INTRODUCTION

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THERE IS INCREASING RECOGNITION that the application of the tools of signal processing to biological signals can teach us a great deal about the organization and operation of biological control systems. The regulation of arterial blood pressure (ABP) is of particular interest in this regard because blood pressure is subject to a number of different physiological control mechanisms. For example, there is a great deal of evidence that neural mechanisms participate importantly in stabilizing mean ABP in the face of fairly rapid perturbations, such as postural adjustments, whereas hormonal and autoregulatory mechanisms are thought to dominate in minimizing the effects of challenges that are slower in onset and more sustained in duration.

In this regard, we recently analyzed the relationship between renal sympathetic nerve activity (SNA) and ABP in the "frequency domain" and found that these two signals were tightly coupled in the unanesthetized rat within a narrow range centered 0.4 Hz (2). At 0.4 Hz, there is a spectral peak in both the ABP spectrum and the SNA spectrum. We explain this 0.4-Hz rhythm as a feedback oscillation inherent within the baroreflex using a simple mathematical model (3, 4). The model predicts this oscillation on the basis of the experimentally measured time delay in the sympathetic limb of the baroreflex. The linearity of the model suggests that there is a strong linear coupling between SNA and ABP. Interestingly, the experimentally measured coupling between SNA and ABP appeared to decrease dramatically at lower frequencies. Our original study, however, was based on 9.56-min long data recordings, which did not permit us to examine this phenomenon critically at frequencies <0.1 Hz. In this paper, we will use the term "very low frequency" range to represent the frequency range f < 0.1 Hz.

Another very interesting observation within the frequency domain is that the power in cardiovascular signals appears to increase as a function of 1/f  for very low frequencies, where  is the slope of the power spectrum curve on a log-log plot (15, 16). The origin of this "1/f noise" is not understood. In particular, the role of the autonomic nervous system, if any, in generating this scaling behavior remains unresolved. On the one hand, sinoaortic denervation (SAD) of the baroreceptors leads to an enhanced blood pressure variability (6) and to a change in the  exponent in the very low frequency range (7). However, Wagner and Persson (20) recently showed that autonomic blockade restored the shape and absolute power of the 1/f noise in the very low frequency range in the SAD animal.

Gaussian fractal noise (or 1/f noise) is stationary. One property of a stationary signal is that its Fourier components have random phases (24). A coherence between phases of different frequencies can be generated by a transient fluctuation (e.g., an artifact or a physiological shift produced by transient physical activity) or by nonlinear dynamics. Intuitively, nonlinear dynamics produce phase coupling because the nonlinear interactions will mix different frequencies together. Classical spectral analysis assumes stationarity so that one can average over data segments to reduce uncertainty in spectral estimates.

In the present experiment we sought to quantify the interactions between SNA and ABP in the awake, undisturbed animal within the very low frequency range. To these ends we recorded SNA and ABP in unanesthetized rats for 2-5 h while they were in their home cages. To measure the linear coupling, we computed the magnitude squared of the coherence between SNA and ABP. Second, we calculated the value of  for both SNA and ABP. To validate our spectral analysis, we tested for stationarity by computing the "spectral coherence" (a generalization of ordinary coherence) between the two signals (8). We found that for f < 0.1 Hz, the magnitude squared of the coherence between the two signals approached zero. Moreover, the power law exponent, , for the power spectrum of SNA differs from that for ABP. Assuming that ABP and SNA in the resting rat are "stationary" signals, these results would indicate that these two signals are not linearly coupled in the very low frequency range with periods >10 s. However, the spectral coherence analysis revealed significant phase coupling (P < 0.01) between different frequencies within SNA and ABP, which indicates that the two signals are not stationary.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Experiments were performed on six Sprague-Dawley rats (Harlan Industries, Indianapolis, IN) weighing between 330 and 450 g. The standards for care and use of animals of the American Physiological Society were observed at all stages of this experiment.

Surgery

Data Acquisition

To obtain accurate measurements from the sympathetic nerve recordings, data were digitized at 10,000 samples/s using a Cache 486 microprocessor and Data Translation DT2821-F analog-to-digital converter. Data were collected for 2-5 h. The initial highly detailed nerve traffic signal was full wave rectified and averaged over every 10 points to produce a 1,000 sample/s signal. This process retains cumulative information from the initial 10,000 sample/s signal. The pressure signals were compressed "online" to a 1,000 sample/s signal by saving every tenth point. Because we were interested in the low-frequency range, the data were further compressed to 50 samples/s by averaging every 20 data points. Using software developed in our laboratory in Visual C++, we carefully removed artifacts from the raw time series.

Data Analysis

4

Auto- and cross-spectral estimates were computed using Welch's method (21). For each variable, a discrete Fourier transform was computed using the fast Fourier algorithm. Before computing the discrete Fourier transform, linear trends were removed from the data, and data were tapered using the Welch window. Squared magnitudes and the products of the computed discrete Fourier transforms were averaged to obtain spectral estimates. Estimates for the magnitude squared of the coherence function between ABP and SNA were computed as the ratio of the magnitude squared of the cross-spectra divided by the product of the autospectra (13) (refer to equation 1). The normalized random error in the magnitude squared of the coherence |C|² (1) depends on both the number of data segments K and the magnitude of the coherence according to

The low-frequency portion of each spectrum was modeled as a power law, namely where P is the power spectral density, and f is the frequency. To aid in characterizing this "fractal" or spectrally self-similar region, we used Yamamoto's coarse-graining technique (5, 23) to minimize the harmonic peaks in the spectrum. The  exponent was then determined from the slope of the least-squares fit to the log-log plot of the coarse-grained power spectrum. The uncertainty in the fitted parameter  is given by the square root of the corresponding diagonal entry in the covariance matrix for the least squares fit. The fit covered frequencies ranging from 0.001 to ~0.5 Hz.

To determine if SNA and ABP are stationary, we calculated the spectral coherence, which is a generalization of the ordinary coherence (8). The coherence C_xy between two time series x(t) and y(t) is defined as

(1)

where X( f ) and Y( f ) are the Fourier transforms of x(t) and y(t), respectively. The symbol X* represents the complex conjugate, and the angular brackets X  denote the expected value of X. This definition of coherence can be generalized to

(2)

for the spectral coherence within a single time series. C_xx( f, g) measures the correlation coefficient between two different frequencies f and g. The numerator X*( f ) X( g) is the generalized spectrum of x(t) (24). Note that |C_xx( f, g)|²  1 and that the diagonal is always 1. So, information is only contained in the off-diagonal terms. The concept of coherence can also be generalized to

(3)

for the spectral coherence between two different time series. Here the numerator is the generalized cross-spectrum. The diagonal term is simply the ordinary coherence defined in equation 1 above.

To compute the spectral coherence, we filtered the compressed time series with an eighth-order Butterworth low- pass filter (cutoff frequency was set equal to 2.0 Hz). We further compressed the two time series to obtain a 5-Hz sampling rate. Each time series was then divided into ~700 segments containing 256 points (or a period of 51.2 s). For each data segment, the Fourier transform was computed as in the classical spectral analysis above. To obtain estimates of the expected values in equation 2 and in equation 3, we formed a matrix by taking the outer product of the appropriate Fourier transform pairs and averaging this matrix over the 700 data segments.

When x(t) is a stationary zero mean, Gaussian noise time series, Goodman (9) gave the probability distribution of equation 2 as where c₀  [0, 1], and K is the number of segments averaged over. In other words, if x(t) is Gaussian noise, then random fluctuations will lead to estimates that exceed c₀ in (1  c₀)^K1 × 100% of the computational values (elements of the matrix). [Note: in our data, the data segments were not statistically independent because we used overlapping segments. So, we adjusted the number of segments according to Welch's correction factor, namely: K  9K/11 (21).]

To determine whether a time series is stationary, we made a contour plot of the magnitude squared of the spectral coherence. To calculate the statistical significance of an off-diagonal contour in a plot, we multiplied the probability P of an exceedance by the number of independent values based on symmetry. For the spectral coherence between two different signals, the number of independent values is 1/2 N², whereas for the spectral coherence within a time series, the number of independent values is 3/8 N².

Simulations

Narrow-band noise. This was a linear model driven by two independent stationary random sources, namely where z₁(t) and z₂(t) are two independent random Gaussian time series with zero mean and unit variance. The linear coupling between x₁ and x₂ leads to a resonance oscillation at 1.0 Hz. This model was integrated using a fourth-order Runge-Kutta method with a time step of 0.001 s. After discarding the first 1,000 time steps to get rid of initial transients, a file was made of 1,048,576 points spaced at a sampling rate of 10 Hz.

Strange attractor. This model was the Rossler attractor (19) and has both linear and nonlinear coupling between variables. The system is defined by The linear coupling between x and y produces an oscillation at 2 Hz, whereas z acts as a variable damping term to the x  y system. The parameter a gives rise to positive damping, which puts energy into the x  y system. As a result of the nonlinear interactions, the z variable exhibits intermittent, positive amplitude bursts.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Filtered time series for SNA and for ABP from rat sds are shown in Fig. 1. These data were passed through an eighth-order Butterworth low-pass filter with a cutoff frequency of 0.2 Hz and compressed to a sampling rate of 0.5 Hz. Both time series display large amplitude, intermittent fluctuations separated by quiet periods consisting of smaller fluctuations. In particular, notice the three prominent bursts in SNA and the corresponding fluctuations in ABP at roughly t = 8,500, 9,000, and 9,500 s. The duration of each of these bursts is close to 1 min.

View larger version (29K): [in this window] [in a new window]   Fig. 1.   Time series for sympathetic nerve activity (SNA; µV) in green and for arterial blood pressure (ABP; mmHg) in red from rat sds. Both time series were passed through a low-pass filter with a cutoff frequency of 0.2 Hz and compressed to a sampling rate of 0.5 Hz. Notice 3 prominent bursts in SNA and corresponding fluctuations in ABP at roughly t = 8,500, 9,000, and 9,500 s. Duration of each of these bursts is close to 1 min.

Spectral Analysis

View larger version (23K): [in this window] [in a new window]   Fig. 2.   Power spectra for SNA (A), ABP (B), and squared coherence (C) for rat sdm. Prominent peak in both SNA and ABP spectra at ~0.4 Hz is mediated by sympathetic limb of baroreflex (3). Notice that coherence between SNA and ABP drops to low values for frequencies <0.1 Hz.

The magnitude squared of the coherence between SNA and ABP corresponding to the power spectra from Fig. 2, A and B, is shown in Fig. 2C. The confidence interval around a computed coherence value depends on the coherence value. For example, the 95% confidence interval for a squared coherence value of 0.8 is 0.74-0.86, whereas the 95% confidence interval for a squared coherence value of 0.2 is 0.06-0.34. For each of the three peaks in spectral power associated with pulse pressure (~8 Hz), respiration (~2 Hz), and the baroreflex (~0.4 Hz), there is a corresponding peak in the coherence. Notice the apparent lack of coherence at very low frequencies <0.1 Hz, where the majority of spectral power within SNA and ABP exists.

The least-squares fit to the coarse-grained power spectrum of SNA from rat sdm is shown in Fig. 3A. Recall that the coarse-graining algorithm minimizes the spectral power attributable to periodic sources, such as the 0.4-Hz rhythm, so that the 0.4-Hz rhythm seen in Fig. 2A is virtually gone. The surviving power is now readily characterized by a power law exponent. However, coarse-graining did not eliminate the shelf in the spectrum beginning at ~0.1 Hz. To obtain an objective estimate for , we performed a nonlinear least squares fit to a combination of two line segments with two different slopes. The fit covered frequencies ranging from 0.001 to ~0.5 Hz and computed the location of the point of intersection so that we did not have to guess where the 1/f trend ended. The slope of the first line segment for SNA (over the very low frequency range) yielded  = 0.69 ± 0.03. For even lower frequencies, we expect there to be different slopes for different frequency ranges, but we did not try to fit to frequencies <0.001 Hz because these frequencies were influenced by detrending.

View larger version (21K): [in this window] [in a new window]   Fig. 3.   Coarse-grained power spectra for SNA (A) and for ABP (B) from rat sdm. Least squares fit to low frequencies is shown as a solid line in each plot. Coarse-graining removed 0.4-Hz rhythm, but not "bend" in each power spectrum near 0.1 Hz. Because of bend in both spectra ~0.1 Hz, we performed a nonlinear least squares fit to a "knee" function consisting of 2 line segments with 2 different slopes. We obtained parameter  from slope of first line segment. For SNA spectrum,  = 0.69 ± 0.03. For ABP spectrum,  = 1.14 ± 0.06. Although coarse graining reduced harmonic power at pulse pressure, it by no means eliminated this strong harmonic component in blood pressure.

Similarly, the least squares fit to the coarse grained power spectrum of ABP from rat sdm is shown in Fig. 3B. Again, although coarse graining reduced the harmonic components in the spectrum, it did not remove the "shelf " in the spectrum at ~0.1 Hz. So, we again performed a nonlinear least squares fit to a combination of two line segments with two different slopes. The slope of the first line segment (over the very low frequency range) yielded  = 1.15 ± 0.06. For this coarse-grained analysis, we divided the data set into 10 overlapping segments each of width 44 min.

We performed a similar analysis to determine the set of s for each remaining rat. As summarized in Table 1, we find that the  exponents for the two variables are significantly different. The (mean ± SD)  exponent for SNA is 0.66 ± 0.12, whereas the  exponent for ABP is 1.12 ± 0.14 (P < 0.01, Student's t-test for paired samples).

View this table: [in this window] [in a new window]   Table 1.   exponents for SNA and ABP

To summarize our classical spectral analysis, in Fig. 4 we show an ensemble average of the coherence curves between SNA and ABP for all six rats. In agreement with earlier work by Brown et al. (2), we find a robust coherence peak between 0.1 and 1.0 Hz, but an apparent lack of coherence <0.1 Hz down to ~0.001 Hz. Near 0.4 Hz the peak coherence was 0.82 ± 0.08, whereas <0.1 Hz the average coherence from all six animals was <0.2.

View larger version (14K): [in this window] [in a new window]   Fig. 4.   Ensemble average of coherence curves between SNA and ABP for all 6 rats. Notice strong coherence peak between 0.1 and 1.0 Hz. There is an apparent lack of coherence below ~0.1 Hz, although the majority of spectral power in both ABP and SNA resides in these frequencies.

Modeling

View larger version (16K): [in this window] [in a new window]   Fig. 5.   A: power spectral density for x1 variable in narrow-band noise simulation. Note harmonic peak at 1.0 Hz. Power spectral density for x2 variable (not shown) is identical. Squared magnitude of ordinary coherence between x1 and x2 is shown in B. Linear coupling in model gives rise to coherence peak. Contour plot of spectral coherence between x1 and x2 is shown in C. This analysis is a generalization of ordinary coherence analysis. Two contour levels were set at 0.00224 and 0.9 for a significance level of P < 0.01. The diagonal peak near 1.0 Hz corresponds to coherence peak seen in B. Lack of off-diagonal contours correctly indicates that underlying dynamical system is strictly linear and that corresponding time series are stationary.

Next we consider the strange attractor simulation, which involves both linear and nonlinear interactions. Contour plots of the spectral coherences for the y(t) and z(t) coordinates of the Rossler attractor are shown in Fig. 6. Notice that each contour plot is unique and complementary to the other two plots. The Rossler attractor does have nonlinear interactions. These non-linear dynamics produce phase coupling between different frequencies, which results in a nonstationary signal. So, we expect to see off-diagonal structure produced by this nonlinear coupling. In Fig. 6, the dramatic pattern in the off-diagonal contours confirms that this analysis is sensitive to the presence of nonlinear dynamics.

View larger version (28K): [in this window] [in a new window]   Fig. 6.   Contour plots for spectral coherences for Rossler attractor: A, spectral coherence between z and y; B, spectral coherence within z; and C, spectral coherence within y. Prominent off-diagonal structure correctly indicates underlying nonlinear dynamics. Contours are color coded with light blue representing highest coherence. All 4 quadrants are displayed to show symmetry of spectral coherence calculation. Diagonal contour has been suppressed in B and C because it contains no information. Notice that quadrant III is image of quadrant I, whereas quadrant IV is image of quadrant II. Finally quadrant IV is not simply an image of quadrant I, but contains novel information.

Generalized Spectral Analysis

View larger version (25K): [in this window] [in a new window]   Fig. 7.   Contour plots for spectral coherences of rat sdm: A, spectral coherence between SNA and ABP; B, spectral coherence within SNA; and C, spectral coherence within ABP. Note significant off-diagonal structure in all 3 plots. Contours are color coded with light blue representing highest coherence. In A, diagonal contour formation represents linear coherence associated with 0.4-Hz frequency band. Diagonal contour has been suppressed in B and C because it contains no information. Notice qualitative similarity between this figure and Fig. 6 for strange attractor.

A summary of our findings for nonlinear phase coupling is shown in Table 2. In three of six rats, the contour plots showed significant (P < 0.05) off-diagonal contours reflecting nonlinear phase coupling of different frequencies in ABP and SNA as well as within ABP and within SNA. Of the three remaining rats, one (rat sdk) displayed a well-defined pattern of nonlinear phase coupling within ABP and another (rat sdt) displayed a well-defined pattern within SNA. In only one (rat sdd) did we fail to detect the presence of nonlinear dynamics. The detection of nonlinear dynamics within ABP is consistent with the finding of nonlinear interactions within heart rate fluctuations reported by Ivanov et al. (10).

	DISCUSSION

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The action of the arterial baroreflex naturally implies a very high (linear) coherence between SNA and ABP. This must be true, of course, because increases in blood pressure reflexly decrease SNA when this reflex is functioning within its normal closed loop. Although changes in parasympathetic activity might be expected to confound this relationship, this limb of the nervous system is apparently modestly involved in cardiac regulation in the Sprague-Dawley rat (18). We have, in fact, demonstrated that the natural periodicity in both SNA and ABP centered ~0.4 Hz can be explained in this animal by using a simple linear mathematical model involving only the sympathetic limb of the baroreflex (3). Therefore, the present findings of a very high coherence between SNA and ABP centered ~0.4 Hz dramatically affirm this fundamental physiological principle.

Classical spectral analysis can be misleading, however, when the time series being analyzed is nonstationary. The 0.4-Hz rhythm is relatively stationary, and the ordinary coherence correctly identifies the linear coupling between SNA and ABP mediated by the baroreflex. However, the coupling between SNA and ABP <0.1 Hz is obscured in the classical spectral analysis by the intermittent nature of the time series for both variables. Visual inspection of the raw data shows that SNA comes in intermittent bursts (Fig. 1; see also Ref. 2). During many of these bursts there is a corresponding fluctuation in ABP. Because the bursts in SNA are intermittent, they are present in some data segments, but are absent in many others; as a result, the direct coupling between SNA and ABP is washed out by averaging. Such intermittent behavior can be produced by simple nonlinear dynamics.

Our finding that SNA and ABP are not stationary implies that their time series cannot be characterized by the  exponents of their corresponding power spectra. That is, at the start of the very low frequency range, SNA and ABP cannot legitimately be modeled as fractal Gaussian noise.

Limitations

Perspectives

The underlying dynamics of the baroreflex and sympathetic nervous system are imprinted on SNA and ABP regardless of the coupling (linear and/or nonlinear) that exists between them. Specifically, the intermittent nature of SNA reflects some sort of underlying nonlinear dynamics that produces the observed phase coupling between different frequencies within SNA. Such intermittent behavior occurs in simple nonlinear models. Notice the qualitative similarity between Fig. 6 for the Rossler attractor and Fig. 7 for rat sdm. The z variable is analogous to SNA, whereas the y variable is analogous to ABP. Both the z variable and SNA occur in "bursts." In the Rossler attractor, the z variable exerts nonlinear control over the x  y system through energy dissipation. The y variable is a controlled variable in the sense that it fluctuates within a limited range. Just as in the Rossler attractor, the sympathetic nervous system exerts control over the cardiovascular system through concentrated bursts of activity, which result in energy dissipation in the arterioles.