quantal neurotransmitter secretion rate exhibits fractal behavior

Quantal Neurotransmitter Secretion Rate Exhibits Fractal Behavior

Steven B. Lowen,1 Sydney S. Cash,2 Mu-ming Poo,3 and Malvin C. Teich4

1Department of Electrical and Computer Engineering, Boston University, Boston, Massachusetts 02215, 2Department ofBiological Sciences, Columbia University, New York, New York 10027, 3Department of Biology, University of California atSan Diego, La Jolla, California 92093, and 4Departments of Electrical and Computer Engineering, Physics, BiomedicalEngineering, and Cognitive and Neural Systems, Boston University, Boston, Massachusetts 02215

The rate of exocytic events from both neurons and non-neuronal cells exhibits fluctuations consistent with fractal (self-similar) behavior in time, as evidenced by a number of statisticalmeasures. We explicitly demonstrate this for neurotransmittersecretion at Xenopus neuromuscular junctions and for rat hip-pocampal synapses in culture; the exocytosis of exogenouslysupplied neurotransmitter from cultured Xenopus myocytesand from rat fibroblasts behaves similarly. The magnitude of thefluctuations of the rate of exocytic events about the meandecreases slowly as the rate is computed over longer andlonger time periods, the periodogram decreases in power-lawmanner with frequency, and the Allan factor (relative variance ofthe number of exocytic events) increases as a power-law func-

tion of the counting time. These features are hallmarks ofself-similar behavior. Their description requires models thatexhibit long-range correlation (memory) in event occurrences.We have developed a physiologically plausible model that ac-cords with all of the statistical measures that we have exam-ined. The appearance of fractal behavior at synapses, as wellas in systems comprising collections of synapses, indicatesthat such behavior is ubiquitous in neural signaling.

Key words: quantal secretion; vesicular exocytosis; miniatureendplate currents; fractal; long-term correlation; lognormal pro-cess; Xenopus neuromuscular junction; myocyte autoreception;fibroblast; hippocampal synapse

Communication in the nervous system is mediated by action-potential-initiated exocytosis of multiple vesicular packets (quan-ta) of neurotransmitter (Katz, 1966). Even in the absence of suchaction potentials, however, many neurons spontaneously releaseindividual packets of neurotransmitter (Fatt and Katz, 1952). Apacket may contain from 7000 to 10,000 molecules of acetylcho-line (ACh), if we use the neuromuscular junction as an example(Kuffler and Yoshikami, 1975). On arrival at the postsynapticmembrane, the ACh molecules induce elementary endplate cur-rents (EECs), which take the form of nonstationary two- (ormulti-) state on–off sequences (Sakmann, 1992). Current flowswhen the ACh channel is open (i.e., when its two binding sites areoccupied by agonist) and ceases when the channel is closed. Apostsynaptic miniature endplate current (MEPC) comprisessome 1000 EECs (Sakmann, 1992). It was shown by Del Castilloand Katz (1954) that superpositions of MEPC-like events com-prise the postsynaptic endplate currents elicited by nerveimpulses.

It generally has been assumed that the sequence of MEPCsforms a memoryless stochastic process (Fatt and Katz, 1952).However, Rotshenker and Rahamimoff (1970) discovered thatexocytosis in the frog neuromuscular junction can exhibit corre-lation (memory) over a period of seconds, provided that extra-cellular Ca21 levels are elevated above their normal values.

Cohen et al. (1974b) subsequently found such correlation (eventclustering) even in the absence of elevated Ca21 levels.

In this paper we study the statistical properties of exocyticevents over a far larger range of time scales than previouslyexamined. MEPCs were recorded from innervated myocytes inXenopus nerve–muscle cocultures and from rat hippocampal neu-rons in cell culture. MEPCs from non-neuronal preparations alsowere examined: the quantal secretion of ACh from isolated myo-cytes (autoreception) and from rat fibroblasts, both exogenouslyloaded with ACh (Dan and Poo, 1992; Girod et al., 1995).

We direct particular attention toward those statistical measuresthat reveal the presence of memory. Our analysis reveals that thetime sequences of the MEPCs, and therefore of the underlyingexocytic events, exhibit memory that appears to decay away slowlyin both neuronal and non-neuronal cells. This long-durationcorrelation is present over the entire range of time scales inves-tigated, which stretches to thousands of seconds. The occurrenceof an MEPC, therefore, makes it more likely that another MEPCwill occur at some time thereafter. The analysis of long MEPCdata sets reveals that the rate of exocytic events behaves in amanner consistent with a fractal process, exhibiting fluctuationsover multiple time scales. Fractals are objects that possess a formof self-similarity: parts of the whole can be made to fit to thewhole by shifting and stretching. The hallmark of fractal behavioris power-law dependence in one or more statistical measures overa substantial range of the time (or frequency) scales at which themeasurement is conducted (Lowen and Teich, 1995; Thurner etal., 1997). Because multiscale fluctuations are at the heart of thisbehavior, selecting short data segments that exhibit minimalfluctuations will dilute whatever fractal characteristics might bepresent in a given data set, as we illustrate. The classic work ofFatt and Katz (1952) is revisited in light of these findings.

Received June 20, 1996; revised April 8, 1997; accepted May 13, 1997.This work was supported by Grants from the Whitaker Foundation to S.B.L., from

the National Institutes of Health (NS-31923) to M-m.P., and from the Office of NavalResearch (N00014-92-J-1251) to M.C.T. We thank Conor Heneghan and EricSchwartz for helpful suggestions.

Correspondence should be addressed to Professor Malvin C. Teich, Department ofElectrical and Computer Engineering, Boston University, Boston, MA 02215.Copyright © 1997 Society for Neuroscience 0270-6474/97/175666-12$05.00/0

The Journal of Neuroscience, August 1, 1997, 17(15):5666–5677

MATERIALS AND METHODSXenopus nerve–muscle cocultures. Cultures were prepared by followingpreviously reported methods (Spitzer and Lamborghini, 1976; Andersonet al., 1977; Tabti and Poo, 1991). In brief, the neural tube from 1-d-oldembryos (stage 20–24; Nieuwkoop and Faber, 1967) was dissociated inCa 21/Mg 21-free Ringer’s solution supplemented with EDTA, plated onclean glass coverslips, and incubated at 20–22°C for 1 d before recording.Recording and culture medium consisted of 50% (v/v) Leibovitz’s me-dium (Life Technologies, Gaithersburg, MD), 1% (v/v) fetal calf serum(Life Technologies), and 49% (v/v) Ringer’s solution (115 mM NaCl, 2mM CaCl2 , 2.5 mM KCl, and 10 mM HEPES, pH 7.3).

Hippocampal cultures. Hippocampal cultures were prepared by follow-ing protocols previously described (Goslin and Banker, 1991), but withthe following modifications. Hippocampi from newborn Sprague Dawleyrats (postnatal day 0) were dissociated and plated at low density, withoutglial support cells. Cells from 7- to 14-d-old cultures were used forexperiments. The recording solution consisted of (in mM) 140 NaCl, 5KCl, 1 CaCl2 , 1 MgCl2 , 10 HEPES, 24 D-glucose, and 0.01 TTX (Sigma,St. Louis, Mo), pH 7.4.

Fibroblast cultures. Parenteral 3Y1 cells, a line of rat skin fibroblasts(Sternberg et al., 1993) were kindly provided by Paul Greengard (Rock-efeller University, New York, NY). The cells were cultured in DMEMand 10% fetal calf serum at 37°C with 5% CO2 and were used 2–3 d aftersubculture. For detection of ACh secretion from fibroblasts, the latterwere treated with trypsin–EDTA solution (Life Technologies); the sus-pension of the fibroblasts was added to the recording chamber thatcontained cultured Xenopus myocytes.

Electrophysiology. Recordings of miniature endplate currents at Xeno-pus neuromuscular junctions were made with the gigaohm seal, nystatinperforated-patch technique (Horn and Marty, 1988). Conventional heat-polished patch pipettes were filled with pipette saline containing (in mM)150 KCl, 1 NaCl, 1 MgCl2 , 10 HEPES, pH 7.4, and nystatin (Sigma).Nystatin was added at a final concentration of 460 mM (final concentra-tion of DMSO, 1%). Nystatin stock (46 mM; 1 mg/ml in DMSO) wasprepared before each experiment, stored at room temperature in alightproof container, and used for up to 6 hr after preparation. Aconventional gigaohm seal was formed by pressing the pipette gentlyagainst the myocyte and providing light suction. Spherically shapedmyocytes were selected.

For all other experiments, including hippocampal cells and non-neuronal cells, the conventional gigaohm seal whole-cell recordingmethod was used (Hamill et al., 1981). For experiments in which MEPC-like events were recorded from an isolated myocyte loaded intracellularlywith ACh (autoreception), the intrapipette solution containing (in mM)150 KCl, 1 NaCl, 1 MgCl2 , and 10 HEPES, pH 7.4, was supplementedwith 20 mM AChCl (Dan and Poo, 1992). The intrapipette solution usedfor recording from hippocampal neurons consisted of (in mM) 155K-gluconate, 1 MgCl2 , 10 HEPES, 1 sodium gluconate, 5 MgATP, 0.5NaGTP, and 0.1 leupeptin, pH 7.4. The internal solution in the patchpipette for fibroblasts contained (in mM) 105 K-gluconate, 1 Na-gluconate, 10 HEPES, 5 MgATP, and 0.5 NaGTP, pH 7.4, supplementedwith 50 mM AChCl. The detached fibroblast was patched before beingmanipulated into contact with an isolated Xenopus myocyte. The record-ing pipette for the myocyte contained (in mM) 150 KCl, 1 NaCl, 1MgCl2 , and 10 HEPES, pH 7.4.

In all experiments, currents were recorded at room temperature undervoltage-clamped conditions (holding voltage Vh 5 270 mV for myocytes,280 mV for fibroblasts, and 265 mV for hippocampal neurons), with anAxopatch 1D amplifier (Axon Instruments, Foster City, CA). The cur-rents were filtered at 1 kHz, digitized, and stored on videotape for laterplayback. Computer analysis was performed with the SCAN programkindly provided by Dr. J. Dempster (Strathclyde University, UK). Thisanalysis resulted in a series of exocytosis event times, which were quan-tized to 55 msec: multiple events falling within a single 55 msec windowwere registered as a single event. To guarantee sufficient statisticalaccuracy for estimating parameters of the fractal behavior, we retainedonly data sets with N $ 400 events for analysis.

RESULTSSpontaneous vesicular exocytosis from neuromuscularjunctions, hippocampal synapses, andnon-neuronal cellsWithin 1 d of plating, Xenopus embryonic spinal neurons establishfunctional synaptic transmission with cocultured myocytes, exhib-

iting stable spontaneous and evoked release characteristics(Chow and Poo, 1985; Xie and Poo, 1986). Spontaneous pulsatilemembrane currents resembling MEPCs (Fig. 1) developed in awhole-cell voltage-clamped myocyte within several minutes offorming a tight seal with a pipette containing normal intracellularsolution supplemented with nystatin. The configuration was per-mitted to stabilize for 10 min before experiments were begun.The observed current pulses were virtually identical in their rateand amplitude distributions to events recorded by the conven-tional whole-cell recording configuration.

These pulsatile current events represent spontaneous exocyto-sis of ACh-containing synaptic vesicles at the developing neuro-muscular junction, because their rate and amplitude distributionare not affected by the addition of TTX (data not shown; but seeXie and Poo, 1986). The large amplitude variability presumablyresults from immature filling of the synaptic vesicles (Evers et al.,1989). (Vesicles containing an unusually small amount of AChcould result in a current event with a magnitude that lies belowthe threshold of detectability of the recording system, therebyescaping detection.) In the low-density cultures we use, eachmyocyte is innervated by a single neuron. Thus, the neurotrans-mitter release that is detected arises from a single synapse, which,in general, comprises a number of individual release sites.

Similar spontaneous pulsatile inward currents were observedfrom isolated myocytes exogenously loaded with ACh (Dan andPoo, 1992). MEPC-like events appeared within several minutesafter establishing the whole-cell configuration. The average am-plitude and rate of these events increased with time thereafter,ceasing to exhibit systematic changes after ;10 min. Analysis ofspontaneous release from the cells was begun after a 15 minloading period to ensure that stability had been established. Inthis preparation the inward currents result from the spontaneous,quantal release of ACh packets from the myocyte and the subse-quent detection of this release by activation of its own surfaceACh receptors (autoreception).

Figure 1. Spontaneous ACh neurotransmitter secretion obtained from atypical developing Xenopus neuromuscular junction. The mean miniatureendplate current (MEPC) rate for this particular neuron is 0.55 events/sec. A, Inward MEPCs, shown as downward deflections, recorded from amyocyte by using a nystatin perforated-patch whole-cell voltage clamp.The MEPCs result from quantal ACh secretion from the spinal neuron.B, A section of the recording in A shown on a magnified time scale. C, Asection of the recording in B, which has been magnified further. D, Adifferential interference contrast photomicrograph of a typical neuron(N ) innervating a myocyte (M ).

Lowen et al. • Fractal Quantal Neurotransmitter Secretion J. Neurosci., August 1, 1997, 17(15):5666–5677 5667

Spontaneous quantal ACh secretion from exogenously loadedfibroblasts also was observed (Girod et al., 1995) with the help ofthe ACh detection system inherent in Xenopus myocytes. Whole-cell voltage-clamped recordings from a myocyte in contact withthe fibroblast displayed transient inward currents resemblingMEPCs. The frequency and amplitude of these events also in-creased gradually with time during a period of ;30 min. Theevents were recorded after this period to permit stability to beachieved. It appears that the secretion events result from thesequestration of ACh into vesicles and the subsequent exocytosisof those vesicles. It is likely that the vesicles are components ofthe constitutive endocytosis or membrane recycling pathways.

The analysis and comparison of the secretion in the three formsof vesicular release discussed above support the view that thebasic machinery for secretion, both in neuronal and non-neuronalcells, is relatively similar (Girod et al., 1995).

Inward currents also were observed in 7- to 14-d-old hippocam-pal neurons cultured at low density. These currents are also theresult of spontaneous neurotransmitter release, because they areapparent even in the presence of TTX. In this case, however, theneurotransmitter was most likely glutamate, because the additionof the AMPA receptor blocker CNQX and the NMDA receptorblocker AP-5 abolished the currents. Unlike the spontaneouscurrents observed at neuromuscular junctions in Xenopus cellculture, it is likely that these MEPCs arise from excitation byseveral presynaptic neurons. Even at low densities it is not pos-sible to trace exact pre- and postsynaptic neuronal cell pairs.Moreover, by 7 d in culture, an extensive network of neuritesforms, making it likely that each neuron receives input from amultitude of neurons.

We have analyzed the statistical patterns of the spontaneoussecretion events generated in all of these preparations and haveformulated a suitable model for the exocytic behavior: the fractallognormal-noise-driven doubly stochastic Poisson process(FLNDP). We begin by examining various statistical measures ofthe sequence of MEPCs observed in the Xenopus neuromuscularjunction.

Interevent-interval histogram (IIH)The solid curve in Figure 2 is a semilogarithmic plot of the MEPCIIH, arising from spontaneous vesicular release activity observedin a typical Xenopus neuromuscular junction. The IIH is a measureof the relative frequency of the times between successive events.The sequence of events from which this histogram was constructedhas a duration L 5 8164 sec and contains N 5 2644 intereventintervals, thereby exhibiting a mean interevent interval E[t] 5 3.09sec and a mean rate l 5 1/E[t] 5 0.324 events/sec.

Traditional mathematical descriptions of vesicular exocytosisgenerally assume that the MEPC sequence forms a renewalprocess. Renewal processes are memoryless; successive intervalsare all independent and are drawn from a single distribution.They are, therefore, characterized completely by the IIH, which isan estimate of the interevent-interval probability density p(t).

The simplest renewal model is the homogeneous Poisson pointprocess (HPP; Cox and Lewis, 1966). The HPP is characterizedby a single constant quantity, its rate l, which is the number ofvesicular release events expected to occur in a unit time interval.The HPP interevent-interval probability density function p(t)behaves as a decreasing exponential function p(t) 5 lexp(2lt),t $ 0, where t is the interevent interval and E[t] 5 1/l is the meaninterevent time. Incorporating the effects of dead time (absoluterefractoriness) or sick time (relative refractoriness) in the process

preserves the exponential tail of the interevent-interval distribu-tion while suppressing p(t) for shorter times. Because the expo-nential function plotted on semilogarithmic coordinates is astraight line, the HPP model clearly does not provide a good fit tothe experimental IIH presented in Figure 2. Cohen et al. (1974a)reached this same conclusion from their studies of the frogneuromuscular junction.

A reasonable fit can be obtained if a slightly more complexrenewal process, the gamma renewal process (GRP; Cox andLewis, 1966), is used. This is the approach taken by Hubbard andJones (1973). The theoretical IIH is then the two-parametergamma density function p(t) 5 t a21 exp(2t/t)/G(a)ta, where t is acharacteristic time, a is a parameter known as the order of thegamma process, and G(z) is the gamma function. The mean inter-event interval for this distribution is E[t] 5 at; its variance is Var[t]5 at2. (The particular case a 5 1 corresponds to the exponentialdistribution, illustrating that the HPP is a special case of the GRP.)The gamma IIH, with its mean and variance set equal to those ofthe data, is shown as the dashed curve in Figure 2. The fit is verygood. Thus, were the MEPC sequence renewal in nature, it wouldbe describable by a GRP, and nothing more need be said about it.

However, we demonstrate that the MEPC sequence is not re-newal using three statistical measures that are sensitive to thepresence of memory in a point process: vesicular release-ratemeasurements over different time scales, the Allan factor (AF),and the periodogram (PG). The dependencies among the inter-event intervals evidenced by these measures reveal that a fractal-rate stochastic point process (FRSPP) (Teich et al., 1996a; Thurneret al., 1997) represents the sequence of MEPCs. A fractal-rate

Figure 2. Semilogarithmic plot of the interevent-interval histogram (IIH)versus interevent-interval t for spontaneous vesicular release obtained froma Xenopus neuromuscular junction (solid curve). The sequence of eventsfrom which this histogram has been constructed has a duration L 5 8164sec and contains N 5 2644 interevent intervals, thereby exhibiting a meaninterevent interval E[t] 5 3.09 sec and a mean rate l 5 1/E[t] 5 0.324events/sec. The bin width is 0.275 sec or five times the clock period used toacquire the data. The best-fitting theoretical gamma density function(dashed curve, corresponding to the GRP) has parameters t 5 12.4 and a 50.249 (and therefore a mean interevent time E[t] 5 at 5 3.09 sec). Thesimulated IIH using the FLNDP model, which also has a mean intereventtime E[t] 5 3.09 sec, is shown as the dotted curve. Both models provide goodfits to the experimental IIH, but only the FLNDP leads to results thataccord with the statistical measures provided in Figures 3–5.

5668 J. Neurosci., August 1, 1997, 17(15):5666–5677 Lowen et al. • Fractal Quantal Neurotransmitter Secretion

process is required because the IIH does not follow a power-lawform and hence does not exhibit scaling. This indicates that thesequence of exocytic events itself does not form a fractal in time.However, the rate of event occurrences is consistent with scalingbehavior, leading to the FRSPP model for the MEPC sequence.

Indeed a particular FRSPP, the FLNDP, provides an excellentrepresentation for all of the statistical measures of the exocyticevents that we have investigated, as discussed at the very end ofthis section. The associated three-parameter simulatedinterevent-interval density function is displayed as the dottedcurve in Figure 2; it clearly provides an excellent fit to the IIH,even a bit superior to that of the GRP. Because the MEPCs donot form a renewal process, however, it is clear that the IIH aloneis inadequate for choosing among alternative models.

Self-similarity of vesicular release ratesPerhaps the simplest measure of neuronal activity is the estimateof the rate: the number of events registered per unit time (Teich,1992). For vesicular release, even this straightforward measure isconsistent with the presence of fractal properties; the magnitudeof the fluctuations of the rate (e.g., its standard deviation) de-creases more slowly, as the counting time used to compute itincreases, than would be expected for independent-event counts.

In Figure 3a we illustrate the vesicular release rate for the sameXenopus neuromuscular data as those analyzed in Figure 2. Twodifferent counting times were used to compute the rate: T 5 25sec (solid curve) and T 5 250 sec (dashed curve). The totalduration of the solid curve is 775 sec (31 consecutive samples,each of 25 sec), whereas that of the dashed curve is 7750 sec (31consecutive samples, each of 250 sec). Evidently, increasing theaveraging time by a factor of 10 reduces the magnitude of thefluctuations only slightly. This indicates that long-duration fluc-tuations are present in the train of vesicular release events,consistent with a fractal rate.

This behavior derives from dependencies among the intereventintervals, as confirmed by using a surrogate data set; the fractalproperties of the rate estimate are destroyed by shuffling (randomlyreordering) the intervals. This operation removes the dependen-cies among the intervals while exactly preserving the interevent-interval histogram. With all dependencies among the intervalseliminated by shuffling (aside from those inherent in retaining thesame IIH), the resulting surrogate essentially behaves as a renewalpoint process. The vesicular release rate for these same data, aftersuch shuffling, is illustrated in Figure 3b. The T 5 250 sec shuffleddata (dashed curve) now exhibits noticeably smaller fluctuationsthan does the T 5 25 sec shuffled data (solid curve). This morerapid reduction in the magnitude of the fluctuations with largeraveraging time is typical for nonfractal rates.

The quantitative behavior of the magnitude of the rate fluctu-ations with counting time is considered more conveniently interms of the AF, which is discussed next.

Power-law behavior of the AFA highly useful measure that is sensitive to correlations in a pointprocess is the AF, a relative variance based on a particularwavelet transform (Teich et al., 1996a). Its definition and prop-erties are presented in Appendix A. The AF calculated at aparticular counting time T provides a quantitative measure of thevariability exhibited by the rate estimates displayed in Figure 3a.For general well behaved processes, the AF A(T) is a function ofthe counting time T; the unique exception is the HPP, for whichA(T) 5 1 for all counting times T. Useful values of the countingtime T range from one-half of the minimum interevent interval toapproximately one-tenth of the duration of the recording.

The solid curve in Figure 4 is the AF for the same sequence ofXenopus neuromuscular junction MEPCs, the IIH of which isshown in Figure 2 and the rate functions of which are shown inFigure 3. The AF is seen to increase steadily for counting timesgreater than ;10 sec, exceeding a value of 100 at a counting timeT 5 400 sec. For sufficiently large counting times, the AF is wellapproximated by a straight line on this doubly logarithmic plot, sothat it is well fit by an increasing power-law function of thecounting time, A(T) } TaA, with aA ;1.5 for this particularjunction. A monotonic, power-law increase indicates the presenceof fluctuations on many time scales. The quantity aA is identifiedas an estimate of the fractal exponent of the point process(Lowen and Teich, 1995; Thurner et al., 1997).

A plot of the AF alone does not reveal whether its substantialmagnitude arises from the distribution of the interevent intervals(the IIH) or from their ordering. This issue is addressed byplotting the AF for the shuffled intervals. AFs constructed fromshuffled data retain information about the relative sizes of theintervals, but all correlations and dependencies among the inter-vals are destroyed by the shuffling process, as discussed earlier.The curves designated “shuffled data” in Figure 4 illustrate theAFs obtained by this method; the long dashes indicate the meanvalue obtained from 100 shufflings, and the short dashes delineatea range of 6 1 SD about this mean value. The lack of observedfractal behavior in the AFs of the shuffled data indicates that it isthe ordering of the intervals that gives rise to the power-lawgrowth of the AF for those records. This confirms that theoriginal data cannot be modeled as a renewal process.

The long-dashed curve behaves very much like the AF for aGRP, approaching a maximum value close to 1/a 5 4.01 for largecounting times, as expected from theory (Teich et al., 1997).Indeed, AF simulations for the GRP that best fits the IIH (long-

Figure 3. Semilogarithmic plot of the rate estimates for original andshuffled data. a, Rate of spontaneous vesicular exocytosis for the same dataset illustrated in Figure 2. Two different counting times were used tocompute the rate: T 5 25 sec (solid curve) and T 5 250 sec (dashed curve).The fluctuations in the estimate do not diminish appreciably as the aver-aging time increases, although some reduction must occur. b, The sameplots computed after shuffling (randomly reordering the intervals of) thedata shown in a. The fluctuations diminish much more rapidly with increas-ing averaging time. In all cases the mean rate l 5 1/E[t] ;0.3, as expected.


dashed curve in Fig. 2) closely resemble the long-dashed curve inFigure 4. It is clear, therefore, that the GRP provides a good fit tothe AF only for the shuffled data and therefore cannot possiblydescribe the original (unshuffled) data. In contrast, the simulatedFLNDP AF shown in Figure 4 (dotted curve) follows the originalAF much more closely; moreover, a shuffled version of theFLNDP also provides a good fit to the shuffled-data AF (shuffledsimulation not shown).

Power-law behavior of the PGFractal variability of the exocytic events also manifests itself inother statistical measures, perhaps the most familiar of which isthe PG, which is an estimator of the power spectral density. Muchas for continuous-time processes, the power spectral density com-puted for the exocytic events reveals how power is concentratedin various frequency bands. In Figure 5 we present a doublylogarithmic PG plot for the same Xenopus data sequence (solidcurve) examined in Figures 2–4. For low frequencies f (corre-sponding to long time scales T), the PG is well fit by a decreasingpower-law function of the frequency, S( f) } f2aS. Thus, MEPCactivity exhibits 1/f-type noise. The quantity aS provides analternative means of estimating the fractal exponent of the vesic-ular exocytosis process. For this particular neuron, the PG yieldsaS ;1.6, which is in close accord with the value aA ;1.5 obtainedfrom the AF, as expected (Teich et al., 1996a; Thurner et al.,1997).

The PG computed from a shuffled version of the data (dashedcurve), in contrast, is quite flat at low frequencies, providing

further evidence that it is the ordering of the intervals, ratherthan their relative magnitudes, that is responsible for the fractalaspects of the rate of vesicular activity. Again, the simulatedFLNDP PG shown in Figure 5 (dotted curve) provides a good fitto the experimental PG, and shuffled versions of the FLNDP alsolead to results that accord with the shuffled-data PG (shuffledsimulations not shown).

Because the PG is the Fourier transform of the joint coinci-dence rate (a measure of correlation used for a process of events),the results presented here are not inconsistent with those ob-tained by Rotshenker and Rahamimoff (1970; their Fig. 1), whoshowed excess correlation to 5 sec in preparations subjected toextracellular Ca21 levels above their normal values. Evidence forexcess power at low frequencies in the absence of elevated Ca 21

also was provided by Cohen et al. (1974b). In our case, however,the power-law form for the PG shown in Figure 5 reaches downto ;1.24 3 1024 Hz, indicating that excess correlation extends toat least 8164 sec (the reciprocal of 1.24 3 1024), which is the fulllength of the data set. We see evidence of power-law behavior inthe PG of data from all of the preparations we have examined,including the Xenopus neuromuscular junction (with and withoutadded KCl).

Alteration of the exocytosis pattern induced bydepolarization with KClThe addition of KCl to the bath solution depolarizes the nerveterminal, thereby resulting in an elevation of the cytosolic Ca21

Figure 4. Doubly logarithmic plot of the Allan factor (AF) versus count-ing time T for the same vesicular exocytosis data (solid curve) that wereexamined in Figures 2 and 3. For counting times larger than ;10 sec, thiscurve approximately follows a straight line, which represents a fractionalpower-law increase of the AF with T on these doubly logarithmic coor-dinates. The fractal exponent aA ;1.5 was estimated by using a least-squares fit of the functional form A(T) 5 C 1 (T/To )aA, in which C is aparameter related to the refractoriness, and To is a fractal onset time(Lowen and Teich, 1997). The range of counting times used to obtain thefit was L/10 4 # T # L/10, in which L is the duration of the recording. Thelong-dashed curve represents the AF after the data were shuffled 100 timesand the AFs computed for each shuffling were averaged; also displayedare the 6 1 SD limits about the AF for the shuffled data (medium-dashedcurves). The simulated FLNDP AF (dotted curve), using parametersidentical to those used in Figure 2, agrees well with the experimental data(solid curve); the shuffled version of the FLNDP agrees well with theshuffled data (shuffled simulation not shown).

Figure 5. Doubly logarithmic plots of the periodogram (PG) for thesame vesicular exocytosis data (solid curve) examined in Figures 2–4. ThePG is an estimate of the power spectral density of the vesicular releaseactivity. For sufficiently low frequencies, the PG approximately follows astraight line on these coordinates, representing a fractional power-lawdecreasing function of f. The data were divided into NFFT 5 4096 equallyspaced adjacent bins, and the number of MEPC events registered in eachbin was recorded. A PG was computed for this sequence of counts.Smoothing was achieved by averaging PG values corresponding to fre-quencies within a factor of 1.02. For this particular junction the PG yieldsaS ;1.6, which is in close accord with the value aA ;1.5 obtained from theAF. The fractal exponent aS was estimated over the frequency range1/L # f # 10 3/L, using a procedure similar to that used to estimate aAfrom the AF, as described in the caption of Figure 4. The long-dashedcurve represents the PG of the shuffled data; it shows no such power-lawbehavior. The FLNDP PG simulation results (dotted curve), using param-eters identical to those used in Figure 2, agree well with the data, andshuffled versions agree with the shuffled data (shuffled simulations notshown).


concentration and a consequent increase in the rate of spontane-ous vesicular exocytosis (Katz, 1962). Normalized IIHs for theXenopus neuromuscular junction in the presence of added KClare shown in Figure 6a (dashed curve corresponds to 10 mM KCl;dotted curve corresponds to 20 mM). The IIH in the absence ofadded KCl, which is a normalized version of the solid curve inFigure 2, is presented for purposes of comparison (solid curve).Normalized IIHs were used to facilitate direct comparison of thecurves. Each recording was obtained from a different neuromus-

cular junction because long data sets are required to obtainaccurate statistics, and individual preparations typically do notremain viable long enough to permit more than a single recordingto be obtained from a given preparation. The IIH curves allfollow an exponential decay for long interevent intervals; how-ever, the value near t 5 0 is suppressed in the presence of KCl,and the coefficient of variation for the intervals is decreased.

The same data used to construct the IIHs in Figure 6a wereused to generate the AF curves in Figure 6b. The three AFs showevidence of power-law behavior at long counting times, althoughthe minimum value of the AF is reduced under stimulation. Theincreased rate of vesicular release serves to regularize the processand thereby leads to a reduction of the Allan variance over timescales where refractoriness is operative (Lowen and Teich, 1997).The presence of KCl also seems to reduce the strength of thelong-term correlation present in the data (see also Cohen et al.,1974b), thus increasing the counting time at which power-lawbehavior becomes apparent.

Data selection: dilution of fractal-rate behaviorFatt and Katz (1952) were the first to investigate the statisticalbehavior of sequences of MEPCs, finding that the IIH wasexponentially distributed. For a renewal process, this implies thatthe sequence can be described by a memoryless HPP. The IIHsmeasured in subsequent studies often have been variants of theexponential, which sometimes has fostered the (erroneous) no-tion that the event sequences are describable by processes akin tothe Poisson, such as the gamma-renewal (Hubbard and Jones,1973) or dead-time-modified Poisson (Vere-Jones, 1966)processes.

Fatt and Katz (1952) were very careful to note that the segmentof data they selected for analysis (their Figs. 11–13) was suffi-ciently short (duration L 5 176.8 sec comprising N 5 800MEPCs; E[t] ;0.221 sec) so as to exclude, as they put it, the“occasional occurrences of short high-rate bursts” of events, andto avoid “progressive changes of the mean,” present in their data.The observation of fractal-rate behavior requires long data sets,and burstiness and apparent trends are at its very core, existing asnatural components of exocytic behavior. We therefore wouldlike to believe that the MEPCs observed by Fatt and Katz indeeddid exhibit fractal-rate fluctuations but that these researchersremoved most traces of it by selecting relatively short segments ofdata for analysis and moreover by choosing precisely those seg-ments that exhibited minimal fluctuations.

Indeed, an analysis by Cox and Lewis (1966, page 220) of eventhe special segment selected by Fatt and Katz reveals a departurefrom Poisson behavior that takes the form of a “relatively long-term effect.” Similar conclusions were reached by Cohen et al.(1973, 1974a,b) and by Van der Kloot et al. (1975).

We proceed to explicitly demonstrate the consequences ofselecting such segments of data with our own measurements. Wechoose the 20 mM KCl data set (16358 events; duration 4451 sec;E[t] ;0.272 sec) because it has a large number of events and itsrate is comparable with that of Fatt and Katz’s classic data set.

The PG for our full data set is displayed as the solid curve inFigure 7. We now select a segment from the center of the full dataset (N 5 800 events, excising both the 7779 events preceding itand the 7779 events after it; L 5 162 sec; E[t] ;0.203 sec) that iscomparable with the segment analyzed by Fatt and Katz innumber of events, data duration, and mean interevent interval.The PG for this truncated MEPC segment is shown as the dashedcurve in Figure 7. Because of its limited length, it is clear that the

Figure 6. Effects of KCl-induced depolarization on Xenopus neuromus-cular junction activity. a, Normalized interevent-interval histograms(IIHs) with various levels of KCl in the bath solution. To facilitatecomparison among the IIH plots, we normalized the interevent intervalsfor each data set to unity mean before we computed the histograms. TheIIH obtained with no added KCl, shown in Figure 2, is replotted innormalized form (solid curve) for the purposes of comparison. The meaninterevent intervals were E[t] 5 3.09, 0.624, and 0.272 sec, respectively, for0, 10, and 20 mM added KCl. Although the shapes of the IIH plots aresimilar for all three levels of KCl, the probability density in the vicinity oft 5 0 is reduced in the presence of this agent; there is a concomitantdecreased coefficient of variation for the intervals. b, AF plots for thesame three data sets used in a. The curve in the absence of KCl isreplotted from Figure 4 for the purposes of comparison (solid curve). TheAF plots for all three KCl levels show an increase at larger counting times,albeit starting at different values of the abscissa.


lowest frequency available to this PG is fmin 5 1/L 5 1/162 ;6 31023 Hz, the left-hand endpoint of the dashed curve. Althoughthe agreement of the two curves is reasonable over the rangewhere they coexist, it is plain that the fractal behavior in the fulldata set cannot be accessed in the truncated version. Similarly, theAF of the truncated MEPC data set cannot be estimated reliablyfor T . 16 sec, assuming that a minimum of 10 samples isrequired for statistical accuracy. The dotted curve in Figure 6breveals that the AF begins to depart from simple renewal behav-ior only for counting times .16 sec. Moreover, selecting a par-ticular short segment of data on the basis of lack of variability (i.e.,lack of burstiness or lack of progressive changes of the mean)serves to reduce further any manifestations of fractal-ratebehavior.

We conclude that fractal-rate behavior, although it well mayhave been present in the original data set collected by Fatt andKatz, could not be discerned in the 176.8 sec segment that theyanalyzed.

Power-law behavior of the AF for rat hippocampalsynapses, Xenopus myocytes, and rat fibroblastsAlthough our attention thus far has been directed principallytoward the Xenopus neuromuscular junction, we also have ob-served vesicular exocytosis consistent with fractal-rate behaviorfrom other neuronal and non-neuronal preparations, as men-tioned earlier. Figure 8 displays the AFs for spontaneous vesicu-lar release from two neuronal and two non-neuronal cells: theXenopus neuromuscular junction (solid curve; reproduced fromFigs. 4, 6b), the rat hippocampal synapse in the presence of TTX(long-dashed curve), Xenopus–myocyte autoreception (short-dashed curve), and the exogenously loaded rat fibroblast broughtinto synaptic contact with a Xenopus myocyte (dotted curve). Thecurves presented in Figure 8 are representative of the data setsthat were sufficiently long to merit analysis (N $ 400); these

comprise four Xenopus neuromuscular junctions, four hippocam-pal synapses, one Xenopus myocyte, and two fibroblasts.

For large counting times, all of the eleven vesicular-exocytosisdata sets examined to date exhibit AFs that increase with thecounting time, in a form consistent with the presence of fractalbehavior. Fractal exponents estimated from the AF plots were inthe range aA 5 0.1–2.7 (mean 5 1.23), whereas those from the PGplots were in the range aS 5 0.2–4.0 (mean 5 1.79). The fractalexponents calculated from the AF and PG were in general agree-ment (Teich et al., 1996a; Thurner et al., 1997); the overallcorrelation coefficient was 10.62, with substantially superioragreement for the longer data sets. Improved correlation would,no doubt, be obtained if the recordings were of yet longer dura-tion so that asymptotic power-law behavior could be attained.Nevertheless, fractal-rate activity seems to be ubiquitous in exo-cytic events.

Biophysical origin of the fractal behaviorA number of possible origins exist for the observed fractal be-havior. One plausible scenario is that exocytosis is governed byfractal Ca21-ion channel activity and that this activity ultimatelyderives from 1/f-type fluctuations of the membrane voltage. Weproceed to provide a biophysical description of this process. Themathematical formulation, which leads to the FLNDP, is devel-oped in Appendix B.

It generally is accepted that voltage-gated Ca21-ion-channelopenings are responsible for vesicular exocytosis (Zucker, 1993).For a fixed membrane voltage V near the resting potential, cal-cium flow is negligible. Occasionally, however, random thermallyinduced channel openings occur, which often lead to spontaneousexocytic events for nearby vesicles. Such spontaneous behavior isalmost completely memoryless and is therefore well modeled byan HPP, with rate l given by the Arrhenius equation (Berry et al.,1980) l 5 !exp{2[EA 2 zFV]/R7}, as given in Equation B1.Here ! is a rate constant (often called the frequency factor), EA

is the constant activation energy associated with the ion-channel

Figure 7. Doubly logarithmic plot of the PG for the entire 20 mM KCldata set (solid curve). The IIH for this data set is shown in Figure 6a(dotted curve), and the AF is shown in Figure 6b (dotted curve). The datawere divided into NFFT 5 4096 equally spaced adjacent bins, and thenumber of MEPC events registered in each bin was recorded. A PG wascomputed for this sequence of counts. Smoothing was achieved by mul-tiplying the estimated correlation function by a triangular window. Thedashed curve represents the smoothed PG for a truncated segment of thedata, which shows no such power-law behavior. Short data sets generallydo not exhibit fractal behavior.

Figure 8. Comparison of AFs for four biological preparations exhibitingin vitro fractal spontaneous vesicular release: the Xenopus neuromuscularjunction (solid curve, reproduced from Figs. 4, 6b; E[t] ;3.09 sec), a ratcultured hippocampal synapse (long-dashed curve; E[t] ;1.95 sec), Xeno-pus myocyte autoreception (short-dashed curve; E[t] ;3.44 sec), and a ratfibroblast (dotted curve; E[t] ;12.1 sec). For longer counting times, all ofthe AFs suggest the presence of a power-law increase.


opening, z is the valence of the charge involved in the channelopening, F is the Faraday constant (coulombs/mol), R is thethermodynamic gas constant, and 7 is the absolute temperature.Channel-opening events that do not lead to exocytosis are ac-counted for in the values of ! and EA. According to this picture,different fixed membrane voltages V lead to spontaneous exocyticpatterns that differ only in their average rates; all are HPPs.These rates are exponential functions of the membrane voltage,as prescribed by the Arrhenius equation. Indeed, the low-variability sections of data selected for analysis by Fatt and Katz(1952) likely would be associated with regions for which the rateis relatively constant so that the sequence of events could be wellapproximated by an HPP, as they found.

However, the membrane voltage is not fixed but, rather, variesrandomly in time. Denoted V(t), it has a gaussian amplitudedistribution and a 1/f-type spectrum (Verveen and Derksen,1968). The rate l(t) of the Poisson process, therefore, also variesin time, as described by Equation B2, which shows that the rate isthe exponential transform of the voltage. Because the latter has agaussian (normal) amplitude distribution, the rate is described bya lognormal amplitude distribution (Saleh, 1978) with a closelyrelated spectrum. (A lognormal random variable is one for whichthe logarithm is normally distributed.) The rate process thereforeis called fractal lognormal noise (FLN) and is described inAppendix B. In short, the channel openings are characterized bya doubly stochastic Poisson process (Cox and Lewis, 1966) with arate that is FLN: the FLNDP.

The net result is that the calcium-flow events, and therefore theexocytic events, are described by the FLNDP, the properties ofwhich are provided in Appendix B. Unlike the HPP, the FLNDPhas memory. Thus, the fluctuating membrane voltage impartsfractal correlations to the rate of exocytic events so that theobservation of a short (long) interevent interval, for example,signifies a locally high (low) rate l(t), which in turn indicates thatthe next interevent interval is also likely to be short (long). TheFLNDP is clearly a nonrenewal process. However, the sequenceof events generated by the FLNDP model does not form a properfractal in time because it does not itself scale. Rather, the rate ofevent occurrences scales so that the FLNDP belongs to the familyof fractal-rate stochastic point processes.

Analytical predictions and computer simulations based on thismodel were compared with the exocytic-event data for a variety ofstatistical measures. We performed 100 simulations of theFLNDP, using parameter values obtained from the data set thatis displayed in Figures 2–5. IIH plots were computed for eachsimulation; these differed only in the seeds used for the randomnumber generator. These 100 plots were averaged together toyield an aggregate IIH plot. This same process was used toconstruct AF and PG plots, using the same simulations (andtherefore the same parameters and random seeds).

The resulting theoretical curves, denoted FLNDP, are pre-sented as the dotted curves in Figures 2, 4, and 5. The theo-retical results shown in all of these figures used a single set ofparameters derived from the experimental data. Agreementwith the data is excellent over all time scales. The slightdeviations between the behavior of the FLNDP and the ex-perimental data evidenced in the AF (Fig. 4) and in the PG(Fig. 5) would diminish no doubt were refractoriness includedin the simulation. Indeed, it is known that refractoriness pro-duces a dip in the AF for counting times in the vicinity of therefractory period (Lowen and Teich, 1997), the very regionwhere the agreement is least satisfactory in Figure 4. More-

over, the AF and PG calculated using shuffled FLNDP simu-lations are in excellent accord with those of the shuffled data(long-dashed curves) in Figures 4 and 5, respectively. We con-clude that, aside from its physiological plausibility, the FLNDPprovides an excellent mathematical model for characterizingsequences of MEPCs observed in our experiments.

It is, however, possible that the fractal-rate behavior mani-fested in exocytic data derives from other mechanisms. Wideranges of conformational states and time scales seem to beubiquitous in large proteins (Liebovitch and Toth, 1990) sothat fractal-rate exocytic behavior could originate from fractalbehavior of the specialized proteins directly involved in vesic-ular exocytosis rather than via mediation by calcium. Or,fractal Ca 21-ion-channel openings could lead to average in-tracellular Ca 21-ion concentrations that behave in a fractalmanner, in turn modulating global docking- and transport-protein behavior (Zucker, 1993). Even in these cases, however,the FLNDP model would provide a useful mathematical de-scription of the vesicular release process, albeit with a differentbiological interpretation.

DISCUSSION

As indicated above, activity consistent with fractal-rate behavioris present in every in vitro spontaneous vesicular-secretion prep-aration of sufficient length (N $ 400 events) that we have exam-ined, neuronal and non-neuronal alike. Because vesicular exocy-tosis at the synapse shares many features and proteins in commonwith exocytosis and intracellular trafficking in all eukaryotic cells(Bennett and Scheller, 1993), it may be that such behavior ispresent in these systems as well.

Fractal and fractal-rate behavior are also present in excitable-tissue recordings for various biological systems in vivo, from themicroscopic to the macroscopic (Bassingthwaighte et al., 1994;West and Deering, 1994). Examples include the openings andclosings of ion channels (Lauger, 1988; Millhauser et al., 1988;Liebovitch and Toth, 1990; Teich et al., 1991; Lowen and Teich,1993a–c); patterns of action-potential firings in the auditory sys-tem (Teich, 1989, 1992; Teich et al., 1990; Powers and Salvi, 1992;Kumar and Johnson, 1993; Kelly et al., 1996; Lowen and Teich,1996, 1997), visual system (Turcott et al., 1995; Teich et al.,1996a,b, 1997), somatosensory cortex (Wise, 1981), and mesen-cephalic reticular formation (Gruneis et al., 1993); and even thesequence of human heartbeats (Kobayashi and Musha, 1982; Saulet al., 1988; Turcott and Teich, 1993, 1996). In almost all of thesecases, the upper limit of the observed time over which fractalcorrelations exist is imposed by the duration of the recording.The appearance of fractal-rate behavior at synapses, as well as insystems comprising collections of synapses, indicates that suchbehavior is ubiquitous in neural signaling.

The connection between fractal-rate fluctuations and informa-tion encoding and transmission in neurons, if there is one, re-mains unclear. Fractal noise exhibits larger fluctuations at lowerfrequencies and thereby generally renders difficult the detectionof the slowest, most gradual changes in a signal. Thus fractalexocytic activity could represent a fundamental source of noiseubiquitous in living cells, to which natural systems must adapt.However, many natural signals are themselves fractal (Voss andClarke, 1978), and it may be that fractal activity in neuronsprovides some advantages in terms of matching the detectionsystem to the expected signal (Teich, 1989, 1992).

Fractal-rate activity also represents a form of memory, because


the occurrence of an event at a particular time increases thelikelihood of another event occurring later, with the strength ofthis memory persisting for some time. Fractal-rate synaptic ac-tivity therefore provides a distributed network for memory andmay provide a mechanism for potentiation.

Although it is difficult to ascribe definitively the observedlong-term correlation in neurotransmitter release events tofractal-rate behavior for a single data set of the limited sizeavailable in these experiments, it is gratifying that the FLNDPmodel, which relies on only three free parameters (interevent-interval mean and variance, and fractal exponent), provides goodagreement with the exocytic data. That activity consistent withfractal-rate behavior exists in all of the 11 data sets that we haveexamined indicates that the FLNDP is a useful model for de-scribing our observations. Further confidence in the use of thismodel is engendered by the success of its close relative, therefractoriness-modified fractal binomial-noise-driven doubly sto-chastic Poisson process (FBNDP), for modeling action-potentialactivity of primary afferent auditory-nerve fibers in the cat (Lo-wen and Teich, 1995, 1996; Thurner et al., 1997). For voltagefluctuations small enough so that the exponential transformin Equation B2 can be approximated by a linear function, theFLNDP becomes nearly the same as the FBNDP. Moreover, theFLNDP and another related process, the fractal binomial-noise-driven doubly stochastic gamma process (FBNDG), have beenused successfully to model retinal-ganglion-cell and lateral-geniculate-cell action-potential activity in the visual system of thecat (Teich et al., 1997).

In conclusion, it is clear that traditional renewal models treat-ing vesicular exocytosis as a memoryless stochastic process arewholly inadequate for representing many of its salient features.Rather, a new class of models that rely on fractal-rate stochasticpoint processes is required.

APPENDIX A: DEFINITION AND PROPERTIESOF THE AFThe Allan factor (AF) is defined as the ratio of the Allan varianceto twice the mean of the event count (Teich et al., 1996a). TheAllan variance, in turn, is the average variation in the differenceof adjacent counts (Allan, 1966). To compute the Allan varianceat a specified counting time T, the data record of duration L firstis divided into L/T contiguous counting windows, each of dura-tion T. Much as in the procedure used to calculate the rateestimate, the number of events Zk(T) falling within the k-thwindow is registered for all indices k corresponding to windowslying entirely within the data record. The difference between thecount numbers in a given window [i.e., Zk(T)] and the one afterit [Zk11(T)] is then computed for all k. The mean square of thisquantity, E{[Zk11(T) 2 Zk(T)]2} , is the Allan variance. Dividingthe Allan variance by twice the mean yields the AF:

A~T! 5E$@Zk11~T! 2 Zk~T!#2%

2 E@Zk~T!#. (A1)

This process is performed for a set of different counting times T(leading to different sequences of counts {Zk(T)}), to generateplots of the functional form of the AF versus counting time. Forgeneral sequences of events, A(T) varies with T; the uniqueexception is the HPP, for which A(T) 5 1 for all T. Sequences ofevents also may be represented by a counting process, N(T), equalto the number of events registered between the time origin and atime t, so that:

Zk~T! 5 N@~k 1 1!T# 2 N~kT!. (A2)

For an arbitrary sequence of events, useful values of the count-ing time T typically range from one-half the minimum intereventinterval, below which A(T) 5 1, to approximately one-tenth theduration of the recording (L/10), above which the statisticalaccuracy becomes poor as a result of an insufficient number ofsamples L/T. The behavior of the AF plot at various interveningtime scales reveals important information about the behavior ofthe underlying process. One example is provided by absolute orrelative refractoriness, which, if present, causes a dip in the AFplot for counting times near (and somewhat larger than) therefractory period. This arises because refractoriness imposes aminimum spacing between events, which serves to regularize thenumbers of events Zk(T) in each counting window. This, in turn,reduces the Allan variance, thereby leading to a diminished AF inthe vicinity of those counting times.

An increase in the AF near a specific time scale occurs ifevent clusters of that particular scale are present in the data;the AF then will reach a plateau beyond the largest time scalepresent. Such behavior would be manifested, for example, by aBartlett–Lewis cluster process such as that used by Cohen et al.(1974b) to model the MEPC sequence in the frog neuromus-cular junction. A fractal-rate stochastic point process, in con-trast, generates a hierarchy of clusters of different durations,which leads to an AF plot that continues to rise as each clustertime scale is incorporated in turn. The net result is an AF thatrises in power-law manner with increasing counting time T(straight line on a doubly logarithmic plot). For such processesthe AF begins to rise above its asymptotic value of unity at acounting time that depends on the relative strength of thefractal component of its rate.

It is important to note that the random fluctuations inherent inany finite data set lead to AF plots that exhibit variability aboutthe values predicted for exactly defined (nonrandom) point pro-cesses (Thurner et al., 1997). For data sets of sizes comparablewith those used in this paper, such fluctuations can prove signif-icant for any single plot, and conclusive proof of fractal behavioris not always possible. However, given a number of data sets ofthis size, uncertainty is greatly reduced. A similar argumentapplies for the PG.

The AF is preferred to the count index of dispersion (Fanofactor), a similar measure constructed from the ordinary vari-ance–time curve, for the analysis of fractal-rate stochasticpoint processes because of its greater generality and freedomfrom bias (Lowen and Teich, 1996; Teich et al., 1996a; Thurneret al., 1997). A particular advantage of the use of the AF (or,equivalently, the Allan variance–time curve, which containsthe same information) over the ordinary variance–time curve(Cox and Smith, 1953; Cohen et al., 1974b) lies in its insensi-tivity to linear trends, a result of the fact that it relies on afirst-order difference. This is a very important feature, whichstems from the close relation of the AF to wavelet theory and,in particular, to the Haar wavelet. Generalizations of the AF,based on other wavelets, are insensitive to higher order trends(Teich et al., 1996a).

APPENDIX B: MATHEMATICAL FORMULATION OFTHE FLNDPFor a membrane voltage near the resting potential, calcium-ionflux into the cell generally remains well below the value neededto trigger an exocytic event (Zucker, 1993), because voltage-


gated Ca 21-ion channels tend to remain closed under theseconditions. Random thermal fluctuations, however, occasion-ally cause a few nearby channels to open in the same timeframe, although the membrane voltage greatly favors theclosed state. It turns out that only a few such channels needopen to provide sufficient calcium to trigger vesicular exocyto-sis (Zucker, 1993). Were the membrane voltage fixed, thisspontaneous behavior would be essentially memoryless; knowl-edge of all previous event occurrence times would yield noadditional information about the future beyond that given bythe average rate l of the events. The mathematical model fordiscrete events in this case is the HPP, which is specified by asingle parameter: the expected rate l of nearby, nearly simul-taneous Ca 21-ion-channel openings and, therefore, of the as-sociated exocytic events.

Transition-state theory (Berry et al., 1980) describes the de-pendence of this expected rate l on various parameters of the ionchannels (Hille, 1992) and predicts that it is given by the Arrhe-nius equation:

l 5 !exp$2@EA 2 zFV #/R7%. (B1)

Here ! is a rate constant (often called the frequency factor), EA isthe constant activation energy associated with the ion-channelopening, z is the valence of the charge involved in the channelopening, F is the Faraday constant (coulombs/mol), R is the ther-modynamic gas constant, and 7 is the absolute temperature. Thatsome channel-opening events do not lead to exocytosis is incorpo-rated into the values of ! and EA. Different fixed membranevoltages V lead to spontaneous exocytic patterns that differ only intheir average rates; all are HPPs. These rates are exponentialfunctions of the membrane voltage, as prescribed by Equation B1.

However, the resting voltage V of an excitable-tissue membraneis not fixed but, rather, exhibits fractal (1/f-type) fluctuations witha gaussian amplitude distribution (Verveen and Derksen, 1968).We therefore replace V by V(t) in Equation B1 to accommodatethese voltage fluctuations:

l~t! 5 !exp$2@EA 2 zFV~t!#/R7%. (B2)

For a stationary process with a gaussian amplitude distribution,the mean and spectrum suffice to describe the process completely(Saleh, 1978). The power spectral density SV( f) for a 1/f-typeprocess follows the form:

SV ~ f ! 5 c/f a (B3)

over a range of frequencies for some positive constants c and a. Arate function with a power spectral density of the form of Equa-tion B3 is fractal, because changing the frequency (or time) scaleis tantamount to changing the amplitude. For example, replacingf by 2 f yields the same result as changing c to 22ac.

Thus, three parameters completely describe the membranevoltage: mV [ E[V ], c, and a. (The membrane voltage variance,sV

2 [ Var[V ], is expressible in terms of these three parameters.)We could, alternatively, have used a formulation in terms of theautocorrelation function RV(t) [ E[V(t)V(t1t)], which is theFourier transform of the power spectral density.

Because the voltage is gaussian (normal), and the rate is theexponential transform of the voltage in accordance with EquationB2, the rate has a lognormal amplitude distribution [the expo-nential of a gaussian is defined to be lognormal (Saleh, 1978)].The random process describing the rate l(t) therefore is given theappellation fractal lognormal noise (FLN); its amplitude distri-

bution is lognormal, and its spectrum is derived from 1/f-typenoise. The mathematical model for the discrete events in this caseis therefore a Poisson process with a rate described by FLN, i.e.,the FLNDP.

It remains for us to determine the relevant statistics of l(t) forthe FLN described by Equation B2. These, in turn, will be used todetermine the statistics of the exocytic events themselves, whichare described by the FLNDP. We proceed by defining an auxil-iary process X(t), which is a normalized version of the membranevoltage V(t):

X~t! ; @ zFV~t! 2 EA#/R7, (B4)

so that the rate in Equation B2 becomes:

l~t! 5 !exp@X~t!#. (B5)

X(t) also has a gaussian amplitude distribution, with associatedmean mX [ E[X] 5 (zFmV 2 EA)/R7, variance sX

2 [ Var[X] 5(zF/R7)2 sV

2 , and autocorrelation function RX(t) 5 mX2 1 (zF/

R7)2[RV(t) 2 mV2 ]. Straightforward application of probability

theory yields the moments of the rate:

E@ln# 5 !nE$exp@nX~t!#%

5 !n~2ps X2 !21/2 E

2`

`

exp@nx2~ x 2 mX!2/ 2s X2 # dx

5 !nexp~nmX 1 n2sX2 / 2! (B6)

by completing the square. In particular, the rate l(t) has a meanE[l] 5 !exp(mX 1 sX

2 /2) and a variance Var[l] 5 !2 exp(2mX)3 [exp(2sX

2 ) 2 exp(sX2 )].

We now turn to the autocorrelation function of the rate:

Rl~t! ; E@l~t!l~t 1 t!# 5 !2E$exp@X~t! 1 X~t 1 t!#%.(B7)

To proceed, we split X(t 1 t) into two portions, one of which isproportional to X(t) and the other uncorrelated with it. Wetherefore write:

X~t 1 t! 5 rX~t!@X~t! 2 mX# 1 mX 1 Y~t, t!, (B8)

in which the correlation coefficient rX(t) is given by:

rX~t! ; @RX~t! 2 mX2 #/s X

2 , (B9)

and where Y(t,t) is defined implicitly by Equation B8. Since:

RX~t! 5 E@X~t! X~t 1 t!#

5 rX~t!$E@X2~t!# 2 mX2 % 1 mX

2 1E@X~t!Y~t,t!#

5 RX~t! 1 E@X~t!Y~t,t!#, (B10)

we see that E[X(t)Y(t 1 t)] 5 0 so that X(t) and Y(t,t) areuncorrelated and Y(t,t) has zero mean. Because X(t) is a gaussianprocess and Y(t,t) is linearly related to it, it is apparent that Y(t,t)is also gaussian and, in fact, is jointly gaussian with X(t). There-fore, X(t) and Y(t,t) are independent (Saleh, 1978). RearrangingEquation B8 leads to the variance of Y(t,t):

E@Y 2~t,t!# 5 E@$rX~t!@X~t! 2 mX#2@X~t 1 t! 2 mX#%2#

5s X2 @1 2 rX

2 ~t!#. (B11)


We now use conditional expectation to determine the autocorre-lation function of l(t):

Rl~t! ; E@l~t!l~t 1 t!#

5 !2E$exp@X~t! 1 X~t 1 t! # %

5 !2E@exp$@1 1 rX~t!#X~t! 1 @1 2 rX~t!#mX 1 Y~t,t!%#

5 !2E@exp$@1 1 rX~t!#X~t!%#exp$@1 2 rX~t!#mX%E@exp$Y~t,t!%#,

(B12)

where the independence of X(t) and Y(t,t) permits the replace-ment of the expectation of the product by the product of theexpectations. Using a relation analogous to that used in EquationB6 to evaluate the first and third factors on the right-hand side ofEquation B12 yields the final result:

Rl~t! 5 !2exp$@1 1 rX~t!#mX 1 @1 1 rX~t!# 2s X2 / 2%

3exp$@1 2 rX~t!#mX% 3 exp$@1 2 rX2 ~t!#s X

2 / 2%

5 !2exp$2mX 1 @1 1 rX~t!#s X2 %

5 E2@l#exp@RX~t! 2 mX2 #

5 E2@l#exp$~ zF/R7!2@RV~t! 2 mV2 #%. (B13)

The associated power spectral density Sl( f) of l(t) is the Fouriertransform of Rl(t). The exponential transformation in the right-most portion of Equation B13 renders the relationship between theautocorrelation functions of the voltage V(t) and the rate l(t)nonlinear; in particular, Sl( f) will not follow an exact power-lawdecay as does SV( f). However, for relatively small sX

2 , whichappears to apply for the vesicular-release events we have recorded,the forms of the two power spectral densities do not differ greatly.

Finally, we consider the process of the exocytic events them-selves, which we denote N(t), as in Appendix A. These events aredescribed in terms of a Poisson process driven by a fractallognormal rate function l(t). The sequence of the coincidentcalcium-flow events, and therefore of the vesicular release eventsthemselves, is then described by a FLNDP, as promised. Thestatistics of the FLNDP process are computed readily from thoseof the FLN rate. If the results for a general Poisson process(Saleh, 1978; Lowen, 1996) are used, the power spectral densitySN( f) of the events becomes:

SN~ f ! 5 Sl~ f ! 1 E@l#. (B14)

The AF is related to the power spectral density by (Lowen, 1996):

A~T! 5 1 18

pE@l#T E0

`

Sl~v!sin4~vT/ 2!v22 dv.

(B15)

For a power spectral density that takes the form of Equation B3,we have A(T) ; Ta.

If we assume further that the rate l(t) [or equivalently thevoltage V(t)] exhibits fluctuations that are slow in comparisonwith the average rate of channel openings E[l], closed-formexpressions for the moments of the times t between channelopenings also can be obtained, again with the help of generalPoisson-process theory (Saleh, 1978):

E@tn# 5 n! E@l12n#yE@l# 5 n! !2n exp@2nmX 1 ~n2 2 2n!sX2 /2#.(B16)

In particular, t has a mean:

E@t# 5 1/E@l# 5 !21exp~2mX 2 s X2 / 2!, (B17)

and a variance:

Var@t# 5 !22 exp~22mX!@2 2 exp~2s X2 !#. (B18)

In short, a membrane voltage with a 1/f-type power spectraldensity ultimately leads to a sequence of exocytic events with anapproximately 1/f-type spectrum. The argument is summarized asfollows. The voltage V(t) has an amplitude distribution that isgaussian and a spectrum SV( f) that decays as 1/fa (Eq. B3). Theautocorrelation function of the voltage RV(t) (which is the inverseFourier transform of the spectrum) does not itself scale for thevalues of a determined from most exocytic recordings but none-theless contains all of the information that resides in the spectrumSV( f). The normalized voltage X(t) (Eq. B4) is linearly related toV(t) and therefore has statistics simply related to those of V(t). Therate l(t) is obtained by exponentially transforming the voltage V(t)(Eq. B2) or, equivalently, the normalized voltage X(t) (Eq. B5); ittherefore exhibits a lognormal amplitude distribution and an au-tocorrelation function Rl(t) obtained via the exponential trans-form of RV(t) (Eq. B13). The spectrum of the rate Sl( f) is obtainedby Fourier transformation of Rl(t). For the parameter values thatemerge from the data, the exponential transform is roughly linearso that RV(t) and Rl(t) are approximately proportional to eachother, so that Sl( f) essentially follows the same 1/fa form as SV( f).The spectrum SN( f) of the exocytic event sequence itself, N(t),differs from Sl( f) only by a constant (Eq. B14) so that it also thenvaries as 1/fa. Finally, for such processes the AF A(T) increaseswith counting time T as Ta (Eq. B15 and following). We concludethat the spectrum of the membrane voltage, of the rate, and of theexocytic events all decay as 1/fa with the same power-law exponenta, which is identical to the exponent that appears in the AF.

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quantal neurotransmitter secretion rate exhibits fractal behavior

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