle SNA is elevated in borderline hypertensives (16). With regard to blood pressure variability, when one considers general variability using a simple SD of blood pressure over 24 h, the variability becomes progressively greater from normotensive to borderline, mild, and more severe essential hypertensive subjects (31). Understanding the origin and effect of this variability is likely to be of considerable clinical importance as previous studies have shown altered blood pressure and heart rate variability to be associated with increased risk of cardiovascular mortality (15, 32, 34), raising the possibility of a diagnostic test using measurement of blood pressure variability.

Whereas the mechanisms responsible for overall blood pressure variability are not yet defined, there has been the suggestion that the amplitude of the 0.1- to 0.4-Hz oscillation in blood pressure reflects either the mean level of sympathetic drive and/or changes in gain along the circuit (6, 26). In rabbits, stimuli that increase the mean level of renal SNA, such as hypoxia and hemorrhage, have been shown to increase the strength of 0.3-Hz oscillations in SNA (22, 28). In the past, we proposed that the increase in the power of the oscillation was due to the increase in the mean SNA level. Analysis of the nonlinear model reveals that this is probably an association rather than a causal effect. Thus it is changes in baroreflex gain (via k_p, r, or ) that give rise to changes in the power of the oscillation with an increase in mean SNA levels having no effect on its strength. We previously showed that hypoxia increases the gain of the MAP-SNA baroreflex curve as well as the mean SNA level (27). The model predicts that a stimulus that changes the gain along the reflex loop will result in increases or decreases in the strength of the 0.3-Hz oscillation. While this change in gain can most easily be detected by determining the MAP-SNA baroreflex relationship, our model indicates that an alteration in the gain in the vasculature response to sympathetic activity would also change the power of this oscillation.

In recent years, it has become popular to refer to the 0.1-Hz oscillation in heart rate as a marker of sympathetic tone. In comparison with the power of the faster oscillation in heart rate associated with respiration, there have been numerous papers reporting changes in sympatho-vagal balance in such varied conditions as anesthesia (20), sleep (3), and the menstrual cycle (36). A mounting number of studies indicates that this hypothesis is flawed on several points, and it is more probable that the 0.1-Hz oscillation in heart rate provides an index of baroreflex gain (5, 38, 39). Our model supports this hypothesis and may explain why some stimuli such as coronary occlusion, which increases mean SNA levels, was associated with reductions in power at 0.1 Hz in heart rate (19). Studies in humans add further support to this hypothesis, where stimulation of carotid baroreceptors by neck suction at two frequencies (0.1 and 0.2 Hz) induced a low-frequency oscillation in heart rate or blood pressure only if baroreflex sensitivity was normal, and that low baroreflex sensitivity was associated with reduced variability at this low frequency (38). Our nonlinear model provides the framework for predicting the changes in the power of the oscillation in blood pressure based on known changes in baroreflex gain. Though operation is normally in the "linear" portion of the sigmoid curve (35), the presence of an inflexion point has been demonstrated as sufficient to induce oscillations. Thus the oscillation amplitude can be well within the range of the sigmoid curve.

Limitations

The extension of the model to include all of these components not only represents a considerable body of work, but also contains significant experimental difficulties in measurement of SNA to multiple organs to parameterize such a model. The purpose of this paper is to highlight the fundamental model structure that can explain low-frequency oscillations, while not accounting for all possible physiological scenarios. In essence, we conclude that a slow oscillation in blood pressure is best described by a nonlinear feedback model.