## Abstract

Muscle sympathetic nerve activity (MSNA) in resting humans is characterized by cardiac-related bursts of variable amplitude that occur sporadically or in clusters. The present study was designed to characterize the fluctuations in the number of MSNA bursts, interburst interval, and burst amplitude recorded from the peroneal nerve of 15 awake, healthy human subjects. For this purpose, we used the Allan and Fano factor analysis and dispersional analysis to test whether the fluctuations were time-scale invariant (i.e., fractal) or random in occurrence. Specifically, we measured the slopes of the power laws in the Allan factor, Fano factor, and dispersional analysis curves. In addition, the Hurst exponent was calculated from the slope of the power law in the Allan factor curve. Whether the original time series contained fractal fluctuations was decided on the basis of a comparison of the values of these parameters with those for surrogate data blocks. The results can be summarized as follows. Fluctuations in the number of MSNA bursts and interburst interval were fractal in each of the subjects, and fluctuations in burst amplitude were fractal in four of the subjects. We also found that fluctuations in the number of heartbeats and heart period (R-R interval) were fractal in each of the subjects. These results demonstrate for the first time that apparently random fluctuations in human MSNA are, in fact, dictated by a time-scale-invariant process that imparts “long-term memory” to the sequence of cardiac-related bursts. Whether sympathetic outflow to the heart also is fractal and contributes to the fractal component of heart rate variability remains an open question.

- Allan factor
- dispersional analysis
- Fano factor
- heart rate variability
- long-range correlations
- time-scale invariance

muscle sympathetic nerve activity (MSNA) in resting humans is characterized by cardiac-related bursts of variable amplitude that occur sporadically or in clusters (1, 9, 10, 18, 20). In some subjects, cardiac-related burst amplitude waxes and wanes on the time scale of the respiratory cycle (9, 10, 18, 23). Whether respiratory-related or not, a hallmark of human MSNA is the variability of cardiac-related burst amplitude and interburst interval. In general, the muscle nerve fails to generate a burst in a variable number of cardiac cycles, thus leading to an MSNA burst-to-cardiac cycle ratio that is typically <1.0 in healthy human subjects.

The primary aim of the present study was to define the basis for the fluctuations in the number of MSNA bursts, interburst interval, and burst amplitude. Two possibilities were considered. First, the variabilities might be best described by a random process in which events are uncorrelated (6, 27). The second possibility is that long-range correlations exist among events. If such correlations extend over more than one time scale, the fluctuations would be best modeled as a time-scale-invariant (i.e., fractal) process in which the present value of the measured property is related not only to recent events but also to much more remote events (3, 22).

A second aim of the present study was to compare the characteristics of the fluctuations in MSNA with those in the heart rate in the same subjects. Heart rate variability (HRV) in healthy humans is characterized not only by respiratory-related and lower-frequency fluctuations but also by a fractal component (15, 16, 29). A potential link between fractal sympathetic nerve activity and fractal HRV is important to consider, because the latter is diminished in cardiovascular diseases such as low-output congestive heart failure (16, 17).

## METHODS

### Subjects

All recordings were performed at the University of Texas Southwestern Medical Center (Dallas, TX). The Institutional Review Board approved the protocols, and each of the 15 subjects (10 men and 5 women) provided informed written consent to participate in the study. The subjects were free of any known cardiovascular or respiratory disease and were instructed to refrain from smoking cigarettes and drinking alcohol or caffeine-containing beverages for ≥12 h before the recording session.

### Recordings

Recording sessions lasted 1–2.5 h, with the subject in a supine position, relaxed, and breathing spontaneously. A lead II ECG recording was used to determine heart period (R-R interval), and respiratory movements were recorded with a strain-gauge pneumograph placed over the upper abdomen. Blood pressure was measured with an automated sphygmomanometer (Welch Allen). Postganglionic MSNA was recorded using standard microneurographic techniques (1, 9, 18, 28). A tungsten microelectrode with an uninsulated tip diameter of 1–5 μm was inserted into the right or left peroneal nerve near the fibular head. A reference electrode with a larger uninsulated tip was inserted subcutaneously ∼2 cm from the recording electrode. The nerve signal was processed by a preamplifier and an amplifier (model 662C-3, Nerve Traffic Analyzer, Dept. of Bioengineering, University of Iowa, Iowa City, IA) with a total gain of 90,000. MSNA was band-pass filtered (700–2,000 Hz), rectified, and then integrated by a resistance-capacitance circuit (time constant = 0.1 s). As previously reported (30), MSNA was characterized by cardiac-related bursts that increased in frequency with end-expiratory breath holds and Valsalva maneuvers, but MSNA was not affected by arousal or skin stroking.

### Data Analysis

Data originally saved using a recording adaptor (model 4000 PCM, Vetter, Rebersberg, PA) and a videocassette recorder (model SLV-750 HF, Sony) were played back and acquired (5-ms sampling intervals) with software and an analog-to-digital converter board (Axon Instruments, Union City, CA). The analyses were performed on a computer (Optiplex GX 400, Dell) using software developed by Gebber et al. (14) and Lewis et al. (21) at Michigan State University.

*Frequency-domain analysis.* Fast Fourier transform was used to construct autospectra of MSNA, ECG, and the respiratory signal and coherence functions (normalized cross spectra) relating pairs of these signals (14, 19). The spectra had a resolution of 0.05 Hz/bin. The autospectra and coherence function were averages of 180–420 20-s data windows with no overlap. We used a coherence value of ≥0.5 to signify a strong linear relation between two signals within specified frequency bands (8). A coherence value of 0.1–0.5 signified a weak, but statistically significant, relation (5).

*Fractal analysis.* After the ECG and integrated MSNA recordings were demeaned to remove slow baseline shifts, programs written by Gebber et al. (14) were used to detect the peaks of MSNA bursts and R waves of the ECG and measure *1*) the intervals between MSNA bursts, *2*) the amplitude of MSNA bursts, and *3*) R-R intervals in the ECG. Figure 1 shows an integrated MSNA before and after the data were subjected to demeaning. Such high-quality recordings are typical of those obtained in the University of Texas Southwestern Medical Center laboratory (1, 12). The threshold above which MSNA bursts were counted was initially set just above background noise, as gauged by visual inspection of the time series. Final adjustment of the threshold was made to that direct-current level at which the peak in the interburst interval histogram corresponding to the R-R interval was most prominent. The rationale for this adjustment is that much of the power in the autospectrum of MSNA is at the frequency of the heartbeat (1). This reflects the entrainment of MSNA to the cardiac cycle by pulse-synchronous baroreceptor nerve activity (1). The MSNA interburst interval histograms also contained peaks corresponding to multiples of the R-R interval, because bursting did not occur in every cardiac cycle. In all cases, the average number of MSNA bursts per minute derived by using the computer program was within 10% of that determined by hand counting the bursts in selected data segments taken from the beginning, middle, and end of the time series.

Three methods were used to determine whether fluctuations in the number of MSNA bursts and heartbeats, MSNA interburst interval, and R-R interval were fractal. The first method involved calculation of the Allan factor, A(*T*), for window sizes of different lengths. Thurner et al. (26) and Turcott and Teich (29) define A(*T*) as the ratio of the event-number Allan variance to twice the mean number of events (R waves or MSNA bursts) in a window size of specified length (*T*) where *N _{i}*(

*T*) is the number of events in the

*i*th window of length

*T*. The Allan variance is expressed in terms of the variability of the difference in the number of events in successive windows. This measure was first introduced in connection with the stability of atomic-based clocks (2).

The Allan factor curve is constructed by plotting A(*T*) as a function of the window size on a log-log scale. For a data block of length *T*_{max}, the window size, *T*, is progressively increased from a minimum of a single bin (5 ms) to a maximum of *T*_{max}/6 so that ≥6 nonoverlapping windows are used for each measure of A(*T*). For a random process in which fluctuations in the number of events are uncorrelated, A(*T*) = 1 for all window sizes (25, 29). For a periodic process, the variance decreases and A(*T*) approaches 0 as the window size is increased (25, 29). For a fractal process, A(*T*) increases as a power of the window size and may reach values >1.0 (25, 29). This reflects the greater variance in number of events with increasing window size. The increase in variance occurs because long window lengths are more apt to reveal rarer clusters of events. Such clusters characterizing fractal time series (3, 22) markedly affect the variance of the measured property when the data block has been divided into a relatively small number of windows of large size. The power law relation appears as a straight line with a positive slope, α, on the log-log scale. The scaling exponent, α, is the power to which fluctuations in number of events on one time scale are proportional (i.e., statistically self-similar) to those on longer time scales. The correlation coefficient (*r*) is used to test for linearity on the log-log scale, and linear regression is used to calculate α, which is bounded in a range of 0–3 and is used to calculate the Hurst exponent (*H*, range 0–1). As described by Eke et al. (11) and Thurner et al. (26), *H* is calculated using the following formula when 0 < α < 1 and using the following formula when 1 < α < 3

As explained by Feder (13) and Bassingthwaighte and Raymond (4), *H* = 0.5 for a time series in which events are uncorrelated, whereas H ≠ 0.5 implies that the time series is fractal providing that the power law extends over more than one time scale (decade on log scale). When *H* > 0.5, events are positively correlated [persistence; values larger (smaller) than the mean tend to be followed by values also larger (smaller) than the mean]. When *H* < 0.5, events are negatively correlated (antipersistence; values larger than the mean tend to be followed by values smaller than the mean and vice versa).

The Allan factor curve for the original time series is routinely compared with those of 10 surrogate data sets in which the intervals between successive MSNA bursts or heartbeats have been shuffled. Specifically, we assigned random numbers to the intervals in the original time series and then sorted the random numbers by size (7). This creates a randomized data set, the mean, variance, and interevent interval distribution of which are identical to those of the original spike train, with no correlations among the intervals (7, 21, 25, 29). If shuffling of the data eliminates the power law relation in the Allan factor curve, it can be concluded that fluctuations in the number of events were fractal. Moreover, elimination of the power law implies that the intervals in the original time series were ordered and interrelated (25, 26, 29). This was tested directly by using dispersional analysis (DA).

The algorithm for DA developed by Bassingthwaighte and Raymond (4) was used to calculate the standard deviation (SD) of the mean values of the MSNA interburst interval or R-R interval for groups of data points of a specified number (*m*). In similar fashion, we used DA to test whether fractal fluctuations in MSNA burst amplitude were time-scale invariant. Specifically, the mean value for each group of *m* data points is obtained, and the SD of these values is calculated for the total number of groups. The process is repeated each time *m* is increased progressively from a minimum of one data point to a maximum of one-quarter of the total number of data points. SD is then plotted against *m* on a log-log scale, yielding a straight line with a negative slope. For a random process with no correlations among events, the slope of the DA plot is –0.5 (4). For a fractal process, the slope is different from –0.5 over a range of *m* extending more than one decade (4). As with Allan factor analysis, the DA plot for the original time series is compared with those for 10 surrogates.

We also performed DA on first differences derived from the original time series. As described by Das et al. (7), a new time series of the absolute differences between successive intervals or MSNA burst amplitudes is constructed. This procedure removes slow trends in the data such as progressive increases or decreases in the value of the measured parameter (16). The removal of such nonstationarities is ascertained by viewing the time series of the first differences. We refer to DA performed on the original time series as ordinary DA and that performed on first differences as detrended DA.

Although *H* can be calculated as the difference between the negative slope of the DA curve and 1.0 (p. 67–70 in Ref. 3), Eke et al. (11) showed that the estimate provided is highly erroneous in cases when the scaling exponent of the power spectrum of the signal (β) is >1.0. This applies as well to α > 1.0, because the scaling exponent of the power law in the Allan factor curve is equivalent to β in the power spectrum (26). When Eke et al. (11) applied DA to blood cell perfusion time series known to have an antipersistent *H* of 0.24, an erroneous persistent *H* of 0.89 was obtained. Because of such potential errors with use of DA, the estimates of *H* in the present study are based only on the scaling exponent (α) of the power law in the Allan factor curve. We calculated *H* for α = 0.24–1.57. As described in results, α < 1.0 yielded persistent *H* values, whereas α = 1.04–1.57 yielded antipersistent values.

The third method used to test for time-scale-invariant behavior was Fano factor analysis. Teich (24) and Turcott and Teich (29) define the Fano factor, F(*T*), as the ratio of the variance of the number of events to the mean number of events in time windows of specified length, *T* where *N _{i}*(

*T*) is the number of events in the

*i*th window of length

*T*. The Fano variance is expressed in terms of the variability of the number of events in individual windows, rather than in terms of the difference in the number of events in successive windows. As described in earlier reports (7, 21), the Fano factor curve is constructed similarly to the Allan factor curve, and surrogate data are used to test whether the power law in the Fano factor curve for the original time series reflects fractal fluctuations in the number of events.

*Histograms.* Distributions of R-R intervals, MSNA interburst intervals, and MSNA burst amplitudes were constructed as described by Lewis et al. (21) and Das et al. (7). Coefficient of variation (CV) of these distributions is defined as Values are means ± SE.

## RESULTS

### Patterns of MSNA

Coherence values relating MSNA to the ECG at the frequency of the heartbeat and coherence values relating MSNA to the respiratory signal at the frequency of breathing for each of the 15 subjects are listed in Table 1, along with other subject characteristics such as mean blood pressure, heart rate, and respiratory rate. As reflected by a coherence value ≥0.5, MSNA was strongly cardiac related in 12 subjects and strongly respiratory related in 3 subjects. Figure 2*A* shows recordings from *subject 12,* in whom MSNA was strongly cardiac and respiratory related. The amplitude of cardiac-related bursts of MSNA was greatest during the inspiratory phase (downward deflection) of the respiratory cycle. In this case, the coherence values relating MSNA to the ECG at the heartbeat frequency and to the respiratory signal were 0.88 and 0.61, respectively.

Figure 2*B* shows recordings from *subject 3,* in whom MSNA was strongly correlated to the cardiac cycle but not to the respiratory signal. The clusters of cardiac-related bursts of MSNA generally exceeded the duration of the respiratory cycle. In this case, the coherence value relating MSNA to the ECG at the frequency of the heartbeat was 0.75. In contrast, the coherence value relating MSNA to the respiratory signal at the frequency of breathing was only 0.04.

### Fractal Fluctuations in Number of MSNA Bursts and Heartbeats and Interevent Intervals

Fluctuations in the number of MSNA bursts and heartbeats were fractal in each of the 15 subjects. The results from *subject 8*, in whom the fractal fluctuations in the number of MSNA bursts and heartbeats were antipersistent (negatively correlated data), are shown in Figs. 3 and 4. MSNA interburst intervals and R-R intervals are plotted as time series (8,400 s long) in Figs. 3*A* and 4*A*, respectively. These time series contained 5,729 MSNA interburst intervals and 10,408 R-R intervals, yielding an MSNA burst-to-R wave ratio of 0.55. The horizontal bands in the MSNA interburst interval time series were separated by approximately the period of the cardiac cycle (807 ms). This reflects the absence of bursts in a variable number of heartbeats, which is indicated as well by the secondary peaks in the distribution of MSNA interburst intervals (Fig. 3*B*). As expected, the distribution of R-R intervals was unimodal (Fig. 4*B*).

The Allan factor curve for MSNA is shown in Fig. 3*C*. Note the appearance of a power law extending over more than one time scale (decade on log scale) that began at a window size of ∼67 s. At this point, the Allan factor curve for the original time series crossed those for the surrogates and reached a maximum A(*T*) near 20 at the largest allowable window size, which was one-sixth of the total duration of the time series. In contrast, the curves for the 10 surrogates were essentially flat at window sizes >67 s. The slope of the power law in the Allan factor curve for the original data was 1.20, yielding an antipersistent *H* of 0.1 (see methods). Thus the fluctuations in the number of MSNA bursts were negatively correlated over a wide range of window sizes.

Fractal fluctuations in the number of heartbeats were also antipersistent in *subject 8*. The power law in the Allan factor curve for the original data began at ∼51 s and extended over more than one time scale (Fig. 4*C*). *H* calculated from its slope (1.15) was 0.08. In contrast to the curve for the original time series, A(*T*) for the surrogates approached zero with increasing window size. The progressive dip in A(*T*) is attributable to the strong periodic component in the time series of R-R intervals, which is little affected by shuffling the data (25, 29).

In *subject 8*, ordinary and detrended DA revealed the presence of long-range correlations among MSNA interburst intervals and among R-R intervals. The detrended DA plots for the time series of MSNA interburst interval and R-R interval are shown in Figs. 3*D* and 4*D*, respectively. The slope of the MSNA plot for *m* ≥ 10 was –0.09, and the slope for the R-R plot was –0.16. In contrast, the slopes of the plots for the 10 surrogate data sets were close to –0.5, as expected for time series of randomly occurring events (4).

Data from *subject 3*, in whom the fractal fluctuations in the number of MSNA bursts and heartbeats were persistent (positively correlated data), are shown in Figs. 5 and 6. Except for the addition of Fano factor curves, the format of Figs. 5 and 6 is the same as that of Figs. 3 and 4. In *subject 3*, the length of the time series was 7,390 s, during which there were 3,259 MSNA interburst intervals and 7,076 R-R intervals, yielding an MSNA burst-to-R wave ratio of 0.46. The slope of the power law in the Allan factor curve for the original MSNA time series was 0.86 for window sizes ≥100 s, yielding a persistent *H* of 0.93 (Fig. 5*C*). In contrast, the curves for 10 surrogates were essentially flat. Nevertheless, there was a considerable range of window sizes over which the Allan factor curve for the original data crossed the curves for the surrogates. Moreover, the range of the power law in the curve for the original time series was restricted to only about one decade of time. Thus we also performed Fano factor analysis (Fig. 5*D*), which presented a clearer picture of the divergence of the curve for the original MSNA time series from those of its surrogates over a range of at least two decades. This can be attributed to the fact that the window size at which the power law begins is usually much smaller in the Fano factor curve than in the corresponding Allan factor curve (25, 26, 29). As a consequence, F(*T*) may be greater than A(*T*) at the largest allowable window size. The slope (0.75) of the power law in the Fano factor curve was close to that for the power law in the corresponding Allan factor curve. The existence of long-range correlations among the original MSNA interburst intervals is supported by the results of DA (Fig. 5*E*). The detrended DA curve for the original data deviated from the curves for the surrogates beginning at *m* = 19. From this value of *m* to its largest allowable value (*m* = 815), the slope of the curve for the original data was –0.15.

In *subject 3*, the Allan factor (Fig. 6*C*), Fano factor (Fig. 6*D*), and detrended DA (Fig. 6*E*) curves for the original time series of R-R intervals deviated from those of the surrogates. The slopes of the power laws in the Allan factor and Fano factor curves for the original data were 0.42 and 0.52, respectively. *H* calculated from the slope of the power law in the Allan factor curve was 0.71. In *subject 3*, the slope of the detrended DA plot was –0.27.

In seven subjects, the nature of the fractal fluctuations in the number of MSNA bursts was different from that of the number of heartbeats. Data from *subject 13*, in whom the fluctuations in the number of MSNA bursts were antipersistent whereas fluctuations in the number of heartbeats were persistent, are shown in Figs. 7 and 8. In *subject 13*, the time series were 7,395 s long and contained 2,366 MSNA interburst intervals and 7,209 R-R intervals, yielding an MSNA burst-to-R wave ratio of 0.33. The slope (1.57) of the power law in the Allan factor curve for the original time series of MSNA (Fig. 7*C*) is clearly greater than that for the 10 surrogates at window sizes ≥200 s. *H* calculated from the slope was 0.29. However, the power law was limited to less than one decade of time because of its late takeoff and the maximal allowable window size, which was 1,230 s in this case. Nonetheless, Fano factor analysis (Fig. 7*D*) revealed that the fluctuations in the number of MSNA bursts were time-scale invariant (i.e., fractal). Note the divergence of the Fano factor curve for the original time series from those of its surrogates at window sizes ≥30 s. In this case, the slope (0.89) of the power law in the Fano factor curve was <1.0, whereas that in the Allan factor curve was >1.0. This difference was expected, because mathematical constraints prevent F(*T*) from increasing faster than ∼*T*1, thereby limiting the range of the slope to 0–1.0 (25). For this reason, the slope of the Fano factor curve was not used to estimate *H*. The detrended DA curve for the same data had a slope of –0.05 for *m* ≥ 15, which fell outside the range of the slopes of the curves for the surrogates (Fig. 7*E*). Thus fluctuations in MSNA interburst intervals also were time-scale invariant.

In *subject 13*, fluctuations in the number of heartbeats were persistent, as demonstrated by Allan and Fano factor analyses. The slopes of the power laws in the Allan and Fano factor curves for the original data were 0.66 and 0.71, respectively. *H* calculated from the slope of the power law in the Allan factor curve was 0.83 (Fig. 8*C*). As is typically the case (25), the power law component of the Allan factor curve is not as smooth as that of the Fano factor curve. Nonetheless, the correlation coefficient (*r* = 0.84) used as a test of linearity was highly significant (*P* < 0.01) for the range of window sizes (19–1,200 s) over which the slope of the power law in the Allan factor curve was measured. Fluctuations in R-R intervals also were fractal, as demonstrated by comparison of the detrended DA curve for the original time series (slope = –0.1 for *m* ≥ 7) with those of its surrogates (Fig. 8*E*).

The results obtained with Allan factor analysis and DA are summarized in Table 2. On the average, the time series were 6,728 ± 464 s long, and the MSNA burst-to-R wave ratio was 0.42 ± 0.05 (range 0.19–0.78; Table 1). On the average, the slope of the power law in the Allan factor curve for the original time series was measured over window sizes of 79–916 s for MSNA and 50–735 s for the heartbeat.

For MSNA, the slope (α) of the power law in the Allan factor curve was <1.0 in eight subjects and >1.0 in seven subjects (Table 2). For the heartbeat, α < 1.0 in 11 subjects and α > 1.0 in 4 subjects (Table 2). H calculated from α was persistent (>0.5) when α < 1.0, and *H* was antipersistent (<0.5) when α > 1.0. In three subjects, the range of window sizes used to calculate α for MSNA was less than one decade (Table 2). Such was also the case for the heartbeat in one subject (Table 2). Nonetheless, these time series were considered fractal, because the power law in the corresponding Fano factor curves extended well over one decade of window sizes.

*H* values calculated from the slopes of the power laws in the Allan factor curves were used to divide the subjects into four groups. In six subjects, *H* was persistent for MSNA and the heartbeat; in two subjects, *H* was antipersistent for both signals (Table 2). In five subjects, *H* for MSNA was antipersistent, whereas *H* for the heartbeat was persistent. In the remaining two subjects, *H* was persistent for MSNA and antipersistent for the heartbeat.

Table 2 also lists the slopes of the ordinary and detrended DA curves for both signals. In each case, the slope fell outside the range of those for the curves for the surrogates. Thus long-range correlations existed among the MSNA interburst intervals and among the R-R intervals in every subject. On the average, the slope of the detrended DA plot was measured at *m* = 10–328 for MSNA and *m* = 5–756 for the heartbeat.

### Fractal Fluctuations in MSNA Burst Amplitude

DA was used to test for fractal fluctuations in MSNA burst amplitude (normalized on a scale of 0–1). The time series of burst amplitudes and the distribution of burst amplitudes (γ-like in shape) obtained from *subjects 7* and *6* are shown in Fig. 9. In *subject 7*, the detrended DA curve for the original time series deviated from the curves for 10 surrogates beginning at *m* = 10. The slope of the curve was –0.09 for *m* ≥ 30. In contrast, the detrended DA curve for the real data for *subject 6* fell within the range of the curves for its surrogates. Thus fluctuations in MSNA burst amplitude were fractal in *subject 7* but random in occurrence in *subject 6*.

The results obtained with DA of MSNA burst amplitude are summarized in Table 3. The fluctuations in MSNA burst amplitude were fractal in five subjects as determined by ordinary DA and in four subjects as determined by detrended DA. The times series was considered to be fractal when the slope of the DA plot for the original data fell outside the range of the slopes of the plots for 10 surrogates. On the average, the slope of the detrended DA plot was measured over *m* = 7–381.

## DISCUSSION

The present study is the first to demonstrate time-scale-invariant (i.e., fractal) fluctuations in human sympathetic nerve activity. The major findings of the study are as follows. First, as demonstrated with Allan and Fano factor analyses, fluctuations in the number of MSNA bursts were determined to be fractal in each of 15 awake normotensive subjects. This was the case independent of whether MSNA was strongly or weakly cardiac related and whether it was respiratory related. Importantly, shuffling of the intervals in the original time series eliminated the power law in the Allan and Fano factor curves and drove the *H* value derived from the slope of the power law in the Allan factor curve to that (∼0.5) expected for a random process with no correlations among events (4, 13). This implies long-range correlations among MSNA interburst intervals (25, 26, 29). This was shown to be the case directly by using DA to test for fractal fluctuations in interevent intervals. Second, as also demonstrated by DA, fluctuations in cardiac-related MSNA burst amplitude were fractal in approximately one-fourth of the subjects. Thus fluctuations in burst amplitude were more likely to occur randomly. Third, in agreement with the work of others (15–17, 29), we found that fluctuations in the R-R interval and in the number of heartbeats were fractal in each of the subjects. Thus fluctuations in MSNA interburst interval and heart rate were fractal in the same subjects.

The fluctuations in the number of MSNA bursts and heartbeats could be positively (persistence) or negatively (antipersistence) correlated. Classification was based on the value of *H* calculated from the slope (α) of the power law in the Allan factor curve: α < 1.0 yielded a persistent *H*, whereas α > 1.0 yielded an antipersistent *H*. On this basis, there were more cases of positively correlated than negatively correlated fluctuations for MSNA and the heartbeat (Table 2). Nonetheless, some cases may have been misclassified. This would be most apt to occur when the estimated value of α was close to 1.0, which is at the border between persistent and antipersistent behavior (11).

The predominance of persistent long-range correlations among fluctuations in the number of heartbeats differs from the findings of Turcott and Teich (29), who also used Allan factor analysis to test for fractal fluctuations in the number of heartbeats in healthy humans. The slope of the power law in the Allan factor curve was >1.0 but <2.0 in 9 of 12 of their subjects; thus antipersistent correlations predominated in their study. When *H*≠ 0.5, its value measures the smoothness of a fractal time series, with antipersistent values reflecting a high degree of roughness (i.e., increases in interval more apt to be followed by decreases and vice versa) and persistent values reflecting relative smoothness (3). We are unable to explain why the fractal fluctuations in the number of heartbeats were more apt to be positively correlated in our subjects than in those studied by Turcott and Teich.

Because Allan factor analysis is based on counting the number of events (MSNA bursts or R waves) in windows of specified length, this method cannot be used to analyze fluctuations in MSNA burst amplitude. Thus we relied on DA for this purpose. As revealed by detrended DA, fluctuations in MSNA burst amplitude were fractal in approximately one-fourth of the subjects. In only one case did ordinary and detrended DA yield divergent results (Table 3). Thus, in four of five subjects, the difference between the slope of the ordinary DA curve and the slopes of the curves for the surrogates could be attributed to long-range correlations among MSNA burst amplitudes, rather than to slow trends (nonstationarity) in the time series. For reasons given in methods, we did not calculate *H* from the negative slope of the ordinary or detrended DA curve.

As demonstrated here for the first time, fluctuations in R-R interval and MSNA interburst interval were fractal in the same subjects. This finding raises the possibility that fractal sympathetic nerve activity contributes to the fractal component of HRV. Alternatively, fractal fluctuations in R-R interval might be reflexly imposed on central networks governing sympathetic nerve activity via baroreceptor feedback. It is also possible that fractal HRV and sympathetic nerve activity are independently generated and unrelated. Regarding this possibility, the nature (persistence vs. antipersistence) of the fractal fluctuations in the number of MSNA bursts was the opposite of that for the number of heartbeats in seven of the subjects.

Of the possibilities listed above, the first has potential clinical relevance, because the fractal component of HRV is diminished in cardiovascular diseases such as low-output congestive heart failure (16, 17). As such, fractal measures of HRV are now used as indexes of the state of cardiac function (16, 17). Thus the mechanisms responsible for fractal HRV warrant investigation.

A potential role of fractal sympathetic activity in generating the fractal component of HRV remains problematic. First, no direct recordings of cardiac sympathetic nerve activity have been made in humans. Thus, whereas we have demonstrated that human MSNA exhibits fractal properties, the same may not be the case for cardiac sympathetic outflow. Second, Yamamoto and Hughson (31) reported only small changes in fractal measures of human HRV after β-adrenergic receptor blockade. In contrast, Yamamoto et al. (32) found that the fractal component of HRV in humans is diminished appreciably after cardiac vagal blockade with atropine. These results suggest that cardiac vagal outflow is more important than cardiac sympathetic outflow in generating the fractal component of HRV in humans. Nonetheless, the effects of β-adrenergic receptor blockade have not been studied in the presence of vagal blockade. Goldberger et al. (16) hypothesized that fractal HRV arises from an interaction of cardiac sympathetic and vagal outflows to the sinoatrial node.

Whereas the consequences of the fractal fluctuations of human sympathetic nerve activity on end organ function remain obscure, time-scale-invariant behavior clearly reflects the occurrence of complex, nonlinear interactions of the elements comprising the neuronal networks controlling visceral function. What significance can be attached to the long-range correlations arising from the nonlinear interactions? Because the fluctuations on one time scale are proportional by some power of those on other scales, fractal processes are of value in predicting future behavior of the system under study. This is because time-scale-invariant behavior represents a form of memory, in that the present value of the measured parameter (e.g., number of MSNA bursts or interburst interval) is related to past values in a time frame defined by the range of window sizes comprising the power law in the Fano and Allan factor curves and/or that portion of the DA curve that deviates from the curves for the surrogates. In contrast, such memory is absent for a random process in which events are uncorrelated. Determining whether fractal fluctuations are positively or negatively correlated reveals more specific information as to the nature of future performance. The smoothness of persistent time series reflects a state in which the present value of the measured parameter is likely to be close to the preceding value. On the other hand, the roughness of antipersistent time series tells us that the present value is less likely to be close to the preceding value (3).

Time-scale-invariant processes may confer important adaptive properties to the system under study. Goldberger et al. (16) suggested that linkage of processes on one temporal scale to those on other scales might prevent excessive mode locking, which would restrict the functional responsiveness of the organism to unexpected challenges. As an example, in patients with heart failure, a breakdown of fractal fluctuations in heart rate is accompanied by the emergence of a dominant cardiac frequency mode near 0.02 Hz (16). Such patients are at a high risk for sudden cardiac death. Whether a similar breakdown of the fractal component of sympathetic nerve activity occurs in heart failure and other diseases remains to be determined. Nonetheless, our findings reinforce the view of Goldberger et al. that maintaining constancy is not the goal of physiological control. Rather, they suggest that the classical theory of homeostasis according to which physiological systems seek to reduce variability should be modified to account for dynamical, far from equilibrium, behavior characteristic of fractal processes.

It is of practical importance to distinguish between fractal and random processes, because their basic properties are qualitatively different. For example, unlike a Gaussian distribution, in which variance converges to a stable value as the number of events measured is increased, the variance of a fractal distribution continues to change as more and more data are analyzed. As such, the absence of a stable variance leads to the power law relation in the Allan factor curve for a fractal process. Thus, as pointed out by Bassingthwaighte et al. (3), a simple measurement made using a particular resolution does not provide a meaningful description of the long-range correlations characteristic of a fractal time series. Rather, one must determine how the values of the measured properties change as a function of the resolution used to make the measurements.

Das et al. (7) described the fractal properties of the discharges of the postganglionic vertebral sympathetic nerve in Dial-urethane-anesthetized cats. This nerve provides sympathetic outflow to the forelimb. The profile of fractal properties of vertebral nerve activity differed from that reported here for MSNA in awake humans. In contrast to human MSNA, fluctuations in the interval between cardiac-related bursts in the cat vertebral nerve were random in occurrence. On the other hand, fluctuations in burst amplitude were more likely to be fractal in the anesthetized cat than in the awake human. Whether these differences are species dependent or attributable to the use of anesthesia in the cat remains to be determined. As in the awake human, HRV in the anesthetized cat contained a fractal component.

## Acknowledgments

The authors thank L. M. Braybrook for typing the manuscript.

GRANTS

This study was supported by National Institutes of Health Grants HL-13187 (to G. L. Gebber), HL-33266 (to S. M. Barman), HL-06296 and DA-10064 (to R. G Victor), HL-69648 (an Individual National Research Service Award to P. J. Fadel), and K23-RR-016321 (to W. Vongpatanasin). In addition, the American Physiological Society supplied a Research Career Enhancement Award to P. J. Fadel.

## Footnotes

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- Copyright © 2004 by the American Physiological Society