Am J Physiol Heart Circ Physiol 291: H2816-H2824, 2006.
First published July 14, 2006; doi:10.1152/ajpheart.00135.2006
0363-6135/06 $8.00
Empirical and theoretical analysis of the extremely low frequency arterial blood pressure power spectrum in unanesthetized rat
David R. Brown,1
Lisa A. Cassis,2
Dennis L. Silcox,3
Laura V. Brown,1,3 and
David C. Randall1,3
1Center for Biomedical Engineering, Wenner-Gren Laboratory, University of Kentucky; 2Graduate Center for Nutritional Sciences, University of Kentucky; and 3Department of Physiology, University of Kentucky College of Medicine, Lexington, Kentucky
Submitted 6 February 2006
; accepted in final form 7 July 2006
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ABSTRACT
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The slope of the log of power versus the log of frequency in the arterial blood pressure (BP) power spectrum is classically considered constant over the low-frequency range (i.e., "fractal" behavior), and is quantified by 
in the relationship "1/f." In practice, the fractal range cannot extend to indefinitely low frequencies, but factor(s) that terminate this behavior, and determine , are unclear. We present 1) data in rats (n = 8) that reveal an extremely low frequency spectral region (0.083–1 cycle/h), where approaches 0 (i.e., the "shoulder"); and 2) a model that 1) predicts realistic values of within that range of the spectrum that conforms to fractal dynamics (
1–60 cycles/h), 2) offers an explanation for the shoulder, and 3) predicts that the "successive difference" in mean BP (mBP) is an important parameter of circulatory function. We recorded BP for up to 16 days. The absolute difference between successive mBP samples at 0.1 Hz (the successive difference, or 
) was 1.87 ± 0.21 mmHg (means ± SD). We calculated for three frequency ranges: 1) 0.083–1; 2) 1–6; and 3) 6–60 cycles/h. The for all three regions differed (P < 0.01). For the two higher frequency ranges, indicated a fractal relationship (6–60/h = 1.27 ± 0.01; 1–6/h = 1.80 ± 0.16). Conversely, the slope of the lowest frequency region (i.e., the shoulder) was nearly flat (0.083–1 /h = 0.32 ± 0.28). We simulated the BP time series as a random walk about 100 mmHg with ranges above and below of 10, 30, and 50 mmHg and with from 0.5 to 2.5. The spectrum for the conditions mimicking actual BP time series (i.e., range, 85–115 mmHg; , 2.00) resembled the observed spectra, with in the lowest frequency range = 0.207 and fractal-like behavior in the two higher frequency ranges ( = 1.707 and 2.057). We suggest that the combined actions of mechanisms limiting the excursion of arterial BP produce the shoulder in the spectrum and that contributes to determining .
power spectra; circadian rhythm; computer model
SPECTRAL ANALYSIS TECHNIQUES, such as the fast Fourier transform, quantify the distribution of power for a given variable across a range of frequencies. A fast Fourier transform of the arterial blood pressure (BP) signal in rat, in particular, yields a spectrum with concentrations of power at the pulse frequency (6 Hz) and at the respiratory frequency (2 Hz) that reflect, respectively, the cyclic (i.e., "periodic") rise and fall in BP with the heart beat and with the act of breathing. Energy expenditure is also focused around 0.4 Hz, where there is no obvious oscillator (5, 12); this spectral peak can be explained as a "resonance-like" periodic process that is the consequence of the time delays, etc., inherent within the baroreflex (8, 14, 23). These regularly repeating processes have a clear "scale" (i.e., a characteristic frequency or period). In addition to the power associated with these periodic processes, the arterial pressure power spectrum increases profoundly within the "low-frequency range" (generally <0.1 Hz, depending somewhat on the species). In fact, the increase in power between those relatively high frequencies, where periodic processes dominate, and this low-frequency range is so large that one typically plots the power spectrum using logarithmic scales for power (ordinate) as well as frequency (abscissa).
Over the heretofore studied range of low frequencies, a graph of the log of arterial BP spectral power versus the log of frequency increases linearly as frequency decreases. In fact, unlike the focused spectral peaks at the pulse and respiratory rates, there is no obvious, specific concentration of power in the low-frequency range, excepting a peak at 1 cycle/day, the circadian rhythm (0.042 cycles/h). This log-log/linear increase in power constitutes the so-called "1/f" behavior and indicates that, unlike a periodic process, there is no characteristic scale for frequencies <0.1 Hz: the statistical characteristics of the signal are the same no matter over what frequency range it is examined. This low-frequency power is often called fractal "noise" in accordance with concepts developed by Mandelbrot (17). Timescale invariant fractal behavior such as this is characteristic of a broad range of natural phenomena (3, 24).
As first demonstrated by Kobayashi and Musha (16), the inverse slope, or "," is one means of characterizing that region of the power spectrum over which the log-log plot of power versus frequency is linear; more precisely, the relationship is given as "1/f." For example, if the amplitude of the fluctuation in mean arterial BP did not change across frequency (i.e., "white noise"), the power spectrum would be flat (i.e., horizontal), or = 0. The = 2 if the noise at a given time depends on its value a moment earlier (e.g., as in Brownian motion); this is equivalent to saying that those forces controlling the behavior of the variable have a "memory" or, in more physical terms, an "inertial mass," which limits the rate at which the variable can change. A number of investigative teams have published values for within the fractal range. Holstein-Rathlou et al. (13), for example, reported an average value of within the range 0.1–10 mHz of 1.45 ± 0.04 (SE) based on 4–6 days of BP recordings in rats via telemetry. Others (25) have reported two frequency ranges with significantly different values for . Irrespective of the number of fractal ranges, those factors that determine the actual value of are not at all clear. Holstein-Rathlou et al. (13) also rightly pointed out that spectral power cannot increase indefinitely at ever lower frequencies; if it did, daily or weekly fluctuations in power would be enormous: there must be some frequency, however low, below which a steep, progressive, 1/f increase in power no longer dominates the spectrum. The process that mitigates fractal (i.e., 1/f) behavior at some undetermined, but extremely low, frequency is unknown.
We now present data that show a marked flattening of the BP power spectrum starting at 1 cycle/h in rat. Next, we present a model that offers an explanation for these empirical observations. Specifically, our model predicts that limitations in the allowed range of excursion in BP (e.g., via the combined actions of the baroreflex, hormonal controls, and autoregulatory mechanisms) mandate a limit on the frequency range over which classical 1/f behavior occurs. The model also suggests that a previously unappreciated parameter, which we call the "successive difference" (see MATERIALS AND METHODS), is important in predicting the value of . Finally, we show that constraining the model with physiologically realistic values for the range of pressure excursion, and a value for the successive difference that approximates the empirically determined value, yields a "power spectrum" that mimics the actual BP spectrum. Taken together, we believe these findings provide new insights into the control(s) of arterial BP. Preliminary reports of these findings have been published (6, 7).
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MATERIALS AND METHODS
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Subjects.
Data are reported for eight Sprague-Dawley rats (Harlan Industries, Indianapolis, IN) weighing 230–380 g. The experiments were performed in accordance with the guidelines for animal experimentation described in "Guiding Principles for Research Involving Animals and Human Beings" of the American Physiological Society (1) and were approved by the Animal Care and Use Committee of the University of Kentucky.
Surgery and postoperative care.
Procedures appropriate for rat survival surgery were followed. The animals were anesthetized with pentobarbital sodium (65 mg/kg ip). The abdominal aorta was exposed via a laparotomy. The sensory element of the Data Science International (DSI, St. Paul, MN) probe (model TA11PA-C40) was placed into the aorta via puncture such that its tip pointed toward the heart. The body of the probe that contains the necessary circuitry and battery was secured to the interior abdominal wall. The incision was closed, and the rat was allowed to recover from the anesthetic. Although BP monitoring commenced immediately, no data were used for two or more weeks.
Experimental procedures and protocol.
The rats were maintained in an isolated, sound-shielded room with a computer-controlled 12:12-h light-dark cycle. The animals were fed ad libitum and observed briefly once daily during the morning (i.e., light-on portion of the daily cycle) but with minimal additional interference. Each animal’s cage was placed atop a platform that contained the necessary circuitry for receiving the telemetry signal. The phasic arterial BP was recorded continuously (see below) for 10 (n = 4) or 16 days (n = 4).
Data acquisition and analysis.
Data were acquired and analyzed using software developed by ViiSoftware (Lexington, KY). Mean BP and heart rate (HR) were calculated from the pulsatile BP signal digitized at 500 Hz. The resulting file was "compressed" to 0.1 Hz by block averaging and was divided into 24-h segments (noon to noon). Any linear trends were removed, and the power spectra for BP and for HR were calculated by using a fast Fourier transform for each 24-h data set. The individual 24-h spectra were ensemble averaged for each rat. The spectra were divided into three ranges: 1) 2 cycles/day (0.083 cycles/h) to 1 cycle/h; 2) 1–6 cycles/h (1 cycle/10 min); and 3) 6–60 cycles/h (1 cycle/min.). The best-fit inverse slope (i.e., ) was calculated within each range by linear regression. In addition, the successive difference, or , was determined from the compressed (i.e., 0.1-Hz data file) arterial pressure time series by measuring the absolute change in mean BP from one 10-s sample to the next; as such, is effectively the rate of change of mean BP (mBP) from sample to sample. The data in Table 1 and all statistical tests were based on the procedures described immediately above. Alternatively, the Fourier transform was applied to the continuous 10-day (or 16-day) data set to produce spectra extending down to 0.1 cycle/day. Data were examined by analysis of variance (ANOVA) with repeated measures or by a t-test, as appropriate. Statistical significance was accepted for P < 0.01. Results are shown as means ± SD.
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Table 1. Average ± SD best-fit and correlation coefficient ( ) for mean arterial BP and heart rate for defined frequency ranges
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Model.
We developed a computer simulation of the mBP time series, Yt, as a random walk [i.e., Brownian motion; see Voss (24) for an exposition]; simulations were performed in Microsoft’s Visual Studio using C++ . Our choice of a random walk is not without precedent. Butler et al. (11), for example, reported that = 2.31 for the arterial BP spectrum in resting, healthy men; this suggested to them that the BP variability "was similar to Brownian motion." Moreover, Pilgram and Kaplan (20) speculated that "HR might be allowed to drift (Brownian motion)." Finally, a random walk imitates a naturally occurring process that has previously been utilized to model fractal BP dynamics (20).
To maximize physiological relevance, we imposed specific constraints on the random walk. First, the model was initialized at 100 "mmHg" so that
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The computer generated a sequence of random numbers (R) between 0 and 1 such that
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and
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where , as above, represents the rate of change of mBP from sample to sample. In this way "pressure," or Yt, randomly increased or decreased from beat to beat by a value equal to starting at 100 mmHg. was initially set at 2 to mimic the empirically observed value of 1.87 (see Table 2); we also specifically tested values of that were above and below this empirical value (cf. Tables 3 and 4). Second, we specified the maximum (MAX) and minimum (MIN) allowable value of pressure so that if, for any given iteration
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and
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These conditions constrained the simulation to remain within specified pressure limits. To test an unbounded condition (e.g., 
Fig. 5A) MAX and MIN were set at +
and –, respectively. Each iteration or data point occurred every 10 s, thereby resulting in 8,640 data points per "day." Each simulation consisted of a 4-day period or 34,560 data points. No circadian rhythm was incorporated in the model. A second model was also tested in which mBP was represented as white noise, rather than a random walk, around a value of 100; maximum and minimum bounding values were specified, as above. Again, white noise ( = 0) was specifically chosen because it imitates a naturally occurring process that has also been utilized previously to model the fractal nature of the BP spectrum (20).
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Table 2. Effects of limitation of arterial BP excursions on model predictions of for each of three frequency ranges with fixed at 2
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Table 4. Effects of variations in on for each of three frequency ranges with arterial BP excursions limited to range of 85–115 mmHg
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Fig. 1. Heart rate (HR; top), mean arterial blood pressure (mBP; middle), and light cycle (bottom) for a single rat across 10 days of continuous, beat-by-beat recording. Animal was housed singly in a standard cage in a room separate from the general laboratory environment with ambient 12:12-h light cycles controlled by computer. Phasic BP via telemetry was digitized at 500 Hz. From this signal, HR was determined as well as mBP. Plot shown here is based on files compressed to 0.1 Hz. Note clear circadian rhythm with lowest pressure and lowest HR during light-on condition.
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Fig. 2. Arterial BP power spectra for each of 8 rats. Each spectrum was computed from a continuous 10-day data set, such as that shown in Fig. 1. Both ordinate and abscissa are in logarithmic units, though the range in the ordinate (10–10 to 100) was too extensive to show an inclusive scale. Values are given on abscissa in cycles per day, so the peak in spectra at 1 is the circadian rhythm. Ordinate for each animal was normalized against the peak power in that subject’s spectrum. Four vertical lines demarcate three frequency regions of interest (0.083–1 cycle/h; 1–6 cycles/h; 6–60 cycles/h). Power spectrum is almost flat (slope 0) across the lowest designated frequency range. Linear increase in power within the higher two frequency ranges conforms to a "1/f" pattern expected of fractal behavior.
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Fig. 3. BP power spectrum for a single rat computed from 10 days of recordings. These data were partitioned into 24-h segments, subject to fast Fourier transform, and the resulting spectra ensembles were averaged to produce the plot shown here. Ordinate is log of power normalized against value at 0.042 cycles/h (i.e., 1 cycle/day). Abscissa is log of frequency in cycles per hour. Best-fit slope is shown for three frequency ranges (i.e., yellow, 6–60 cycles/h; green, 1–6 cycles/h; red, 0.083–1 cycle/h). Note slope for lowest frequency range 0.
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Fig. 4. HR power spectra for each of 8 rats computed from a continuous 10-day data set. Data are plotted analogous to Fig. 2, except the ordinate is from 10–12 to 100. Ordinate for each animal has been normalized against the peak power in that subject’s spectrum (i.e., at circadian frequency). Four vertical lines demarcate three frequency regions of interest (0.083–1 cycle/h; 1–6 cycles/h; 6–60 cycles/h).
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RESULTS
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Time series and average BP, HR, and mBP successive difference.
Figure 1 portrays 10 days of HR (top) and mean arterial BP (middle) recordings for a single rat, together with the computer-controlled light-dark cycle. Notice the clear circadian rhythm with the lowest HR and lowest BP values during the light-on portion of the daily cycle. The 24-h average for mBP computed over the 10-day recordings for each rat and then averaged across rats was 107 ± 11 mmHg. The corresponding value for HR was 361 ± 18 beats/min. The absolute value of the difference between successive mBP samples (i.e., the successive difference, or ) averaged 1.87 ± 0.21 mmHg.
Empirically determined spectra for BP and HR.
Figure 2 is a compilation of the mBP power spectra for each of the eight rats; both the power (ordinate) and frequency (abscissa) are plotted on a logarithmic scale. The frequency range extends down to 1 cycle/10 days (i.e., 0.0042 cycles/h; recall that the continuous 10-day data set was used to produce these plots). The scale for power in each individual spectrum was normalized against its peak value in that spectrum. This peak occurred at 1 cycle/day (i.e., 0.042 cycles/h) in seven animals; the single exception was the animal whose spectrum is at bottom right in Fig. 2; in this animal, the circadian peak was the second highest recorded value. The cutoff frequencies for the three ranges designated in MATERIALS AND METHODS, including the upper limit of the abscissa at 60 cycles/h, are indicated by the vertical lines. Note particularly in each animal that the spectrum appears to be essentially flat (i.e., the best-fit line approaches horizontal) for the region 0.083–1 cycle/h (i.e., the "shoulder" region). For the higher two regions, the spectra display the expected log-linear behavior.
Figure 3 displays the slopes for each of the three frequency regions for a single animal; this spectrum was derived from the recordings shown in Fig. 1 by dividing the data set into ten 24-h data sets; the 10 individual spectra were then ensemble averaged to produce this plot. Power has been normalized relative to the circadian peak. Note, therefore, that at the low-frequency end of this spectrum (i.e., 0.042 cycles/h = 1 cycle/day), power increases to the highest value occurring in the spectrum. Although this logarithmic plot seems to minimize the magnitude of this peak, it is, in fact, the pronounced circadian rhythm that is so obvious in Fig. 1. The yellow line represents the best-fit slope for the highest frequency range (i.e., 6–60 cycles/h); the green line is the slope for the range 1–6 cycles/h, while the red line is for the lowest range (i.e., 0.083–1 cycle/h). As expected from Fig. 2, there is a broad shoulder in power within this lowest frequency range where the best-fit line approaches horizontal. The slopes for the two higher frequency regions have a finite value >0, and there appears to be a modest difference between for the two regions.
Table 1 presents the group average (n = 8) values of for each of the three frequency ranges for mBP; the corresponding correlation coefficients () are also given. The ANOVA for repeated measures disclosed a significant difference between frequency ranges for the BP spectrum (f2,14 = 103.2); post hoc tests revealed that all three slopes were significantly different from one another. The for the shoulder region (i.e., 0.083–1 cycle/h) ranged from –0.22 to 0.71, with an average of 0.32 (95% confidence limits: 0.08 and 0.55). Its value was normally distributed and significantly (t-test) exceeded 0 (i.e., 
white noise).
Figure 4 is analogous to Fig. 2, except that it shows the eight HR spectra. The peak in each spectrum occurs at the circadian frequency. A shoulder region is overall less evident in these spectra, though the slope is shallow in the lowest frequency range in several rats. A quantitative summary of the HR spectra is given in Table 1. The ANOVA for the HR spectrum also showed a significant difference for frequency range (f2,14 = 23.6). In this case, however, whereas the two higher frequency ranges differed from one another, the lowest frequency range differed only from the 1–6 cycles/h range. Finally, statistical comparison of the values for the BP versus HR spectra revealed a significant difference (only) for the lowest frequency range (i.e., 0.083–1/h).
Model predictions.
Figure 5A (left) shows the time series generated by the model for the situation where "BP" was permitted to vary as a random walk without limits (i.e., MAX and MIN set at + and –, respectively) but with set at 2 mmHg to approximate the observed value of 1.87 mmHg. This simulation yielded the "time series" in the left panel and the power spectrum in the right panel of Fig. 5A. As one would expect for Brownian motion, this spectrum is monotonically linear across the extent of the abscissa (i.e., with no differences across the three frequency ranges). The model was run 50 times for several allowed ranges in BP excursion to generate the data in Table 2. Average values of are shown here for each of the three standard frequency ranges. Row 1 of Table 2, for example, corresponds to the unbounded random walk illustrated in Fig. 5A; as expected for this situation, the average value for was 2 for each frequency range. The prediction changed notably, however, when a maximum and minimum allowable BP were established. In Fig. 5B, for example, BP was restricted to a range within 75 to 125 mmHg (left). The corresponding spectrum (Fig. 5B, right) had a clear shoulder for the lowest frequency range, with a that differed between the lowest and higher frequency ranges (Table 2, row 2). In fact, when the BP was restricted to within a physiologically realistic 85- to 115-mmHg excursion (Fig. 5C), the values of the model for for the frequency ranges 2/day to 1/h and for 1/h to 6/h (Table 2, row 3) were not noticeably different from those empirically determined values reported in Table 1 for the corresponding frequency ranges. Restricting the possible excursion in BP to (only) 95–105 yielded the spectrum in Fig. 5D with an exaggerated shoulder region; corresponding slopes are given in row 4 of Table 2.
Figure 6 shows a BP time series and corresponding BP spectrum from one of our rats plotted by using scales identical to those in Fig. 5C. Figs. 5C and Fig. 6 are remarkably similar. This is a visual confirmation of the realistic predictions of the model.
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Fig. 6. Left: mean BP from telemetry data for an unanesthetized rat recorded beat by beat for 4 days. Right: log-log plot of BP power spectrum data. Peak at 0.042 cycles/h is the circadian rhythm. Data are plotted on same scale as in Fig. 3 for ease of comparison of empirically determined time series and power spectrum with model predictions. Note in both data and model the shoulder in BP power spectrum beginning at 1 h/cycle.
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Table 3 documents the effects of different values of on the predictions of the model for in the case where mBP was allowed to vary without restriction. The actual range of pressure for each value of is indicated in column 2. In these predictions where there were no restrictions on BP excursion, the values for were uniformly 2 and the value of had no effect whatever on these slopes. Restated, the limitation of the possible range of the excursion in mBP is a key element in predicting the existence of a shoulder within the extremely low frequency range and is an essential condition for having any influence on . Conversely, Table 4 shows the results of the model when mean pressure was restricted to a physiologically realistic range between 85 and 115 mmHg and was set at the value indicated in column 1. Under these circumstances, manipulating the rate of change in mBP per 10-s interval (i.e., ) had remarkable effects on the output of the model and, in fact, was a key element in realizing the physiologically realistic findings shown in row 3 of Table 4.
Table 5 summarizes the results of the simulation when the model was driven by a white noise signal around 100 instead of the random walk used in all previous simulations. As before, the values of the slopes for the three standard frequency ranges are the averages derived from 50 iterations for each of the allowed ranges in pressure. Slopes were uniformly 0 for each frequency range irrespective of the bounding values. In other words, there were no manipulations of the parameters of the model that altered the slope from the classic value of zero for any given frequency range or for any given allowed range of mBP excursion.
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Table 5. for each of three frequency ranges when BP is represented as a white noise around 100 and bounded by indicated high and low values
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DISCUSSION
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Our empirical findings indicate that the slope of the power spectrum flattens remarkably for frequencies below 1 cycle/h (i.e., 0.0003 Hz). This confirms Holstein-Rathlou et al.’s (13) contention that spectral power cannot increase indefinitely at ever lower frequencies. This unexpected finding prompted us to seek potential explanations for the shoulder using a model. The model hypothesizes that this failure of power to increase indefinitely as frequency decreases is the inevitable consequence of the fact that BP must remain within certain limits, irrespective of those mechanisms (e.g., the baroreflex, renin-angiotensin-aldosterone system) that stabilize it. Our model also hypothesizes that influences the actual value of within a given frequency range, but only under the physiologically realistic conditions where BP 1) is constrained to remain within limits and 2) has a memory (i.e., the previous BP influences the possible range of the current value, as modeled via Brownian motion). We believe that these findings provide useful clues as to the nature of arterial BP regulation.
We specifically report a shoulder in the arterial pressure power spectrum for the first time. The shoulder was previously unnoticed, we believe, because very long data recordings are required if one wishes to examine the distribution of power within a spectrum at extremely low frequencies. Although common practice until recently has been to record the variable of interest for a few hours, earlier experiments have used creative data acquisition/analysis techniques to probe the low-frequency ranges of the BP power spectrum. For example, as early as 1989, Broten and Zehr (4) recorded arterial pressure in conscious dogs via telemetry. Their 2-Hz sampling rate allowed them to identify an ultradian rhythm in BP with a peak at 0.76 cycles/h; although this is close to the upper frequency limit of our shoulder region, this team’s interests focused on autonomic control of these periodic rhythms and not self-similar behavior. Wagner and Persson (25) recorded BP for 4 h; they then divided these recordings into two 2-h segments and averaged the power spectra obtained from each to yield their final composite spectrum. These spectra extended over a sufficiently large frequency range to reveal the two fractal regions. We now affirm that, as in dog (25), there are also two regions within the low-frequency range of the BP power spectrum that conform to fractal dynamics in the unanesthetized rat. In addition, however, our very broad frequency spectra clearly contain a region extending from 1 cycle/h down to 2 cycles/day (i.e., 0.083/h), wherein the slope of the spectrum is almost flat. Although the beginning of such a region is retrospectively discernable in previous work (9, 13, 18, 19) and Yamamoto et al. (27) reported a region in the human HR power spectrum below 0.01 Hz, where = 0.53, little or no attention has otherwise been given to this feature of the BP power spectrum. Holstein-Rathlou et al. (13), however, noted that the lower limit of the 1/f pattern was 2.8 h/cycle, and that the 1/f process did not extend below the circadian peak. They presented a power spectrum derived from BP telemetry recordings in Sprague-Dawley rats that showed at least the initial portions of a shoulder starting at about the same frequency as we observed.
The difference in between the two identified frequency ranges was not dramatic in neurally intact dogs (25) but became clear after baroreceptor denervation that interrupted aspects of normal BP regulation. Likewise, the difference in the slopes characterizing the two higher frequency regions was not visually dramatic in our rat spectrum, though the values differed significantly. Wagner and Persson (25) also computed the power spectrum in dogs after they sectioned all branches of both vagosympathetic nerves between the thoracic aperture and the aortic arch. The for both regions increased after these surgical manipulations, and the two regions became more clearly demarcated. These findings, they reasoned, are consistent with the probability that both regions are influenced by the altered autonomic control, but the withdrawal of the short-term BP buffering seemingly had the largest impact on the higher frequency range. Likewise, Oosting et al. (19) found that various pharmacological manipulations of the autonomic nervous system changed . Our model is consistent with the possibility that the baroreflex, among other mechanisms, could influence the dynamics (e.g., ) of the fractal region, as described by Wagner and Persson (25), via its influence on the range of BP excursion and on . Figure 4 and Table 1 reveal that the shoulder is less characteristic of the HR power spectrum. It is interesting, therefore, that the dynamic range of HR typically exceeds that of BP, which, by our model, would extend the log-linear power range to lower frequencies.
Given the nearly ubiquitous appearance of "1/f" behavior in cardiovascular signals, there is a special curiosity about the nature of fractal noise (3, 24). One might well seek insights into such behavior from the value of . In their study of 1/f dynamics in canine BP signal, Marsh et al. (18) reasonably assumed that "each of the regulatory systems that participate in blood pressure regulation has a preferred frequency of operation; the absence of a characteristic time scale may be taken to imply that no single regulatory system dominates the regulation of blood pressure in the dog." That is, although any individual regulatory mechanism is able to operate over only a circumscribed frequency range, the combined actions of many systems are distributed over a very broad range of frequencies. The restriction that we placed in our model on BP excursion, in fact, operates uniformly over an indefinite frequency range. Although unrealistic for any single regulatory system, this broad-frequency restriction mimics the combined actions of multiple individual systems. The success of our model in replicating specific characteristics of the power spectrum (e.g., the shoulder) within the extremely low frequency is, therefore, consistent with the "distributed" hypothesis of BP control at low frequencies.
The factor(s) that determine are not known. Pilgram and Kaplan (20) asked whether is frequency independent and whether its value was constant over time. Their statistical analysis of 24-h HR records indicated that the 1/f structure, in fact, fluctuates with time and that the value of might vary with timescale. More particularly, they wondered whether a value for 1, which they feel is generally characteristic within the typical fractal frequency range, might be due to a rapid shifting between the relatively simple theoretical case where 0 (i.e., white noise) and where 2 (i.e., Brownian motion). Although technical and/or theoretical limitations preclude the experimental demonstration of any such shifts, their computer simulation showed that it was possible to produce 1-h segments where was close to 1 by conjoining very short segments with = 0 or = 2. Our model, however, indicates that switching between multiple regulatory systems is not necessary to explain the observed value of , at least if one accepts the physically mandatory condition that the current value of BP influences succeeding values. This may explain why a previously unrecognized parameter, , in conjunction with the bonding limits, actually helps determine the value of . As we speculate in PERSPECTIVES, the value of and the findings of our model may, indeed, have physiologically realistic and interesting implications.
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PERSPECTIVES
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Pilgram and Kaplan (20) posited that, from a control systems perspective, Brownian motion corresponds to control being turned off, whereas a signal the behavior of which conforms to a white noise pattern (i.e., = 0) is under strict control: the latter system may be displaced from its set point by an external perturbation, but it quickly returns toward its set point value only to be knocked off again. We do not propose that BP is behaving as a white noise in the shoulder region (i.e., empirically, 0), and, in fact, we show that a realistic model of the low-frequency region of the arterial BP power spectrum can be built on a Brownian process (Tables 2 and 4) but not on a white noise process (Table 5). If so, what causes the BP fluctuations that ultimately constitute "fractal noise" and what processes are engaged in shaping the beat-by-beat pressure profile? Regarding the first question, multiple origins for fractal behavior of BP have been proposed (11), including neurohumoral mechanisms, chemoreflexes/pH fluctuations, changes in cardiac afterload and preload, local vascular autoregulation, and changes in vasomotor tone associated with thermoregulation (25, 26, 28). Possible contributions from sources within the central nervous system have also been discussed (15, 28). Recall from the Introduction that the baroreflex imposes a strong periodic (i.e., nonfractal) rhythm on BP at 0.4 Hz (5, 8, 12). Elimination of the descending sympathetic outflow from the brain by complete spinal cord transection in the rat (2) causes the power at 0.4 Hz to conform to a scale-independent behavior (22); this finding argues that fractal behavior is the "default" condition even at 0.4 Hz. Regarding the second issue, certainly physical constraints (i.e., analogous to mass in a physical system) limit the rapidity of any BP change: pressure, indeed, has at least a short memory, as is required in any Brownian process (i.e., its value at time t = n is influenced by its value at t = n – 1). Moreover, it seems likely that the baroreflex and other closed-loop regulatory mechanisms influence the low-frequency region of the power spectrum via altering the range of allowed BP variation and by determining the successive difference in mean BP.
What, if any, insights into the regulation of BP dynamics within the extremely low frequency range do our data and model afford? Although the following is clearly speculative, we raise the interesting possibility that the baroreflex does not continuously and uncompromisingly dictate beat-by-beat pressure behavior but is aggressively called into play when pressure rises above or falls below allowable limits. That is, our empirical findings and theoretical model seem to be consistent with the possibility that those closed-loop systems impinging on arterial pressure, including, but by no means limited to, the baroreflex, are relatively disengaged from beat-by-beat pressure control so long as pressure "wanders" within appropriate limits. In this sense, the biofeedback systems would act as "sloppy controllers" that allow BP to fluctuate moment to moment as directed by those hydraulic features endowing the cardiovascular system with a memory of its immediate past history (i.e., the present value is constrained by the immediately previous values), and as influenced by external perturbations, so long as its value remains within certain constraints. If so, what nonbiofeedback factor(s) govern BP when it is within allowed boundaries? We reported earlier (21) that the initial arterial pressure response to an acute behavioral challenge is produced by a "sudden burst" of sympathetic nervous activity that precedes the initial pressor response and concluded that this represented an open-loop process. It seems reasonable that such open-loop control, or "central command," is active in innumerable natural conditions and plays a more important role than perhaps is typically recognized in moment-to-moment BP control. We have suggested elsewhere (10) that autonomic regulatory mechanisms involving many levels of central integration (i.e., in addition to the brainstem baroreflex pathways) play a demonstrable role in ongoing cardiovascular control.
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GRANTS
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This study was supported by National Institutes of Health Grants HL-64121 (to L. Cassis) and NS-39774 (to D. Randall).
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ACKNOWLEDGMENTS
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We gratefully acknowledge the assistance in performing statistical analysis provided by Dr. Helena Truszczynska, Statistical Services, University of Kentucky Computer Center.
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FOOTNOTES
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Address for reprint requests and other correspondence: D. C. Randall, Dept. of Physiology, Univ. of Kentucky College of Medicine, Lexington, KY 40536-0298 (e-mail: randall{at}uky.edu)
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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