Small Numbers res - Rieke Lab

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Small Numbers res JOHN M. BEKKERS AND CHARLES F. STEVENS ivision of Neuroscience, John Curtin School of Medical Research, Australian National University, Canberra, ACT 2601, Australia; and The Salk Institute, Howard Hughes Medical Institute, La Jolla, Calijhrnia 92037 SUMMARY

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CONCLUSIONS

1. We have studied the statistical properties of excitatory postsynaptic currents (EPSCs) measured at small numbers of synaptic contacts between pairs of hippocampal neurons maintained in dissociated cell culture. Synaptic transmission at few synapses was enabled by microperfusion of a small region of the postsynaptic cell with Ca-containing solution, while blocking transmission at all other synaptic boutons by bathing them in low-Ca solution. Frequency histograms of the amplitudes of EPSCs recorded in this way showed no clear quantization. Numbers of active synapses, estimated immunohistochemically with the use of light microscopy, ranged from 4 to I4 in different experiments. 2. Miniature EPSCs (mEPSCs), originating in the same small pulation of synapses as produced the evoked EPSCs, were elicd by microperfusion of bath solution made hypertonic by the addition of sucrose. These “sucrose-evoked” mEPSCs appeared to be identical to “spontaneous” mEPSCs in every respect except control over their frequency and site of origin. Sucrose-evoked mEPSCs originating in few synapses still exhibited a broad amplitude distribution. Thus, if mEPSCs constitute the postsynaptic response to a single quantum of neurotransmitter (the “‘quanta1 amplitude”), their broad amplitude distribution would tend to obliterate evidence of quantization in evoked EPSC amplitudes, even if evoked release was, indeed, quantal. 3. This idea, which is a corollary of the Katz model of quanta1 transmission, was tested quantitatively by assuming 1) neurotransmitter release obeys uniform binomial statistics, and 2) the quanta1 amplitude has a distribution given by the observed distribution of sucrose-evoked mEPSCs. The expected distribution, calculated on the basis of these two assumptions, was fitted to the observed distribution of evoked EPSC amplitudes by varying two free parameters, the binomial parameters N and p. In five cells out of six that were fully analyzed, the Poisson limit of the binomial model (Iv large, p small) provided a very good fit to the data. This and other evidence suggests that the release probability at a single presynaptic terminal is low. In two out of the six cells, the binomial model, with IV constrained to the histochemically determined bouton count, yielded acceptable fits; for the remaining cells the constrained binomial model could be rejected. 4. It is concluded that the Katz model of quantized release of neurotransmitter gives an adequate description of excitatory synaptic transmission in hippocampal cultures, when one assumes the broad distribution of mEPSC amplitudes reflects the distribution of the postsynaptic effect of a single quantum of transmitter.

Altho some of its details have been challenged, the quanta1 ry of synaptic transmission introduced by Katz and his collaborators (Del Castillo and Katz 1954; Katz 1.969)is widely accepted as a valid model of transmission at the neuromuscularjunction (Martin 1977; McLachlan 2978). 0022-30771’95

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According to the simplest version of this theory, the release of neurotransmitter from the presynaptic terminal is probabilistic and quantized, and the postsynaptic effects of separate neurotransmitter quanta (the “quanta1 amplitudes”) are identical and add approximately linearly. The quanta1amplitude is, by this theory, equivalent to the spontaneousminiature synaptic current or voltage (the “miniature”) that is often resolvable in postsynaptic cells (Fatt and Katz 1952). At the neuromuscular junction the observed discrete size steps in the amplitude of postsynaptic currents (Martin 1977), together with the properties of miniatures and the known architecture of the end plate (Rash et al. 1988), make the quanta1 theory immediately believable. On the other hand, the application of this theory to synapsesin the CNS is often fraught with technical difficulties (Korn and Faber 1991; Redman 1990). The main problem is that a central neuron may receive many thousands of synaptic inputs over its entire dendritic surface, so it is often difficult to study in isolation the fluctuations in transmission at a small number of synaptic contacts, as is required for a convincing test of the quanta1model. If the number of active synapses is not reduced in some way, there is a risk of obliterating any quanta1structure in the recorded responses either because of inherent differences between synapsesor because of their different electrotonic locations. Several groups have sought to overcome these difficulties by studying central neurons under conditions where relatively few synaptic contacts are active: this hasbeen done for motoneurons and dorsal spinocerebellar tract neurons in cat spinal cord (Jack et al. II98 1; Walmsley et al. 1988), Mauthner cells in goldfish (Korn et al. 1982), and dentate granule cells in rat hippocampus (Edwards et al. 1990). Evoked synaptic responsesrecorded in these neurons exhibited more or less clear amplitude quantization, and, in the case of the Mauthner and granule cells, spontaneous miniatures were also recorded, the mean amplitudes of these being comparable with the quanta1 spacing of the evoked responses(Edwards et al. 1990; Korn et al. 1987). Thus these results support the applicability of the basic quanta1theory to some central neurons. We decided to test this theory in cultured pyramidal neurons from the CA1 region of rat hippocampus, for two main reasons. First, the phenomenon of long-term potentiation (LTP) in the hippocampus makes it particularly important to apply quanta1 analysis to excitatory afferents onto CA1 pyramidal cells (Bliss and Collingridge 1993). Second, by using a culture system we hoped to be able to overcome some of the difficulties associated with applying \quantal

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analysis to this synapse. A number of groups have explored the quantization of neurotransmission at the Schaffer collateral-pyramidal cell synapse in slices (e.g., Kullmann and Nicoll 1992; Larkman et al. 1991; Liao et al. 1992; Voronin et al. 1992), but some aspects of this work remain controversial (e.g., Clements 1991; Korn and Faber 1991; Voronin 1993). For example, it was unclear whether a single afferent fiber was being fired reliably, and the number of active synapses was unknown. Hence or otherwise, the occurrence of quantization was found to be sporadic, sometimes dependent on subjective selection procedures. Furthermore, different groups have used different criteria to distinguish genuine biological quantization from sampling error (Stricker et al. 1994), and a rigorous comparison of these criteria has not yet been done. In addition, controversy surrounds the choice of an appropriate model for the statistics of neurotransmitter release. Some groups have analyzed their data on the assumption of uniform binomial release statistics (e.g., Bekkers and Stevens 1990; Larkman et al. 1992; Voronin et al. 1992), whereas others have questioned this constraint (e.g., Faber and Korn 199 1; Redman 1990). In summary, therefore, quanta1 analysis applied to hippocampal slices is contentious. Our approach was to use dissociated cell cultures as a model system in which to satisfy as many as possible of the requirements for a ‘ ‘proper’ ’ quanta1 analysis (Korn and Faber 199 1). Recordings were made from isolated pairs of CA1 pyramidal cells while monitoring both pre- and postsynaptic currents. The number of active synapses was reduced by using localized external calcium applications to restrict transmission to a small subset of the total number of synaptic contacts on a neuron. In addition, miniature synaptic currents originating from approximately the same subset of active synapses were measured. Finally, an estimate of the number of synaptic terminals participating in each experiment was obtained by immunohistochemical means. Surprisingly, we found no clear evidence of quantization of postsynaptic current amplitudes under the conditions of our experiments. However, after taking into account the observed wide variability in the amplitudes of the miniatures, our results were generally well-described by the simple Katz model with Poisson release statistics. METHODS

Neurons were isolated from the CA1 field of the hippocampi of newborn rats (Long-Evans) and maintained in culture on “microdots” as described (Bekkers and Stevens 1991). Briefly, coverslips were coated with a film of agarose, then a mist of substrate solution was sprayed onto the coverslips, yielding a random array of dots lOO- 1,000 ,um diam. Dissociated neurons plated onto these coverslips restricted their growth to the dots of permissive substrate. Experiments used lone pairs of neurons occupying single dots; this was done to exclude the possibility of polysynaptic innervation, and because such neurons were almost invariably connected, which improved the success rate for experiments.

Solutions Two principal kinds of external solution were used: 1) “5 Ca external, ’ ’ which contained (in mM) 137 NaCl, 5 KCl, 5 CaCl,, 1 MgCl;?, 10 glucose, and 10 N-2-hydroxyethylpiperazine-N’-2-

C. F. STEVENS

ethanesulfonic acid (HEPES) adjusted to pH 7.4 with NaOH, and adjusted to 3 10 mosM with sorbitol; and 2) “low Ca external,” which was the same as 5 Ca external except that it contained 10 mM MgCl* and no added calcium. The internal (pipette) solution contained (in mM) 150 potassium methylsulfate, 5 KCl, 5 KBAPTA [ 1,2-bis(o-aminophenoxy)ethane-N,N,N’,N’-tetraacetic acid], and 10 HEPES adjusted to pH 7.2 with KOH. In experiments in which current-voltage data for miniature synaptic currents was measured, Cs replaced K in the internal solution.

Electrophysiology Whole cell recordings were made from isolated cell pairs 7- 11 days after plating; at these ages connectivity was still relatively sparse, which increased the likelihood of finding regions of the dendritic tree that contained few synapses. Patch electrodes, which were Sylgarded, had resistances of l-3 M0 when filled with internal solution, and soma access resistance was 0.6 SD from the average amplitude of all epochs. The fit of theory to experiment was done in the following series of steps. 1) The (unsmoothed) amplitude histogram for mEPSCs was numerically convolved with itself n times, yielding a set containing one each of a 2-fold, 3-fold * 0 0(12 + I)-fold convolution. For IZ > 15 the convolved distribution was approximated by a Gaussian function whose mean and variance were (n + 1) times the mean and variance of the original mEPSC data. Strictly speaking, the noise-free mEPSC distribution should be used here. However, the noise distribution was narrow compared with the mEPSC distribution, and deconvolving the noise in one typical experiment had little effect. 2) Starting values for the binomial parameters p and N were estimated (McLachlan 1978) from the expressions p = 1 - v”lm and N = mfp, where v2 = (s2 - ms$)ly’ (the variance of the release process) and m = A/q (the mean quanta1 content). Here A and s2 are the mean and variance of the (noise-free) evoked EPSC amplitudes, and g and sz are the corresponding quantities for the (noisefree) mEPSCs; these were all approximated by the corresponding measured (noisy) quantities. In practice a range of different N and 1~ starting values were used to confirm the robustness of the fit. 3) The starting values for p and N were used to calculate the probabilities of release of 1, 2, 3 0quanta, and the probabilities were used to scale the amplitudes of the 1 -fold, 2-fold, 3-fold 0 0 0convolutions, respectively, of the mEPSC histogram, calculated in 1). These scaled histograms were then added together to give a theoretical histogram of EPSC amplitudes. 4) The failures peak was added by scaling the noise histogram l

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by the probability of failures calculated from the binomial distribution and adding this histogram to the one calculated in 3). 5) Steps 3) and 4) were repeated with the use of the Simplex algorithm to adjust N and p for best fit between the theoretical histogram and the experimentally measured histogram of EPSC amplitudes, with the use of minimization of the ,$ statistic as the fit criterion. x2 was calculated after merging bins in the tails of the distribution so that the predicted counts per bin was not or 0.98 (&, indicating an excellent fit. Using the sucrose-evoked mEPSC distribution instead gave a similar, but noisier, fit. These and other parameters concerning the analysis are summarized in Table 1 (cell I). Four to six boutons were present in the region of microperfusion for this cell pair. This must be an upper limit to the number of synaptic contacts contributing EPSCs, because some of them may have been inhibitory autapses. (The postsynaptic neuron was inhibitory.) On the other hand, if a synaptic contact may contain two or more functionally independent release sites, the bouton count may underestimate the appropriate N value. In any case, when the binomial N in the fit was constrained to either 4 or 6, the optimal fit still could not be rejected at a = 0.05. Despite the fact that the microperfusion solution contained a high Ca concentration (5 mM), a relatively large number of stimuli resulted in failures of synaptic transmission (566 out of 983). This gives a probability of 0.58 that release failed at all active synaptic contacts. If we assume that none of the 4-6 identified boutons are autapses, and that they are all independent and identical, this implies a probability of 0X7-0.91 that release failed at any 1 synaptic site. Data from another cell pair are presented in Fig. 4. Because the release of components that constituted the EPSC was not synchronous in this case (Fig. 4A, inset), we have used the time integral of EPSCs (and of mEPSCs), the charge transfer, rather than the peak amplitude, as a measure of

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FIG. 4. 0 uan t’d 1 analysis applied to cull 2 in Table 1. A: frequency histogram of the charge carried by evoked EPSCs (371 events), with the fitted distribution (heavy line) superimposed. The distribution of auantal amnlitudes was assumed to be Gven bv the histogram of. sucrose-evoked mEPSCs (i), and Poisson statistics was found to adequately describe release. B and C: frequency histograms of the charge carried by sucrose-evoked (B) and poststimulus (C) mEPSC amplitudes. ln,s~~ in A shows examples of evoked EPSCs in this experiment. Note the asynchronous release, necessitating the use of charge as a measure of transmission.

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EPSC size. The advantage of this procedure is that all EPSC components are counted equally (which would not be the case for a peak amplitude measurement when release is asynchronous), but the disadvantage is that the measurement is less accurate because slight uncertainty in the baseline can, with the integration, produce larger errors than is the case litude measurement. In this cell pair, again, nd observed histograms were in best agreement at the Poisson limit of the binomial release model. This was the case whatever measure of x2 was used (a = 0.29 for XT, a = 0.97 for &), and whether the optimization was done with the use of the poststimulus or sucrose-evoked mEPSC distributions (Table 1, cell 2). Note that, in this case, ability of failure at all 7- 10 identified boutons was .04; however, this implies (on the same assumptions as before) a failure probability at a single bouton of -0.7. For this cell pair, however., constraining binomial N to the bouton count (7- 10) gave optimal fits that could be rejected at c8 = 0.05. A final example is shown in Fig. 5. As before, an unconstrained binomial fit tended to the Poisson limit, which gave a satisfactory fit (h = 2.14; (x = 0.21 for XT, cy = 0.98 for xz), as did a binomial fit while constraining N to the bouton count (7-14). The figure illustrates the fit constraining N = 7, which gave an optimal p = 0.29 (a = 0.13 for xtY cy = 0.98 for x;). Analysis of two additional cell pairs yielded comparable results: the observed distribution of mEPSCs together with the Poisson release model gave statistically satisfactory fits (&y > 0.05; Table 1, cells 4 and 5). In a sixth cell pair the Katz model could be rejected (a < 0.05; Table 1, cell 6). DISCUSSION

The experiments described here used microperfusion techniques to study the statistical nature of neurotransmission in

small populations of hippocampal synapses in culture. We have found that mEPSCs originating in these subpopulations exhibit a rather broad distribution of amplitudes, and their form appears to be similar whether release is spontaneous or evoked by hypertonic solution. Furthermore, we have shown that synapses in culture commonly operate as predicted by the Katz quanta1 theory if 1) release is assumed to obey uniform binomial statistics (in fact, the Poisson limit), and 2) the mEPSCs are assumed to constitute the unitary postsynaptic response, the quanta1 amplitude. Our analysis reinforces the notion that the Katz model need not be synonymous with visible quantization of EPSC amplitudes if the quanta1 variance is large. Precautions

in the quanta1 amlysis

One of our objectives in this work was to remove as many as possible of the uncertainties associated with quanta1 analysis applied to central neurons (Kern and Faber 199 1; Redman 1990). Thus the following precautions were taken in our experiments. 1) The presynaptic spike (strictly, the sodium “action current”) was monitored for reliable firing throughout. Furthermore, experiments used lone pairs of neurons on microdots, so there was no possibility of polysynaptic connectivity. We had no way, of course, of knowing if all axon collaterals issuing from the presynaptic neuron tired reliably. 2) Noise levels were kept low (standard deviation I .52.1 pA at 2 kHz) by using relatively young (7- 1 1 day in vitro), small (20- to 3O+m soma diameter) neurons. Thus mEPSCs were easily resolved, and the evoked EPSC amplitude histograms always showed a “dip” between the failures peak and the remainder of the histogram (A, Figs. 3-5). 3) To minimize cable filtering, microperfusion was used

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5) Bouton counts were made by visualizing peroxidase staining at the limits of light microscopy. However, this does not address the question of the number of potential release sites per bouton. Preliminary analysis of electron microscopy (EM) serial sections from comparable cultures reveals that each bouton typically possesses a single active zone (R. A. Jacobs and C. F. Stevens, unpublished observations). In summary, our approach has addressed many of the requirements of a rigorous quanta1 analysis but still falls short of the ideal. Two obvious improvements would be to obtain data from a single identified bouton and to serially reconstruct that same bouton with the use of EM (e.g., as done recently by Gulyas et al. 1993, in the hippocampal slice). Furthermore, our analysis assumed that transmitter release obeys uniform binomial statistics, whereas a more rigorous approach would be to compare alternative models (Stricker et al. 1994) (and see below).

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Quan tal ana lysis applied to cell 3 in Table 1. A : frequency histoof ev raked EPSC amplitudes (300 events), with the fitted distri bution line) superimposed. The distributi on of quanta1 amplitudes was as(heavy sumed to be given by the histogram of sucrose-evoked mEPSCs (B). In this fit, uniform binomial release statistics were assumed, with N constrained to the lower limit of the histochemically determined number of boutons. Poisson statistics also gave an acceptable fit. Arrow indicates the height of the experimental failures peak. B and C: frequency histograms of sucroseevoked (B) and poststimulus (C) mEPSC amplitudes. This is the same cell pair as is illustrated in Fig. I. FIG. 5.

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to activate synaptic co ntacts that were restricted to the soma or proximal dendrite. 4) A further objective of the microperfusion approach was to reduce the number of active synapses, ideal ly to just 0 ne, to minimize possible heterogeneity between synapses or, conversely, to look fo r possible variability in quanta1 amplitude at a single site. However , the small est bouton count within the microperfused region was four to six (Table I), owing to the rather high density of synaptic contacts, even between young, isolated cell pairs. This point is further discussed below.

An objective of our experiments was to look for heterogeneity of quanta1 amplitude between synapses. The idea was that, if synapses are markedly heterogeneous, amplitude histograms made from small populations of them should exhibit peaks or other fine structure that is normally obscured when measuring across many synapses (Edwards et al. 1990; Redman 1990). The smallest bouton count was four to six (Table I), but convincing peaks were not apparent in either the evoked EPSC or mEPSC amplitude distributions (Fig. 3). Although there is a suggestion of peaks at -40 pA and -80 pA in Fig. 5A, we think these arise in sampling error because the bin counts are low and their presence was sensitive to the choice of binwidth. This lack of peaks is surprising, because evidence of peaks in evoked postsynaptic current histograms has been reported for inhibitory connections between hippocampal granule cells (Edwards et al. 1990) and cultured neurons from the superior colliculus (Kraszewski and Grantyn 1992) and for excitatory connections in the hippocampal slice (Kullmann and Nicoll 1992; Larkman et al. 199 1; Liao et al. 1992). The discrepancy was not due to greater noise in our system, because our noise levels were similar to those reported by the above workers. Qne possible reason for the discrepancy is that four to six boutons is still too many if each contains several release sites, or if they are very heterogeneous. The number of active release sites in the slice experiments mentioned above is not known (but see Sorra and Harris 1993, who suggest greater numbers and complexity than previously thought). However, quanta1 size may be more tightly constrained in the slice than in culture, perhaps because developmental cues are lacking in culture (but see Manabe et al. 1992; Raastad et al. 1992). Thus quanta1 spacing may inherently be more apparent in the slice even if many release sites are involved. On the other hand, Kraszewski and Grantyn (1992), working with cultured superior collicular neurons, showed clear peaks in several of their amplitude histograms obtained for GABAergic synaptic currents, despite the fact that their estimate of the number of synapses ranged from 8 to > 100. Under similar conditions, without microperfusion to restrict the number of active synapses, our EPSC amplitude histograms did not exhibit clear quantization of the kind reported bv Kraszewski and Grantvn (unpublished results: y1 = 23

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cell pairs, each in a range of Ca concentrations). Perhaps excitatory and inhibitory synapses, or neurons from different regions of the brain, develop differently in culture insofar as the precision of their quanta1 size is concerned. Not only the quanta1 variability, but also the mean quanta1 amplitude, may be larger in culture than in sliced tissue. We found a mean conductance of -550 pS for spontaneous mEPSCs and -760 pS for sucrose-evoked mEPSCs in culture. In contrast, in the hippocampal slice much lower mean values have been reported: 50- 150 pS for spontaneous mEPSCs (Manabe et al. 1992) and 210 pS for sucroseevoked mEPSCs (Bekkers et al. 1990). Moreover, in slice experiments in which clear EPSC quantization was resolved, the quanta1 spacing corresponded to only -70 pS (Kullmann and Nicoll 1992; Larkman et al. 1991; Liao et al. 1992). It is possible that estimates for the slice are smaller because the mEPSCs experience greater dendritic filtering than in culture (Ulrich and Luscher 1993). In any event, it is clear that caution is needed when comparing neurotransmission in the two preparations.

Distribution

of mEPSC amplitudes

Both our sucrose-evoked and poststimulus mEPSC distributions had a broad, skewed shape, as previously reported for hippocampus (Bekkers and Stevens 1989; Finch et al. 1990; Manabe et al. 1992; McBain and Dingledine 1992; Raastad et al. 1992) and other central neurons (Kraszewski and Grantyn 1992; Liu and Feldman 1992; Silver et al. 1992; Ulrich and Luscher 1993). One interpretation of this is that mEPSCs can originate in synapses located anywhere on the dendritic tree and so are subject to differing amounts of dendritic filtering, which distorts their amplitudes as measured at the soma (Ulrich and Luscher 1993). However, this cannot explain our results. By microperfusing regions close to the soma, we restricted our measurements to synapses that should have been under good voltage control or, at least, subject to the same amount of distortion (see also Kraszewski and Grantyn 1992). Nevertheless, the strength of our conclusions depends in large measure on the claim that sucrose-evoked mEPSCs are identical to spontaneous mEPSCs in every way except control over their frequency of occurrence and site of origin. We have given evidence that this is the case: the two are similar in their selectivity, pharmacology, and kinetics, after making allowance for differential cable filtering (Fig. 2). We were still concerned that sucrose application might drive vesicle release too fast, producing residual desensitization of postsynaptic nonNMDA receptors when a second release followed closely upon a first. By calculating serial correlograms of sucroseevoked mEPSC data, we have found that this is not the case, at least under the conditions employed here (Bekkers and Stevens 1994). Although some small and systematic differences between spontaneous mEPSCs and those evoked by hypertonic solution may well exist, any such differences are small enough that they are obscured by statistical fluctuations in sample sizes up to several hundred. If differential dendritic filtering cannot account for the variability in mEPSC amplitudes, what else could explain this result? If we consider the amplitude histogram in Fig. 3 as an example, one interpretation is that all active synapses were functionally similar, each contributing mEPSCs with

C. F. STEVENS

similar amplitude distributions. In this view, the coefficient of variation (CV = SD/mean) of mEPSC amplitudes for any one synaptic contact is the same as the population CV, namely, 0.49 for this particular cell (Table 1). In fact, an even larger CV (0.65) was reported for sucrose-evoked mEPSCs from an apparent single synapse in the same culture system (Bekkers et al. 1990), supporting this idea that all synapses have a similarly broad mEPSC amplitude distribution. This might be due to variations in the amount of neurotransmitter released per quantum, on the assumption that the postsynaptic response is never saturated (Bekkers et al. 1990; cf. Tong and Jahr 1994), or it may be due to stochastic fluctuations in postsynaptic channel gating (Faber et al. 1992; Jonas et al. 1993; Kullmann 1993). On the other hand, an alternative view is that the net distribution is made up of much narrower distributions with their means displaced from each other and representing the responses at the different synapses. Then their envelope would resemble the histogram in Fig. 3r. A similar idea has been suggested for the neuromuscular junction (Kriebel 1988) and for some central synapses (Edwards et al. 1990; Jonas et al. 1993; Liu and Feldman 1992; Ulrich and Luscher 1993) where peaks have been reported in histograms of miniature amplitudes. This situation might arise if, say, different synapses contain different numbers of glutamate receptors in their postsynaptic membrane and most receptors are activated during the release of a single vesicle of transmitter (Clements et al. 1992; Edwards 1991; Jack et al. 1981; Tong and Jahr 1994). According to this model, quantization of miniatures (and of evoked EPSCs) arises because receptors are inserted into the postsynaptic membranes of different synapses in integral multiples of a receptor “module” that has a fixed size. This model has implications for the statistics of the release process that are further discussed below. An alternative interpretation of miniature quantization is that it occurs at a single release site and is due to the cooperative fusion of various numbers of vesicles, without postsynaptic saturation (Bekkers et al. 1990). On the basis of the evidence presented in this paper, we cannot distinguish between these possibilities. To address this issue convincingly, one would need to collect mEPSCs from a single identified release site.

Statistics of the release process A simple statement of the quanta1 hypothesis is that EPSCs are made up of summed mEPSCs. Extensions of the basic hypothesis differ in how they account for the details of this summation. Here we have tested one extension [that of Del Castillo and Katz (1954)] in which it is supposed that one unit of transmitter released at any one synapse can give rise to the full, measured range of mEPSC amplitudes (‘ ‘intrasynapse quanta1 variance”), and the release of transmitter units obeys a uniform binomial rule. We have chosen this interpretation because of our previous finding of a large variance in mEPSC amplitudes at an apparent single synaptic site in culture (Bekkers et al. 1990). An alternative extension of the basic quanta1 model is that the measured mEPSC distribution reflects heterogeneity between synapses (‘ ‘intersynapse quanta1 variance’ ‘). For example, this might occur under the receptor module model mentioned above, where the mEPSC amplitude represents an assay of the number of postsvnantic glutamate receptors, and this number varies

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ANALYSIS

between synaptic contacts. In this case the miniature distribution confounds two factors: the response of the postsynaptic cell to a unit of transmitter, and the (generally unknown) distribution of postsynaptic receptor numbers. According to this model, even if transmitter release does obey uniform binomial statistics, the analytic approach taken here would not necessarily give a good fit to the EPSC data. The fact that our analysis does provide a good fit shows that the postsynaptic saturation model is not required to describe our data, but it does not rule out this possibility. A rigorous test would involve analyzing transmission at a single release site. As mentioned in the INTRODUCTION, previous quanta1 analyses have been criticized for presupposing a particular statistical model for neurotransmitter release, such as the Poisson model (e.g., Yamamoto 1982) or the uniform binomial model (e.g., Larkman et al. 1992; Voronin et al. 1992). In this paper we, too, have made a starting assumption about transmitter release, namely, that it obeys uniform binomial statistics. Our initial reason for doing this was that our data sets were either small or contained many failures of transmission, and we felt that this precluded a unique fit to more complex release models with more free parameters. The uniform binomial model has two free parameters (N and p), whereas the next most complex model, the compound binomial, requires N + 1 free parameters (1 for N plus 1 for each of the different ps at each of N release sites). Very recently, powerful techniques for the statistical ranking of alternative release models have been described (Stricker et al. 1994). However, our goal in this work was not to rank alternatives, but simply to test whether the simplest, most parsimonious model currently available could describe our data. In five cells out of six, we found, in fact, that the Poisson limit of the binomial model provided a statistically acceptable fit. Although more complex models could be fitted to our data, there would be no gain in statistical significance in most cases and a potential cost in increased numbers of arbitrary free parameters. An acceptable model should be not only parsimonious, but also physically plausible. How plausible is the uniform binomial model (and its limit, the Poisson model) for excitatory synapses in culture? It has frequently been reported that the binomial N obtained from quanta1 analysis is always less than or equal to the number of morphologically defined release sites (Korn and Faber 199 1; Redman 1990). This finding has given rise to the “one site-one vesicle” hypothesis: each release site contributes no more than a single quantum to the net response to presynaptic stimulation. A result of the analysis presented here was that fits to the binomial model, with N constrained to the number of histochemically identified boutons (i.e., presumed release sites), yielded fits that were not significantly better than Poisson fits, and in some cases could be rejected. The Poisson approximation to the binomial distribution is usually excellent with moderate N and small p (as we find here). Thus the one site-one vesicle hypothesis could still apply to our results. An alternative interpretation is that a more complex release model (e.g., the compound binomial) actually applies in culture, and the Poisson model is an adequate limiting case. In fact, from physiological and structural considerations it seems likely that something like the compound binomial model would apply in culture, but the number of free parameters required to describe this model precluded mean-

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ingful analysis of our data. Testing of this interpretation would also require data of a kind not provided by these experiments. A second issue concerns the assumption of the uniform binomial model that the release probability p is constant across all release sites. It has recently been reported that p at NMDA receptor-mediated synapses in culture can vary by up to eightfold (Rosenmund et al. 1993). On the other hand, the Poisson model is insensitive to heterogeneity in p, provided thatp is always small (Del Castillo and Katz 1954). In fact, in most of our cell pairs, the release probability per bouton was indeed very low. This was apparent even without performing a full quanta1 analysis, from the observation that many failures of synaptic transmission occurred. However, our estimate of release probability per bouton is probably an underestimate of the true value, because some of the counted boutons may have been autapses (Bekkers and Stevens 1991). Even if we assume the extreme case of just one functional bouton, however, release probability per bouton was often still a low number, despite the high Ca/Mg ratio (5 mM/l mM) in the microperfusate. Our results suggest that excitatory synapses in culture operate well below their maximum possible capacity and, indeed, imply that synaptic transmission is quite unreliable. On the other hand, this would provide synapses with considerable reserve for increased synaptic efficacy. We have previously shown that LTP in our cultures is expressed by means of an increase in the presynaptic parameter, p (Bekkers and Stevens 1990). Others have since claimed in hippocampal slices that LTP tends to be expressed presynaptically in synapses for which p happens initially to be low, but postsynaptically otherwise (Larkman et al. 1992; Liao et al. 1992). Thus it is perhaps not surprising that cultures, with their initially low p, express LTP presynaptically. In summary, we have found that, within the uncertainties imposed by sampling errors for populations of several hundred observations, the Katz theory (extended to include the observed distribution of mEPSC sizes) provides an adequate description of central synaptic transmission as represented by our hippocampal neurons in culture. The extent to which this conclusion applies to synapses in vivo has yet to be clarified. We thank J. D. Clements for much stimulating discussion, and F. A. Edwards and S. J. Redman for comments on the manuscript. This work was supported by the Howard Hughes Medical Institute (C. F. Stevens) and a Queen Elizabeth II Research Fellowship from the Australian Research Council (J. M. Bekkers). Address for reprint requests: J. M. Bekkers, Division of Neuroscience, John Curtin School of Medical Research, GPO Box 334, Canberra, ACT 260 1, Australia. Received

7 December

1993; accepted

in final form

4 November

1994.

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