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1Department of Pharmacology and Toxicology, Michigan State University, East Lansing, Michigan; and 2Department of Pharmacology, Faculty of Medicine, Hacettepe University, Ankara, Turkey
Submitted 28 September 2005; accepted in final form 12 November 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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, of the power law in the Allan factor curve. fGn is defined as 0 < < 1 and fBm as 1 < < 3. The process responsible for the fractal spike trains of 11 of 12 classifiable LTF neurons with sympathetic nerve-related activity was fGn. In contrast, the process responsible for the fractal spike trains of eight of nine classifiable RVLM presympathetic neurons was fBm. The time series of simultaneously recorded vertebral sympathetic nerve discharge and the arterial pulse also were fBm-based signals. Because a fBm signal is the cumulative sum of the elements comprising the corresponding fGn signal, these results show smoothing of fractal time series in a feedforward direction from medullary presympathetic neurons to postganglionic sympathetic neurons. This may involve integration by RVLM neurons of their LTF inputs or independent fractal processes acting at different levels of the network controlling sympathetic nerve discharge. Whether feedforward smoothing of fractal signals is a feature in other neural systems is open to investigation. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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, 1985, 1987, 2000) identified single neurons in the cat medullary lateral tegmental field (LTF) and rostral ventrolateral medulla (RVLM) with activity correlated to the cardiac-related or 2- to 6-Hz pattern of postganglionic sympathetic nerve discharge (SND). Such LTF and RVLM neurons are referred to as presympathetic neurons in this paper. These neurons do not fire during every rhythmic cycle of SND, and the number of missed cycles is highly variable. As such, the distribution of their interspike intervals (ISIs) takes the form of an exponential, gamma, or multimodal distribution.
In the study by Lewis et al. (2001), Fano factor analysis was used to test whether time series of the highly variable ISIs of LTF and RVLM neurons are best described by a renewal (random) or a fractal-based point process. For a renewal process, the data are uncorrelated, i.e., ISIs are independent of each other (Cox and Lewis 1966; Lowen and Teich 2005). In contrast, fractal spike trains are characterized by long-range correlations among the ISIs (Bassingthwaighte et al. 1994; Lowen and Teich 2005). Specifically, fractal-based time series are statistically self-similar in that fluctuations in ISIs or the number of spikes counted over brief periods of time are proportional to those fluctuations measured over longer periods. Because the features measured using different temporal resolutions are related, self-similarity implies that the data are correlated. That is, the current value of the measured parameter is dependent on values in the distant as well as recent past (Bassingthwaighte et al. 1994; Liebovitch 1998). The spike train has fractal properties when the correlations extend over more than one time scale, as reflected by a power law relationship between some index of the variance of the measured parameter and the temporal resolution used to make the measurement. In the study by Lewis et al. (2001), the fractal nature of the spike trains of LTF and RVLM presympathetic neurons was shown. However, for reasons given in METHODS, Fano factor analysis did not allow separation of two major classes of fractal processes, fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). For this purpose, we used a more powerful method (Allan factor analysis) in this study.
As explained by Eke et al. (2000, 2002), cumulative summation of the elements of a fGn signal yields fBm, whereas the differences between the elements of fBm yield fGn. Thus the two classes of fractal signals are interconvertible with fBm representing the integral of fGn. The rationale for testing whether the fractal spike trains of LTF and RVLM presympathetic neurons fall into different classes is based on our past findings concerning their interrelations. First, the axons of LTF neurons classified as sympathoexcitatory project to the region of the RVLM containing neurons with spinal axons that project to the thoraco-lumbar intermediolateral sympathetic nucleus (Barman and Gebber 1987). Second, the LTF neurons fire earlier during cardiac-related bursts of postganglionic SND than do their counterparts in the RVLM (Barman and Gebber 1983), and the difference in the timing of their naturally occurring discharges corresponds to the conduction time between the two regions as determined by antidromic activation of LTF neurons by stimuli applied to the RVLM (Barman and Gebber 1987). Third, chemical inactivation of either the LTF or RVLM of the cat significantly reduces postganglionic SND (Barman et al. 2000; Orer et al. 1999). These findings suggest that feedforward connections from the LTF to the RVLM are involved in generating basal SND (Barman and Gebber 2000). However, whether RVLM simply relay their fractal inputs from the LTF to preganglionic sympathetic neurons or act on them in a more complex way is unknown. The results of this study are consistent with the latter possibility.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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100 mmHg with an infusion of dextran in saline; body temperature was kept near 38°C with a heat lamp. A surgical state of anesthesia was indicated by the failure of noxious stimuli (pinch) to desynchronize the frontal-parietal EEG (Gebber et al. 1999).
Recordings of LTF and RVLM unit activity were available from an earlier study from this laboratory (Lewis et al. 2001). One- to 3-h recordings of femoral arterial pressure and postganglionic vertebral SND were newly collected from nine vagotomized cats with intact baroreceptor nerves.
The methods used for baroreceptor denervation and/or vagotomy and to record femoral arterial blood pressure, medullary unit spikes (extracellular), and multiunit population activity from the postganglionic vertebral or inferior cardiac sympathetic nerve are described in detail elsewhere (Barman and Gebber 1983, 1985, 1987). Briefly, action potentials of single medullary neurons (preamplifier band-pass 300–3,000 Hz) recorded with metal or glass microelectrodes were isolated from background using analog window discrimination or digital spike sorting (Das et al. 2003; Orer et al. 2003). The criteria used to establish the unitary nature of the recordings can be found in the original papers (Barman and Gebber 1983, 1985, 1987). Postganglionic SND was recorded with a preamplifier band-pass of 1–1,000 Hz. The synchronized discharges of populations of postganglionic fibers appear as slow waves (envelopes of spikes) with this band-pass (Barman and Gebber 1983, 1985, 1987). Cardiac-related SND was present in baroreceptor-intact cats, and non–cardiac-related 2- to 6-Hz slow waves appeared in baroreceptor-denervated cats.
Software written in our laboratory (Gebber et al. 1999; Lewis et al. 2001) was used to detect standardized pulses (1-ms duration) representing medullary unit action potentials, the peaks of a digitally band-pass filtered version of the sympathetic nerve slow waves, and the peak systolic phase of the femoral arterial pulse (AP). The symmetric, nonrecursive digital filter with a Lanczos smoothing function produced minimal amplitude and phase distortion of SND (see Fig. 5 in Das et al. 2003). The width of the band-pass was between 4 and 6 Hz, with the center frequency matched to that of the primary peak in the autospectrum of SND (Gebber et al. 1999). Digital filtering of SND aided in the accurate detection of peaks of the slow waves.
Time series and histograms showing the distributions of interevent intervals were constructed using a bin resolution of 1 and 10 ms, respectively. Spike-triggered averaging was used to determine whether medullary unit activity was correlated to cardiac-related or 2- to 6-Hz SND, and AP-triggered averages of SND and histograms of medullary unit spike occurrence were constructed to show the cardiac-related components of these signals. Power density spectral analysis was used to characterize the frequency components of SND and the strength of correlation of SND to the AP. These methods have been described in earlier reports from this laboratory (Barman and Gebber 1983, 1985, 1987).
Fractal analysis
Unit spike trains and time series of digitally filtered postganglionic SND (see Das et al. 2003) and the AP were subjected to two methods of fractal analysis, the first of which was Allan factor analysis.
Thurner et al. (1997) and Turcott and Teich (1996) define the Allan factor, A(T), as the ratio of the event-number Allan variance to twice the mean number of events (unitary spikes, slow waves, or heart beats) in a window size of specified length (T)
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,b).
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 Tmax, the window size, T, is progressively increased from a minimum of a single bin (1 ms) to a maximum of Tmax/4 so that at least four nonoverlapping windows are used for each measure of A(T). As explained by Thurner et al. (1997) and Turcott and Teich (1996), A(T) is 1.0 at all window sizes for a random process in which fluctuations in the number of events are uncorrelated. For a periodic process, the variance decreases and A(T) approaches zero as the window size is increased. However, for a fractal process, A(T) increases as a power of the window size. This reflects the greater variance in number of events with increasing window size. The power law relationship between A(T) and window size appears as a straight line with a positive slope, , on the log-log scale. The , also known as the scaling exponent, is the power to which fluctuations in the number of events on one time scale are proportional (i.e., statistically self-similar) to those measured on other time scales. The Pearson correlation coefficient (r) is used to test for linearity on the log-log scale, and linear regression is used to calculate and its 95% CIs.
Because , derived using Allan factor analysis, is bounded in a range of 0–3, it can be used to distinguish between fGn and fBm (Eke et al. 2000, 2002; Thurner et al. 1997; Turcott and Teich 1996). In theory, fractal time series with 0 < < 1 are fGn, whereas the range 1 < < 3 denotes fBm. Thurner et al. (1997) used artificially generated signals of known to assess the accuracy to which Allan factor analysis estimates . For example, from 100 estimates of for a signal with a known of 0.8, the mean ± SD determined by these workers was 0.795 ± 0.072 for a power law extending over a range of window sizes between 62.5 and 625 s and 0.807 ± 0.114 for a range of window sizes between 125 and 1,250 s. The ranges for signals with known of 0.2 and 1.5 were similarly restricted. As such, we treated our physiological time series in the following way. We used twice the largest SD of the estimated values for a signal with a known of 0.8 (see Thurner et al. 1997) to establish a range (0.77–1.23) around 1.0, outside of which fGn and fBm could be distinguished. When the estimate of was <0.77, the signal was classified as fGn. When the estimate of was >1.23, the signal was classified as fBm. When the estimate of fell within the range of 0.77–1.23, the signal could not be classified.
The second method used to test for fractal fluctuations was Fano factor analysis. Teich (1992) and Turcott and Teich (1996) define the Fano factor, F(T) as the ratio of the variance of the number of events, var[Ni(T)], to the mean number of events, mean [Ni(T)], in a window size of specified length (T)
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; Turcott and Teich 1996). This is likely because of the greater variance of counts per window than for the first differences between windows. As such, Fano factor analysis may reveal a power law relationship extending over more than one time scale (implying fractal behavior) when the data block is too short to show this with Allan factor analysis. Thus we used Fano factor analysis to establish the fractal nature of the time series when the power law in the Allan factor curve did not extend beyond one full time scale (Fadel et al. 2004a). It should be noted, however, that mathematical constraints prevent F(T) from increasing faster than 
T1, thereby limiting the range of the slope () of the power law in the Fano factor curve to 0–1 (Thurner et al. 1997). Thus Fano cannot be used to distinguish between fGn and fBm. Surrogate data
The Allan and Fano factor curves for the original time series are routinely compared with those of 10 or 20 surrogates that are constructed by shuffling the order of the original interevent intervals. Specifically, we assign random numbers to the intervals and then sort the random numbers by size (Lewis et al. 2001). This creates a randomized data set for which the mean, variance, and frequency distribution are identical to those of the original time series, but with no correlations among events. If shuffling of the data eliminates the power law in the Allan and Fano factor curves, it can be concluded that long-range correlations are present in the original time series.
Values in the text are means ± SD.
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RESULTS |
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Figure 1A shows a time series (1,001 s in length) of 1,713 ISIs for an RVLM neuron with activity correlated to the cardiac-related rhythm in postganglionic inferior cardiac SND of a baroreceptor-intact cat. The distribution of the ISIs as well as the original recordings of unit activity (inset) appear in Fig. 1B. The ISI histogram was gamma-like in shape with a long tail to the right of the mode. The CV of the distribution was 0.83. The mean ISI was 587 ms and the mode was 230 ms. The longest ISI exceeded 2.5 s, indicating that, on occasion, the unit missed firing for >10 consecutive cardiac cycles. The spike-triggered average (STA) of inferior cardiac SND in Fig. 2A was constructed using 1,714 spikes of this RVLM neuron. The average shows SND for 500 ms before and after RVLM unit spike occurrence at time 0. Note that the peaks in the STA are much larger than those in the "dummy" average of SND. The latter was constructed using a series of randomly generated pulses of the same number and frequency as for the RVLM neuronal spike train. The discharges of medullary neurons were considered to be correlated to SND if the amplitude of the first peak to the right of time 0 in the STA was at least three times that of the largest deflection in the "dummy" average. In the case shown, the RVLM neuron was most apt to fire near the onset of the rising phase of the cardiac-related slow wave of inferior cardiac SND. The AP-triggered average of inferior cardiac SND and histogram of RVLM unit activity in Fig. 2B also show the cardiac-related components of these signals. Note again that unit discharge was most apt to occur near the onset of the rising phase of the sympathetic nerve slow wave.
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). That is, for window sizes smaller than the shortest ISI, either zero or one event is counted, with the former outcome being more likely to occur. F(T) and A(T) then dipped below 1. The dip that is most pronounced in the Allan factor curve can be attributed to the rhythmic (i.e., cardiac-related) component of unit activity. Most importantly, a power law began at a window size near 3 s in the Fano factor curve and near 25 s in the Allan factor curve. Whereas the power law in the Fano factor curve extended over more than two time scales (2–250 s), that in the Allan factor curve just reached one decade (25–250 s). Each time scale is represented by a decade on the log scale. The upper limit of these ranges was the largest allowable window size (Tmax/4). The scaling exponent, = 0.95 of the power law in the Fano factor curve, was near its maximum theoretical value of 1.0, whereas the Allan was higher (1.76). This value allowed us to classify the spike train as a fBm-based point process. Because = 2.0 in the Allan factor curve signifies a special case (random walk) of fBm with no long-range correlations (Goldberger et al. 1996), we used twice the largest SD reported by Thurner et al. (1997) for the estimated values of a signal with known of 1.5 to establish a range (1.80–2.20) around 2.0 over which we considered such special cases to occur. Although our estimate of the Allan for this RVLM neuron was close to the lower limit of this range, the power laws in the Fano and Allan factor curves were eliminated by shuffling the ISIs. Note the flatness of the superposed curves for 10 surrogates (Fig. 1, C and D, gray regions). Thus the original time series contained long-range correlations among the fluctuations in spike number.
The results of fractal analysis of the spike train (421 s in length containing 1,635 action potentials) of an LTF neuron with activity correlated to cardiac-related SND are shown in Fig. 3. The ISI histogram was gamma-like in shape (CV = 0.38) with a long tail to the right of the mode (Fig. 3A). The range of window sizes over which the power law in the Fano factor curve extended was 0.4–105 s (Fig. 3B; black trace) and that in the Allan factor curve was 10–105 s (Fig. 3C; black trace). The scaling exponents in the two curves were equivalent ( = 0.75). The Allan value allowed us to classify this spike train as a fGn-based point process. As was typically the case, the Fano and Allan factor curves for the surrogates (gray regions) did not contain a power law. Thus the original time series contained long-range correlations.
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LTF NEURONS.
Table 1 lists the properties of the spike trains of 15 LTF neurons with sympathetic nerve–related activity. Eleven of these neurons in nine baroreceptor-intact cats had activity correlated to the cardiac-related rhythm in SND, and the other four neurons in three baroreceptor-denervated cats had activity correlated to 2-to 6-Hz SND. Using the procedure described in METHODS, 11 of the LTF spike trains were classified as fGn-based point processes and 1 as a fBm-based point process. The spike trains of the other three LTF neurons could not be classified. In each case, the power law in the Allan factor curve extended over no less than one time scale (decade of window sizes). As might be expected for the 11 spike trains classified as fGn-based point processes, Allan and Fano values were similar and positively correlated. The Allan and Fano values averaged 0.54 ± 0.17 and 0.50 ± 0.14, respectively. The Pearson correlation coefficient relating these two variables was highly significant (r = 0.76; P = 0.007). Similar results have been reported for the spike trains of neurons in the cat lateral geniculate nucleus (Teich et al. 1997) and rat suprachiasmatic nucleus (Kim et al. 2005).
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> 1.0). The Allan value for one of these neurons (fBm*) was not different from 2.0. The Fano values for these eight neurons were very close to the maximum theoretical value of 1.0. The spike train of only one RVLM neuron could be classified as a fGn-based point process. The Allan values for the other eight spike trains were not different from 1.0; their spike trains could not be classified. Of those nine spike trains classified either as fBm- or fBm-based point processes, the power law in the Allan factor curve for six extended over at least one complete time scale. Nonetheless, the other three spike trains were classified as fractal in nature because the power law in the corresponding Fano factor curve extended close to or beyond two time scales.
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The time series of SND and AP recorded simultaneously with medullary unit activity were too short to allow them to be classified. This is because the window size at which the power law in the Allan factor curves for these parameters began generally was much larger than that for the unit spike trains (Tables 1–4). Thus we performed nine new experiments on vagotomized cats in which postganglionic vertebral SND and the AP were recorded for a period of 1–3 h. The vagus nerves were sectioned bilaterally in the neck to eliminate the effect of cardiac vagal outflow on heart rate variability (Yamamoto et al. 1995).
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) of the power law was 2.42 for SND and 2.21 for AP. These values were different from 2.0. Thus the fluctuations in the number of sympathetic nerve slow waves and heart beats occurring over time were derived from fBm processes with long-range correlations. Indeed, the Allan factor curves for 10 surrogates (Fig. 4, C and D, gray regions) did not contain a power law.
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value not different from 2, the time series of vertebral SND fell into the class of fBm-based point processes with long-range correlations (Table 3). All nine of the time series of the AP could be classified as fBm-based point processes, with one case of Allan not different from 2.0 (Table 4). As expected, Fano was close to 1.0 for these time series. |
DISCUSSION |
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, of the Fano factor curve to a range of 0 to 1.0 (Thurner et al. 1997), this method could not be used. As shown by Eke et al. (2000, 2002), dispersional analysis and rescaled range analysis provide reasonable estimates of the scaling exponent for fGn, but not fBm. Like Allan factor analysis, the scaling exponent, 
, obtained with power spectral analysis, has a range encompassing both fGn and fBm signals. However, the work of Thurner et al. (1997) and Eke et al. (2000) suggests that the range of over which classification of fractal-based signals is uncertain is wider than that for Allan . As such, we chose Allan factor analysis for the classification of our fractal time series.
Our most striking finding is the prevelance of fBm-based point processes for postganglionic SND and RVLM presympathetic unit activity versus fGn-based point processes for LTF presympathetic unit activity. Taking into account the range of Allan values over which classification was uncertain, seven of the time series of vertebral SND fell into the fBm class and none in the fGn class. For RVLM presympathetic neurons, the spike trains of eight could be classified as fBm-based point processes and only one as a fGn-based point process. In contrast, 11 LTF presympathetic unit spike trains fell into the fGn class and only one could be classified as fBm.
There are at least two ways to explain the prevalence of fGn in the LTF and fBm in the RVLM and postganglionic SND. One possibility is that the outputs of LTF neurons directed to the region of the RVLM containing bulbospinal presympathetic neurons (Barman and Gebber 1987) are cumulatively summed by the latter group to form an integral of the former. As such, the RVLM would function as an integrator rather than as a simple relay station of its fractal inputs from the LTF. The consequence of such integration in the RVLM would be a smoother fractal time series transmitted to preganglionic sympathetic neurons and ultimately to postganglionic nerves. Regarding this point, the average Allan for vertebral SND (1.67 ± 0.52) was not significantly different (unpaired t-test) from that (1.35 ± 0.50) for RVLM unit activity. However, the average window size (92 ± 37 ms) at which the power law in the Allan factor curve for vertebral SND began was significantly larger than that (14 ± 10 ms) for RVLM unit activity. This may reflect the relatively small variance of the interval between sympathetic nerve slow waves compared with that of medullary unit interspike intervals.
The prevalence of fGn in the LTF and fBm in the RVLM might also be explained by independent fractal processes acting in the two regions. Such might be attributable to differences in the intrinsic membrane properties of LTF and RVLM presympathetic neurons. It is also possible that the fractal behavior of RVLM neurons might depend on excitatory and/or inhibitory synaptic inputs from sources other than the LTF.
Currently, the physiological consequences of converting fGn to fBm or vice versa remain obscure. The problem is worthy of future study on two accounts. First, the properties of fGn and fBm are distinctly different. Because cumulative summation of the values making up a fGn signal yields fBm, the time series of the latter is smoother and the data points are positively correlated, i.e., persistent (Eke et al. 2000). That is, values larger (smaller) than the mean tend to be followed by values also larger (smaller) than the mean. fGn signals are by nature rougher than fBm because they are composed of the differences in the values making up fBm. As a consequence, the data points making up fGn can be either negatively correlated (antipersistence) or positively correlated (Eke et al. 2000). When the data are negatively correlated, values larger than the mean tend to be followed by values smaller than the mean and vice versa. Despite the relative roughness of fGn time series, such signals are considered stationary because the variance approaches a constant value as the window size used to make the measurements is increased (Eke et al. 2000, 2002). In contrast, despite their relative smoothness, fBm is nonstationary because the variance continues to increase as the window size is lengthened.
A second reason for the classification of fractal signals relates to the possibility that the character of the signal under study may be dependent on species, experimental conditions, and/or the presence or absence of pathological conditions. A case in point relates to the results of an earlier study from our laboratory (Fadel et al. 2004b) in which Allan factor analysis was used to characterize the fractal properties of postganglionic peroneal muscle sympathetic nerve activity and heart period in awake human subjects free of cardiovascular disease. The Allan values listed in Table 2 of that study present a picture quite different from that reported here for anesthetized cats. In contrast to the prevalence of fBm in time series of cat SND and heart period, the data from humans showed the prevalence of fGn both for SND and the AP. Whether this reflects species differences, the presence or absence of anesthesia, and/or forebrain-mediated influences in the awake state remains to be determined. The extent to which fractal fluctuations in heart period are influenced by those in cardiac sympathetic nerve activity or by vago-sympathetic interactions also deserves further study (Goldberger et al. 1996; Yamamoto and Hughes 1994).
In summary, in the anesthetized cat, we showed smoothing of fractal time series in a feedforward direction from medullary presympathetic neurons to a postganglionic sympathetic nerve. This may involve cumulative summation of LTF fractal inputs by RVLM neurons so as to form an integral of the former. Alternatively, independent processes may generate one class (fGn) of fractal signal in the LTF and another class (fBm) in the RVLM and postganglionic nerves.
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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Address for reprint requests and other correspondence: G. L. Gebber, Dept. of Pharmacology and Toxicology, Michigan State Univ., East Lansing, MI 48824-1317 (E-mail: gebber{at}msu.edu)
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REFERENCES |
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