## Abstract

We have previously shown that there are two oscillating components in spontaneously fluctuating single-nephron blood flow obtained from Sprague-Dawley rats (Yip K-P, Holstein-Rathlou NH, and Marsh DJ. *Am J Physiol Renal Physiol* 264: F427–F434, 1993). The slow oscillation (20–30 mHz) is mediated by tubuloglomerular feedback (TGF), whereas the fast oscillation (100 mHz) is probably related to spontaneous myogenic activity. The fast oscillation is rarely detected in spontaneous tubular pressure because of its small magnitude and the fact that tubular compliance filters pressure waves. We detected myogenic oscillation superimposed on TGF-mediated oscillation when ambient tubular flow was interrupted. Two well-defined peaks are present in the mean power spectrum of stop-flow pressure (SFP) centering at 25 and 100 mHz (*n* = 13), in addition to a small peak at 125–130 mHz. Bispectral analysis indicates that two of these oscillations (30 and 100 mHz) interact nonlinearly to produce the third oscillation at 125–130 mHz. The presence of nonlinear interactions between TGF and myogenic oscillations indicates that estimates of the relative contribution of each of these mechanisms in renal autoregulation need to account for this interaction. The magnitude of myogenic oscillations was considerably smaller in the SFP measured from spontaneously hypertensive rats (SHR, *n* = 13); consequently, nonlinear interactions were not observed with bispectral analysis. Reduced augmentation of myogenic oscillations in SFP of SHR might account for the failure in detecting nonlinear interactions in SHR.

- bispectrum
- autoregressive
- bicoherence
- nonlinear interactions
- hypertensive
- tubuloglomerular feedback

arterial blood pressure is highly variable over many time scales, and renal blood flow must be autoregulated to minimize the effects of this variability on filtration patterns (8, 18). The consensus view is that autoregulation is mediated by at least two mechanisms, tubuloglomerular feedback (TGF), which is sensitive to the flow rate in the loop of Henle, and the myogenic mechanism, which is sensitive to the local vascular hydrostatic pressure. The frequency response of TGF has been studied and is reasonably well understood, but little is known of the dynamics of the myogenic mechanism. The measured transfer function of the renal circulation has a component active in the frequency band of 100–150 mHz, which is not compatible with TGF and was attributed to the myogenic mechanism (11). With Doppler velocimetry (25, 31), we detected two distinct oscillations in the spontaneous variation of single-nephron blood flow (31). One operates in the 20- to 50-mHz range, and the other at 100–200 mHz. The slower oscillation is TGF mediated, and the faster one is of intrinsic vascular origin and has been suggested as the signature of the myogenic mechanism in renal autoregulation (31). Both oscillations are pressure sensitive, and the myogenic oscillations are enhanced when TGF was inhibited by furosemide (31). Nonlinear interactions between TGF and myogenic oscillations have been detected in whole kidney blood flow when a broadband forcing was applied to the arterial pressure (5). If a nonlinear interaction between TGF and myogenic oscillation is a normal process, it should be detected at the nephron level without external arterial forcing. The purpose of this study was to test for the presence of these nonlinear interactions in measurements of tubular pressure under free-flow and stop-flow conditions. The presumption is that interrupting flow to the macula densa will change the TGF input to the myogenic mechanism. Nonlinear interactions at the nephron level can be quantified with the technique of bispectral analysis (13, 21) by measuring the tubular pressure, which has a higher signal-to-noise ratio than does single-nephron blood flow data measured with laser Doppler velocimetry (25, 31).

However, myogenic oscillation is detected with difficulty in spontaneous tubular pressure, most likely because of its small magnitude and the fact that the intrinsic vascular oscillations may not become a significant portion of the feedback signal, owing to low-pass filtering arising from thick ascending limb transport and the mechanical compliance of the tubule (14, 23). Because inhibition of TGF activity by furosemide enhances myogenic oscillation in single-nephron blood flow, we conjectured that myogenic oscillation might be augmented and become detectable in tubular pressure when TGF is inactivated by blocking tubular flow (31). The goals of the present study were *1*) to determine whether myogenic oscillation in tubular pressure could be detected when TGF is inactivated by interrupting tubular flow, *2*) to test whether there are nonlinear interactions between TGF and myogenic oscillation in tubular pressure, and *3*) to test whether similar observations can be found in spontaneously hypertensive rats (SHR).

## METHODS

#### Animal preparation.

Experiments were carried out in accordance with guidelines for the care and use of research animals. All experiments were performed under protocols approved by the Institutional Animal Care and Use Committee at the State University of New York at Stony Brook and the University of South Florida, in accordance with Public Health Service Policy on Human Care and Use of Laboratory Animals. Experiments were performed in male Sprague-Dawley rats (SDR) (250- to 300-g body wt) and 10- to 12-wk-old male SHR (250- to 304-g body wt). All rats were purchased from Harlan Farms. The rats had free access to food and tap water before the experiments. Anesthesia was induced by placing each rat in a chamber containing 5% halothane administered in 25% oxygen and 75% nitrogen through a Fluotec Mark-3 vaporizer. A tracheotomy was performed, and the rats were placed on a servo-controlled heated operating table, which maintained body temperature at 37°C. The tracheostomy tube was connected to a small-animal respirator (Harvard model 683) adjusted to maintain arterial blood pH between 7.35 and 7.45, with a mixture of 25% oxygen-75% nitrogen. Tidal volume ranged from 1.9 to 2.5 ml, depending on body weight, with a frequency of 57–60 breaths/min. The final concentration of halothane needed to maintain sufficient anesthesia was ∼1%. A polyethylene catheter (PE-50) was placed in the right jugular vein for infusions. After a priming dose of 6-mg gallamine thiethiodide (Flaxedil) in 1 ml of 0.9% saline, a continuous infusion of 60-mg iv gallamine triethiodide in 10 ml of 0.9% saline was given at 20 μl/min. The left kidney was exposed through a flank incision, immobilized with a Lucite ring, and superfused with saline preheated at 37°C. The renal capsule was left intact. Experiments started after a recovery period of 45 min.

#### Stop-flow pressure measurement.

A pressure-measuring micropipette filled with 0.5 M NaCl and colored with Fast Green was inserted into a surface proximal tubule to determine the sequence of surface loops traversed by an injected dye bolus. The pipette was attached to a servo-nulling pressure circuit (Instrumentation for Physiology and Medicine, La Jolla, CA). A second micropipette filled with bone wax and attached to a hydraulic microdrive was inserted into the tubule just distal to the pressure-measuring pipette, and tubular flow was obstructed by injecting wax into the tubule lumen. The pipette containing wax was then removed from the tubule.

#### Data acquisition.

Arterial pressure was measured in the left carotid artery with a Statham-Gould P23dB pressure transducer connected to a transducer amplifier (TBM4, WPI). Proximal tubular pressures were measured with the servo-nulling method (Instrumentation for Physiology and Medicine, San Diego, CA). Micropipettes for pressure measurement had a tip diameter of 1–3 μm and were filled with 0.5 M saline stained with Fast Green. The pressure was measured with a Statham P23dB pressure transducer connected to the transducer amplifier. The output was displayed on a Gould oscillographic recorder, using a display sensitivity of 6 mmHg/cm. Arterial pressure and proximal tubular pressure were recorded simultaneously on a TEAC R-61 four-channel cassette data recorder for off-line analysis. The interval of recording ranges from 15 to 20 min. The recorded data were replayed through an electronic low-pass filter with a roll-off frequency of 1.5 Hz and sampled digitally at 4 Hz. Arterial blood pressure was monitored continually throughout every experiment to verify that changes in arterial pressure did not account for the observed changes in stop-flow pressure (SFP).

#### Data analysis.

We further limited the data bandwidth to 0.5 Hz from the experimentally obtained sampling rate of 4 Hz (down-sampling by a factor of 4 after low-pass filtering at 0.5 Hz) to focus more on the frequency band of interest to autoregulation. We have analyzed 13 tubular pressure data records for SDR and 13 for SHR. Each time series, consisting of 1,000 data points, was subjected to linear trend removal, demeaned (by subtracting out the mean value), and normalized to unit variance.

Spectral analyses for tubular pressure data were performed by using the modified covariance autoregressive (AR) spectrum (S_{AR}), defined as: where ρ_{w} is an estimate of the driving noise variance, and *a*(*n*) are the AR parameter estimates. The model order, *P*, was determined by using a model-order criterion recently developed in our laboratory, known as the optimal parameter search (16).

To determine whether nonlinear interactions occur between the myogenic mechanism and TGF when the proximal tubular flow is blocked by wax, bispectral analysis was performed. Unlike the power spectrum, the bispectrum is a useful tool for detecting and quantifying the presence of quadratic phase coupling (QPC), a phenomenon in which two waves interact and generate a third wave with a frequency equal to the sum and/or difference of the two waves. The power spectrum, however, is not able to discriminate whether the third energy component is due to the QPC between the two waves and, consequently, fails to detect the presence of phase coupling. QPC can arise only among harmonically related components (13, 21). The bispectrum has successfully detected QPC in other physiological signals, most notably in electroencephalograms obtained from rats during various sleep states (22).

The bispectrum can be estimated by using either Fourier transform (FT) approaches or model-based approaches such as the AR process. For our analysis, we have utilized an AR bispectrum because the AR approach has been shown to provide higher resolution QPC detection than do FT-based methods.

To estimate the AR bispectrum, let [*y*(1),*y*(2), …,*y*(*N*)] be the given tubular pressure data set. The AR process of the given tubular pressure data can be expressed as (1) The *e*(*n*) are independent and identically distributed with *E*[*e*(*n*)] = 0, *E*[*e*^{3}(*n*)] = β ≠ 0, and *y*(*m*) is independent of *e*(*n*) for *m* < *n*. Multiplying both sides of *Eq. 1* by *y*(*n* − *j*)*y*(*n* − *k*) and then taking the expectation, the following is obtained (2) where *R*(*j*,*k*) are the third-order correlation functions, and δ(*j*,*k*) is the two-dimensional impulse-response function. *Equation 2* can be written in matrix form as *Ra* = *b*, where (3) Because the true third-order correlation functions, as defined above, are not known, they are estimated by the following procedure.

First, form the third-order moment estimate, i.e., *1*) segment the data into *K* records of *M* samples each, i.e., *N* = *KM*. Let {*y*^{i}(*l*); l = 1, 2,…, *M*} be the data samples of the *i*th record. *2*) Form the third-order moment estimate: (4) where *i* = 1, 2,…, *K*; *p*_{1} = max(0, −*m*, −*n*); *p*_{2} = max(*M* − 1, *M* − 1 − *m*, *M* − 1 − *n*). *3*) Average *r*^{i}(*m*,*n*) over all segments (5) Second, compute the following matrix equation (6) with the same matrix and vector notation shown in *Eq. 3*. The desired AR parameters, *a*, in *Eq. 6* are obtained by using the least squares method.

Once the AR parameters, *a*, are estimated using *Eq. 6*, the next goal is to compute the bispectrum estimate as (7) where (8) We have used the optimal parameter search model order search criterion to determine the model order *P*. For our analysis, *K* = 10 (the number of data segments), *M* = 100 (the number of data samples), and *P* = 50 (the AR model order). It should be noted that the following symmetry exists for the bispectrum: *B*(ω_{1}, ω_{2}) = *B*(ω_{2}, ω_{1}) = *B** (−ω_{1} − ω_{2}, ω_{2}) = *B*(ω_{1}, −ω_{1} − ω_{2}). Furthermore, *B*(ω_{1}, ω_{2}) is periodic in ω_{1} and ω_{2} with period 2π. Thus knowledge of the bispectrum in the region ω_{2} ≥ 0, ω_{1} ≥ ω_{2}, ω_{1} + ω_{2} ≤ π is sufficient for a complete description of the bispectrum (21).

Bispectrum analysis, as defined by *Eq. 9*, provides detection of QPC, but it does not provide any information about the degree of the coupling. To quantify the phase coupling, the bicoherence can be estimated: (9) where *B*(ω_{1}, ω_{2}) is the AR-based bispectrum estimate as defined above, and *P*(ω) is the AR power spectrum. Note that a bicoherence value of 1 indicates a strong QPC, a value of 0.5 indicates a borderline QPC, and a value of 0 indicates an absence of QPC.

The bispectrum is based on the assumption that the data are stationary. To ascertain whether the tubular pressure data are stationary so that we can properly utilize the bispectrum, we have applied to every data record one of the better time-frequency representation (TFR) techniques known as the smoothed pseudo-Wigner-Ville (SPWV). The TFR provides both time and frequency content of the signal, thus characterizing whether the data are stationary or nonstationary.

The general TFR has the following form: (10) where φ(θ,τ) is a two-dimensional function called the kernel, and *s*(*u* − τ/2) and *s**(*u* + τ/2) represent the real and complex conjugate of the signal, respectively. Given the TFR in *Eq. 10*, if the kernel φ(θ,τ) is equal to 1, we obtain the well-known Wigner distribution. The Wigner TFR provides one of the best time and frequency resolutions. However, the main drawback with the Wigner TFR is that, when there are multiple frequencies in a given signal, the time-frequency spectrum will also show artificial frequencies in addition to the true ones present in the signal. Thus, given the fact that we do not know what frequencies exist in the spectrum, the artifact ones will create havoc in deciphering the signal. To alleviate the problem of artificial frequencies in the case of a multicomponent signal, many different kernels have been designed. The kernel we used in this case for analyses of renal data is the SPWV. The SPWV has a kernel of the form: (11) where η(τ/2) and *G*(θ) are two windows whose effective lengths independently determine the time and frequency smoothing, respectively. This kernel reduces artificial frequencies yet retains higher TFR than most available methods.

## RESULTS

The spontaneous variation of tubular pressure in SDR oscillates at 1.2–2 cycles/min (Fig. 1, *top*), and its AR power spectrum shows a peak at 20 mHz (Fig. 1, *bottom*). This oscillation is due to the operation of TGF. By interrupting the ambient tubular flow with a wax block, the mean tubular pressure was increased from 11.9 ± 1.7 to 37.1 ± 2 mmHg (*n* = 13, *P* < 0.05). Despite increased tubular pressure, TGF-dependent oscillations persisted in the SFP mainly because TGF-initiated rhythmic vasoconstriction from the adjacent nephrons could propagate along the vasculature to the nephron without tubular flow (3, 12, 30). In addition to the persistence of TGF oscillations in the SFP, there was a faster oscillation (6–8 cycles/min) superimposed on TGF oscillation. The magnitude of this fast oscillation varied among data records. In some data records, the fast oscillation was dominant. Two typical SDR data records of SFP and their AR power spectral densities are shown in Figs. 2 and 3. These observations in SFP are consistent with those in single-nephron blood flow, in which myogenic oscillation is augmented when TGF is inhibited by furosemide (31). Of the four well-defined peaks at 20, 100, 120, and 140 mHz (Fig. 2) in the AR power spectrum (frequency resolution = 0.00195), we suggest that the first (20 mHz) and second (100 mHz) peaks are due to the autoregulatory mechanisms, and the third frequency peak (120 mHz) results from nonlinear interactions between the two autoregulatory mechanisms. It is also possible that the fourth (140 mHz) peak arises from the addition of the first and third (20 and 120 mHz) peaks and that the second (100 mHz) peak results from interaction of the first and third peaks. The other spectral peaks may arise due to harmonics and nonlinearity in the dynamics of the autoregulatory mechanisms, as clearly demonstrated by Layton et al. (14). Similarly, of the first three main peaks at 20, 110, and 130 mHz in Fig. 3, the third peak (130 mHz) may arise due to nonlinear interactions between the first (20 mHz) and the second (110 mHz) peaks. The presence of harmonic peaks in the spectra are expected because the tubular pressure signals are not purely sinusoidal. The use of the bispectrum will demonstrate whether these peaks are spontaneously excited normal modes, coupled modes, or due to harmonic generation.

To determine whether any data being analyzed meet the stationarity requirement of the bispectrum, the SPWV time-frequency spectra, as described in *Eqs. 10* and *11*, can be computed. To help readers in identifying stationary vs. nonstationary signals, we have simulated examples of both time-invariant (TIV) and time-varying (TV) signals in Fig. 4 for illustration purposes only. We created signals with oscillatory components at 20 and 100 mHz. However, for the TIV signal, the two oscillatory frequencies appear at all times, whereas, for the TV signal, the two frequencies occur only at certain time points (time *t* = 0–200 s for 20 mHz and *t* = 300–512 s for 100 mHz). Figure 4, *A* and *B*, shows time-frequency plots obtained by using the SPWV technique for the TIV and TV signals, respectively. Stationary signals will have frequencies appearing across all times, which is exactly what is observed in the contour plot of Fig. 4*A*.

Representative SPWV time-frequency spectra (shown in contour plots) of the SFP data obtained from SDR and SHR are shown in Fig. 5, *A* and *B*, respectively. Similar to TIV AR spectra, we observe frequency peaks associated with the myogenic and TGF mechanisms in the contour plots of Fig. 5. Moreover, the advantage of the time-frequency spectrum over the stationary AR spectrum is that we have the added benefit of observing how these frequencies behave over the course of time, which provides information as to whether the signal analyzed is TIV or TV. Clearly, both 20- and 100-mHz frequencies occur at all times, suggesting that the tubular pressure signals are TIV for SDR. For SHR, only the frequency peak associated with the TGF peak is present, and it appears consistently across all time. However, it should be noted that, in some of the SHR data records, we do observe small-magnitude myogenic peaks that appear only in brief amounts of time.

To further test the stationarity of the data, we performed a statistical test called the “run test,” as suggested by Bendat and Piersol (1). The run test involves testing for invariance of both mean and variance values over many time segments. Similar to the time-frequency analysis, the run test also confirmed stationarity of both SDR and SHR data.

As both time-frequency spectra and a statistical test indicated stationarity of the data, the use of the TIV bispectrum for detecting QPC remains valid for the tubular pressure data of both SDR and SHR. Again, we have created an example to assist readers in interpreting QPC phenomena as detected by the bispectrum. We simulate the following sinusoidal frequencies: *f*_{1} = 0.04 Hz, *f*_{2} = 0.1 Hz, *f*_{3} = *f*_{1} + *f*_{2} = 0.14 Hz, *f*_{4} = 0.2 Hz, and *f*_{5} = *f*_{1} + *f*_{4} = 0.24 Hz with phases such that θ_{1} + θ_{2} = θ_{3} and θ_{1} + θ_{4} ≠ θ_{5}. According to the theory of the bispectrum, the QPC will only occur at the frequency pair (*f*_{1} = 0.04 Hz, *f*_{2} = 0.1 Hz) in the bispectrum plot when the following situation occurs: the third frequency *f*_{3} = 0.14 Hz is the addition of the two other frequencies *f*_{1} and *f*_{2} and the phase θ_{3} = θ_{1} + θ_{2}. Indeed, as shown in the bispectrum plot of Fig. 6, we observe a QPC at (*f*_{1}, *f*_{2}) with a maximum value of 1.0. Note that *f*_{5} = *f*_{1} + *f*_{4} and *f*_{1} = *f*_{3} − *f*_{2} relationships also exist; thus we have the possibility of QPC peaks at the frequency pairs (*f*_{1}, *f*_{4}) and (*f*_{2}, *f*_{3}), but we do not observe substantial peaks (peak values are 0.01 and 0.05, respectively) at these frequency pairs, because the phases were such that θ_{5} ≠ θ_{1} + θ_{4} and θ_{1} = θ_{3} − θ_{2}.

An intuitive example of the QPC is the amplitude modulation (AM) signal. The general form of the AM signal is described by the following: *s*(*t*) = *A*_{c} cos(2 π *f*_{c} *t* + θ_{c})[1 + β cos (2π *f*_{m} *t* + θ_{m})]. The above multiplicative expression can be simplified by the use of the trigonometric product relationship and yields which leads to the appearance of sinusoidal term that satisfies the phase criterion θ = θ_{c} ± θ_{m}. In the above expression, *f*_{c} and *f*_{m} are the carrier and modulating frequencies, respectively, and *A*_{c} and β represent the amplitude and the modulation index of the AM signal, respectively. Based on the above AM expression, there are three distinct peaks: *f*_{c}, *f*_{c} + *f*_{m}, and *f*_{c} − *f*_{m}. A time series tracing and spectrum of the AM expression with *f*_{c} = 0.1 Hz and *f*_{m} = 0.04 Hz are shown in Fig. 7, *A* and *B*, respectively. Indeed, as expected, three peaks at 0.1, 0.14, and 0.06 Hz are observed in the spectrum. Based solely on the spectrum plot, it is unclear whether any of these frequencies are coupled. The bispectrum plot shows similar QPC only at the frequency pair (0.1, 0.04) Hz, similar to the example shown in Fig. 6. Note that *f*_{m} is not present in the power spectrum because of the multiplicative expression of the AM signal. However, the QPC is between *f*_{c} and *f*_{m}; thus this QPC peak at (0.1, 0.04) Hz is seen in the bispectrum. Normally, all oscillatory peaks should appear in the spectrum, and the QPC peak will be among these spectral peaks. The above-noted absence of *f*_{m} (0.04 Hz) spectral peak is unique to the example of the AM signal.

Thus bispectral analysis is well suited to test whether the frequency modes observed in the spectrum are spontaneously excited normal modes of the system or coupled modes. Furthermore, we can use the bispectrum to determine whether a frequency mode was due to harmonic generation (e.g., *f*_{x} = 2*f*_{y}). In addition, if two oscillations in the tubular pressure data are interacting in the form of phase coupling, then there should be a peak in the three-dimensional plot of the bispectrum at frequencies *f*_{1} = 20 and *f*_{2} = 110 mHz, corresponding to the TGF and myogenic mechanisms, respectively. On the other hand, if there are no interactions, no such peaks should be detected. A typical bispectrum plot showing an indication of phase coupling is shown in Fig. 8*B*. For this SFP record, shown in Fig. 8*A*, we observe dominant spectral peaks at *f*_{1} = 20,*f*_{2} = 40, *f*_{3} = 70, *f*_{4} = 95, *f*_{5} = 130, *f*_{6} = 150, and *f*_{7} = 170 mHz. To determine whether these frequencies are spontaneously excited normal modes of the system or whether they are interacting modes, the bispectrum was computed. The bispectrum plots (contour: Fig. 8*B*, *top*; magnitude: Fig. 8*B*, *bottom*) exhibits two dominant peaks at the frequency pairs (*f*_{1}, *f*_{1}) and (*f*_{5}, *f*_{1}). Thus we conclude that the primary modes, *f*_{1} and *f*_{5}, interact and generate modes at *f*_{6} = *f*_{1} + *f*_{5}. Note that this coupled mode corresponds to nonlinear interactions between the myogenic and TGF mechanisms. In addition, modes at *f*_{1} = 20 mHz and *f*_{6} = 150 mHz interact and generate a mode at*f*_{7} = *f*_{6} + *f*_{1} = 170 mHz, and a frequency pair at (*f*_{1}, *f*_{1}) is the self-interacting mode.

To test whether the nonlinear phase coupling in the SFP time series is significant, a bicoherence value was calculated from each time series and was used as an index to measure the degree of coupling. A bicoherence value of 1 indicates that the nonlinear phase coupling is highly significant. A value of 0.5 is borderline significant (21). The distribution of bicoherence of 13 SFP time series is shown in Table 1. The mean bicoherence was found to be 0.74 ± 0.32 (*n* = 13).

To ensure that the bicoherence index does indeed provide statistical evidence of QPC, we employed a surrogate data technique (26). This approach may provide statistical evidence of QPC, even for those data with bicoherence values that are <0.5. All of the tubular pressure data were subjected to surrogate data analysis. The goal of the surrogate data transformation is to destroy the nonlinear dynamics in the data. This leaves a time series with only linear properties; thus no QPC should be detected. If the bispectrum proceeds to find QPC, then we know that the bispectrum itself is not accurate in that it introduces spurious QPC phenomenon.

We employed the amplitude-adjusted FT algorithm to generate 10 surrogate data sets from each of the original tubular pressure data sets (26). The surrogate data are created by rescaling the original time series and taking the FT of the rescaled time series. The phases of the FT rescaled time series are then randomized, which results in elimination of nonlinearity in the signal, if present. Finally, the surrogate time series is rescaled once more by applying the inverse FT, so that it has the same amplitude distribution as well as the same Fourier spectrum as the original time series.

To test the statistical significance of the surrogate data, we computed a dimensionless quantity, *S*, defined as (12) where *x*_{d} is the bicoherence value for the real data, and *x̄*_{surr} and σ_{surr} denote the mean and the standard deviation of the surrogates' bicoherence values, respectively. A value of *S* > 2 implies that the null hypothesis is rejected, meaning that true QPC exists in the experimental data. The mean and standard deviation values of the bicoherence values of the surrogates, as well as the original data, are provided in Table 1. We consistently found a near-zero value of bicoherence for all of the surrogate data sets, meaning that the coherence values found in the original data sets (Table 1) were due to inherent nonlinear dynamics governing the system. This is confirmed by the fact that, for most of the data sets, the significance factor *S*, as determined using *Eq. 12*, is greater than the value of 2. As indicated in Table 1, only 2 data sets of 13 did not have QPC.

Free-flow tubular pressure in SHR fluctuates aperiodically as previously reported (28, 29). No fast oscillation in the range of 100–200 mHz was augmented in the SFP of SHR. On the contrary, the fluctuation of SFP became more periodic in the frequency range of TGF oscillation. A SFP record from SHR and its corresponding power spectrum are shown in Fig. 9. Note that, similar to SDR's spectra, the dominant oscillation is located mainly in the frequency range typical for the TGF. However, compared with the dominant TGF oscillations, only small-magnitude oscillations are seen in the frequency range of 100–200 mHz for SHR. The magnitude of these myogenic-related oscillations in Fig. 9 is a factor 10 times smaller than SDR's stop-flow spectra shown in Figs. 2 and 3. Consequently, bispectral estimates do not show a statistically significant indication of nonlinear phase coupling in the SFP obtained from 13 data records of SHR.

## DISCUSSION

Spontaneous periodic variations in tubular pressure received minimal attention in the literature until Holstein-Rathlou et al. (9, 10) demonstrated that these periodic variations are due to nonlinearities and time delays in the TGF feedback loop. In the present study, the spontaneous variation of SFP was examined to search for the presence of myogenic oscillation, which has been observed from measurements of single-nephron blood flow. We found that TGF oscillation persists in SFP. Because tubular flow was interrupted by a wax block, it is unlikely that this oscillation is due to the operation of TGF within the nephron. The most likely explanation is that TGF-mediated rhythmic constriction propagates along the vasculature from the adjacent nephrons supplied by the same cortical radial artery (2). Entrainment of TGF oscillations in nephrons supplied by the same cortical radial artery has been demonstrated previously (7, 12, 30) and is consistent with our observation that a TGF oscillation is found in SFP.

Transfer function analysis of renal autoregulation with a broadband forcing applied to arterial pressure showed that there is another oscillator operating in the range of 100–200 mHz (11). Nonlinear modeling of whole kidney blood flow autoregulation based on kernel analysis further suggests that TGF oscillation interacts nonlinearly with an oscillator in the frequency range of 100–200 mHz (5, 17). However, both of these studies were based on experiments with external forcing applied to arterial pressure and measurements of whole kidney blood flow. It was not known whether this observation also held at the level of the single nephron and when the kidney is perturbed only with spontaneous arterial pressure fluctuations (8, 18). Moreover, kernel analysis requires broadband forcing of pressure data, which cannot be obtained from this type of spontaneous tubular pressure fluctuation. Using a noninvasive laser Doppler velocimetry technique, we were able to detect a small-amplitude oscillation at ∼120 mHz superimposed on the TGF oscillation in the spontaneous variations of single-nephron blood flow (25, 31). However, time series obtained from the single-nephron blood flow have low signal-to-noise ratios, which makes quantitative detection of nonlinear interaction more difficult. Our laboratory (23) has previously found that time series of tubular pressure are more robust against noise contamination because tubule compliance acts as a low-pass filter, and thus tubular pressure data may provide better quantification of nonlinear interactions at the nephron level. The amplitude of the fast oscillations in single-nephron blood flow is increased when TGF is inactivated by furosemide (31). These observations suggest that the fast oscillation can be changed by manipulating TGF activity. Indeed, oscillations of tubular pressure in the frequency range of 170–180 mHz were induced by intraluminal tubular perfusion in halothane-nitrous oxide-anesthetized rats (15). In the present study, we were able to amplify this fast oscillation in tubular pressure when the afferent arteriole was dilated by interrupting the ambient tubular flow. The fast oscillation is enhanced in single-nephron blood flow when TGF is inhibited by furosemide and also in tubular pressure when TGF is inactivated by interrupting tubular flow. This amplification of myogenic oscillation most likely reflects interactions between TGF and the spontaneous myogenic oscillation at the afferent arteriole and not the resonance generated through the tubular structure. Application of bispectral analysis successfully detects TGF oscillation and myogenic oscillation as two independent oscillations.

Our laboratory (33) has recently shown that blood pressure and flow data obtained from the whole kidney as well as single nephrons exhibit nonstationary characteristics. It is unclear why the tubular pressure data exhibit quasi-stationary behavior, as evidenced by the TV spectra shown in Fig. 5. As stated above, this may be due to better signal-to-noise ratio with tubular measurement than both whole kidney and single-nephron blood flow measurements. It should be noted that, in some SHR SFP data, we observed very-low-magnitude myogenic peaks appearing only at certain time points. Although our statistical “run” test provided stationarity of SHR data, we speculate that, in some SHR data, the stationarity assumption may not be entirely correct.

Computation of bicoherence values and bispectra indicate that there is nonlinear phase coupling between these two oscillators. The detection of nonlinear interactions between TGF and the myogenic oscillation at the nephron level without artificial forcing applied to arterial pressure suggests that nonlinear interactions between these two oscillators operate under normal physiological conditions. QPC between the two mechanisms regulating afferent arteriolar resistance suggests the occurrence of interactions at the cell and or subcellular level. Because vascular smooth muscle cells have a single contractile mechanism, the convergence of the two mechanisms on a single cell requires that they interact at some point in the chain that leads from an increase in intracellular calcium to a contraction. There are insufficient data to speculate about the exact point of interaction. We have previously suggested that myogenic oscillation serves as a dither, which linearizes the dynamic response of the TGF-myogenic ensemble and that improves the overall efficiency of renal autoregulation (31).

Spontaneous variations of free-flow tubular pressure in SHR fluctuate aperiodically. A similar bifurcation of tubular pressure dynamics is observed in renovascular hypertensive rats (29). However, the cause of the bifurcation has not been identified. Measurements of SFP in SHR indicate that no myogenic oscillation was enhanced, and the variations in SFP became more periodic compared with that in free-flow condition. We hypothesized that TGF and myogenic mechanism are complementary in normal operating conditions and are synergistic when challenged by acute increase of renal perfusion pressure (6, 19, 20, 31). We have found that TGF-mediated nephron-nephron interactions are stronger in SHR than in SDR (4, 30). The absence of augmentation of myogenic oscillations in SFP of SHR suggests that residual TGF activity derived from the adjacent nephrons is sufficient to regulate afferent arteriolar tone without activating the complementary myogenic mechanism. It might explain why nonlinear phase coupling was not detected in SFP of SHR, as observed in SDR. The observation that variations in tubular pressure become more periodic in SFP than in free-flow conditions in SHR suggests that nephron-nephron coupling via TGF might have a key role in the bifurcation of renal hemodynamics in hypertension.

Another reason for the apparent absence of nonlinear interactions may be that TIV bispectral analysis is not appropriate for SHR data. Our laboratory (33) has recently shown, using TV spectral techniques, that SHR is more TV than SDR. Indeed, our preliminary results using the TV bispectrum show nonlinear interactions in SHR, but these interactions are highly transient, and their duration is very short compared with those in SDR.

In the AR power spectrum, because the TGF oscillating mechanism is active while the magnitude of myogenic oscillations is negligible, it is expected that no interactions can be detected with the bispectrum. This result does not exclude the fact that, in the SHR, an interaction might occur, if a second mechanism were active. Furthermore, due to the possible nonstationary behavior of SHR in some of the data records, the TIV bispectrum may not be the appropriate method for detecting the QPC phenomenon.

The myogenic oscillations observed in SFP seem to be different from the spontaneous vasomotion observed in other vascular beds, in which the magnitude is increased when perfusion pressure is reduced (27). Inhibiting TGF with furosemide or blocking tubular flow does not reduce the perfusion pressure in afferent arterioles, while the amplitude of the myogenic oscillation is drastically increased. The explanation may be that myogenic oscillations interact closely with TGF oscillations, and the increased amplitude of myogenic oscillation is the direct result of perturbing TGF activity.

In conclusion, both TGF oscillations and myogenic oscillations were found in the SFP of SDR, as was observed in single-nephron blood flow. Bispectral analysis suggests that there are two independent oscillations and that there is nonlinear phase coupling between them. The results indicate that interruption of TGF activity permits the amplitude of myogenic oscillations to increase. Under normal free-flow conditions, it appears that TGF exerts a moderating effect on myogenic activity. Estimates of the relative contribution of each of these mechanisms need to account for this interaction.

## GRANTS

The present study was supported by National Institutes of Health Grants HL-69629, HL-59156, DK-60501, and DK-15968.

## Footnotes

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