Nonlinear identification of the total baroreflex arc: chronic hypertension model

Abstract

The total baroreflex arc is the open-loop system relating carotid sinus pressure (CSP) to arterial pressure (AP). Its linear dynamic functioning has been shown to be preserved in spontaneously hypertensive rats (SHR). However, the system is known to exhibit nonlinear dynamic behaviors. The aim of this study was to establish nonlinear dynamic models of the total arc (and its subsystems) in hypertensive rats and to compare these models with previously published models for normotensive rats. Hypertensive rats were studied under anesthesia. The vagal and aortic depressor nerves were sectioned. The carotid sinus regions were isolated and attached to a servo-controlled piston pump. AP and sympathetic nerve activity were measured while CSP was controlled via the pump using Gaussian white noise stimulation. Second-order, nonlinear dynamics models were developed by application of nonparametric system identification to a portion of the measurements. The models of the total arc predicted AP 21–43% better (P < 0.005) than conventional linear dynamic models in response to a new portion of the CSP measurement. The linear and nonlinear terms of these validated models were compared with the corresponding terms of an analogous model for normotensive rats. The nonlinear gains for the hypertensive rats were significantly larger than those for the normotensive rats [−0.38 ± 0.04 (unitless) vs. −0.22 ± 0.03, P < 0.01], whereas the linear gains were similar. Hence, nonlinear dynamic functioning of the sympathetically mediated total arc may enhance baroreflex buffering of AP increases more in SHR than normotensive rats.

the carotid sinus baroreflex plays a central role in maintaining arterial pressure (AP) in the face of fast-acting, exogenous disturbances and may also contribute to long-term AP regulation (7, 20, 21). This system responds to increases in carotid sinus pressure (CSP) by decreasing efferent sympathetic nerve activity (SNA), which, in turn, decreases AP. We refer to the aggregate, open-loop system relating CSP to AP as the total baroreflex arc, the “controller” subsystem relating CSP to SNA as the neural arc, and the “effector” subsystem relating SNA to AP as the peripheral arc.

We and others have identified the linear dynamics of the three baroreflex arcs in the form of transfer functions (i.e., gain and phase as a function of frequency) (3–5, 16). These linear models can capture the dynamic behavior of the systems to a significant extent. We previously showed that the linear dynamics of the total arc are preserved in spontaneously hypertensive rats (SHR) despite resetting of mean AP (4). However, the nonlinear dynamics of this system, which have been less investigated, could possibly respond differently to the chronic hypertension model.

In a recent study (9), we employed the Gaussian white noise approach for nonlinear system identification to develop a second-order, nonlinear dynamic model of the total arc in normotensive Wistar-Kyoto rats (WKY). The model predicted AP 12% better than a linear dynamic model in response to new Gaussian white noise and important nonlinear behaviors, including baroreflex thresholding and mean responses to input changes about the mean. The validated model revealed that the structure of the total arc is a linear dynamic system in parallel with a cascade combination of a squaring system and a different linear dynamic system, as shown in Fig. 1. This structure falls within the category of “Uryson” models (10).

Fig. 1.

Second-order Uryson model. x[n], Input; y[n], output; h₀, h₁[n], and h₂[n], 0th-, 1st-, and 2nd-order system kernels.

In this study we aimed to likewise establish second-order, nonlinear dynamic models of the total arc, as well as its subsystems, in SHR and to compare these models with our previously published models for WKY. Our results indicate that the second-order nonlinear dynamics of the total arc in SHR, which showed the same structure as in WKY, are augmented significantly more than the linear dynamics. Hence, nonlinear dynamic functioning of the total arc may enhance baroreflex buffering of AP increases more in SHR than WKY.

METHODS

Data collection.

Experiments were performed in eight SHR (22.2 ± 4.5 wk of age) under approval of the Animal Subjects Committee of the National Cerebral and Cardiovascular Center of Japan. The procedures are described elsewhere (4). Briefly, under general anesthesia and mechanical ventilation, the bilateral vagal and aortic depressor nerves were sectioned to eliminate the confounding effects of the aortic and cardiopulmonary baroreflexes but at the expense of only being able to study the carotid sinus baroreflex under sympathetic nervous mediation. The carotid sinus regions were isolated from the circulation and interfaced to a servo-controlled piston pump via catheters to control CSP. AP was measured via a fluid-filled catheter in a femoral artery. SNA was obtained via electrodes on a postganglionic branch of the splanchnic sympathetic nerve. The preamplified SNA was band-pass filtered with cutoff frequencies of 150 and 1,000 Hz and then full-wave rectified and low-pass filtered with cutoff frequency of 30 Hz. All signals were continuously recorded at a sampling rate of 200 Hz while CSP was controlled using two different Gaussian white noise stimulations. 1) To establish models for SHR at the same CSP level as the previous models for WKY, the Gaussian white noise CSP stimulation was at mean of 120 mmHg and standard deviation of 20 mmHg for ∼15 min (with values >3 SDs from the mean being skipped). So the signal ranged from ∼90 to 150 mmHg for 90% of the stimulation. A different signal realization was used for each animal. The switching interval of the noise was 500 ms to produce relatively flat CSP spectral power up to 1 Hz (Fig. 2). 2) To establish models for SHR at the normal CSP level of SHR, the Gaussian white noise CSP stimulation was at mean of 160 mmHg but the same otherwise. Hereafter, we refer to the former stimulation as SHR₁₂₀ and the latter stimulation as SHR₁₆₀.

Fig. 2.

Gaussian white noise training data from 1 animal. SHR₁₂₀ and SHR₁₆₀, spontaneously hypertensive rats during Gaussian white noise carotid sinus pressure (CSP) stimulation with mean of 120 and 160 mmHg, respectively. AP, arterial pressure; SNA, sympathetic nerve activity; AU, arbitrary unit.

Data preprocessing.

The signals were preprocessed as described in our previous report on nonlinear identification of the total arc in WKY (9). As described in our previous study (9), all signals were decimated to 2 Hz. For SHR₁₂₀ and SHR₁₆₀ of each animal, a 6-min segment of stationary data after linear detrending was selected to develop or train the models, and a distinct 3-min segment of such data was chosen to test the models. SHR₁₂₀ data from one animal and SHR₁₆₀ data from three animals were highly nonstationary and, thus, excluded from further analysis. AP was low-pass filtered with cutoff frequency of 0.7 Hz to remove any spectral peaks caused by spontaneous respiratory effort without impacting the modeling results. Since the SNA magnitude depended on the electrode contact, SNA was calibrated per animal, such that the average gain of the transfer function of the neural arc was unity for frequencies <0.03 Hz in the SHR₁₂₀ training data.

Nonlinear model and identification.

Nonlinear models of the total arc and its subsystems were developed as outlined in our previous report (9), where each system was assumed to be represented by a second-order Volterra series as follows

$$y\left[n\right]=h_0+{\displaystyle \sum _{k_1=0}^{\text{M}}{h}_1\left[k_1\right]x\left[n-k_1\right]}+{\displaystyle \sum _{k_1=0}^{\text{M}}{\displaystyle \sum _{k_2=0}^{\text{M}}{h}_2\left[k_1,k_2\right]x\left[n-k_1\right]x\left[n-k_2\right]}}$$ (1)

where n is discrete time, x[n] and y[n] are the measured input and output (i.e., CSP and AP for the total arc, CSP and SNA for the neural arc, and SNA and AP for the peripheral arc), with x[n] precisely denoting the input after removing its mean value, and h₀, h₁[n₁], and h₂[n₁,n₂] are the system kernels, with memory M, for estimation. The zeroth-order kernel h₀ is simply the mean value of y[n]. The first-order or linear kernel h₁[n₁], which is the time-domain version of the conventional transfer function, indicates how the present and past input samples (e.g., x[n] and x[n − 3]) affect the current output sample y[n]. The second-order nonlinear kernel h₂[n₁,n₂] indicates how the interaction between, or product of, two input samples that are n₁ and n₂ samples in the past (e.g., x²[n − 1] or x[n − 2]·[x − 4]) impact y[n]. While this model neglects higher-order nonlinearity, many physiological systems can be well represented with a second-order Volterra series (8).

The kernels of the total, neural, and peripheral arcs were estimated from the preprocessed SHR₁₂₀ and SHR₁₆₀ training data using a nonparametric, frequency-domain method (22). The memory M was set to 25 s, which is twice the length of the linear kernel of the total arc reported in our previous study (4). This value captured the memory of all systems (see results). The second-order kernel estimates were then visually examined to ultimately arrive at reduced nonlinear models with potentially more accurate kernel estimates (see results). Note that the second-order Uryson model of Fig. 1 is one example of a reduced nonlinear model. In this simpler model, the product of the present and past input samples of the same lag (e.g., x²[n − 1]) affect y[n] but not the product of past input samples of different lags (e.g., x[n − 2]·[x − 4]). Hence, while the second-order Volterra kernel is a function of two variables (i.e., h₂[n₁,n₂] as indicated in Eq. 1), the second-order Uryson kernel is a function of only one variable (i.e., h₂[n] as indicated in Fig. 1). Furthermore, in a second-order Uryson model, the second-order kernel (i.e., h₂[n]) differs in shape from the first-order kernel (h₁[n]), as implied in Fig. 1. This means that a second-order Volterra model may be reduced to a second-order Uryson model, if 1) the off-diagonal values of the second-order Volterra kernel are zero (i.e., h₂[n₁,n₂] = 0 for n₁ ≠ n₂) and 2) the diagonal values (i.e., h₂[n₁,n₂] for n₁ = n₂) are not simply proportional to the first-order kernel (i.e., h₂[n,n] ≠ a·h₁[n], where a is an arbitrary constant).

Model testing.

The resulting nonlinear models with the first- and second-order kernel estimates and linear models with only the first-order kernel estimates were evaluated. First, the inputs from the SHR₁₂₀ and SHR₁₆₀ training and testing data were applied to the models. Then R² values between the predicted and measured outputs were computed. Finally, R² values from linear and nonlinear models were compared after logarithmic transformation via paired t-tests with Holm's correction for multiple comparisons (2).

Model comparison with WKY.

The resulting validated models for SHR₁₂₀ and SHR₁₆₀ were compared with previously published models for WKY (9). The models for WKY were developed and validated by application of similar methodology to 10 age-matched WKY, except the Gaussian white noise stimulation was employed only at a mean of 120 mmHg, which is the normal CSP level of WKY. In particular, first- and second-order kernel estimates for SHR₁₂₀, SHR₁₆₀, and WKY were characterized in terms of three parameters: 1) area, to indicate the steady-state gain, 2) absolute peak amplitude, to indicate the maximal gain, and 3) dominant time constant [via a robust rectangular method (6)], to indicate the speed in reaching steady state. The area and absolute peak amplitude of the second-order kernel estimates were scaled by the standard deviation of the input, so that they could be meaningfully related to the corresponding parameters of the first-order kernel estimates. The kernel parameters for SHR₁₂₀, SHR₁₆₀, and WKY were compared after logarithmic transformation using unpaired t-tests, again with Holm's correction.

RESULTS

Gaussian white noise data.

Figure 2 shows the preprocessed CSP, AP, and calibrated SNA from the SHR₁₂₀ and SHR₁₆₀ training data of one animal. The group average (mean ± SE) mean and standard deviation of the preprocessed AP in the training data were 176.3 ± 15.5 and 9.4 ± 1.1 mmHg, respectively, for SHR₁₂₀ and 143.7 ± 15.1 and 10.6 ± 1.7 mmHg, respectively, for SHR₁₆₀. The corresponding values for WKY from our previous report were 97.1 ± 4.4 and 6.6 ± 0.2 mmHg (9). The mean of AP for SHR₁₂₀ and SHR₁₆₀ was significantly higher than that for WKY, which suggests baroreceptor resetting in SHR. The standard deviation of AP for SHR₁₂₀ and SHR₁₆₀ tended to be higher than that for WKY, which hints at enhanced total arc dynamics in SHR. The group average mean and standard deviation of the preprocessed SNA in the training data were 123.6 ± 17.5 and 25.0 ± 2.0 arbitrary units (AU), respectively, for SHR₁₂₀, 90.6 ± 16.7 and 28.9 ± 2.8 AU, respectively, for SHR₁₆₀, and, again from our previous report, 80.7 ± 11.8 and 20.7 ± 0.6 AU, respectively, for WKY. (Note that SNA cannot be compared between different animals because of the SNA calibration step.) These mean values with corresponding CSP levels and the standard deviation values define the operating point and range of applicability, respectively, of the models of the baroreflex arcs reported here.

Total arc model.

Figure 3 shows the group average first- and second-order kernels of Volterra models of the total arc estimated from the SHR₁₂₀ and SHR₁₆₀ training data. Note that the inputs for the first- and second-order kernels are CSP and CSP², respectively, while the output for both kernels is AP. Hence, in discrete time, the units are mmHg/mmHg (unitless) for the first-order kernel and mmHg/mmHg² (mmHg⁻¹) for the second-order kernel.

Fig. 3.

Group average 1st-order (linear) and 2nd-order kernel estimates (means ± SE) of complete Volterra models (see Eq. 1) of the total baroreflex arc in SHR₁₂₀ and SHR₁₆₀. Inputs for 1st- and 2nd-order kernels are CSP and CSP², respectively; output for both kernels is AP. Hence, in discrete time, units are mmHg/mmHg (unitless) for the 1st-order kernel and mmHg/mmHg² (mmHg⁻¹) for the 2nd-order kernel. Front view, view with azimuth of 135° and elevation of 0° with respect to the axis origin.

The kernels are qualitatively similar to those for WKY in two ways (9). 1) The second-order kernels revealed small off-diagonal values. In fact, the off-diagonal energies (i.e., sums of squares of h₂[n₁,n₁ + 1] and h₂[n₁,n₁ + 2],…) were typically <10% of the diagonal energy (i.e., sum of squares of h₂[n₁,n₁]), and all the off-diagonal energies were statistically smaller than the diagonal energy (P < 0.0003, by t-tests). Hence, the second-order kernels were approximately diagonal. 2) The shape of the diagonals of the second-order kernels was different from that of the first-order kernels (Fig. 4). As described above, these two attributes of the second-order kernel mean that the Volterra model may be reduced to the Uryson model of Fig. 1 for both SHR₁₂₀ and SHR₁₆₀. The kernels of the reduced models were reestimated using a nonparametric frequency-domain method (9) with M again set to 25 s. (This procedure yielded somewhat different and likely more accurate second-order Uryson kernel estimates than the diagonal of the second-order Volterra kernel estimates.) The resulting group average Uryson kernel estimates are shown in Fig. 4, while the numerical values of the kernel estimates, including those for WKY, are provided in the appendix. The four kernels were similar in shape to each other, as well as to the corresponding kernels for WKY (9). These kernels indicated low-pass characteristics and dynamics that were similar to each other (i.e., an impulse increase in CSP at time 0 would cause AP to decrease and then return to baseline without oscillation).

Fig. 4.

Group average 1st- and 2nd-order kernel estimates of reduced Uryson models of the total arc in SHR₁₂₀ and SHR₁₆₀.

Table 1 shows the group average R² values for the AP predicted by the individual animal models when stimulated by the Gaussian white noise CSP in the SHR₁₂₀ and SHR₁₆₀ training and testing data vs. the measured AP. The training data results actually indicate model-fitting ability, whereas the testing data results truly indicate model prediction capacity. In the testing data, the linear models achieved an R² value of only 0.45 ± 0.04 for SHR₁₂₀ but 0.59 ± 0.06 for SHR₁₆₀. The Uryson models significantly improved on these values by 43% for SHR₁₂₀ and 21% for SHR₁₆₀. The linear and Uryson model predictive capacities for WKY were more similar to those for SHR₁₆₀ than SHR₁₂₀ (9). Note that the predictive capacity of the Volterra models was worse than that of the Uryson models in the testing data because of overfitting in the training data.

Table 1.

Group average R² values for AP predicted by models of the total baroreflex arc vs. measured AP

	Model
	Linear	2nd-Order Volterra	2nd-Order Uryson
SHR120
Training data	0.53 ± 0.05	0.81 ± 0.02*	0.69 ± 0.04*
Testing data	0.45 ± 0.04	0.52 ± 0.07	0.64 ± 0.04*
SHR160
Training data	0.63 ± 0.05	0.82 ± 0.03*	0.71 ± 0.05*
Testing data	0.59 ± 0.06	0.63 ± 0.06	0.71 ± 0.05*

Values are means ± SE. AP, arterial pressure; SHR₁₂₀ and SHR₁₆₀, spontaneously hypertensive rats (SHR) during carotid sinus pressure stimulation with mean of 120 and 160 mmHg, respectively.

^*Statistical significance for paired t-test comparison with corresponding linear model after Holm's correction for 3 comparisons.

Table 2 shows group average parameters of the first- and second-order kernels of the validated Uryson models of the total arc for SHR₁₂₀ and SHR₁₆₀, as well as the corresponding values for WKY from our previous report (9). The absolute peak amplitude and time constant of the linear kernel for SHR₁₆₀ were nearly twice as large and almost half as small as those for WKY, respectively. These differences were either significant or close to significant. However, the area (or gain) of the linear kernel for SHR₁₆₀ and all three parameters of this kernel for SHR₁₂₀ were not significantly different from those for WKY. The gains and absolute peak amplitudes of the second-order kernels indicated a 20–60% magnitude of effect relative to their linear kernel counterparts, whereas the time constants of the second-order kernels generally indicated a slower effect than the linear kernels. The parameters of the second-order kernels for SHR₁₂₀, SHR₁₆₀, and WKY were more significantly different than those of the linear kernels. In particular, the gains and absolute peak amplitudes of the second-order kernels for SHR₁₂₀ and SHR₁₆₀ were ∼170–270% larger than those for WKY. In addition, the time constant of the second-order kernel for SHR₁₂₀ was ∼60% smaller than those for WKY and SHR₁₆₀. All these differences were significant. In sum, the second-order kernel of the total arc was augmented significantly more than the linear kernel for SHR than WKY.

Table 2.

Group average parameters of the kernels of the validated Uryson model of the total arc

	WKY	SHR120	SHR160
Linear kernel
Area/gain, unitless	−0.70 ± 0.11	−0.61 ± 0.15	−0.76 ± 0.09
Absolute peak amplitude, unitless	0.085 ± 0.011	0.127 ± 0.022	0.165 ± 0.029
Time constant, s	4.1 ± 0.5	2.4 ± 0.4	2.4 ± 0.2
2nd-Order Uryson kernel
Area/gain, unitless	−0.22 ± 0.03	−0.37 ± 0.05	−0.38 ± 0.04
Absolute peak amplitude, unitless	0.020 ± 0.004	0.053 ± 0.007	0.037 ± 0.006
Time constant, s	6.2 ± 0.8	3.5 ± 0.4	5.5 ± 0.6
	P Values
	WKY vs. SHR120	WKY vs. SHR160	SHR120 vs. SHR160
Linear kernel
Area/gain, unitless	0.49	0.41	0.23
Absolute peak amplitude, unitless	0.12	0.021	0.31
Time constant, s	0.10	0.007*	0.57
2nd-Order Uryson kernel
Area/gain, unitless	0.019*	0.006*	0.65
Absolute peak amplitude, unitless	<0.001*	0.013*	0.091
Time constant, s	0.006*	0.75	0.014*

Values are means ± SE. WKY, Wistar-Kyoto rats during Gaussian white noise carotid sinus pressure stimulation with mean of 120 mmHg. WKY values are from Ref. 9. P values were obtained via unpaired t-tests.

^*Statistical significance after Holm's correction for 3 comparisons.

Neural arc and peripheral arc models.

Similar to our earlier finding in WKY (9), the first- and second-order kernels of Volterra models of the neural arc estimated from the SHR₁₂₀ and SHR₁₆₀ training data suggested reduced Uryson models (results not shown). Figure 5 shows the group average kernels of the Uryson models of the neural arc estimated from these data. The four kernels appeared similar in shape to each other, as well as to the corresponding kernels for WKY (9). These kernels indicated high-pass characteristics and dynamics that were similar to each other (i.e., an impulse increase in CSP at time 0 would cause SNA to decrease and then return to baseline with oscillation) and were much faster than their total arc counterparts (Fig. 4).

Fig. 5.

Group average 1st- and 2nd-order kernel estimates of Uryson models of the neural arc in SHR₁₂₀ and SHR₁₆₀.

Reliable second-order kernels of Volterra models of the peripheral arc could not be estimated, because the input of this system (SNA) was not Gaussian white noise. Figure 6 shows the group average kernels of linear models of the peripheral arc estimated from the SHR₁₂₀ and SHR₁₆₀ training data. The two kernels were similar overall to each other and to the corresponding kernel for WKY (9). These kernels indicated low-pass characteristics and expected open-loop dynamics (i.e., an impulse increase in SNA at time 0 would cause AP to increase and then return to baseline without oscillation) and were similar in speed to their total arc counterparts.

Fig. 6.

Group average 1st-order kernel estimates of linear models of the peripheral arc in SHR₁₂₀ and SHR₁₆₀.

Table 3 shows the group average R² values for the SNA predicted by the individual subject models of the neural arc when stimulated by the Gaussian white noise CSP in the SHR₁₂₀ and SHR₁₆₀ training and testing data vs. the measured SNA. In the testing data, the linear models achieved high R² values of 0.65 ± 0.03 for SHR₁₂₀ and 0.82 ± 0.01 for SHR₁₆₀. The Uryson and Volterra models significantly improved on the value for SHR₁₂₀ by 8–9% but did not appreciably improve the value for SHR₁₆₀ (1%). So the neural arc for SHR₁₆₀ was approximately linear. Table 3 also shows the group average R² values for the AP predicted by the individual animal linear model when stimulated by the SNA in the SHR₁₂₀ and SHR₁₆₀ training and testing data vs. the measured AP. For both the training and testing data, the R² values were high. In particular, for the testing data, the R² values were 0.83 ± 0.03 for SHR₁₂₀ and 0.78 ± 0.10 for SHR₁₆₀. Hence, the peripheral arc was approximately linear. The predictive capacities of the models of the neural arc for SHR₁₆₀ and the models of the peripheral arc for both SHR₁₂₀ and SHR₁₆₀ were similar to those for WKY (9).

Table 3.

Group average R² values for efferent SNA/AP predicted by models of the neural and peripheral arcs vs. measured SNA/AP

	Neural Arc
	Linear	2nd-Order Volterra	2nd-Order Uryson	Peripheral Arc (Linear)
SHR120
Training data	0.65 ± 0.03	0.76 ± 0.02*	0.73 ± 0.02*	0.82 ± 0.04
Testing data	0.65 ± 0.03	0.71 ± 0.03*	0.70 ± 0.03*	0.83 ± 0.03
SHR160
Training data	0.82 ± 0.01	0.85 ± 0.01*	0.84 ± 0.01*	0.83 ± 0.07
Testing data	0.82 ± 0.01	0.83 ± 0.01	0.83 ± 0.01*	0.78 ± 0.10

Values are means ± SE. SNA, sympathetic nerve activity.

^*Statistical significance for paired t-test comparison with corresponding linear model after Holm's correction for 3 comparisons.

As described in our previous, extensive report on the linear dynamics of the baroreflex arcs in SHR (4), the parameters of the kernels of the validated linear models of the neural and peripheral arcs for SHR₁₂₀ and SHR₁₆₀ were mostly not significantly different from those for WKY to the extent that they could be compared (i.e., the gains and absolute peak amplitudes of these kernels cannot be compared between different subjects because of the SNA calibration step). Finally, the linear and second-order kernels of the validated Uryson model of the neural arc for SHR₁₂₀ were characterized by gains of −0.55 ± 0.09 and −0.27 ± 0.06 AU/mmHg, absolute peak amplitudes of 0.93 ± 0.02 and 0.27 ± 0.04 AU/mmHg, and time constants of 0.30 ± 0.05 and 0.48 ± 0.06 s, respectively.

DISCUSSION

The major finding of this study is that nonlinear dynamic functioning of the sympathetically mediated carotid sinus baroreflex is enhanced significantly more in SHR than its linear dynamic functioning. We arrived at this result by developing and validating dynamic models of the total baroreflex arc and its subsystems in SHR via Gaussian white noise CSP stimulation and nonlinear system identification and then comparing these models with our previously established models for age-matched WKY (9).

Nonlinear identification method.

This study falls within the large body of literature on system identification analysis of cardiovascular variability interactions (11, 12, 14). Among the various identification methods that have been employed, we chose a frequency-domain method (22) to estimate the kernels of the nonlinear models. This nonparametric method assumes that the input is Gaussian and broadband but makes no assumptions on the form of the kernels. The method yields the best unbiased estimates of the linear and nonlinear kernels in the least-squares sense. Hence, the linear term of the nonlinear model and the optimal linear model are one and the same. Parametric identification methods, which have been widely employed in this area (11, 13, 14), could provide better estimates of the kernels by assuming a particular kernel form, so as to trade off bias for precision. This possibility could especially hold in the identification of the peripheral arc, the input of which was not as broadband as the other investigated systems. However, our goal was to determine the form of the kernel. Even so, we did apply conventional autoregressive exogenous input identification to estimate the linear kernels, and this method did not yield more predictive linear kernels than the frequency-domain method (results not shown).

Total arc model in SHR.

We applied the frequency-domain method to develop second-order nonlinear dynamic models of the total arc for both SHR₁₂₀ (SHR with Gaussian white noise stimulation at the normal CSP level for WKY) and SHR₁₆₀ (SHR with the same stimulation but at the prevailing CSP level for SHR) (Fig. 2). These models were qualitatively similar to the corresponding model for WKY. In particular, they generally indicated that the total arc may be represented as a linear dynamic system in parallel with a cascade combination of a squarer and a slower, linear dynamic system (Figs. 1, 3, and 4). Hence, total arc nonlinearity, which was captured by the squaring and slower, linear dynamic systems, was in the low-frequency regime.

These Uryson models significantly improved AP prediction over standard linear models by 43% for SHR₁₂₀ and 21% for SHR₁₆₀ (Table 1). The predictive capacity of the Uryson model relative to a linear model for SHR₁₂₀ was superior to those for WKY and SHR₁₆₀. The reason for the relatively stronger nonlinearity in SHR₁₂₀ may pertain to the operating point. That is, the CSP levels for WKY and SHR₁₆₀ are in the linear regimes of their respective static sigmoidal CSP-AP relationships, while the CSP level for SHR₁₂₀ is near the nonlinear thresholding regime of its relationship (15). In this sense, comparisons between WKY and SHR₁₆₀, as opposed to SHR₁₂₀, may actually be more appropriate.

The validated models of the total arc for SHR₁₂₀ and SHR₁₆₀ were, however, quantitatively different from the corresponding model for WKY (Table 2). More specifically, the linear kernel (time-domain version of the transfer function of the faster, dynamic system) for SHR₁₆₀ showed significantly augmented transient dynamics in terms of magnitude and speed compared with the corresponding kernel for WKY. The linear kernel for SHR₁₂₀ showed some tendency for similarly enhanced transient dynamics. However, the linear kernels for SHR₁₂₀, SHR₁₆₀, and WKY revealed steady-state gains, which are more meaningful, of very similar values. By contrast, the nonlinear kernels (time-domain version of the transfer function of the slower, dynamic system) for SHR₁₂₀ and SHR₁₆₀ showed enhanced steady-state gains by ∼170% and significantly augmented transient dynamics in terms of magnitude (and speed for SHR₁₂₀) relative to the corresponding kernel for WKY. Note that while the linear kernels change AP in the opposite direction of CSP, the nonlinear kernels always reduce AP because of the squaring of CSP (Fig. 1). Hence, in normal, closed-loop conditions, the nonlinear steady-state gain of the total arc may augment buffering of AP increases, while blunting buffering of AP decreases, to a greater extent in SHR than WKY. Furthermore, the nonlinear steady-state gain of the total arc may decrease mean AP in response to increases in the AP variance to a greater extent in SHR than WKY.

Neural arc and peripheral arc models in SHR.

We also developed second-order Uryson models of the neural arc for SHR₁₂₀ and SHR₁₆₀ (Fig. 5). However, linear models of the neural arc showed good-to-excellent SNA prediction, so the Uryson model for SHR₁₆₀ did not improve on the SNA prediction (Table 3). Similarly, a linear model of the neural arc sufficed for WKY (9). While the neural arc model for SHR₁₂₀ was able to improve SNA prediction, perhaps due to stronger nonlinearity at the different operating point (Table 3), the 8% improvement was small compared with the 43% AP prediction improvement attained by the Uryson model of the total arc for SHR₁₂₀. Similar to WKY, we could only develop linear models of the peripheral arc for SHR₁₂₀ and SHR₁₆₀ (Fig. 6), as the SNA input to this system was not Gaussian white noise. However, these standard models showed excellent AP prediction. As we described in a previous, extensive report (4), the kernels of the validated linear models of the neural and peripheral arcs for SHR₁₂₀ and SHR₁₆₀ were mostly not significantly different from those for WKY. However, comparisons of these kernels were substantially limited because of the SNA calibration step.

In sum, the total arc models showed nonlinear behaviors, while the models of the neural and peripheral arc subsystems, especially for WKY and SHR₁₆₀, showed approximately linear behaviors. As we discussed earlier (9), one explanation for this puzzling result is that the neural arc was nonlinear, but its nonlinearity was not well identified because of a linearizing effect caused by confounding SNA from higher brain centers. Note that the improved predictive capacity of the Uryson model of the neural arc for SHR₁₂₀, wherein nonlinearity may have been stronger, supports the contention that the neural arc was nonlinear.

Potential physiological mechanisms.

This study does not reveal the mechanisms underlying the more significant enhancement of nonlinear dynamic functioning of the total arc in SHR. However, previous studies provide a bit of insight. 1) The carotid artery stiffens in SHR (1). Since the carotid sinus baroreflex precisely responds to stretch, such stiffening alone would suggest blunted total arc functioning in SHR. Hence, enhanced functioning downstream in the total arc must have occurred in terms of linear dynamics and, to a greater extent, nonlinear dynamics. 2) Baroreflex control of heart rate is blunted in SHR after vagal block (4, 18). Therefore, downstream dynamic functioning pertaining to the control of vascular properties and/or cardiac contractility must have specifically been enhanced. 3) It may be possible that the linear component (faster dynamic system) and nonlinear component (squaring and slower dynamic systems) of the Uryson model correspond to myelinated and unmyelinated fiber pathways of the total arc (17, 19) and that these pathways are differentially impacted in SHR relative to WKY.

Study limitations.

As we outlined previously (9), this study has experimental and mathematical limitations. More specifically, our experimental procedures included the use of anesthesia, opening of the baroreflex loop, and elimination of the vagal component of the baroreflex, while our mathematical procedures neglected higher-order nonlinearity. These procedures are limitations for sure. At the same time, they may have permitted as accurate an identification of the sympathetically mediated total arc as possible without assuming any form for the nonlinearity.

Perspectives and Significance

The carotid sinus baroreflex is known to exhibit nonlinear dynamic behaviors, including thresholding and saturation and mean responses to pulsatile input changes. Yet a simplifying assumption of many studies of this system has been that it obeys the linearity principle. The present results indicate the pitfall in ignoring baroreflex nonlinearity: the nonlinear gain of the sympathetically mediated total baroreflex arc was enhanced in SHR relative to WKY, while the linear gain was preserved. Hence, the nonlinear dynamic functioning of this system may enhance steady-state baroreflex buffering of AP increases more in SHR than WKY, perhaps to compensate for malfunctioning of other regulatory systems. If the common linearity assumption were invoked here, the story would have been different. This study is not the first to demonstrate the significance of baroreflex nonlinearity in hypertension. In Thrasher's chronic baroreceptor unloading model of hypertension (20, 21), mean CSP did not change but carotid sinus pulse pressure decreased, which, in turn, led to a sustained, baroreflex-mediated increase in mean AP. This nonlinear behavior of the carotid sinus baroreflex, by contrast, played a causative, rather than protective, role in the hypertension model. Future investigations of baroreflex nonlinearity in hypertension may improve our understanding of the role of the baroreflex in this prevalent disease process.

GRANTS

This work was supported in part by National Institutes of Health Grants AG-041361 and EB-018818.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

M.M. analyzed the data; M.M., T.K., K.S., M.S., and R.M. interpreted the results of the experiments; M.M. prepared the figures; M.M. drafted the manuscript; M.M., T.K., K.S., M.S., and R.M. approved the final version of the manuscript; T.K. performed the experiments; T.K., M.S., and R.M. edited and revised the manuscript; R.M. developed the concept and designed the research.

APPENDIX

The numerical values specifying the group average kernel estimates of the Uryson model of the total arc for SHR₁₂₀ and SHR₁₆₀ in Fig. 4, as well as for WKY in our previous study (9), are provided in Table A1.

Table A1.

Numerical values of kernel estimates

	Kernel of Uryson Model of the Total Arc
	WKY		SHR120		SHR160
Time, s	1st-order, unitless	2nd-order, mmHg−1	1st-order, unitless	2nd-order, mmHg−1	1st-order, unitless	2nd-order, mmHg−1
0	0.0053441	0.0000888	0.0106978	0.0004215	0.0141826	0.0003052
0.5	−0.0113258	−0.0000093	−0.0317528	−0.0005482	−0.0305524	0.0001441
1	−0.0606674	−0.0004854	−0.1130225	−0.0024845	−0.1424666	−0.0005875
1.5	−0.0846313	−0.0009561	−0.1186430	−0.0032550	−0.1650641	−0.0015852
2	−0.0690056	−0.0010924	−0.0644120	−0.0027865	−0.1032691	−0.0021265
2.5	−0.0594982	−0.0010686	−0.0475934	−0.0024280	−0.0786241	−0.0021722
3	−0.0595114	−0.0010640	−0.0530959	−0.0022385	−0.0783448	−0.0021070
3.5	−0.0531564	−0.0010605	−0.0433527	−0.0019858	−0.0578963	−0.0020268
4	−0.0487779	−0.0010096	−0.0361096	−0.0016735	−0.0485531	−0.0018949
4.5	−0.0458009	−0.0009050	−0.0308038	−0.0012624	−0.0435933	−0.0017367
5	−0.0385619	−0.0007954	−0.0182875	−0.0010134	−0.0269673	−0.0015018
5.5	−0.0335319	−0.0007345	−0.0118544	−0.0009023	−0.0166916	−0.0013480
6	−0.0287025	−0.0006717	−0.0086836	−0.0007416	−0.0123483	−0.0012249
6.5	−0.0227861	−0.0005876	−0.0047388	−0.0006102	−0.0073592	−0.0009416
7	−0.0198903	−0.0005125	−0.0026564	−0.0004946	−0.0053071	−0.0007900
7.5	−0.0164859	−0.0004362	0.0015160	−0.0003245	−0.0015483	−0.0007129
8	−0.0117555	−0.0003785	0.0052325	−0.0002953	0.0016648	−0.0004655
8.5	−0.0089331	−0.0003444	0.0029448	−0.0002883	0.0016336	−0.0003715
9	−0.0063109	−0.0002790	0.0018112	−0.0001505	0.0026040	−0.0004274
9.5	−0.0033516	−0.0002317	0.0018556	−0.0000881	0.0023051	−0.0003040
10	−0.0010963	−0.0002079	−0.0005557	−0.0000920	0.0017833	−0.0001827
10.5	0.0016547	−0.0001710	−0.0010800	−0.0000583	0.0040070	−0.0002130
11	0.0020230	−0.0001363	−0.0019769	−0.0000554	0.0051331	−0.0001643
11.5	0.0008485	−0.0001073	−0.0036355	0.0000160	0.0042096	−0.0000540
12	0.0018865	−0.0000624	−0.0031750	0.0000851	0.0046813	−0.0000425
12.5	0.0020187	−0.0000454	−0.0030819	0.0000508	0.0038249	−0.0000237
13	−0.0001120	−0.0000472	−0.0040712	0.0000144	0.0027484	−0.0000296
13.5	−0.0011690	−0.0000349	−0.0024378	0.0000264	0.0029665	−0.0000446
14	−0.0014246	−0.0000237	−0.0006728	0.0000344	0.0022949	−0.0000434
14.5	−0.0019218	−0.0000164	−0.0005096	0.0000342	0.0016824	−0.0001204
15	−0.0017010	−0.0000233	−0.0005132	0.0000203	0.0011306	−0.0001845
15.5	−0.0018849	−0.0000378	−0.0015177	0.0000214	0.0003206	−0.0001653
16	−0.0023700	−0.0000380	−0.0025900	0.0000599	−0.0000272	−0.0001353
16.5	−0.0021489	−0.0000316	−0.0028518	0.0000561	−0.0001561	−0.0000861
17	−0.0020097	−0.0000260	−0.0031153	0.0000217	−0.0000403	−0.0000537

Only 17 s of values were needed to capture these kernels.