Abstract
A simple mathematical model was used to investigate the spectral properties of the tubuloglomerular feedback (TGF) system. A perturbation, consisting of smallamplitude broadband forcing, was applied to simulated thick ascending limb (TAL) flow, and the resulting spectral response of the TGF pathway was assessed by computing a power spectrum from resulting TGFregulated TAL flow. Power spectra were computed for both open and closedfeedbackloop cases. Openfeedbackloop power spectra are consistent with a mathematical analysis that predicts a nodal pattern in TAL frequency response, with nodes corresponding to frequencies where oscillatory flow has a TAL transit time that equals the steadystate fluid transit time. Closedfeedbackloop spectra are dominated by the openloop spectral response, provided that γ, the magnitude of feedback gain, is less than the critical value γ_{c} required for emergence of a sustained TGFmediated oscillation. For γ exceeding γ_{c}, closedloop spectra have peaks corresponding to the fundamental frequency of the TGFmediated oscillation and its harmonics. The harmonics, expressed in a nonsinusoidal waveform for tubular flow, are introduced by nonlinear elements of the TGF pathway, notably TAL transit time and the TGF response curve. The effect of transit time on the flow waveform leads to crests that are broader than troughs and to an asymmetry in the magnitudes of increasing and decreasing slopes. For feedback gain magnitude that is sufficiently large, the TGF response curve tends to give a square waveshape to the waveform. Published waveforms and power spectra of in vivo TGF oscillations have features consistent with the predictions of this analysis.
 kidney
 renal hemodynamics
 nonlinear dynamics
 mathematical model
the power spectrum, which quantifies the relative prevalence of frequency components in a signal, has emerged as an important tool in the analysis of experimental time series derived from renal hemodynamic variables. Spectral analysis helped establish that singlenephron oscillations in intratubular pressure, flow, and distal tubule chloride concentration, with frequency of 20–50 mHz, arise from an intrinsic instability in the tubuloglomerular feedback (TGF) loop (7); spectral analysis showed that significant spectral power was distributed over a range of frequencies in tubular flow that appears to exhibit deterministic chaos (22); and spectral analysis has been used to distinguish oscillations arising from TGF from those of intrinsic myogenic origin (3, 11, 23).
In this study we used a simple mathematical model to investigate the spectral properties of the TGF system. Previously, we have used the same model framework to elucidate the emergence of TGFmediated oscillations (13, 17), to distinguish steadystate gain from instantaneous gain (14), and to characterize the nonlinear filter properties of the thick ascending limb (TAL) (15).
We first summarize the model equations and associated physiological parameters, and we review the TGFmediated oscillation that may be exhibited by the model when γ, the magnitude of gain of the TGF pathway, exceeds a critical value γ_{c}. Then, using an openfeedbackloop configuration, we compute power spectra that characterize the lowpass filter of the model TAL and the transmission of the TGF signal from the macula densa (MD) to the afferent arteriole (AA). These results confirm the nodal structure identified in the companion study (15) and provide additional insight into the spectral properties of the TGF system components when operating in the absence of feedback.
The key new results of this study, however, are obtained from the closedfeedbackloop configuration. Power spectra, computed for selected values of feedbackloop gain magnitude γ, characterize the evolution of the TGF system as γ increases through the regime that will not support sustained TGFmediated oscillations, γ < γ_{c}, and into the regime that does, γ > γ_{c}. For γ < γ_{c}, the power spectra are dominated by the openloop spectral characteristics, with the fundamental resonant frequency of the TGF oscillation superimposed on the openloop frequency response. For γ > γ_{c}, spectral power is increased, and power spectra are composed of peaks corresponding to the fundamental resonant frequency of the TGFmediated oscillation and its harmonics, which are explained via the principle of Fourier decomposition. The harmonics, which become more pronounced as gain magnitude increases, arise from distortions in TGF oscillations that are introduced by nonlinear properties of the TGF pathway, notably the integrative effect of TAL transit time and the constraints of the TGF response function. The distortions in the waveform of tubular flow arising from nonlinear transit time consist of crests that are broader than troughs and a slope asymmetry, in which the absolute magnitude of the ascending slope is less than the absolute magnitude of the descending slope. For TGF gain magnitude that is sufficiently large, the bounds of the TGF response function tend to give a square waveshape to the waveform. Finally, we observe that published waveforms and power spectra from in vivo measurements of TGFmediated oscillations frequently exhibit features consistent with the predictions of this theoretical analysis.
Glossary Parameters
 C_{o}
 Chloride concentration at TAL entrance (mM)
 C_{op}
 Steadystate chloride concentration at MD (mM)
 k
 Sensitivity of TGF response (mM^{−1})
 K_{m}
 Michaelis constant (mM)
 L
 Length of TAL (cm)
 P
 TAL chloride permeability (cm/s)
 Q_{op}
 Steadystate SNGFR (nl/min)
 ΔQ
 TGFmediated range of SNGFR (nl/min)
 r
 Luminal radius of TAL (μm)
 t_{o}
 Steadystate TAL transit time (s)
 V_{max}
 Maximum transport rate of chloride from TAL (nmol ⋅ cm^{−2} ⋅ s^{−1})
 α
 Fraction of SNGFR reaching TAL
 δ
 Distributed delay interval at the JGA (s)
 τ_{p}
 Pure delay interval at the JGA (s)
Independent variables
 f
 Frequency of flow oscillation (mHz)
 t
 Time (s)
 x
 Axial position along TAL (cm)
Specified functions
 C_{e}(x)
 Extratubular chloride concentration (mM)
 ψ_{δ}(t)
 Kernel function for distributed delay (dimensionless)
Dependent variables
 C(x, t)
 TAL chloride concentration (mM)
 C_{MD}(t)
 Effective MD chloride concentration (mM)
 F(C_{MD}(t))
 TAL fluid flow rate (nl/min)
 P_{f}
 Power spectral density
 S(x)
 Steadystate TAL chloride concentration (mM)
 T(x, t)
 Fluid transit time from TAL entrance (s)
MATHEMATICAL MODEL
Model equations.
The mathematical model for the TGF loop (14, 17) is given by the following system of coupled equations
Equation 1 is a partial differential equation for the chloride ion concentration C in the intratubular fluid of the TAL of a shortlooped nephron. At time t = 0, initial concentrations C(x, 0) (for x ∈ [0, 1]) and C(1, t) (for t ∈ (−∞, 0)) must be specified. We impose the boundary condition C(0, t) = 1, which means that fluid entering the TAL has constant chloride concentration. The rate of change of that concentration at x∈ (0, 1] depends on processes represented by the three righthand terms in Eq. 1. The first term is axial convective chloride transport at the intratubular flow speed F. The second is the transtubular efflux of chloride driven by active metabolic pumps situated in the tubular walls; that efflux is approximated by MichaelisMenten kinetics, with maximum transport rateV _{max} and Michaelis constant K _{m}. The third term is transtubular chloride backleak, which depends on a specified fixed extratubular chloride concentration profile C_{e}(x) and on chloride permeability P.
Equation 2 describes fluid speed in the TAL as a function of the effective luminal chloride concentration C_{MD} at the MD (see below). This feedback relation is an empirical equation well established by steadystate experiments (20). The constant C_{op} is the steadystate chloride concentration obtained at the MD when F ≡ 1. The positive constantsK _{1} and K _{2} describe, respectively, the range of the feedback response and its sensitivity to deviations from the steady state.
Equation 3
represents time delays in the feedback pathway between the luminal fluid chloride concentration at the MD, C(1,t), and an effective MD concentration C_{MD}(t), which is used to calculate the flow response that is modulated by smooth muscle of the AA. In quasisteady state, Eq. 2
provides a good description of the TGF feedback response. However, dynamic experiments (1) show that a change in MD concentration does not significantly affect AA muscle tension until after a pure delay time τ_{p}, and then the effect is distributed in time so that a full response requires additional time, with, say, greatest weight in the time interval [t − τ_{p} − δ,t − τ_{p}], where δ is a second delay parameter. To simulate the pure delay followed by a distributed delay, we introduced the convolution integral in Eq. 3
to describe the effective signal received by the AA at time t(17). We require that the kernel ψ_{δ} satisfy
A steadystate solution to Eqs. 14 may be obtained by setting F = 1 for 1 unit of normalized time (the transit time of the TAL at flow speed 1), starting at t = 0, to give the steadystate operating concentration C_{op} = C(1, 1) at the MD. If one specifies that C(1, t) = C_{op}for t ∈ (−∞, 1), then the input flow to the TAL,F, is fixed at 1 for all previous time. The feedback loop can then be closed at t = 1. If the system remains unperturbed, the system solution will not vary in time. We denote the resulting steadystate TAL concentration profile C(x, 1) by S(x).
Model parameters.
A summary of parameters and variables, with their dimensional units as commonly reported, is given in the Glossary. The values of model parameters are given in Table 1; the criteria for their selection and supporting references were given in (13). The extratubular concentration is given in nondimensional form by C_{e}(x) = C_{o}(A _{1} exp(−A _{3} x) + A _{2}), where A _{1} = (1 − C_{e}(1)/C_{o})/(1 − e ^{−A3}),A _{2} = 1 − A _{1}, andA _{3} = 2, and where C_{e}(1) corresponds to a cortical interstitial concentration of 150 mM. Graphs of C_{e} and the steadystate luminal profile S were given in figure 1 of Ref. 13. The steadystate operating concentration C_{op} was calculated numerically using the TAL dimensions and transport parameters, with steady flowF = 1 in Eq. 1.
Rather than a pure delay of 4 s in signal transmission at the juxtaglomerular apparatus (JGA), as we used in Ref. 13, a pure delay τ_{p} of 2 s was followed by a transition interval δ of 3 s, providing a distributed delay in approximate agreement with experiments (1).
The steadystate TAL transit time t _{o} is a key parameter that plays a prominent role in this study. This transit time is the interval required for a water molecule to travel up the TAL, from the TAL entrance to the MD, at the steadystate flow rate, assuming plug flow; it is equal to the TAL volume divided by the steadystate TAL flow rate, i.e., t _{o} = πr ^{2} L/(αQ_{op}), and it corresponds to one unit of normalized time.
Bifurcation locus.
For simplified versions of the model given by Eqs. 14, we have previously shown that, for some parameter ranges, the timeindependent steadystate solution is unstable, and subsequent to a perturbation, the solution may take the form of stable, sustained oscillations (13, 14, 17). This parameterdependent behavior probably arises from a Hopf bifurcation.
A bifurcation may occur when γ, the magnitude of the instantaneous gain of the feedback response, exceeds a critical value γ_{c} (13). The instantaneous gain, investigated in detail in Ref. 14, corresponds to the maximum reduction in singlenephron glomerular filtration rate (SNGFR) resulting from an instantaneous shift of the TAL flow column toward the MD, under the assumption that the SNGFR response is also instantaneous. The instantaneous gain is given by −γ = K _{1} K _{2}S′(1), where K _{1} K _{2} is a measure of the strength of the feedback response, and S′(1) (a negative quantity) is the slope of the steadystate chloride concentration profile at the MD. (In a negative feedback loop, the feedback gain is negative by convention; thus we use the phrasing “gain magnitude” when referring to γ.)
The critical gain magnitude γ_{c} and the associated critical angular frequency ω_{c} can be determined from the model’s characteristic equation, given in appendix . The parameters from Table 1 lead to critical gain γ_{c} ≈ 3.24 and critical frequency f _{c} = ω_{c} / (2π) ≈ 45.9 mHz, in dimensional units. The critical frequency f _{c} predicts the frequency of the oscillations arising from the bifurcation.In this study, we assume that all parameters are fixed except for the sensitivity of the TGF feedback response, k; the gain magnitude γ depends on the sensitivity through the equationK _{2} = kC_{o}/2 (see appendix ). Thus, by changing sensitivity k, we change γ. The baseline value of k used previously by us in Ref. 13 is 0.24 mM^{−1}, which results in γ ≈ 3.14, a value below, but near, the critical value γ_{c} ≈ 3.24. We have previously hypothesized that the nearness of physiologically plausible values of γ to γ_{c} may explain the tendency of some shortlooped nephrons to exhibit sustained TGFmediated oscillations while other nephrons do not (13).
Power spectra.
The fast Fourier transform (FFT) was used to obtain power spectra from numerical solutions of the model equations. A power spectrum of a signal, roughly speaking, is a graph that shows the relative strengths of the oscillatory components of differing frequency that combine to make up the signal. A power spectrum is usually represented as “power spectral density” (19); the abscissa in such a power spectrum is frequency, and the ordinate is the sum of the squares of the two Fourier coefficients corresponding to that frequency (seeEqs. EC2 and EC3 in appendix ). We will denote the power spectral density corresponding to frequency fby P _{f} . Although a power spectrum is graphed as a continuous curve, in practice a computed spectrum is discrete, and there tends to be leakage from frequency “bins” corresponding to large Fourier coefficients into adjacent frequency bins (19). appendix summarizes the methods used to compute numerical solutions to the model equations and to compute power spectra based on those solutions. High resolution in space and time is required to compute solutions that are sufficiently accurate to provide acceptable spectral resolution (see discussion andappendix ).
We will investigate the spectral properties of TGF system components by superimposing a broadband forcing on a variable of the system; this variable is considered to be the input of the system. A broadband forcing is a signal that includes many oscillatory components, of nearly uniform amplitude, over a range of frequencies. The effect of the forcing on an output of the system can be used to understand the action of the system components as a function of input signal frequency.
RESULTS
Power spectra for open TGF loop.
We have previously noted that both our model for the TAL and our model for the distributed delay contain a lowpass filter (13, 1517), i.e., qualitatively speaking, lowfrequency oscillations pass through these model components with little reduction in amplitude, but the amplitudes of high frequency oscillations are attenuated. To distinguish the contributions of the two filters to the TGF loop and to test the adequacy of our methods, we computed power spectra for each of the two filters separately, and then we computed the spectrum for the filters operating in series. In all these cases, the TGF loop was open, i.e., TGFregulated flow did not enter the TAL.
The three power spectra arising from model components are shown in Fig.2. In each case there was an input signal, containing broadband forcing, and an output signal, as specified below; each power spectrum was computed from the output signal. The spectra in Fig. 2 (and also Fig. 3) were normalized by dividing by the power spectrum of the input signal; the power spectrum of that input signal, normalized by itself, is the constant valueP _{f} = 1.
For the separate TAL filter, Eq. 1 was solved with input given as steadystate flow F = αQ_{op} plus smallamplitude broadband forcing. The output was given by Eq.2, but as F(C(1, t)), i.e., the flow was determined by the TGF response function, but without the delays that would be introduced by Eq. 3. The sensitivity k was adjusted to provide a gain magnitude of γ = 1. In this case, we obtained the spectrum in Fig. 2 marked “TAL only”; the general trend of decreasing amplitude as a function of frequency indicates that the TAL operates as a lowpass filter. However, the spectrum exhibits local minima, which correspond to nodes, and local maxima, which correspond to antinodes. The nodal structure is explained in the companion study (15), which shows that the range of NaCl excursions at the MD depends on the fluid transit time through the TAL, with nodes corresponding to frequencies where the oscillatory flow has a transit time to the MD that equals the steadystate TAL fluid transit time. In Table 2 we list selected nodes predicted by analytical techniques in the companion study (15) and the corresponding nodes found in this study through numerical calculation of model solutions and subsequent spectral analysis. There is excellent agreement up to 500 mHz, but there is increasing divergence as frequency increases, arising from the approximate nature of the numerical calculations. Nonetheless, there is agreement with error less than 1.3% from 0 mHz through at least 1800 mHz.
In appendix we explain that the power spectrum value for “TAL only” at f = 0, given by P _{0} ≈ 0.8190, indicates a steadystate gain of the TAL that is within 0.3% of a value computed previously by other means (14). This close agreement provides confirmation that the power spectrum has been correctly computed and scaled.
The curve marked “Distributed delay only” in Fig. 2 shows the spectral response of the distributed delay of Eq. 3; in this case, the input C(1, t − τ_{p}) was replaced by broadband forcing with mean value zero, and C_{MD} was taken as the output. This spectrum also indicates a lowpass filter, and this spectrum also exhibits nodes, at frequencies of (2 + n)/3 per second, for n = 0, 1, 2, … These nodal frequencies arise because the integral in Eq. 3 vanishes when the integrand has the form ψ_{δ}(t − s − δ/2) × sin(2π(2 + n)s/3 + φ), with ψ_{δ}given by Eq. 4, for any phase shift φ. A different kernel function ψ_{δ} would, of course, yield a different spectral structure, since it would be composed of different Fourier components, but physiologically reasonable choices of ψ_{δ} are likely to have little qualitative effect below 500 mHz (see appendix ). In this spectrum for “Distributed delay only,” the response at f = 0 isP _{0} = 1, because a constant signal is transmitted undiminished.
The thick shaded curve marked “TAL with distributed delay” in Fig. 2 is the power spectrum of the TAL and distributed delay acting in series. The input was the same as for the case of “TAL only,” and the output was obtained from Eq. 2, via the distributed delay of Eq. 3. Thus, this spectrum is the power spectrum for the openfeedbackloop configuration of the TGF model. We see from this spectrum that the only effect of the distributed delay below 500 mHz is to attenuate the spectral power of the TAL spectrum, especially above 300 mHz.
In the companion study (15) we found that the TAL lowpass filter, in the absence of chloride backleak, exhibited 1/f scaling for frequencies larger than about 64 mHz, i.e., the amplitude of chloride excursions at the TAL decreased inversely with frequency. When the spectra for “TAL only” and “Distributed delay only” are graphed on a loglog plot (not reproduced here), the resulting plots are linear for sufficiently large frequencies, which indicates that both spectra exhibit 1/f scaling. The TAL exhibits 1/fscaling, with chloride backleak present, in about the same range as in the absence of backleak. However, the distributed delay exhibits 1/f scaling only above 300 mHz; consequently, the spectrum produced by the two filters in series exhibits 1/f scaling only above 300 mHz. Thus the combined action of the two filters does not exhibit 1/f scaling in the range that includes much of the frequency domain of TGF and myogenic autoregulation. In particular, the scaling predicted by the model cannot be the source of 1/fscaling observed in experimental records of blood pressure in rat for frequencies ranging from 0.01 mHz up to 3 mHz (10).
Power spectra for closed TGF loop.
Figure 3 gives power spectra for the closedfeedbackloop configuration illustrated in Fig. 1, corresponding to increasing values of instantaneous gain magnitude γ, which is a measure of feedback strength. For each spectrum, the input was smallamplitude broadband forcing added to flow entering the TAL; the output, which was subjected to spectral analysis, was the TAL flow predicted by the feedback response (see Fig. 1). In the model, the spectral characteristics of TAL flow are the same as those of SNGFR, since by assumption TAL flow is a fixed fraction of SNGFR (seeappendix and Ref. 13).
In Fig. 3, dashed lines are the (normalized) power spectrum of the input; solid curves are the closed feedback loop power spectra; the shaded curves, for comparison, are the spectra for open feedback loops (exhibited as “TAL with distributed delay” in Fig. 2). In Fig. 3,A–E, the labels along the curves are frequencies corresponding to nodes or antinodes of the closedloop spectrum; in Fig. 3 F, the extrema of both spectra are labeled.
Figure 3, A–D, shows the development of a resonant frequency, emerging from the openloop spectrum, and increasing from ∼34 to ∼46 mHz as the gain magnitude γ is increased from 1 to 3.24. A second resonant frequency emerges from the openloop spectrum at ∼89 mHz; this frequency can be identified by detailed analysis of the characteristic equation (Eq. EB1 in appendix ). The antinodes in Fig. 3 are slightly to the left of the frequency midpoints between nodes.
As noted in the section describing the mathematical model,a bifurcation may occur at the critical value of gain magnitude, γ_{c} ≈ 3.24. Consequently, for γ exceeding γ_{c}, the small applied perturbation elicits sustained TGF oscillations at a frequency near the critical frequencyf _{c} ≈ 45.9 mHz. This has dramatic effects on the power spectrum, as shown in Fig. 3, E and F. Two features are particularly noteworthy. First, the power at all frequencies is greatly increased, as indicated by the upward shift in the power spectral density curve. This additional power does not arise from the smallamplitude broadband forcing; rather, it arises from sustained flow oscillations of large amplitude. Indeed, additional numerical studies employing a transient perturbation, but no broadband forcing, produced spectra that are almost identical to those in Fig. 3,E and F. Additional studies also showed that the transition to spectra qualitatively like Fig. 3, E andF, occurred for γ < 3.26, confirming, in conjunction with the result of Fig. 3 D, an abrupt transition to sustained oscillatory flow localized within ∼0.6% of γ_{c}.
The second noteworthy feature is the emergence of a series of harmonics of the fundamental resonant frequency of the TGF system, ∼46 mHz. As γ increases, the higher frequency harmonics become stronger, as can be seen through comparison of E with F of Fig. 3. The harmonics in Fig. 3, E and F, arise from the action of the nonlinear elements in the TGF system in shaping the largeamplitude oscillations.
Waveshape distortion in open TGF loop.
If all elements of the TGF system were linear, then oscillations in key variables would be pure (i.e., singlefrequency) sine waves, and the highfrequency components that represent distortion from a pure sine wave (i.e., the harmonics) would not be present in Fig. 3, E and F. The nonlinear elements in our model include the filter and transport characteristics of the TAL (Eq.1 ) and the TGF feedback relationship (Eq. 2 ). The pure and distributed delays in the feedback pathway (which enter throughEqs. 3 and 4 ) are linear elements; indeed, the effect of the distributed delay on a sinusoidal component is to attenuate its amplitude without changing its frequency.
Examples of distortion by nonlinear elements of the TGF pathway are illustrated in Figs. 4 and 5. Columns A and B in Fig. 4 show openfeedbackloop responses to specified sinusoidal input flows. Columns C and D in Fig. 4 exhibit waveforms arising in the closedfeedbackloop case for γ = 5 and γ = 10, which both exceed γ_{c}. [Experimental studies indicate that steadystate, in vivo gain magnitude, which slightly underestimates instantaneous gain magnitude (14), ranges from 1.5 to 9.9 (6).]
In each column of Fig. 4, row 1 shows the input SNGFR (Q_{IN}) as a function of elapsed time (the timescale is at the base of Fig. 4). In the model, Q_{IN} is related to input TAL flow by a constant factor, F _{IN} = αQ_{IN}. Row 2 of Fig. 4 gives TAL fluid transit time T from the TAL entrance to the MD, which is computed fromEq. 2 in the companion study (15) and which is expressed in units of the steadystate transit time t _{o} ≈ 15.7 s. Transit time is an important quantity since theoretical considerations indicate that chloride concentration at the MD depends largely on TAL transit time (15). Row 3 of Fig. 4 gives luminal chloride concentration at the MD. The vertical dashed lines in rows 1–3 of Fig. 4 coincide with a local maximum and a local minimum of the transit time; the shaded bars in row 1 of Fig. 4indicate the corresponding transittime intervals.
The thin solid curves in rows 4–6 of Fig. 4 give the TGF responses, expressed as SNGFR, arising from Eq. 2 (Q = F/α), for the indicated magnitudes of gain γ (C4, C6, D4, and D5 of Fig. 4 are intentionally left blank). The wide shaded curves in Fig. 4, rows 4–6, are sine functions, adjusted to match the frequency of the solid curves and to have a similar amplitude. They are provided so that distortion relative to sine curves will be apparent.
The frequency of the sinusoidal waveforms in A1 and B1in Fig. 4 was specified at 45.9 mHz, the value of the critical frequency f _{c}, to permit comparison with the closed loop results of columns C and D. Column A of Fig. 4 has SNGFR excursions of amplitude ΔQ_{IN} = 6 nl/min, so that SNGFR has lower and upper bounds of 27 and 33 nl/min, respectively. For an input oscillation of this small amplitude, the response of the system, at every stage, and for values of γ up to 10, appears to be essentially linear, i.e., the resulting waveforms inrows 2–6 of column A of Fig. 4 all appear to be nearly sinusoidal. Nonetheless, there is a small degree of distortion, which is apparent in Fig. 4, A4–A6, and in the corresponding power spectra (see, e.g., Fig. 5 A).
In Fig. 4 A1 we observe that the maximum transittime interval, indicated by the top left shaded bar, spans a trough in Q_{IN}, and the minimum transittime interval, indicated by the other shaded bar, spans a crest. These transittime intervals correspond to the maximum and minimum transit times marked by the dashed lines on Fig. 4 A2. In Fig. 4 A3, maximum and minimum transit times correspond to minimum and maximum MD chloride concentrations, respectively, consistent with the analysis of the companion study (15). For the special choice of the critical frequencyf _{c}, the response waveform Q in Fig. 4 A4 is in phase with the input Q_{IN}.
In column B of Fig. 4, ΔQ_{IN} = 9, and oscillations in Q_{IN} are therefore bounded by minimum and maximum flow rates of 21 and 39 nl/min, respectively. As a consequence of these larger amplitude oscillations, two nonlinear features emerge. First, transit time T increases slowly relative to its rate of decrease, an asymmetry arising from the larger time interval corresponding to the maximum T, relative to the minimumT. This asymmetry, which may be seen by comparing the shaded bars in Fig. 4 A1 to those in Fig. 4 B1, and which is apparent in Fig. 4 B2, is reflected in asymmetry in the concentration record in Fig. 4 B3, which leads to the output waveform for Q in Fig. 4 B4. This waveform has two particular features, arising from TAL transit, that distinguish it from the shaded sine wave: the wide crest of the wave, relative to the trough, and a rise to the crest that is slower than the fall from the crest. We call this slope asymmetry “slow up/fast down.”
A second nonlinear element arising through the TGF response given byEq. 2 is apparent in Fig. 4, B5 and B6. Because the TGF response has maximum amplitude of ΔQ = 18 nl, the response is bounded below and above by 21 and 39 nl/min, respectively (seeappendix ); consequently, a “railing” effect occurs for values of MD concentrations that lead to extreme values of effective concentration C_{MD} through Eq. 3. In Fig.4 B5 we observe railing at the lower bound, corresponding to large MD concentration. In Fig. 4 B6, for γ = 10, we observe railing at both extremes, which tends to produce a square waveform.
Waveshape distortion in closed TGF loop.
When the feedback loop is closed, as in columns C and Dof Fig. 4, the nonlinear behavior of the system is compounded by the nonlinear feedback, leading to a broader crest and generally squarer waveshape in the waveform for TAL flow. (With the closure of the loop, the waveforms in Fig. 4, C1 and C5, coincide, as do the waveforms in D1 and D6.) However, with the increasing nonlinearity, the action of the TAL lowpass filter becomes apparent in the waveforms for transit time in Fig. 4, C2 and D2: because the filter integrates flow, the large slopes in Q_{IN}are reduced.
In Fig. 4 C5, with the closure of the loop, the slow up/fast down characteristic is enhanced, relative to Fig. 4 B5, although the distributed delay of Eq. 3 tends to reduce the degree of the effect that could be expected, given the pronounced fast up/slow down waveform in Fig. 4 C3. The flattening in the crest of the waveform in Fig. 4 C5 arises from the broad trough in Fig.4 C3; and small negative slope within the crest of Fig.4 C5 arises from the small positive slope in the trough of Fig.4 C3. Also, by comparison with the peaked maxima in Fig.4 C3, one sees that that railing has clipped the lower range of SNGFR flow in Fig. 4 C5. Thus the case illustrated in column C of Fig. 4 represents the mixed effects of the nonlinear TAL and TGF responses.
In column D of Fig. 4, where the gain magnitude corresponds to the extreme physiological range, the clipping effect of railing dominates the transformation of MD concentration, in Fig. 4 D3, to flow, in Fig. 4 D6. Transit time in Fig. 4 D2 has the largest maximum and the smallest minimum of the cases examined, leading to the most pronounced extrema in MD concentration, in Fig.4 D3. Compared to the other waveforms for flow in Fig. 4, the square waveform in Fig. 4 D6 has the most pronounced deviation from the reference sine wave.
Effect of nonlinear elements on power spectrum.
Figure 5 provides power spectra corresponding to some of the waveforms of Fig. 4. The thin solid curves in Fig. 5, A and B, are the spectra for the sinusoidal, and thus singlefrequency, flow inputs Q_{IN} illustrated in Fig. 4, A1 and B1. The thick shaded curves in Fig. 5,A and B, are the power spectra of the TGF response, corresponding to the solidline curves in Fig. 4, A4 andB4. Although the response illustrated in Fig. 4 A4appears to be substantially linear, the power spectrum for the response shows that the elements of the TGF pathway have introduced substantial spectral structure: the peaks in the gray curve in Fig. 5 Acorrespond to the fundamental frequency of the input, plus a series of harmonics. When the amplitude of the input is increased, as in Fig.4 B1, the clearly nonlinear response observed in Fig.4 B4 corresponds to the more pronounced excitation of harmonics shown in Fig. 5 B.
Figure 5, C and D, give power spectra of the MD chloride concentration C (thin solid curves), corresponding to Fig. 4,C3 and D3, and power spectra of SNGFR Q (thick shaded curves), corresponding to Fig. 4, C5 and D6. These spectra show the substantial increase in the power of the harmonics as the distortion from the sine waveform becomes more pronounced with increasing gain magnitude. Also, these spectra for the MD concentration show the action of the TAL lowpass filter in reducing the strength of the harmonics that are present in TAL flow. These harmonics are then reconstructed in the flow by the TGF response function, largely through the effect of railing.
The pattern of harmonic frequencies in Fig. 5 can be understood in terms of the Fourier components of a periodic wave. A superposition of sine curves, with varying frequencies and amplitudes, is required to represent a nonsinusoidal periodic oscillation, and if that oscillation has frequency f, then the sine curves must have frequencies ofnf, n = 1, 2, 3, …, since the oscillation is periodic. As γ increases, the waveforms of the TGF pathway become more distorted, with the curve segments connecting extrema becoming steeper; consequently, the highfrequency Fourier components make larger contributions to the waveform representation.
DISCUSSION
We have used a mathematical model to investigate the spectral properties of the TGF pathway. For an openfeedbackloop configuration, the results of this study are consistent with the nodal TAL response pattern predicted in the companion study (15). For the closedfeedbackloop configuration, this study predicts that the spectral properties of TGFregulated flow depend largely on whether the gain magnitude exceeds the critical gain required for the emergence of sustained TGFmediated oscillations. The nodal structure of power spectra for subcritical gains is a consequence of the filter properties of the TAL; for sustained oscillations, power spectra exhibit a harmonic structure that arises from nonlinear properties of the model TGF pathway.
Although the results presented here are based on numerical calculations that employ a fixed parameter set (with the exception of TGF sensitivity, which is used to vary feedback gain), the study’s qualitative conclusions are independent of the particular parameter choices in Table 1; indeed, the results and conclusions depend only on the structural characteristics of the model, through its dependence on steadystate transit time t _{o}, and on the combinations of parameters that generate the critical gain magnitude and critical frequency through the characteristic equation (Eq.EB1 ). Thus the nodal patterns and harmonic frequency structure observed in Fig. 3 should arise for any choice of parameters in the physiological range, through a rescaling of the frequency axis.
Effects of idealizations in model formulation.
Factors not included in the model formulation may affect the in vivo spectral characteristics of the TGF system. Several of these factors were considered in the companion study (15), including the elastic compliance of the tubular walls, axial diffusion of NaCl within luminal flow, spatial inhomogeneities in TAL luminal diameter and transport capacity, and the effect of timevarying luminal concentrations on transepithelial transport rate. Other factors, particular to the applicability of this study, include oscillations introduced by respiration, the dynamic properties of absorption by the proximal tubule and descending limb, and spectral characteristics contributed by spontaneous vasomotion of the renal vasculature. However, because rat respiration has a frequency of ∼1 Hz, this factor is not likely to significantly affect tubular flow spectral characteristics at frequencies below 500 mHz (see appendix ). Little research has been conducted on the dynamic properties of glomerulotubular balance (GTB), but existing experimental studies suggest that GTB is robust for flow variations within the physiological range (21). The effect of spontaneous vasomotion awaits further investigation.
A final factor that may impact the applicability of the results is the phenomenological characterization of the delay in feedback response given by Eqs. 3 and 4 . However, theoretical considerations, developed in appendix , indicate that the spectral structure introduced by the delay in feedback, in the range of applicability of the model, will be insensitive to the precise mathematical characterization of the delay, provided that the characterization has certain general features. The insensitivity is a consequence of the short time scale of the delay, relative to the steadystate TAL transit time.
Numerical methodology.
In the course of this investigation, we found that great care must be exercised to obtain power spectra that faithfully represent the nonlinear features predicted by the mathematical model. Numerical solutions to model equations must be computed with sufficient accuracy to preserve the structure inherent in the model equations, and good frequency resolution must be attained in the power spectra that are computed from the numerical solutions. The computation of power spectra, based on given data, has been heavily studied (19); the methodology used in this study is summarized in appendix .
We considered the computation of accurate numerical solutions for dynamic TAL flow in Ref. 18, where we reviewed research which shows that numerical methods may produce approximate solutions to model equations that exhibit artifactual diffusion (which redistributes spectral power) and/or artifactual dispersion of Fourier components (which displaces propagation speed of spectral components as a function of frequency). In this study we used a lowdiffusion, lowdispersion method, combined with high resolution in space and time, to faithfully represent the highfrequency Fourier components in the concentration profiles of the TAL and thus preserve spectral structure (see appendix ). The results for test cases were confirmed by comparison with the analytical results in the companion study (15), which predict a regular nodal pattern (cf. figure 1 in Ref. 15 and Fig.2 in this study).
Three previous model studies that used similar formulations for the TAL (8, 9, 11) appear to have not detected the regular pattern of harmonics predicted by this study, and the waveforms exhibited in Ref. 8 do not exhibit marked nonsinusoidal features. The discrepancy between these studies and our results may be due, at least in part, to a highly dispersive numerical method that was used in the previous studies, coupled with low spatial and temporal resolution.
Comparison with published experimental data: waveforms. This model study predicts specific patterns of waveform distortion in tubular flow, which are associated with specific spectral characteristics. Are the salient characteristics of these waveforms and associated power spectra observable in vivo? The answer to this question speaks directly to the adequacy of our model to represent essential features of the TGF system. Moreover, if waveform distortion is observable in vivo, the results of this model study have important implications for the interpretation of power spectra derived from experimental records.
To determine whether the features of waveform distortion shown in Fig.4 were present in vivo, we examined published tracings of TGF oscillations. The first feature we sought to identify is a broadening of the crest of the flow waveform, relative to the trough, which retains a more pointed shape, as seen in Fig. 4 C5. This feature is most obvious at gain magnitudes of γ ∼ 4–5, before the waveform is large enough to be significantly constrained by the limits of the TGF response function. The second feature was a difference in the magnitude of ascending and descending slopes of the flow waveform, which was also seen in Fig. 4 C5. These features, which arise from the inverse relationship between fluid speed and TAL transit time, are particularly evident in the MD chloride concentration waveforms in Fig. 4 of this study and in figure 3 of the companion study (15), before the distributed delay of AA response has acted to smooth the curves. Note that when examining concentrations, the trough, rather than the crest, is broadened, and the ascending slope magnitude is larger than descending magnitude. (In figure 3 of the companion study (15), the flatness of the trough of the curve corresponding tof = 0.5/t _{o} may be accentuated by solute backleak, which impairs the capability of the TAL to reduce chloride concentration at low flow rates.)
Eight experimental time records for singlenephron pressure, flow, or MD concentration were examined; the results are summarized in Table3. Every record showed differences, in the majority of the displayed periods, in ascending and descending slope magnitudes, with the fall in the recorded variable being discernibly more rapid than the rise (except for the reverse case in distal chloride concentration). Four of the eight records exhibited broadening of the crests of the waveform, whereas the excursions through the minima were sharp. Because the broadening of the crests of the oscillatory flow waveform may be accentuated by increasing TGF gain (see Fig. 4), its appearance in some records but not others may be attributable to internephron heterogeneity.
A typical measured TGF oscillatory waveform of late proximal tubule pressure, from a study by HolsteinRathlou et al. (11) (their figure 1,bottom right), is reproduced in Fig.6 B; an earlier study has shown that the waveforms of flow and pressure have similar shapes (7). In Fig.6 A we exhibit model SNGFR, for γ = 4; the waveform has been scaled temporally to have the same period as the experimentally measured waveform, 28.4 s, which corresponds to a frequency of 35.2 mHz. In Fig. 6 A, the dashed curve, appearing in the fourth through the sixth oscillation, is a sine wave, provided for comparison with the nonsinusoidal model waveform. In Fig. 6, the vertical dashed lines, appearing in the seventh through the tenth oscillations in bothA and B, are drawn to coincide with the points on the model waveform with local maximum slope magnitude. Comparison of the waveforms in Fig. 6, A and B, indicates a close correspondence between the wave shape predictions of the model and the in vivo measurement: both waveforms exhibit broad crests and a smaller ascending slope magnitude relative to descending magnitude.
From this examination of experimental results, we conclude that in vivo TGF oscillations exhibit features consistent with the predictions of our model. However, the mechanisms that operate in the model may not be the sole causes of waveform distortion in vivo. For example, the slope asymmetry could also result from differences in the time constants of the turnon and turnoff transitions of the TGF mechanism acting across the JGA, a feature not represented in the model; indeed, experiments suggest such a hysteresis in TGF responses to manipulations of Henle’s loop flow between zero and high flows (figure 13 of Ref. 20). However, these results are not strictly comparable to our simulation studies, since the TGF ontransition will be principally determined by the washout of the TAL, while the offtransient will be dominated by the kinetics of TAL NaCl absorption, as the epithelium dilutes the stationary fluid column within the TAL. To a lesser extent, all commonly used experimental techniques will include, and may be influenced by, the dynamics of TAL absorption. Hence, there is no definitive evidence, at present, regarding asymmetry in the transmission of the TGF signal across the JGA; but if such asymmetry exists, it would reinforce the asymmetry arising from the transittime dependence of TAL NaCl absorption.
Another alternative cause for waveform distortion is a displacement of the TGF response operating point to a site near the top of the response curve, which could result in the broadening of the waveform crest, but not the trough, by imposing a constraint on the flow increase allowed by the feedback response (in Eq. 2 we assume that the operating point is at the midpoint of the response curve). However, experimental evidence indicates that the TGF operating point is usually near the center of the TGF curve in normal, extracellular volumereplete rats (20), consistent with the model assumptions (13).
Comparison with published experimental results: power spectra.
Regardless of the mechanism of waveform distortion, the TGF waveform in vivo has marked similarities to those shown in Figs. 4 and 6 A, which suggests that the harmonics of the TGF fundamental should be present in power spectra computed from experimental data. In particular, power spectra from in vivo data should have some of the features of the spectra shown in Fig. 3, E and F, especially for low frequencies (for instance, <200 mHz), where confounding factors may be reduced.
The prediction that power spectra of TGF oscillations will be characterized by maxima at the fundamental frequency and its harmonics is supported by spectra in the experimental literature. Studies by HolsteinRathlou and Marsh (7) and Yip et al. (22) appear to show a resonant TGF oscillation in rat proximal tubular pressure and its first harmonic, similar in this respect to our Fig. 3 E. In figure 1B in Ref. 7, an oscillation in pressure of 35–44 mHz appears to have a harmonic in the range of 74–88 mHz. [The higher frequency components at 133–163 mHz in figure 1B of Ref. 7may arise from intrinsic vascular oscillations of the AA (1, 23); also note that the plot in figure 1B of Ref. 7 uses a linear, rather than a logarithmic, ordinate, which may account for the absence of discernible harmonic structure in the power spectrum of distal chloride concentration given in the same figure.] In figure 1D in Ref.22, an oscillation of 33–35 mHz appears to have a harmonic in the range of 66–70 mHz. In figure 1E of Ref. 22, a resonant frequency of ∼23 mHz appears to have a harmonic at ∼47 mHz.
Another study by Yip et al. (23), however, contains power spectra of rat efferent arteriolar flow that appear to exhibit a nodal pattern. For example, figure 5B (dashed curve) of Ref. 23 exhibits local minima at ∼50, 97, 153, and 204 mHz and local maxima at 24, 66, 123, and 172 mHz, patterns that are very similar to the nodal structure observed in our Fig. 3, A–D. However, it appears that these experimental flow measurements already exhibit a resonant TGF oscillation of modest power (see figure 5A of Ref. 23), in which case comparison should be made with our Fig. 3, E–F, where there is better qualitative agreement in shape but somewhat less agreement in nodal pattern. A plausible explanation for the apparent discrepancy is that the nephron examined may have a subcritical gain magnitude, and its oscillations may be driven through coupling with a neighboring nephron that is spontaneously oscillating (5). Indeed, when we perturbed our model configuration with γ = 3, using a largeamplitude signal with a sweeping frequency, from 0 to 1 mHz, we obtained a power spectrum that was in good qualitative agreement with the nodal structure in the closedloop spectrum of our Fig. 3 Cand similar to the dashedline spectrum reported by Yip et al. (23) in their figure 5B.
Finally, studies of whole kidney renal blood flow and arterial pressure by Cupples et al. (3) yielded complex power spectra that exhibit elements suggestive of both nodal patterns and TGF harmonics (see figure 1 in Ref. 3).
Based on these comparisons with experimental data, we conclude that sustained TGFmediated oscillations in vivo can be sufficiently nonsinusoidal to exhibit substantial power in harmonics that lie above the fundamental frequency. The same is true for oscillations in nephrons with subcritical gain magnitude, since the TGF system may express the nodal structure of the TAL filter in response to perturbations. The harmonics and nodal structure may be confounding factors in studies of in vivo power spectra, where detailed analysis has been used to identify and quantify other oscillatory elements, e.g., the intrinsic myogenic oscillation and interactions between TGF and the myogenic oscillation (2, 23). High numerical resolution in computer simulations and careful consideration of experimental design and data analysis are needed in studies of TGF dynamics, because nonlinear characteristics of the system can be easily lost or obscured.
Acknowledgments
We thank Chris Clausen for helpful discussions on the methodology of spectral analysis. We thank John M. Davies for assistance in preparation of Figs. 16.
Footnotes

Address for reprint requests: H. E. Layton, Department of Mathematics, Duke University, Box 90320, Durham, NC 277080320 (Email:layton{at}math.duke.edu)

This work was supported in part by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK42091.

A brief report on this work appeared in Proc. Int. Congr. Industr. Appl. Math. 3rd (Zeitschrift für Angewandte Mathematik und Mechanik, 76: 33–35, 1996).
 Copyright © 1997 the American Physiological Society
Appendix
Normalization of Equations
The dimensional forms of Eqs. 1
and
2
are given by
Appendix
The Characteristic Equation
The characteristic equation for Eqs. 14, obtained by procedures described in Refs. 13 and 17, is given by
Appendix
Spectral Properties of the Distributed Delay
The distributed delay is characterized by the choice of the kernel function ψ_{δ} appearing in Eq. 3 and specified inEq. 4. We have previously shown that the bifurcation locus is unlikely to be much affected by the specific form of the kernel function (17). Here we reason from general considerations that the spectral properties of the TGF pathway, for frequencies below 500 mHz, do not much depend on the form chosen for the kernel function, in the sense that a physiologically reasonable choice for the function is unlikely to introduce significant structure (i.e., pronounced local extrema) in spectral power at frequencies below 500 mHz.
The basis of these general considerations is that the time scale of the distributed MD delay (2–4 s; Ref. 1) is much shorter than that of the transit time of the TAL (15–20 s) or of the sustained oscillations that may be mediated by the TGF pathway (with period 20–33 s; Ref. 7). These time scales correspond to characteristic frequencies of 250–500 mHz for the MD delay, 50–60 mHz for the TAL transit time, and 30–50 mHz for the sustained oscillations. Thus spectral structure below 500 mHz (and particularly below 250 mHz) will be dominated by contributions from TAL transit time or TGFmediated oscillations. It follows as a corollary that tubular compliance, estimated in Ref. 7 to have a characteristic time of 1 s (corresponding to 1000 mHz), will not significantly affect spectral structure below 500 mHz.
The kernel function ψ_{δ} represents the time course of the signal that modulates AA diameter. Experiments show that a step increase in MD chloride concentration produces a sigmoidal decrease in AA diameter like that shown in figure 6 of Ref. 1. To represent this sigmoidal transition, we make the following mathematical assumptions about the kernel function: the function is continuous on the real line; the function is nonnegative in an interval W = [−δ/2, δ/2] of duration δ and equal to zero outside the interval; and inW, the function increases monotonically from a value of zero to a maximum amplitude, near the center of the interval W, and then decreases to a value of zero (thus ψ_{δ} is symmetric, or nearly so, about the center of W). The form of the kernel specified in Eq. 4 is consistent with these assumptions.
A general kernel function, meeting these assumptions for ψ_{δ} on W, can be expressed as a Fourier series (4),
Since we require that a constant input concentration C in Eq. 3
pass through the distributed delay without a change in value (i.e., C_{MD} must equal C if C is unchanging in time), the kernel function must have weight one on the interval W (i.e.,
The choice of a constant function ψ_{δ} produces a linear transition in response to a step change in input C, but the choice of ψ_{δ} given by Eq. 4 produces a sigmoidal transition. The corresponding power spectrum will have minima with the same spacing, but the minima will start at 666 mHz, since only input components of the form sin(2πnu/δ + φ) for n = 2, 3, 4, …, will be orthogonal to ψ_{δ}. These minima occur in the curve labeled “Distributed delay only” in Fig. 2.
For the general kernel, taken to possess the assumed properties of the physiological kernel function, we expect that the cosine terms, which are even functions, will dominate the sine terms, which are odd functions, and that lower frequency terms will dominate high frequency terms. Thus, the largest Fourier coefficients will bea _{o} and a _{1}, as in Eq. 4 , and consequently the most significant orthogonal cancellations will occur for input frequencies of n/δ, n = 2, 3, …, and orthogonal cancellations arising from the input frequency 1/δ will be small. If the distributed delay is spread on an interval of duration 4 s or less, then 2/δ will be no smaller than 500 mHz, and the less significant frequency 1/δ will be no smaller than 250 mHz. It follows that the conclusions of this study are unlikely to be affected by the choice of ψ_{δ}.
Now consider a more general formulation of the MD delay in which the formal distinction between the pure and distributed delays is removed. Let Eq. 3
be replaced by C_{MD}(t) =
Appendix
Numerical Methods
Numerical methods are identified by corresponding figure numbers.
Figures 2 and 3. The power spectra in Figs. 2 and 3 arise from an imposed perturbation of the form I(t) = ς(t)/I_{o}, where ς(t) =
In Fig. 2, for the case designated “Distributed delay only,” C(1,s − τ_{p}) in the integrand of Eq. 3 was replaced by I(t), and C_{MD}(t) was considered to be the output O(t). For the case “TAL only,” F was taken to be 1 + I(t), and O(t) was taken to be the dimensionless SNGFR, computed without any delay, i.e., O(t) = 1 +K _{1}tanh[K _{2}(C_{op} − C(1,t))]. For “TAL with distributed delay,” F was taken to be 1 + I(t), and O(t) =F(C_{MD}(t)), computed from Eqs.14.
Equation 1 was solved using a secondorder essentially nonoscillatory (ENO) scheme, coupled with Heun’s method for the time advance. This algorithm yields solutions that exhibit secondorder convergence in both space and time (12). The integral of Eq. 3 was evaluated by the trapezoidal rule. The numerical time and space steps were Δx = 1/640 and Δt = 1/320 = 3.125 × 10^{−3} s, where Δt, here and below, is given in dimensional units. This high degree of numerical grid refinement is required for sufficiently accurate resolution of oscillations up to 1 Hz, and it provides good qualitative results up to 2 Hz (see Fig. 2 and Table 2). As shown in the companion study (15), flow oscillations produce standing waves in luminal chloride concentration along the TAL; consequently, each frequency component of the broadband forcing having frequency greater than 1/t _{o} will produce one or more nodes, relative to the steadystate concentration profile, at sites along the length of TAL. To obtain valid information about the spectral properties of the model, the standingwave components must be resolved by the numerical methods. For oscillations of 2 Hz, the nodes will be separated (see Ref. 15) by dimensional length (L/t _{o})/(2 Hz) ≈ 0.0159 cm, or nondimensional length 0.0318, and the associated wavelength will be twice this length. Thus the 640 subintervals used for the TAL resolved the wavelength of this highest frequency component on a numerical grid of ∼10 points.
Sampling of model output O(t) for Fig. 2 and for Fig. 3,A–D, began after one period of I(t). In Fig. 3,E –F, the perturbation I(t) allowed sustained oscillatory solutions to develop; in these cases, sampling began after two conditions were met: 1) the oscillations in Freached maximum amplitude, and 2) an integral number of periods of I(t) had elapsed. The output O(t) for all cases was sampled at 5 Hz (i.e., every 64Δt = 0.2 s) for 5 × 2048 points, corresponding to a real time interval of 2048 s, exactly twice the period of the perturbation I(t). The mean of the output O(t) was computed and subtracted from O(t) to prepare the data for spectral analysis.
The spectra displayed in Figs. 23 are estimates of power spectral density, called periodograms, which are computed from the discrete Fourier transform of the demeaned output O(t). In our implementation, the periodograms were computed via a FFT and a supplementary algorithm from Ref. 19, both adapted to double precision arithmetic. The supplementary algorithm minimizes spectral variance per data point by averaging periodograms obtained from overlapping data sets. We used four overlapping sets of 4096 points, and we chose the Welch window for the FFT. The periodograms obtained from O(t) were normalized through division by the periodogram of I.
The domain of the periodogram values corresponds to the frequenciesf _{n} = 2f_{N} n/4096, where n = 0, 1, …, 2048, and where the Nyquist frequencyf_{N} is given by (2 × 64Δt)^{−1}. Thus, domain values are spaced at intervals of about 1.221 mHz, from 0 to 2.5 Hz.
For n > 0, the ordinate values of the periodogram,P_{n}
, approximate the sum of the squares of the Fourier coefficients (corresponding to the f_{n}
) of the response to the input signal I(t). However,P
_{0} corresponds to the square of the constant term of the Fourier series, and
Figure 4. The steadystate TAL profile S(x) corresponding to flow F = 1 was computed from Eq. 1 via the ENO scheme. Normalized versions of the oscillations in Fig. 4,A1 and B1, were then introduced through F. The waveforms in rows 2–6 were recorded after at least one period of the flow oscillation to ensure that the initial profile S(x) had been expelled. The transittime integral was evaluated by the trapezoidal rule. To elicit the oscillations in Fig. 4,columns C and D, TAL flow was initially perturbed by a square pulse (10% of steadystate flow) lasting for one transit time interval t _{o}. Waveforms were recorded after oscillations reached full amplitude.
Figure 5. The signals in Fig. 5 were processed as described for Figs. 23, including normalization by the spectrum of the broadband perturbation used in Figs. 23, to allow magnitude comparisons among the figures. The spectra for MD concentration C (1,t), which corresponds to the signal that is magnified by TGF, were multiplied by 10^{2} to permit comparison with the spectra for SNGFR.
Figure 6. The waveform in Fig. 6 A, for γ = 4, was computed like the waveforms in columns C and D of Fig.4. The waveform in Fig. 6 B was digitally scanned at 300 dots per inch from a reprint of Ref. 11 and stored as a PostScript file. To clearly exhibit the shape of the waveform, the image was stretched horizontally by a factor of about 2.5 (by scaling within the PostScript file), and the axes and legends were redrawn.