Abstract
Fano factor analysis and dispersional analysis were used to characterize time series of single and multifiber spikes recorded from the preganglionic cervical sympathetic nerve and cardiacrelated slowwave activity of the whole postganglionic sympathetic vertebral nerve (VN) in anesthetized cats. Fluctuations in spike counts and interspike intervals for single preganglionic fibers proved to be fractal (i.e., timescale invariant), as reflected by a power law relationship between indices of the variance of these properties and the window size used to make the measurements. Importantly, random shuffling of the data eliminated the power law relationships. Fluctuations in spike counts in preganglionic multifiber activity also were fractal, as were fluctuations in the height and of the area of cardiacrelated slow waves recorded from the whole postganglionic VN. These fractal fluctuations were persistent (i.e., positively correlated), as reflected by a Hurst exponent significantly >0.5. Although fluctuations in the interval between cardiacrelated VN slow waves were random, those in the interval between heart beats were fractal and persistent. These results demonstrate for the first time that apparently random fluctuations in sympathetic nerve discharge are, in fact, dictated by a complex deterministic process that imparts “longterm” memory to the system. Whether such timescale invariant behavior plays a role in generating the fractal component of heart rate variability remains to be determined.
INTRODUCTION
Whereas the naturally occurring action potentials of single neurons in the cat rostral ventrolateral medulla (RVLM) are correlated to the cardiacrelated and 10Hz rhythmic components of postganglionic sympathetic nerve discharge (SND), these neurons miss firing in a variable number of rhythmic cycles and exhibit periods of quiescence lasting several seconds or longer (Lewis et al. 2001). As such, the interspike intervals (ISIs) of these RVLMspinal presympathetic neurons are distributed exponentially or in a gammalike fashion. Traditionally, these distributions have been modeled as random Poisson processes in which the ISIs are uncorrelated (Cox and Lewis 1966; Tuckwell 1988). Nevertheless, exponential and gammalike distributions are also characteristic of time series in which longrange correlations exist (Teich 1989, 1992). Such time series are best described as fractal point processes. Fractal point processes are characterized by longrange correlations among events extending over multiple time scales (Bassingthwaighte et al. 1994; Liebovitch 1998; West 1990). Such statistical selfsimilarity or timescale invariant behavior is revealed by methods that test for a power law relationship between the variance of some property and the resolution (e.g., window length) used to make the measurements. In a study from our laboratory (Lewis et al. 2001) on brain stem presympathetic neurons, we used Fano factor analysis to demonstrate such a relationship, as reflected by the proportionality to a power of fluctuations in spike counts measured over short periods (approximately 1 s) to those measured on longer time scales (≥100 s). The longrange correlations so revealed reflect a form of memory in that the current value of the measured property is dependent on events in the “distant” past (i.e., largest window size).
Longrange correlations are also characteristic of time series of the intervals between heart beats in healthy humans (Goldberger 1992; Goldberger et al. 1996; Ivanov et al. 1999). This has attracted the interest of cardiologists, since the fractal component of heart rate variability (HRV) is diminished with aging and in patients with congestive heart failure (Goldberger et al. 1996; Goldberger and Rigney 1991). Whereas it has been suggested that autonomic outflows to the sinoatrial node in some way contribute to fractal HRV (Goldberger et al. 1996; Yamamoto and Hughson 1994; Yamamoto et al. 1995), it is not known whether SND and/or cardiac vagal nerve activity themselves contain fractal components. The current study was designed to test this possibility for SND in view of our past findings on brain stem unit activity. Specifically, we have addressed the following questions.1) Do longrange correlations exist among the ISIs of individual preganglionic sympathetic neurons (PSNs)? 2) Are fluctuations in activity recorded from populations of PSNs and whole postganglionic sympathetic nerves fractal in nature? 3) What is the nature of the longrange correlations in SND? In particular, is fractal activity characterized by positive (persistent) or negative (antipersistent) correlations? 4) Does fractal SND coexist with fractal HRV in the same cats?
METHODS
General procedures
The experimental protocols described below were approved by the AllUniversity Committee on Animal Use and Care of Michigan State University. The experiments were performed on 11 spontaneously breathing, baroreceptorintact cats anesthetized by intraperitoneal injection of a mixture of diallylbarbiturate (60 mg/kg), urethan (240 mg/kg), and monoethylurea (240 mg/kg). A surgical state of anesthesia was indicated by the failure of noxious stimuli (pinch, muscle cauterization) to desynchronize spindles and deltaslow wave activity in the frontalparietal electroencephalogram (Gebber et al. 1999; Steriade and Llinas 1988). Blood pressure was measured from a femoral artery, and 6% dextran in normal saline was infused into a femoral vein at a rate of 6 ml/h. Body temperature was maintained near 37°C with a heat lamp, and endtidal CO_{2} was in the range of 3.5 to 5% (Transverse Medical Monitors Capnometer, model 2200).
Nerve recordings
Multiunit activity was recorded from thin strands teased from the left preganglionic cervical sympathetic nerve using the method described by Koley et al. (1989). The recordings were made from the central end of the strand with bipolar platinum electrodes and a preamplifier bandpass of 300–3,000 Hz. Online analog window discrimination was used to isolate multiunit spikes from background noise. Spikesorting software (RUN Technologies, Mission Viejo, CA) was used offline to separate the spikes of different preganglionic fibers making up the multiunit recording field. Spikes were grouped into separate files based on similarities in spike height, width, shape, depolarization velocity, and other characteristics. A minimum ISI of ≥60 ms was taken as an indication that the spikes in a file arose from a single fiber (Mannard and Polosa 1973).
Bipolar platinum electrodes were used to record monophasically from the central end of the cut whole postganglionic vertebral sympathetic nerve (VN) near its exit from the left stellate ganglion. VN recordings were made with a preamplifier bandpass of 1–1,000 Hz. Bursts of multiunit activity (envelopes of spikes) appear as slow waves when this preamplifier bandpass is used (Cohen and Gootman 1970;Gebber et al. 1994).
Spike train analysis
The action potentials of single PSNs (spikesorted files) or groups of preganglionic cervical sympathetic fibers were represented by standardized 5V squarewave pulses (2 ms in duration). From time series of these pulses, we counted the number of spikes in bins of designated length (single and multiunit activity) and measured ISIs using software written in our laboratory by C. D. Lewis (Gebber et al. 1999). Tests were performed to determine whether fluctuations in these parameters were fractal or random in nature.
The first test involved calculation of the Fano factor for window sizes of different lengths. The Fano factor, F(T) as used by Teich (1992) and Lewis et al. (2001), is defined as the variance of the number of spikes divided by the mean number of spikes in a time window of lengthT
Whether a power law relationship in the Fano factor curve truly reflects a fractal process, and thus longrange correlations of events, is tested by constructing surrogate data sets in which ISIs have been randomly shuffled. Specifically, we assigned random numbers to the ISIs in the original time series and then sorted the random numbers by size. This creates a randomized data set whose mean ISI, ISI variance, and ISI histogram are identical to those of the original spike train, but with no correlations among events (Teich and Lowen 1994). If shuffling of ISIs eliminates the power law relationship, then it can be concluded that the ISIs in the original time series were ordered and interdependent. We routinely compared the Fano factor curve for the original spike train with those for 10 or 20 surrogates, thereby approximating 90 or 95% confidence limits.
Dispersional analysis (DA) was also used to test for fractal properties of singleunit spike trains. The algorithm developed byBassingthwaighte and Raymond (1995) involves calculation of the SD of the mean values of the ISI for groups of data points of a specified number (m). Specifically, the mean ISI 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 onequarter of the total number of data points. SD is then plotted against m on a loglog scale yielding a straight line with a negative slope. The slope is used to calculate the Hurst (H) exponent using the formula
Slow trends in the data may lead to erroneous conclusions when using DA. For example, a progressive increase or decrease in ISI would be interpreted as persistence (H > 0.5) even when the spike train lacks true fractal properties. For this reason, we also performed DA on first differences derived from the original time series. In the case of ISIs, a new time series of the absolute differences between successive intervals is constructed. The first difference, D(I) takes the form
HRV and VN activity
Ordinary and detrended DA were used as tests for fractal fluctuations of the intervals between heartbeats and for fractal fluctuations of the following properties of cardiacrelated bursts (slow waves) of postganglionic VN activity: 1) interval between slow waves; 2) troughtopeak slowwave height (normalized on scale of 0 to 1.0); and 3) normalized slowwave area. The cardiac interval was the interval (ms) between the peak systolic phases of successive femoral arterial pulse waves. Time series of properties 1–3 were constructed after digital bandpass filtering of the original recordings of VN activity with software obtained from RC Electronics (Santa Barbara, CA). A symmetric, nonrecursive filter with a Lanczos smoothing function was used. The width of the bandpass was between 4 and 6 Hz, with the center frequency matched to that of the sharp peak at the frequency of the heartbeat in the autospectrum of SND (Gebber et al. 1999). The digital filter had a rolloff slope of 39%/Hz outside of the bandpass. As shown in Fig. 5, the digitally filtered records of VN activity are smoother than the originals, thus aiding in the accurate detection of peaks and troughs for time series analysis. Note that digital filtering produced minimal or no amplitude and phase distortion.
Histograms
Single unit ISI histograms and arterial pulse (AP)triggered histograms of single and multiunit preganglionic cervical sympathetic nerve activity were constructed as described in previous reports from our laboratory (Barman et al. 2002; Lewis et al. 2001). Distributions of the intervals between heartbeats or cardiacrelated VN slow waves as well as slowwave heights and areas were also constructed.
RESULTS
Preganglionic single fiber activity
The spike trains of 24 single fibers from the cervical sympathetic nerve were tested for fractal properties using Fano factor analysis and DA. These files were obtained from 13 multifiber preparations (6 cats) using the spikesorting software. The number of spikes contained in these files ranged from 540 to 7,010. A time series was considered fractal irrespective of the number of spikes it contained providing that 1) a power law relationship was present in the Fano factor curve for the original data, but not in those for ≥10 surrogate data blocks and 2) the H exponent derived by detrended DA fell outside of the range of values for the surrogate data blocks. The spike train was considered not to have fractal properties if conditions 1 and 2 were not met when the number of spikes in the time series was ≥2,000. The rationale for using this number is based on our work with brain stem presympathetic neurons showing that the percentage of spike trains with fractal properties is dependent on sample size and reaches 100 when the time series contains ≥2,000 spikes. As such, we were able to classify 15 of 24 files of singlefiber activity as fractal. There were 9 files that could not be classified because the spike train contained <2,000 spikes and the Fano factor and detrended DA curves for the original spike train did not differ from those of the surrogates. There were no files containing ≥2,000 spikes (n = 7) that were nonfractal.
Figure 1 shows the results for the spike train of a single PSN with cardiacrelated activity. The time series of ISIs was nearly 2,000 s in length (Fig. 1 C) and contained 3,130 spikes, which are superposed in Fig. 1 A. Note the clustering of relatively long intervals near the beginning of the time series. Clusters of long and/or short intervals are characteristic of fractal point processes (Teich 1989) and may appear at any position in the time series (compare Figs. 1 C with 2C). Such clusters account for the increase in variance of spike counts with increasing window size, which, in turn, leads to a power law relationship in the Fano factor curve. The APtriggered histogram of PSN activity is shown in Fig. 1 B. Note that the probability of PSN discharge was highest during the diastolic phase of the AP. The ISI histogram was gammalike in shape with a coefficient of variation (CV) of 0.74 (Fig. 1 D). The minimum and modal ISIs were 136 and 302 ms, respectively. The mean ISI was 630 ms; thus PSN discharge rate averaged 1.6 Hz. The Fano factor curve for the original spike train (Fig. 1 E, single black line) shows a power law relationship with a slope of 0.75 (α, scaling exponent) beginning at a window size near 3 s. The scaling exponent was calculated for the range of window sizes between 3 and 187 s, thus meeting the 10 nonoverlapping window requirement for determiningF(T) accurately (see methods). Within this range, the curve for the real data clearly deviated from the superposed curves for 10 surrogate data blocks (Fig. 1 E, gray region). Therefore the power law relationship can be attributed to longrange correlations among ISIs. For window sizes < 3 s, the values of F(T) for the real data were the same as those for the surrogate data blocks. F(T) was close to 1.0 for window sizes less than the minimum ISI. This feature of the Fano factor curve is consistent with a Bernoulli process with a low probability of success (Teich 1992). This is explained by the fact that, for very small windows, the spike count can be either zero or one, with the former more likely to occur. For window sizes between the minimum ISI and approximately 3 s,F(T) dipped below 1.0 and reached its lowest value near the modal ISI. The extent of the dip is related to skewness of the ISI histogram (Teich 1992). In general, the more symmetric the histogram, the greater the dip (compare Figs. 1,D and E, with 2, D and E). The skewed distribution of the Fano factors at large window sizes in the curves for the surrogates is more apparent than real because of the loglog scaling. Nonetheless, some skewness toward values < 1.0 is expected because the distribution of ISIs did not fit a pure exponential (Teich and Lowen 1994).
The results of detrended DA (Fig. 1 F) also point to the fractal nature of the spike train of this PSN. Note that the slope of the curve for the real data (single black line) was relatively flat as compared with those of the superposed curves for 10 surrogates (gray region). The range over which the slopes of the curves for the real data and surrogates differed was for groups (m) of between 10 and 280 data points. In this case, the H exponent calculated from the negative slope of the curve for the real data was 0.88. The H exponent for the surrogates ranged from 0.42 to 0.55.
The results in Fig. 2 are for the spike train of a PSN that lacked cardiacrelated activity (Fig.2 B). For window sizes ≥ 1 s, the slope of the power law relationship in the Fano factor curve for the real data was 0.71 (Fig. 2 E, single black line). In contrast, the curves for 10 surrogate data blocks were flat, with F(T) hovering near 1.0 (gray region). The H exponent derived by detrended DA for the real data was 0.89 for m of between 5 and 720 data points (Fig. 2 F). The DA curve in this range clearly fell far from the curves for the surrogate data blocks. The number of spikes in this time series was 2,888 and the mean discharge rate was 1.7 Hz. The CV of the ISI histogram (Fig. 2 D) was 1.2, and the minimum ISI was 72 ms. The highly asymmetric shape of the ISI histogram is consistent with the virtual absence of a dip in the Fano factor curve for window sizes < 1 s.
The properties of the fractal spike trains of 15 single PSNs, 8 with and 7 without cardiacrelated activity, are summarized in Table1. Note that the properties were generally similar for the two groups. However, mean blood pressure was significantly lower (P < 0.05; unpairedttest) for the group of PSNs lacking cardiacrelated activity. Also note that the H exponents derived by detrended DA were not significantly different from those obtained using ordinary DA. The H exponent was persistent in every case, indicating that the longrange correlations among ISIs were positive.
Preganglionic multifiber activity
Fano factor analysis was used to test for fractal fluctuations in spike counts recorded from 16 multifiber strands teased from the preganglionic cervical sympathetic nerve (6 cats). Each strand contained between two and five active units, some of which only occasionally emitted a spike. Fluctuations in spike counts were fractal for 13 of 16 multifiber fields. The smallest number of spikes in the 16 fields was 2,043. A case of fractal multifiber activity is illustrated in Fig. 3. This field had cardiacrelated activity with peak counts occurring during diastole (Fig.3 A). The nearly 2,000slong time series in Fig.3 B shows the number of spikes counted in 20s bins. In this case, there was a tendency toward increased multifiber activity during the recording period. The total number of spikes in the time series was 5,784. The Fano factor curve for the real data (Fig. 3 C, single black line) shows a power law relationship with a slope (α) of 0.72 beginning at a window size of approximately 2 s. In contrast, after an initial dip below F(T) = 1.0, the curves for 10 surrogates data blocks were essentially flat (gray region). Thus shuffling of the ISIs in multifiber time series effectively randomized the number of spike counts in successive windows of specified length.
Three of 16 multifiber fields did not exhibit fractal fluctuations in spike counts despite the fact that these time series contained no less than 3,071 action potentials. An example is shown in Fig.4. In this case, there was little or no cardiacrelated activity (Fig. 4 A). Note that the Fano factor curve for the original time series (Fig. 4 C, single black line) fell within the range of the curves for 10 surrogate data blocks (gray region).
The properties of the 16 multifiber spike trains are summarized in Table 2. Note that the majority of both the fractal and nonfractal time series exhibited cardiacrelated activity. Mean firing rates were similar for the two groups. The slope of the power law relationship in the Fano factor curves for the 13 multiunit fractal time series was not significantly different from those in the curves for single fiber activity (see Table 1).
DA of preganglionic multifiber activity was not performed so as to avoid dealing with “mixed” populations of ISIs. The intervals would include those between the spikes of the same unit (whether fiber 1, 2, or n) and those between the spikes of different sets of unit pairs (fibers 1 and 2, 2 and n, etc.). The sequence of such “mixed” intervals might be random even when fluctuations in spike counts for the total population of active fibers were proven to be fractal by Fano factor analysis.
Postganglionic SND and HRV
Femoral blood pressure and the activity of the whole postganglionic VN were recorded in five cats. As illustrated in Fig.5, we made cyclebycycle measurements of the cardiac interval (CI), the interval between successive cardiacrelated slow waves of VN activity (SWI), troughtopeak normalized slowwave height (referred to as SWH), and normalized slowwave area (SWA). The measurements of VN activity were made from digitally filtered records (see methods). Time series containing ≥2,000 readings of these parameters were constructed, after which DA was performed.
The results of one of the five experiments are illustrated in Figs.6 and 7. The time series in Fig. 6, A–D(left), are 900 s in length and contain 2,120 (CI and SWI) or 2,121 (SWH and SWA) measurements. The corresponding histograms (Fig. 6, A–D, right) show the distribution of these measurements. Values of the four parameters were distributed normally and mean CI and mean SWI were identical (425 ms). Because slow trends (decrease in CI and SWI, increase in SWH) appeared in the time series, the decision on whether momentary fluctuations in a particular parameter were fractal in nature was made on the basis of detrended DA rather than ordinary DA.
Figure 7 shows the results of detrended DA of the time series illustrated in Fig. 6. The slopes of the curves for CI, SWH, and SWA, but not SWI (real data; single black line) were flatter than those for 10 surrogate data blocks (gray region) over the following ranges ofm: 5–540 (CI); 40–540 (SWH); and 30–540 (SWA).H exponents derived from the negative slopes were 0.67 (CI), 0.67 (SWH), and 0.76 (SWA). H exponents for SWI and the surrogates for each of the parameters were close to 0.5. These results were typical of those observed in four of five cats. None of the parameters were fractal in the other cat.
The results of DA in the four cats showing fractal fluctuations are summarized in Table 3. The following points should be noted. 1) Fractal H exponents for CI, SWH, and SWA obtained with detrended DA were persistent, but lower than those obtained with ordinary DA. 2) The fractal range (detrended DA) began at a smaller value of m for CI than for SWH or SWA. 3) Although ordinary DA yielded highly persistent H for SWI, the H exponents derived with detrended DA were indistinguishable from those for the surrogates.
DISCUSSION
Various methods of fractal analysis have been used to test for timescale invariant behavior of biological signals that appear to be either periodic or aperiodic (Bassingthwaighte et al. 1994; Goldberger et al. 1996; Ivanov et al. 1999; Teich 1992: Yamamoto and Hughson 1994). In the case of an apparently periodic signal such as the heart beat, the question entertained is whether relatively small fluctuations in CI occur randomly or are correlated over multiple time scales. The question is the same for the fluctuations of aperiodic signals. In the current study, time series of pre and postganglionic sympathetic activity and CI were tested for timescale invariant behavior using two established methods of fractal analysis. Fano factor analysis was applied to time series of single and multifiber activity from the preganglionic cervical sympathetic nerve to determine whether fluctuations in spike counts were fractal. DA provided additional information as to whether the ISIs in time series of singlefiber activity were positively (persistence) or negatively (antipersistence) correlated. DA was not applied to preganglionic multifiber activity to avoid calculations based on mixed populations of ISIs (intervals between spikes of the same vs. different fibers), possibly occurring in a random sequence. Nevertheless, DA was used to test for fractal fluctuations in the height, area, and intervals between cardiacrelated slow waves recorded from the whole postganglionic VN and the intervals between heartbeats. The Fano factor curves for these parameters are not presented in this paper. These curves contained large dips of F(T) to values well below 1.0 over a wide range of window sizes due to the highly symmetric (i.e., normal) distribution of these parameters (see Fig. 6). These profound dips act to obscure power law relationships more readily revealed in this study by DA.
Our major findings are as follows. 1) Fluctuations in spike counts and ISIs were fractal over a large range of window sizes for all single PSN time series that contained ≥2,000 spikes. Moreover, the longrange correlations of ISIs were persistent in every case.2) Fluctuations in preganglionic multifiber spike counts were fractal in the majority (13 of 16) of cases. 3) Fluctuations in the height and area but not the interval between cardiacrelated slow waves of postganglionic VN activity were fractal and persistent in most (4 of 5) cats. Fluctuations in CI were fractal in the same experiments. These findings and their implications are discussed in the following text.
The preganglionic cervical sympathetic nerve is functionally heterogeneous in that it contains fibers controlling blood vessel diameter, pupil size, nictitating membrane contraction, pilomotion, and sweating (Bishop and Heinbecker 1932; Eccles 1935). Whether the spike trains of each of these fiber types are fractal cannot be decided on the basis of the currently available data. Although PSNs governing vasoconstriction are reputed to have the strongest cardiacrelated activity (Janig 1988), the fact that the spike trains of single PSNs were fractal independent of whether their discharges were cardiac related turned out not to be helpful. Regarding this issue, mean blood pressure was significantly lower in the cases in which the spike train did not contain cardiacrelated activity (see Table 1). Thus these PSNs may not have been functionally distinct from those with cardiacrelated activity. Rather, baroreceptor input may have been too weak to induce cardiacrelated activity at lower blood pressures.
In a previous study (Lewis et al. 2001) on brain stem neurons with activity correlated to the cardiacrelated or 10Hz rhythmic component of SND, we found that the percentage of fractal spike trains (n = 19) reached 100 when the time series contained ≥2,000 spikes. On this basis, we proposed that all such neurons have fractal firing patterns. We are reticent to propose the same for PSNs because only a small sample of time series containing this number of spikes was obtained in the current study.
Fluctuations in spike counts were fractal for the majority (13 of 16) of time series of multifiber preganglionic activity. This observation raises the possibility that groups of PSNs share common fractal inputs possibly from the brain stem (Lewis et al. 2001) or that timescale invariant behavior generated independently by individual neurons in the brain stem and/or spinal cord is somehow synchronized. These possibilities are deserving of future investigation since synchronization of the fractal discharges of populations of sympathetic neurons might play a role in explaining the fractal component of HRV.
HRV in awake humans is characterized not only by respiratoryrelated and slower thirdorder fluctuations, but also by a timescale invariant component extending over a very low frequency (0.00003 to 0.1 Hz) range (Goldberger et al. 1996; Ivanov et al. 1999; Malliani et al. 1991; Yamamoto and Hughson 1994). In the current study, we found fractal fluctuations in CI recorded from cats anesthetized with dialurethane. DA revealed persistent correlations among CIs for groups (m) of between 7 and 653 data points. Population activity recorded from the whole postganglionic VN in the same cats also exhibited persistent fractal properties over a somewhat shorter range of window sizes (see Table 3). Specifically, fluctuations in the height and area of cardiacrelated VN slow waves were fractal. As was the case for CI,H exponents derived for these properties were persistent but lower in value when detrended DA was substituted for ordinary DA. Although ordinary DA yielded persistent H exponents for fluctuations in the interval between cardiacrelated VN slow waves, those derived with detrended DA were not significantly different from 0.5. Thus it is likely that slow trends in the time series of SWI rather than true fractal behavior accounted for the persistentH derived by using ordinary DA.
The fractal component of HRV in humans is diminished with aging and in cardiovascular diseases such as heart failure (Goldberger 1992; Goldberger et al. 1996; Ivanov et al. 1999). As such, methods of fractal analysis have recently been introduced into the cardiology clinic. Because of the clinical implications attached to the loss of fractal HRV, it becomes even more important to identify the mechanisms responsible for timescale invariant fluctuations in CI. There are conflicting views concerning the role played by autonomic outflows to the heart. WhereasGoldberger et al. (1996) have postulated that the fractal component of HRV in healthy humans arises from a nonlinear interaction of sympathetic and vagal influences on the SA node,Yamamoto and Hughson (1994) reported that the power law relationship in power spectrum of CI in humans is minimally affected by βadrenoceptor blockade. Whether sympathetic outflow is involved in generating fractal HRV in the cat remains to be investigated. This possibility seems more attractive in the light of our finding that both pre and postganglionic sympathetic activities contain fractal components.
Fractal fluctuations in the height and area of cardiacrelated bursts of postganglionic VN activity indicate that a timescale invariant process is involved in determining the number of active neurons and/or their firing rate during each cardiac cycle. At first glance, the fact that fluctuations in the interval between cardiacrelated bursts of VN activity were not fractal seems surprising since fluctuations in CI were fractal in the same cats. However, it should be remembered that the phase angle relating SND to the AP in the cat shows considerable variability on a heart beattobeat basis (Larsen et al. 2000; Lewis et al. 2000). This arises from the fact that the cardiacrelated rhythm in SND is not the simple consequence of constant latency central inhibition of baroreceptor reflex origin. Rather, cardiacrelated bursts are forced nonlinear oscillations whose timing relative to pulsesynchronous baroreceptor input can assume many different values (Larsen et al. 2000; Lewis et al.2000). Thus CI might fluctuate in a timescale invariant manner while fluctuations in the interval between cardiacrelated bursts of SND occur randomly. At any rate, if fractal SND plays a role in generating fractal HRV, the properties of SND so responsible would apparently be SWH and SWA.
In summary, we have demonstrated that apparently random fluctuations in activity recorded from PSNs and the postganglionic VN are, in fact, fractal in nature. Such properties include spike counts, ISIs, and population burst height and area. Time series of these properties should not be modeled as random, stochastic processes comprised of uncorrelated events. Rather, the timescale invariant fluctuations are dictated by a complex deterministic process that imparts a type of “longterm” memory into the system responsible for SND, as reflected by longrange, positive correlations among events. Thus, for example, the current value of the ISI is determined not only by recent events, but also by those in the “distant” past (range over which a power law relationship appears in the Fano factor and DA curves). It remains to be investigated whether the fractal properties of SND play a role in generating the fractal component of HRV that is typical of the “healthy” cardiovascular state.
Acknowledgments
The authors thank L. M. Braybrook for typing the manuscript.
This study was supported by National Heart, Lung, and Blood Institute Grants HL13187 and HL33266.
Footnotes

Address for reprint requests: G. L. Gebber, Department of Pharmacology and Toxicology, Michigan State University, East Lansing, MI 48824 (Email: gebber{at}msu.edu).
 Copyright © 2003 The American Physiological Society