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NEUROHUMORAL CONTROL OF CARDIOVASCULAR FUNCTION
0.4 Hz in rat conforms to self-similar scaling following spinal cord transection
1Department of Physiology, University of Kentucky College of Medicine, Lexington; 2Departments of Biology, and Chemistry and Physics, Asbury College, Wilmore; 3Center for Biomedical Engineering, Wenner-Gren Lab, University of Kentucky, Lexington; and 4Cardinal Hill Rehabilitation Hospital, Lexington, Kentucky
Submitted 22 September 2003 ; accepted in final form 19 October 2004
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ABSTRACT |
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 TOP ABSTRACT MATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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4-h intervals in six rats in the neurally intact state and in the same animals after complete spinal cord transection (SCT) between T4 and T5. In the intact state, power within the frequency range of 0.35–0.45 Hz was 1.53 ± 0.38 mmHg2/Hz (mean ± SD by fast Fourier transform). One week after SCT, power within this range decreased significantly (P < 0.05) to 0.43 ± 0.62 mmHg2/Hz. To test for self-similarity before and after SCT, we analyzed data using a wavelet (i.e., functionally, a digital bandpass filter) tuned to be maximally sensitive to fluctuations with periods of 2, 4, 8, 16, 32, or 64 s. In the control state, all fluctuations with periods of 
4 s conformed to a "self-similar" (i.e., fractal) distribution. In marked contrast, the oscillations with a period of 2 s (i.e., 0.4 Hz) were significantly set apart from those at lower frequencies. One day and seven days after the complete SCT, however, the BP fluctuations at 0.4 Hz now also conformed to the same self-similar behavior characteristic of the lower frequencies. We conclude that 1) an intact sympathetic nervous system endows that portion of the power spectrum centered around 0.4 Hz with properties (e.g., a periodicity) that differ significantly from the self-similar behavior that characterizes the lower frequencies and 2) even within the relatively high frequency range at 0.4 Hz self-similarity is the "default" condition after sympathetic influences have been eliminated. autonomic nervous system; sympathetic nervous system; spinal cord injury; fractal
0.1 Hz were statistically indistinguishable irrespective of their frequency or time scale.
Ivanov et al.'s (10) conclusion regarding "time scale" is more easily understood when one considers that the value of any physiological variable changes to some degree over time: HR, for example, increases and decreases on a time scale as short as the respiratory cycle and as long as the 24-h light-dark cycle. The well-known respiratory sinus arrhythmia is driven by inspiration and expiration and so has a relatively stable frequency. These periodic, or "harmonic," components of the overall HR variability are not self-similar precisely because they clearly do have a characteristic time scale, that is, their period. HR recordings, or "time series," however, include components throughout a frequency range extending below the usual periodic rhythms. We (2, 4–7) and others (reviewed in Ref. 16; see also Ref. 8) have used several mathematical techniques to probe the nature of these lower frequency components of the HR and arterial blood pressure (BP) signals. For example, Yamamoto and Hughson (23) developed a "coarse-grained" spectral analysis technique and reported that the HR power spectrum below a frequency of 0.1 Hz increases as some function of 1/f (where f is frequency), which is characteristic of self-similar signals (e.g., Refs. 3, 21). More recently, our group (7) reported that arterial BP and sympathetic nerve activity (SNA) signals did not conform precisely to stationary 1/f noise, at least within the frequency range between 0.02 and 2.0 Hz in the unanesthetized rat. We found that intermittent "large-amplitude events" are important contributors to the very low-frequency variability in these signals (6). With this caveat, the dynamic behavior of HR, BP, and SNA generally displays self-similar behavior below 0.4 Hz.
Another phenomenon of particular interest regarding the dynamic behavior of BP is the so-called 0.4-Hz rhythm in rats (e.g., Refs. 4, 13), as well as the corollary 0.3 Hz and 0.1 Hz rhythms in rabbit (e.g., Ref. 17) and humans (e.g., Ref. 15), respectively. In rats, the 0.4-Hz cyclicity in arterial BP is highly coherent with a corresponding rhythm in SNA (4) and can be explained as a resonance-like phenomenon inherent within the baroreflex (5, 19). As such, it could be regarded as a periodicity that, in theory, would not be expected to exhibit self-similar behavior. Conversely, if the feedback loop governing this periodicity were interrupted, the probability distribution (see below) for the frequency range centered around 0.4 Hz might meld with those for the lower frequencies or conform to a self-similar distribution.
There are certain potential limitations inherent in the use of "workhorse" techniques like spectral analysis that have been used in the majority of experiments cited above. Fortunately, these limitations can be circumvented by other mathematical procedures. In their study, Ivanov et al. (10) used a wavelet-based time-series analysis in which they rescaled their individual findings to demonstrate a "hidden, possibly universal, structure" in HR recorded over 6 h from midnight to 6 am. Interpreting this statement requires some understanding of the analysis that they used. A wavelet can be thought of as a digital bandpass filter that can be "tuned" to a given frequency region of interest (20). For each given frequency region, one examines the probability (P) of observing a fluctuation of a given amplitude (A) in the HR signal [P(A)] as a function of the amplitude of the fluctuation (i.e., A). In other words, P(A) is the probability distribution for the magnitude of HR increases and decreases within a given frequency range (e.g., see Fig. 1). It is difficult to compare the direct results of this analysis across subjects, however, because of the idiosyncratic nature of each animal's absolute HR fluctuations: each individual subject's probability distribution will have a peak amplitude and variance of the distribution characteristic of that subject. Ivanov et al. (10), however, showed how one could rescale the data via an appropriate normalization to facilitate such comparisons. A formal mathematical definition of self-similarity is that such probability density functions are the same shape irrespective of the resolution at which they are examined (14). Therefore, the self-similar nature of the HR fluctuations becomes immediately obvious because all of Ivanov et al.'s rescaled probability distributions are essentially identical irrespective of their time scale (i.e., frequency).
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0.4 Hz will be distinctly different from those tuned to lower frequencies when normal autonomic regulation of arterial BP is intact, but that transecting the spinal cord in the rat to interrupt the descending sympathetic outflow will result in the 0.4-Hz rhythm's conforming to self-similar behavior. |
MATERIALS AND METHODS |
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Surgery and postoperative care.
Surgical methods and postoperative care have been described in detail elsewhere (2). Briefly, the animals were anesthetized (6 mg/kg ip xylazine and 75 mg/kg ip ketamine), and chronically in-dwelling catheters were placed in the left femoral artery and the left jugular vein. One day later, BP was recorded for four consecutive hours while the animals were awake in their home cages. The following day, the rats were reanesthetized, as above, the appropriate vertebrae were exposed, and the spinal cord was completely transected between T4 and T5. The vertebrae were exposed in two sham-operated animals, but spinal cord transection (SCT) was not performed and a bladder catheter was not implanted. Fluid and buprenorphine hydrochloride (0.01 mg/kg sc) were given as needed during the first 12 h postoperatively. Gentamicin sulfate (5 mg/kg sc) was given daily for the duration of the study. The rats were evaluated daily for hydration, and fluids were supplemented as needed; when their caloric intake was considered to be inadequate, they were offered dietary supplements. The rats were allowed to recuperate for 24 h before initiation of postspinal transection BP recordings.
Data acquisition.
BP was recorded for 4 h while the rats were undisturbed in their home cages. The first recordings were made in the neurally intact state (i.e., 24 h after catheter implantation) and then for 4 h at 1 day and at 7 days (range of 6–8 days) after SCT. The BP recordings were made with a Cobe pressure transducer interfaced with a computerized data acquisition system developed in-house (see below).
Data analysis. Data were digitally sampled at 500 Hz with a National Instruments E-series A/D converter. HR was computed from the pulsatile BP signal. Spectral power within the range from 0.35 to 0.45 Hz was computed with fast Fourier transform procedures described in detail elsewhere (4). Likewise, procedures for the wavelet computation have been described elsewhere (6, 10). Briefly, the wavelet transform of a time series x(t) is defined as
 | (1) |
where 
is the analyzing wavelet, which has a width determined by the scale (s) and is centered at time (t). One "slides" the wavelet through the data series by successive increments in t'-t. We used the Mexican hat wavelet (i.e., the second derivative of a Gaussian function) for which there is a conversion factor of approximately four between the s and the actual period (i.e., in s; 6). For example, applying a wavelet with s = 0.5 to a time series tunes the function to be maximally sensitive for rhythms with a period of 2.0 s or a frequency of 0.5 Hz (i.e., very near the 0.4-Hz frequency of particular interest; Ref. 4). We then determined the amplitudes of the variations in the arterial BP time series (see Ref. 6 for mathematical techniques) and constructed a probability distribution for these amplitudes [P(A); see the introduction] for each value of s. We computed P(A) for variations that had periods of 2, 4, 8, 16, 32, and 64 s. Figure 1 is an example of the results of this stage of the analysis.
Another defining characteristic of a variable that conforms to self-similar scaling is that the P(A) closely conforms to a class of mathematical relationships called "generalized homogenous functions" (see Ref. 10 for details). Once again, we followed the lead of Ivanov et al. (10) and fit the probability distributions to a gamma probability density, namely
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For random events with no memory, P(x;
,
)dx is the probability for events to occur in a time interval x, where is the mean rate at which the events occur. For our analysis, x is the amplitude of a fluctuation instead of a time interval. The rescaling process referred to previously, and explained elsewhere (10), allows one to derive a normalized probability density function:
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This expression allowed us to quantify each probability distribution, since x0 is the location of the peak and characterizes the width of the distribution. We used the "lsqnonlin" function in MATLAB 5.3 (Math Works, Natick, MA) to find the parameter that minimized the least-squares residual error between the fit to the gamma distribution and the scaled P(A). Because the location of the peak x0 after scaling is a complicated function of , it was convenient to fit simultaneously both x0 and .
All data are reported as means ± SD. Appropriate ANOVAs with post hoc tests, as allowed, were performed to test for significance of differences at P < 0.05.
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RESULTS |
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Average BP power within the range from 0.35 to 0.45 Hz in the neurally intact state was 1.53 ± 0.38 mmHg2/Hz; the average peak power was 1.96 ± 0.48 mmHg2/Hz at a frequency of 0.39 ± 0.05 Hz. One day after SCT, the average power within this 0.1-Hz-wide frequency bin was significantly decreased to 0.10 ± 0.04 mmHg2/Hz and remained low 1 wk later (0.43 ± 0.62 mmHg2/Hz, not significant vs. day 1).
Figure 1 shows the normalized wavelet probability distribution for arterial BP from a single, unanesthetized, neurally intact rat (i.e., before SCT). These data have been normalized but have not been fit to the gamma distribution; that is, these are "raw" data showing the actual distribution of the data points for this animal. The scaling factor, s, for the distribution shown in red was 0.5; thus this distribution is tuned to period of 2 s. All the remaining colors in Fig. 1 show distributions tuned by s to a lower frequency (blue = 4 s, green = 32 s, yellow = 64 s). Two observations are immediately obvious from the findings from this rat. First, the distributions for frequencies below 0.4 Hz are virtually identical; that is, they are self-similar. Second, the 0.4-Hz wavelet analysis is clearly offset from the remaining distributions in this animal when the autonomic innervations of the heart and circulation were intact.
The next step in the analysis was to fit the raw distributions to a normalized gamma distribution for each rat for the selected values of s (i.e., for each frequency tested). Finally, the gamma distributions were ensemble averaged across rats for each of the frequencies. Figure 2A shows these average probability distributions for each frequency (by color code) for arterial BP recordings made in the intact state. The clear visual impression again is that the probability distribution for s = 0.5 (i.e., 0.4 Hz) is set apart from the others and that all distributions are identical for frequencies between 0.25 Hz and 0.02 Hz (i.e., the lowest frequency examined). This can be tested quantitatively by determining significance of difference between x0 and across the various distributions (Table 1). Recall that x0 reflects the position of the peak of the distribution along the x-axis, whereas is proportional to the shape (i.e., breadth) of the distribution. The value of x0 for s = 0.5 was significantly different (F5,20 = 6.73) from that for scaling factors of 1, 2, 4, 8, and 16 (i.e., periods ranging from 4 to 64 s); this quantitatively confirms the visual impression that the distribution shown in red in Fig. 2A is offset from all the others. Conversely, the values for did not differ across scales.
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0.4 Hz was no longer offset from those for the lower frequencies, as is confirmed in Table 1 for observations made 1 day and 1 wk after SCT. Conversely, for the sham operations, x0 was not changed for s = 0.5 presurgery vs. 1 wk postsurgery (pre-SCT: 0.59 ± 0.07; 1 wk post-SCT: 0.56 ± 0.12). |
DISCUSSION |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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One could explain the low-frequency rhythms, such as the 0.4-Hz cyclicity in rat, by invoking a central oscillator (e.g., Ref. 18), but the most parsimonious explanation seems to be that the strong coherence is an expected consequence of a "resonance" within the baroreflex (5, 19). The observation that interrupting the baroreflex by sinoaortic denervation, breaking the feedback loop, markedly attenuates or eliminates the 0.4-Hz rhythm (11, 13) is consistent with the latter explanation. Likewise, the most parsimonious explanation for the decrease in BP power within the band 0.35–0.45 Hz that we observed after SCT is that this procedure interrupted the descending input to the sympathetic preganglionic neurons; that is, SCT is another way of "opening" the normal feedback control of sympathetic nervous activity. Therefore, although it is not necessary to evoke the existence of an oscillator to explain the 0.4-Hz rhythm, the resonance-like phenomenon that we favor (5) imposes a firm regulation on BP dynamics centered around this frequency that appears as an oscillation with a stable periodicity. As such, power at this frequency cannot conform to self-similar dynamics.
Our wavelet analysis provided a very visual, as well as thoroughly quantitative, confirmation of the generally self-similar nature of the fluctuations in arterial BP for frequencies at or below 0.2 Hz (i.e., s of 1). In other words, after applying Ivanov et al.'s normalization process (10) to facilitate comparisons, we found the probability distributions at the lower frequencies to be essentially identical. In particular, even the distribution for s = 0.5 melded with the others after SCT. Interrupting the baroreflex is reportedly associated with a reduction in the chaotic behavior of arterial BP (9). Julien et al. (13) noted that there was significant residual power in their midfrequency band (0.273–0.742 Hz) after sinoaortic denervation, although no clear peaks could be discerned (i.e., the 0.4-Hz peak was eliminated). They speculated that, in the midfrequency band, endogenous rhythms "of small amplitude and seemingly random frequency are generated by central nervous structures" (13). We interpret our findings with respect to 1) the spectral power between 0.35 and 0.45 Hz and 2) the wavelet probability distributions as follows. SCT 1) eliminated the normal, stable periodicity produced by the action of the baroreflex exerted via the sympathetic nervous system and thereby 2) allowed those processes responsible for the self-similar behavior characteristic of the lower frequencies to emerge at this heretofore "privileged" frequency.
We had previously reported that the "noise" within the arterial BP signal between at least 0.02 and 2.0 Hz was not stationary (7) and that a component of the relationship between SNA and BP within the very low-frequency range was attributable to intermittent large-amplitude events (6). The wavelet analysis does not require a stationary signal; thus we are now able to examine the low-frequency range from this favored perspective. We noted (data not shown) that the actual incidence of very large-amplitude events (i.e., as would be seen at the right-most extent of the abscissa on the probability distribution) typically exceeded the frequency that was predicted by the best-fit gamma function. It seems likely that these unexpectedly frequent occurrences of large fluctuations are, in fact, our large-amplitude events (6).
Although the rats were quite active physically after SCT, there were alterations in the pattern and extent of this activity that one might expect to result in changes in BP variability and, as such, could be a confounding factor in our analysis. In a previous study, we found that the cross-correlations between HR and BP in neurally intact, chronically maintained, unanesthetized but immobile (i.e., neuromuscular blockade) rats are qualitatively and quantitatively similar to those for unanesthetized and mobile rats (2), that is, positive with a peak that occurs when HR changes lead BP changes in the intact state in both the normal and immobile animals. In contrast, after SCT, results become sharply negative with BP changes leading HR changes. This function, therefore, appears to be quite sensitive to functional changes in the autonomic control of the heart (2). By extrapolation of these earlier findings, and given the relatively high degree of mobility in the SCT animals, it seems unlikely that differences in physical activity account for the melding of the 0.4-Hz probability distribution with those for lower frequencies.
Perspectives
The mathematical model that we offered as an explanation for the 0.4-Hz rhythm (5) includes terms relevant to the nature of the baroreflex, delays inherent within the biofeedback system, and the coupling of changes in SNA to the corresponding cardiovascular responses. Others have stressed the probable importance of nonlinear processes in producing the oscillation (19). It seems very likely that the relative displacement of the probability density distribution for s = 0.5 (i.e., 0.4 Hz) from the distribution common to all the remaining curves, as is reflected in x0, is dependent on the physiological integrity of the individual components of the systems regulating BP. In effect, the SCT can be regarded as "the ultimate sympathetic dysfunction," and eliminating normal sympathetic function in this way dramatically altered the fundamental "behavior" of arterial BP within the specific region centered around this "resonant frequency." The realm of self-similar behavior in arterial BP in rats is generally restricted to frequencies of 0.1 Hz (e.g., Ref. 22), if for no other reason that periodic signals, primarily under the control of autonomic mechanisms, dominate at frequencies that exceed this value; the present analysis does not extend to frequencies below 0.01 Hz. Our findings indicate that self-similarity is a "normal" default condition, even at this relatively higher frequency, when such neural influences are eliminated.
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FOOTNOTES |
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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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This article has been cited by other articles:
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  D. R. Brown, L. A. Cassis, D. L. Silcox, L. V. Brown, and D. C. Randall Empirical and theoretical analysis of the extremely low frequency arterial blood pressure power spectrum in unanesthetized rat Am J Physiol Heart Circ Physiol, December 1, 2006; 291(6): H2816 - H2824. [Abstract] [Full Text] [PDF]
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 C. Julien The enigma of Mayer waves: Facts and models Cardiovasc Res, April 1, 2006; 70(1): 12 - 21. [Abstract] [Full Text] [PDF]
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