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# Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses

**Authors’ present addresses**R. A. Silver: Department of Physiology, University College London, Gower Street, London WC1E 6BT, UK.

A. Momiyama: Department of Physiology, Nagasaki University School of Medicine 1-12-4 Sakamoto, Nagasaki 852-8523, Japan.

**Authors’ email addresses**R. A. Silver: ku.ca.lcu@revlis.a

## Abstract

- EPSCs were recorded under whole-cell voltage clamp at room temperature from Purkinje cells in slices of cerebellum from 12- to 14-day-old rats. EPSCs from individual climbing fibre (CF) inputs were identified on the basis of their large size, paired-pulse depression and all-or-none appearance in response to a graded stimulus.
- Synaptic transmission was investigated over a wide range of experimentally imposed release probabilities by analysing fluctuations in the peak of the EPSC. Release probability was manipulated by altering the extracellular [Ca
^{2+}] and [Mg^{2+}]. Quantal parameters were estimated from plots of coefficient of variation (*CV*) or variance against mean conductance by fitting a multinomial model that incorporated both spatial variation in quantal size and non-uniform release probability. This ‘multiple-probability fluctuation’ (MPF) analysis gave an estimate of 510 ± 50 for the number of functional release sites (*N*) and a quantal size (*q*) of 0.5 ± 0.03 nS (*n*= 6). - Control experiments, and simulations examining the effects of non-uniform release probability, indicate that MPF analysis provides a reliable estimate of quantal parameters. Direct measurement of quantal amplitudes in the presence of 5 mm Sr
^{2+}, which gave asynchronous release, yielded distributions with a mean quantal size of 0.55 ± 0.01 nS and a*CV*of 0.37 ± 0.01 (*n*= 4). Similar estimates of*q*were obtained in 2 mm Ca^{2+}when release probability was lowered with the calcium channel blocker Cd^{2+}. The non-NMDA receptor antagonist 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX; 1 μm) reduced both the evoked current and the quantal size (estimated with MPF analysis) to a similar degree, but did not affect the estimate of*N*. - We used MPF analysis to identify those quantal parameters that change during frequency-dependent depression at climbing fibre-Purkinje cell synaptic connections. At low stimulation frequencies, the mean release probability (
*P*¯_{r}) was unusually high (0.90 ± 0.03 at 0.033 Hz,*n*= 5), but as the frequency of stimulation was increased,*p*_{r}fell dramatically (0.02 ± 0.01 at 10 Hz,*n*= 4) with no apparent change in either*q*or*N*. This indicates that the observed 50-fold depression in EPSC amplitude is presynaptic in origin. - Presynaptic frequency-dependent depression was investigated with double-pulse and multiple-pulse protocols. EPSC recovery, following simultaneous release at practically all sites, was slow, being well fitted by the sum of two exponential functions (time constants of 0.35 ± 0.09 and 3.2 ± 0.4 s,
*n*= 5). EPSC recovery following sustained stimulation was even slower. We propose that presynaptic depression at CF synapses reflects a slow recovery of release probability following release of each quantum of transmitter. - The large number of functional release sites, relatively large quantal size, and unusual dynamics of transmitter release at the CF synapse appear specialized to ensure highly reliable olivocerebellar transmission at low frequencies but to limit transmission at higher frequencies.

Quantal analysis is a classical approach to the investigation of mechanisms underlying synaptic transmission, in which key parameters - the number of functional release sites (*N*), the quantal size (*q*) and the release probability - are extracted from fluctuations in synaptic responses using statistical models. Unlike the situation at the neuromuscular junction (NMJ), application of quantal analysis to central synapses has proved difficult because some of the necessary simplifying assumptions are often not valid. For example, at synapses where release probability is non-uniform, a compound binomial (Jack, Redman & Wong, 1981; Walmsley, Edwards & Tracey, 1988; Stricker, Field & Redman, 1996) rather than a simple binomial statistical model is required, increasing the number of parameters to be estimated from the data set. Variations in quantal size at individual release sites, and between release sites, makes it difficult to identify peaks in distributions of synaptic response amplitude and complicates their interpretation (Walmsley, 1995). Furthermore, the difficulty in testing the accuracy of these methods directly has led, for example, to controversy in attributing long-term changes in synaptic efficacy to a pre- or postsynaptic locus. Recently, several alternative strategies have been developed to examine the quantal properties of transmitter release. These include the use of a non-competitive antagonist to estimate release probability (Rosenmund, Clements & Westbrook, 1993), the use of capacitance measurement to monitor vesicle release (Heidelberger, Heinemann, Neher & Matthews, 1994), and the use of the fluorescent dye FM1-43 to examine vesicle cycling and transmitter release (Ryan, Reuter, Wendland, Schweizer, Tsien & Smith, 1993; Issacson & Hille, 1997; Murthy, Sejnowski & Stevens, 1997). However, these techniques provide only part of the quantal description of the synapse, and there are difficulties in applying them to slice preparations in which synaptic architecture is preserved.

In order to overcome these problems, we have developed a new approach which extends previous quantal analysis methods (del Castillo & Katz, 1954*a*; Miyamoto, 1975; Clamann, Mathis & Lüscher, 1989; Malinow & Tsien, 1990; Bekkers & Stevens, 1990). With this approach, which we term ‘multiple-probability fluctuation analysis’ (MPF analysis), quantal parameters are estimated from synaptic current fluctuations measured over a wide range of experimentally imposed release probabilities at the same input. This method, which is also related to non-stationary fluctuation analysis of ion channels (Sigworth, 1980; Traynelis, Silver & Cull-Candy, 1993; Silver, Cull-Candy & Takahashi, 1996) and to a method recently suggested by Frerking & Wilson (1996), provides estimates for the underlying quantal parameters with minimal assumptions. Furthermore, it can provide estimates of *N*, *q* and the mean release probability when both the quantal size and release probability are non-uniform.

At synaptic connections, the underlying quantal parameters are not static, but change with time as a result of short-term facilitation and depression (Zucker, 1989) or long-term changes in synaptic efficacy. Understanding this dynamic behaviour is central to understanding transmission, since synaptic efficacy, at any given time, is dependent on the recent history of the input. Furthermore, modelling studies have shown that the temporal behaviour of synaptic efficacy (the tendency to exhibit short-term facilitation or depression) is important in determining the pattern of frequency-coded information that is transmitted to the postsynaptic cell (Sen, Jorge-Rivera, Marder & Abbott, 1996; Tsodyks & Markram, 1997). Since the early work of del Castillo & Katz (1954*b*) at the NMJ, synaptic depression is generally assumed to have a presynaptic locus. However, at central glutamate synapses the situation appears less clear. While several studies have demonstrated presynaptic depression (Larkman, Stratford & Jack, 1991; Borst & Sakmann, 1996; von Gersdorff, Schneggenburger, Wies & Neher, 1997), non-NMDA receptors desensitize when exposed to low concentrations of glutamate, so postsynaptic desensitization may persist long after the synaptic current has decayed. Indeed, recent studies have shown that postsynaptic desensitization plays an important role in synaptic depression at some central excitatory synapses (Trussell, Zhang & Raman, 1993; Zhang & Trussell, 1994).

In this study, we have investigated transmission at the climbing fibre-Purkinje cell synaptic connection, which in paired-pulse experiments exhibits substantial EPSC depression (Konnerth, Llano & Armstrong, 1990; Perkel, Hestrin, Sah & Nicoll, 1990). This synaptic connection, as part of the olivocerebellar pathway, is thought to play an important role in the timing of motor tasks (Welsh, Lang, Sugihara & Llinas, 1995) and in motor learning (Ito, 1984). Each Purkinje cell is innervated by a single climbing fibre (CF) which forms a distributed synaptic connection with numerous contacts on the extensive dendritic tree. CF stimulation generates a large glutamate-mediated EPSC, which, under physiological conditions, causes Purkinje cells to fire complex spikes (Eccles, Llinas & Sasaki, 1966). We have investigated the locus of frequency-dependent depression by examining the underlying quantal parameters with MPF analysis. Our results show that this input has unusual quantal parameters for a central synaptic connection. Furthermore, we demonstrate with MPF analysis that the frequency-dependent depression at the CF synapse has a purely presynaptic origin over a physiological range of frequencies. Our results suggest that the temporal dynamics of release probability at this synapse are specialized to ensure highly reliable low-frequency transmission. Some of these findings have been published in this journal in a preliminary form (Silver, Momiyama & Cull-Candy, 1997).

## METHODS

### Experimental preparation and data acquisition

Following decapitation, in accordance with The Animals (Scientific Procedures) Act 1986, the brains from 12- to 14-day-old Sprague-Dawley rats were removed and cooled in ice-cold saline. Parasagittal slices of cerebellum, 200 μm thick, were prepared and incubated at 31°C for 1 h (see Silver *et al.* 1996). The saline solution used for slicing and incubation contained (mm): 125 NaCl, 2.5 KCl, 1 CaCl_{2}, 5 MgCl_{2}, 1.25 NaH_{2}PO_{4}, 26 NaHCO_{3} and 25 glucose (pH 7.3 when bubbled with 95% O_{2} and 5% CO_{2}). Slices were then transferred to the recording chamber and perfused with solutions containing (mm): 125 NaCl, 2.5 KCl, 0-4 CaCl_{2}, 1-5 MgCl_{2}, 1.25 NaH_{2}PO_{4}, 26 NaHCO_{3} and 25 glucose (pH 7.3 when bubbled with 95% O_{2} and 5% CO_{2}). Climbing fibre EPSCs were recorded at room temperature (21-25°C) with 20 μm 7-chlorokynurenic acid (and occasionally 20 μm D-amino-5-phosphonopentanoic acid, d-AP5), 20 μm bicuculline methiodide, 25 μm picrotoxin and 0.5 μm strychnine added to the perfusate to block NMDA, γ-aminobutyric acid and glycine receptors, respectively. The following agents were also added to the external solution in specific experiments: 1 μm 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX), 10-30 μm CdCl_{2}, 5 mm SrCl_{2}, 500 μm (*RS*)-α-methyl-4-carboxyphenylglycine (MCPG), 200 μm (*RS*)-α-methyl-4-tetrazolylphenylglycine (MTPG), 250 μm (*RS*)-α-methyl-4-phosphonophenylglycine (MPPG), and 50 μm l-2-amino-4-phosphonobutyric acid (l-AP4). Drugs were obtained from Tocris Cookson (Bristol, UK) or Sigma.

Patch pipettes were made from thin-wall borosilicate glass (Clark Electromedical). The pipette solution contained (mm): 110 CsF, 30 CsCl, 2 NaCl, 0 or 0.5 CaCl_{2}, 10 Hepes, 5 or 10 EGTA and 2 Mg-ATP (adjusted to pH 7.3 with CsOH). CF inputs were stimulated at 0.033-10 Hz (5-30 V; duration, 0.02-0.2 ms) with a second patch electrode filled with 1 m NaCl placed in or on the surface of the granule cell layer. We restricted analysis to frequencies of 10 Hz and below because this covered the physiological range, and EPSC stimulation became unreliable above this frequency. EPSCs were recorded with an Axopatch 200A (Axon Instruments) and, in dual recording experiments, voltage measurements were made with an L/M-EPC-7 amplifier (List Electronic). Series resistance was estimated from the settings of the Axopatch 200A, and series resistance compensation was used with capacitance compensation and prediction switched off. Signals were recorded on DAT (DTR 1204, Biologic, France) after filtering at 10 kHz. EPSCs recorded on tape were subsequently filtered to 1-5 kHz (Bessel filter) and digitized at 20-25 kHz using AxoTape 2 and a Digidata 1200 interface (Axon Instruments).

### Identification of single climbing fibre inputs and optimization of synaptic current recordings

Climbing fibre EPSCs were distinguished from parallel fibre EPSCs (which exhibit paired-pulse facilitation) by their large size and characteristic paired-pulse depression (Fig. 1*A*). We ensured that no small-amplitude parallel fibre EPSCs contaminated the CF EPSC by examining current records that were just below the CF threshold (Fig. 1*B*). In neonatal rats (postnatal days (P)0-P12), Purkinje cells are innervated by multiple climbing fibres, but after this time only a single climbing fibre remains (Mariani & Changeux, 1981). We ensured that only a single CF was activated in our preparation by examining whether EPSCs showed an all-or-none response to a stimulus of graded intensity (Fig. 1*C*).

The large size of the CF EPSCs and the extensive dendritic arbor of the Purkinje cell make voltage clamp of these currents technically difficult. We adopted several strategies to optimize the quality of voltage clamp and to minimize the errors involved. Firstly, 12- to 14-day-old animals (mainly P13) were used, as at this age the Purkinje cell dendritic arbor is less extensive than in the adult, and CF innervation is located on the soma and proximal dendrites (Altman, 1972). Secondly, the electrode resistance was minimized by using large electrodes (2-3 MΩ) combined with series resistance compensation (70-85%; mean series resistance after compensation, 0.94 ± 0.04 MΩ; *n* = 29). Thirdly, recordings were made at depolarized voltages (-30 mV; with the exception of the Sr^{2+} and Cd^{2+} experiments) so that the amplitudes of synaptic currents were reduced and voltage-gated currents inactivated. The effectiveness of this approach is illustrated in Fig. 1*D* depolarization from -60 mV activated large voltage-gated currents, but at a holding potential of -30 mV these currents were completely inactivated.

Climbing fibre EPSCS recorded under these conditions were still distorted by an escape voltage which arose due to the large size of the CF current. This is likely to have two components; voltage drop down the dendrite (see ‘Consideration of remaining voltage-clamp errors’, below) and voltage drop across the recording pipette (somatic voltage escape). To estimate the latter component, we made recordings from Purkinje cells that were patched with two electrodes, one measuring voltage and the other current (Fig. 1*E*). During the large CF synaptic current the voltage in the cell soma deviated from the command voltage (upper dashed line). This voltage drop was due to current flowing through the current recording electrode and could therefore be calculated from the compensated electrode resistance and the current. This calculation relies on the fact that the EPSC current-voltage relationship was linear (data not shown) and that the change in EPSC kinetics over the range of escape was negligible (change in half-decay time (*t*_{½}), 5.3 ± 0.9% between -30 and -15 mV; *n* = 4). Figure 1*F* shows that the correction using the measured voltage and the correction using the voltage calculated from the compensated electrode resistance agree closely. We further tested the adequacy of our escape-voltage correction by comparing the corrected waveforms of control EPSCs (with typical escape voltages) with corrected EPSCs where the synaptic conductance had been substantially reduced with a sub-maximal concentration of the non-NMDA receptor antagonist CNQX (Fig. 6*A*), so that there was little voltage escape. No change in the half-decay time of the EPSC was observed following reduction of the EPSC to 14 ± 1% of control with CNQX (*n* = 5; *P* = 0.26, Student's *t* test), indicating that our correction procedure restored EPSC waveforms distorted by the voltage escape across the patch electrode. We have therefore used this approach to calculate the somatic voltage escape for each EPSC in this study and have corrected the EPSC waveform to that expected if there were no voltage escape (with the exception of the experiments involving Sr^{2+}, where voltage escape was negligible).

### Consideration of remaining voltage-clamp errors

The time course of the EPSC will be affected by low-pass filtering by the cell circuit. Simulations using a multicompartment model Purkinje cell from a P14 animal with 100 synaptic contacts extending along the primary and secondary dendrites suggests that EPSCs recorded at the soma will yield an underestimate of the synaptic conductance (~65% of the true value; A. Roth & M. Häusser, personal communication). Since the peak variance will also be reduced, quantal size measured directly and that estimated from MPF analysis will be attenuated by cell filtering to a similar degree. However, this distortion of the synaptic current will have relatively little effect on estimates of release probability because attenuation of the EPSC and attenuation of quantal size (and therefore the maximal response) will be similar.

Correction for voltage escape at the soma by compensating for the voltage drop across the recording electrode does not correct for any voltage drop that occurs due to current flow across the spine neck or down the proximal dendrites. Simulations indicate that at high release probabilities, when the synaptic conductance change is maximal, the drop in driving force down the dendrite could be substantial. However, three experimental observations argue against this being a major problem. Firstly, there was no difference between the quantal size estimated at low release probabilities when the voltage drop was minimal, and the quantal size estimated at high release probabilities when the voltage drop was maximal (see Fig. 5*B*). Secondly, there was a positive correlation between the synaptic conductance and estimated quantal size (*P* = 0.01, Spearman rank test). Thirdly, quantal size estimated with MPF analysis was similar to that measured directly with Sr^{2+} and Cd^{2+}. The simplest explanation of these results is that the drop in the driving force down the dendrite was lower in our recordings than suggested by simulations, possibly due to the synapses being located more proximally than assumed in the simulation. However, we cannot rule out the possibility that the quantal conductance increased with release probability, as a result of spillover of glutamate from neighbouring sites, compensating for the drop in driving force.

### Analysis of EPSCs

EPSCs were recorded in solutions containing different [Ca^{2+}] and [Mg^{2+}] that gave a range of different release probabilities for MPF analysis. CFs were stimulated for sufficient time for EPSCs to be recorded at steady state for a particular probability condition. Series resistance and reversal potential were measured at the beginning and end of each epoch and series resistance stability was monitored throughout, from the current response to a voltage step prior to each EPSC. EPSCs were then corrected using these parameters and the command potential. Corrected EPSCs were averaged by aligning on the stimulus artifact (unless there was a frequency-dependent change in latency), and the time of the peak amplitude determined. The amplitude of individual EPSCs was then calculated from the difference between two 0.2 ms windows, one placed just prior to the artifact and the other at the time of the peak of the mean current. Current fluctuations in the background were estimated in the same manner with the two measurement windows shifted into the pre-event baseline. EPSC conductance, variance and *CV* were calculated from the largest number of contiguous EPSCs whose amplitude showed no temporal correlation (Spearman rank test implemented in MathCad, MathSoft International, UK). MPF analysis data were plotted as *CV* or variance against mean synaptic conductance (Fig. 2*B* and *C*).

Data analysis was carried out using Clampfit (Axon Instruments), Origin 4.10 (Microcal, Northhampton, MA, USA) and Statistica (StatSoft, Tulsa, OK, USA). EPSC decays were fitted with exponential functions using a Simplex routine implemented in ‘*N*’ (Stephen Traynelis, Emory University, Atlanta, GA, USA). Plots of *CV* against mean synaptic conductance, variance against mean synaptic conductance, and EPSC amplitude recovery were fitted using a Levenberg-Marquardt algorithm implemented in Mathematica (Wolfram Research, UK) and SigmaPlot (Jandel, Germany). Each data set was fitted several times with a range of different initial parameters to ensure that the fitted equation converged on one solution. In some fits of *CV* and variance against mean synaptic conductance plots, the quantal parameters *N* and *q* were constrained to be > 0. In most cases fits converged on single finite solutions. However, when local minima occurred these were located and the solution was chosen with the lowest sum of squares of the residuals. Each fit was checked ‘by eye’ to ensure the equation fitted well to the data. All data reported are expressed as means ±s.e.m. Groups were compared with Student's *t* test and differences were considered significant at the 5% level.

### Estimation of quantal parameters with binomial and multinomial models

We have used two classes of statistical model to estimate quantal parameters from *CV*-conductance and variance-conductance plots (MPF analysis plots). Simple binomial and multinomial models, which assume that transmitter release probability is uniform, and compound binomial and multinomial models, which incorporate non-uniform release probability.

### Uniform release probability models

We first used the simple binomial model to estimate quantal parameters from MPF analysis plots, which assumes that release probability and quantal size are uniform. This can be expressed in terms of *CV* and conductance, *G*, (for the *CV*-conductance plots) where *q*_{b} (subscript b denotes binomial model) is equal to the quantal size, *q*, and *N*_{b} is equal to the number of functional release sites, *N*, when there is no quantal variability:

Alternatively, the same binomial model can be expressed in terms of variance (σ^{2}) and conductance (for the variance-conductance plots):

It should be noted that if the quantal size is not uniform, *q*_{b} will not be the true arithmetic mean but weighted towards the larger quantal amplitudes.

In the second model we included intersite quantal variability (this could result from different quantal conductances or imperfect voltage clamp), which was measured directly from asynchronous EPSCs evoked in the presence of Sr^{2+} (Fig. 3*A* and *B*). We took an approach similar to that used by Markram, Lübke, Frotscher, Roth & Sakmann (1997) to derive a simplified multinomial equation that included intersite quantal variability as measured at the soma by assuming that the release probability was uniform and that intrasite variance was negligible (though this can be included in the model if known; see eqn (8)). With these simplifications the multinomial model can be expressed in terms of the *CV*, quantal size (*q*_{m}), number of release sites (*N*_{m}) (subscript m denotes multinomial model), and the coefficient of variation of the intersite quantal amplitude, *CV*_{qII}:

Alternatively, this multinomial model can be expressed in terms of variance and conductance:

These simple multinomial and simple binomial functions have an identical shape, and when *CV*_{qII} = 0 the simple multinomial model reduces to the simple binomial. *CV*_{qII} cannot be determined from the fit and must be measured independently. In both models the release probability was calculated from the ratio of the synaptic conductance and the maximal synaptic response (the product of *N* and *q*).

### Effect of non-uniform release probability on the variance-conductance relationship

The simple binomial and simple multinomial models assume a uniform release probability. Since we were unable to measure the release probability at each release site (*P*_{r}), we do not know the shape of the *P*_{r} distribution, or how this would change as a function of mean release proba{fontsize bility (*P*¯_{r}). We have therefore used simulations to examine the effect of non-uniform *P*_{r} on the shape of the EPSC variance-conductance relationship. The effect of non-uniform *P*_{r} was modelled with beta distributions implemented in Mathematica. Beta distributions such as those illustrated in Fig. 4*A* were generated with the density function:
where *B*(α,β) is the beta function. Since the mean of a beta distribution is simply α/(α+β), a particular distribution can be defined in terms of its mean and α. Each simulated *P*_{r} distribution could therefore be defined in terms of α and mean release probability *P*¯_{r}, by substituting for β where β = α/*P*¯_{r} - α. This approach allowed a large set of different beta distributions to be generated for a particular α value by varying *P*¯_{r} from 0 to 1. A wide variety of distribution shapes was covered by using a range of α values (0.1, 0.5, 0.9, 1, 2 and 5) to generate the sets of distributions (i.e. compare α = 1 and α = 5 in Fig. 4*A*). It is therefore likely that some of these simulated distributions will approximate the true distribution of *P*_{r}. This approach is strengthened by the fact that the measured *P*_{r} distribution for hippocampal synapses in culture (Murthy *et al.* 1997) is described by a gamma distribution, which is closely related to the beta distribution. Having constructed a set of distributions for each α value that covers the full range of *P*¯_{r}, the coefficient of variation of release probability (*CV*_{Pr}) was then calculated for each distribution from eqn (5):

This gave a *CV*_{Pr} as a function of *P*¯_{r}, illustrated in Fig. 4*B* for α = 0.1, α = 1, and α = 5. The relationship between the expected EPSC variance and mean release probability was then calculated using eqn (6):

Expressing this relationship in terms of synaptic conductance, *G*:

The EPSC variance expected for non-uniform *P*_{r} is less than that for the uniform binomial with the same *P*¯_{r} by an amount equal to *N*_{bqb}^{2} times the variance of *P*_{r} (σ_{Pr}^{2}). Distortion of the variance-*P*¯_{r} relationship (Fig. 4*C*) and the variance-conductance relationship from the parabolic uniform *P*_{r} case therefore depends on the degree of dispersion of *P*_{r}. Because dispersion of *P*_{r} at the CF synaptic connection and its dependence on *P*¯_{r} are unknown, it is not possible to calculate the precise errors involved. However, because the effect of non-uniform *P*_{r} becomes minimal as *P*¯_{r} becomes small (see eqn (6)), the initial slope of the variance*P*¯_{r} (or conductance) relationship was similar to the uniform *P*_{r} case (Fig. 4*C*) even when non-uniformity in *P*_{r} was relatively large (Fig. 4*B*) and the relationship markedly skewed towards larger values (i.e. less variance in higher *P*¯_{r} regions). In such cases estimates of *q* from low *P*¯_{r} regions are less affected by dispersion in *P*_{r}. Reliable estimates of *N* can therefore be obtained with a uniform model when *P*_{r} is non-uniform by combining such estimates of *q* with estimates of *G*_{max} = *Nq* from high *P*¯_{r} measurements.

The presence of non-uniform *P*_{r} may not always result in a variance-mean release probability plot that is skewed towards larger values, since *P*_{r} distributions are possible which are not described by our model. If the *CV*_{Pr} is large as *P*¯_{r}→ 0 and decays more rapidly as a function of *P*¯_{r} than the examples in Fig. 4*B*, the second term in eqn (6) can become significant at low *P*¯_{r} values. Under these conditions the variance-conductance relationship may be skewed towards smaller values with the initial slope significantly lower than the uniform case, giving an underestimate of *q*. In such cases the steeper slope at high *P*¯_{r} values will provide a more accurate estimate of *q*. This situation can be modelled with beta distributions where α is varied as a function *P*¯_{r} and β is kept constant (see below). We have not used this model (derived below) at the CF synapse as variance-conductance plots that were not symmetrical were skewed towards larger values. It is also possible that the variance-conductance relationship could be parabolic when *CV*_{Pr} > 0 if it is constant over the full *P*_{r} range (see eqn (7)). However, this could only occur if synapses were present with *P*_{r} = 0. Since we define *N* as the number of *functional* release sites, the variation in release probability due to ‘silent’ synapes (i.e. release sites where *P*_{r} remains at 0) can be disregarded. If such sites are excluded, *CV*_{Pr} must become lower when the mean release probability approaches 1 because 0 < *P*_{r}≤ 1. Thus, any non-uniformity in *P*_{r} must be associated with a *CV*_{Pr} that depends on *P*¯_{r}. A symmetrical variance-conductance plot could only occur with non-uniform *P*_{r} if the *CV*_{Pr}-*P*¯_{r} relationship was such that it compensated for the reduced effect of *CV*_{Pr} at low *P*¯_{r} values. This situation seems unlikely.

### Non-uniform release probability models

We have developed compound binomial and compound multinomial models that incorporate non-uniform release probability because this property appears widespread at central synaptic connections (Jack *et al.* 1981; Walmsley *et al.* 1988; Stricker *et al.* 1996; Issacson & Hille, 1997; Murthy *et al.* 1997). The theoretical relationship for a synaptic connection with non-uniform *P*_{r} and non-uniform *q* but with no correlation between these two parameters is shown in eqn (8) (see Frerking &Wilson 1996):

where *CV _{qI}* is the coefficient of variation of intrasite quantal variation. This general equation has too many free parameters to be useful for extracting quantal parameters. However, it can be simplified by assuming that the dispersion in

*P*

_{r}as a function of

*P*¯

_{r}(

*CV*

_{Pr}in the relationship) can be approximated by a family of beta distributions with a particular α value. By substituting

*CV*

_{Pr}in eqn (8) for eqn (5), a compound multinomial relationship can be derived. When

*CV*is known from the mini distribution and intrasite variance is negligible,

_{qII}*q*

_{m},

*N*

_{m}and α are the only free parameters:

If *CV _{qII}* is unknown, the (1 +

*CV*

_{qII}

^{2}) term can be dropped giving a compound binomial expression.

These models have several advantages. Firstly, the level of non-uniformity in *P*_{r} is not assumed. The α value (and therefore the *CV*_{Pr}) is determined by the shape of the variance-conductance data. This approach will therefore be applicable to cases where each cell in the population has a different *P*_{r} distribution. Secondly, the full MPF analysis plot can be fitted even when the variance-conductance relationship is highly skewed, allowing more accurate estimates of the underlying quantal parameters than from initial slope estimates. Thirdly, these models include the simple multinomial model (or the simple binomial model) as one of the solutions (α→∞). Fourthly, this approach provides some information about the relationship between *CV*_{Pr} and *P*¯_{r}, which can be calculated from α and eqn (5). However, it should be noted that the estimated α value indicates the family of beta distributions that can generate the fitted relationship. It does not necessarily imply that the *P*_{r} distributions underlying the EPSCs are beta distributions. Even if the underlying *P*_{r} distributions are not well approximated by a beta distribution, eqn (9) is likely to provide a better estimate of the initial slope of the relationship, and thus *q*_{m} and *N*_{m}, than the uniform *P*_{r} model. If finding a solution to eqn (9) is problematic, initial guesses for the fit can be determined from the variance-conductance plot normalized by the maximal conductance (*G*_{max}~*N*_{m}*q*_{m})by applying a simplified version of eqn (9) where only *q*_{m} and α are free parameters and where *G*_{n} is *G/G*_{max}:

If variance-conductance plots are skewed towards smaller values, eqns (9) and (10) are not appropriate. In such cases, where the *CV*_{Pr} becomes large as *P*¯_{r}→ 0 and the relationship decays rapidly with *P*¯_{r}, the steeper slope of high *P*¯_{r} region will give a more accurate estimate of *q*. Quantal parameters can be estimated under these conditions with a related model, again based on the beta distribution, but where α is varied instead of β. As *P*¯_{r} becomes small, α becomes small and thus *CV*_{Pr} becomes very large. The equation for this model can be derived from the mean and variance of a beta distribution (in a similar way to eqn (9)) but by substituting for α (α = *P*¯_{r}β/(1 - *P*¯_{r}) instead of β, giving eqn (11):

This equation gives variance-conductance relationships that are mirror images of those for eqn (9) when β and α have the same value in the two equations. The large range of solutions to eqns (9) and (11) make these models applicable to synapses with a wide range of underlying *P*_{r} distributions.

### Effect of correlations between release probability and quantal size on the variance-conductance relationship

A correlation between quantal size and release probability is theoretically possible at synapses where both release probability and quantal size are non-uniform. Although a recent study (Nusser, Cull-Candy & Farrant, 1997) suggested that no such correlation existed for inhibitory synapses on cerebellar stellate cells, another study suggests a positive correlation between these parameters at excitatory hippocampal synapses (Murthy *et al.* 1997). We have therefore used simulations to examine how correlations between quantal size and release probability would affect estimated quantal parameters. The release probability was calculated at 100 independent release sites from a cumulative beta distribution (using the InverseBetaRegularized function in Mathematica). This provided a list of 100 ranked *P*_{r} values for each mean release probability, *P*¯_{r}. Simulations with different levels of non-uniformity in *P*_{r} were generated by using a range of α values (5, 1, 0.5 and 0.1). A positive correlation between *P*_{r} and quantal size was produced by ranking quantal amplitudes as a function of site number. A negative correlation was generated by simply reversing the ranking order of *q*. The mean synaptic conductance and variance were calculated by summing *qP*_{r} and mean square conductance-mean conductance squared over the 100 release sites. Quantal parameters were estimated from variance-conductance plots by fitting the compound multinomial model (eqn (9)).

Two quantal distributions were used. In the first model, ranked quantal size was calculated by making the quantal amplitude at each site a linear function of site number. A slope of 1/101 gave a mean *q* = 0.5, *CV*_{qII} = 0.57 and a range of *q* values of 1/101 to 100/101. In the second, biologically more accurate simulation, CF quantal events were used (measured in Sr^{2+}, Fig. 3*B*). A normalized cumulative distribution was made from 1438 quantal current amplitudes pooled from four cells. A list of 100 ranked quantal sizes was generated from these data by interpolating the cumulative probability verses quantal amplitude plot. Figure 4*D* shows an example of a negative correlation between *q* at each site and *P*_{r} for a simulation in which measured quantal events were used. The four different plots show the relationship between *q* and *P*_{r} at different mean release probabilities.

Since fits to the correlated variance-conductance relationships gave qualitatively similar results for the two simulations, we only show results from simulations using measured quanta in Sr^{2+} (Table 1 and Fig. 4). Plots from positive correlations between *q* and *P*_{r} were skewed in a manner similar to those for non-uniform *P*_{r} when *q* was uniform but with an initial slope that was steeper than the weighted mean quantal size (Fig. 4*E*, •). With negative correlations, variance-mean release probability relationships exhibited rather symmetrical shapes (Fig. 4*E*, ^). This presumably resulted from two effects that distort the relationship in opposite ways: a skew towards larger values from non-uniform *P*_{r} (see Fig. 4*C*) and a skew towards smaller values from low *P*_{r} sites with large quantal size which will generate substantial variance at high *P*¯_{r} values. The variance was lower than for the uncorrelated case (Fig. 4*E*, continuous line; eqn (8)) and resulted in underestimates of *q*_{m} and over estimates of *N*_{m}. However, it should be noted that for both simulations the correlation is the strongest possible for the quantal amplitude distribution and the beta distribution of release probabilities. Thus, distortions in the estimates of *q*_{m} and *N*_{m} (Table 1) probably represent an extreme case. Furthermore, both positive and negative correlations gave reliable estimates of *P*¯_{r}.

### Remaining assumptions

Application of these statistical models assumes (1) a constant number of independent release sites, (2) the quantal size does not vary with time, and (3) linear summation of quanta. Several pieces of evidence indicate that these are appropriate at the CF synaptic connection. The very low EPSC variance observed at high release probabilities (*CV* = 0.0103 at 0.033 Hz in 4 mm Ca^{2+}, 0.5 mm Mg^{2+}) suggests that any temporal fluctuations in *N* and release probability must be small. It also argues against any co-ordinated temporal modulation in quantal size at this input. The fact that direct measurement of quantal events recorded in the presence of Sr^{2+} (Fig. 3*A* and *B*; and those in Cd^{2+}) have amplitudes similar to the quantal size estimated from MPF analysis is consistent with release sites being independent and summating linearly. If transmitter released from neighbouring sites overlaps, summation of quantal events might deviate from linear if synaptic receptors were not saturated by a quantum of transmitter. Consistent with transmitter spillover in the CF cleft, the half-decay time of the EPSCs was longer (in 5 out of 6 cells) at high release probabilities (0.96; *t*_{½} = 3.7 ± 0.4 ms) than at low release probabilities (0.06; *t*_{½} = 2.5 ± 0.1 ms, *n* = 6; *P* < 0.05, Student's *t* test). However, these observations would also be consistent with changes in the release process itself giving different transmitter concentration profiles at different release probabilities.

## RESULTS

### Quantal parameters underlying transmission at the climbing fibre-Purkinje cell synaptic connection

Synaptic transmission at CF inputs was investigated with patch-clamp recording from visually identified Purkinje cells in cerebellar slices from young rats. We used several strategies to optimize the quality of voltage clamp and we corrected for deviations from the command voltage at the soma (see Methods). The average waveform of CF EPSCs following such correction had a 10-90% rise time of 0.83 ± 0.03 ms and a peak conductance of 200 ± 23 nS (*n* = 16; as measured at the soma) when evoked at 0.2 Hz. Since the fit to one exponential function was inadequate, the sum of two exponential functions was used to describe the EPSC decay. The mean time constants were 3.2 ± 0.2 and 10 ± 1 ms (*n* = 15), with the fast component constituting 82 ± 2% at the peak.

We investigated the quantal parameters underlying transmission at this synaptic connection with multiple probability fluctuation (MPF) analysis - illustrated in Fig. 2. This figure shows CF EPSCs evoked at 0.2 Hz in solutions with different [Ca^{2+}]/[Mg^{2+}] ratios which gave different probabilities of release. In the presence of high [Ca^{2+}], low [Mg^{2+}], when the release probability was maximal, the EPSC was large. Furthermore, the EPSC amplitude variability, as indicated by the coefficient of variation (*CV*, standard deviation/mean), was small. As the EPSC amplitude was reduced by lowering [Ca^{2+}] and raising [Mg^{2+}], the *CV* increased (Fig. 2*A*). No failures occurred even at the lowest calcium concentration used, ruling out the possibility of contaminating variance arising from failure to stimulate the presynaptic axon. Figure 2*B* shows the relationship between *CV* and EPSC (expressed as conductance) for this cell. The data points, recorded at different release probabilities, were well fitted by a simple binomial model (eqn (1), Methods) indicated by the continuous line. Figure 2*C* shows a plot of variance verses conductance for data from the same cell, fitted with eqn (2) (Methods). This type of plot has been used previously by Clamann *et al.* (1989) to investigate quantal parameters during post-tetanic potentiation of synaptic currents, and is similar to the convention used for non-stationary noise analysis of ion channels (Sigworth, 1980). It is particularly useful for quantifying *q* and mean release probability (*P*¯_{r}), and showing changes in the latter. The relationship was parabolic with variance minima occurring at both low and high probabilities of release and maximal variance at an intermediate *P*¯_{r}.

By fitting the variance-conductance data with a simple binomial model (eqn (2), Methods) it was possible to estimate the underlying quantal parameters. This approach gave a mean of 510 ± 70 (range, 285-740; *n* = 6) for the number of functional release sites (*N*_{b}; subscript indicates the estimate of *N* is from the binomial model), and a mean of 0.50 ± 0.04 nS (range, 0.34-0.61 nS) for the quantal size (*q*_{b}). As the underlying non-NMDA channels are relatively impermeable to calcium (Häusser & Roth, 1997), our estimates of *q*_{b} and *N*_{b} are unlikely to be affected by the changes in Ca^{2+} that we have used to alter *P*¯_{r}.

### Extension of the simple binomial model to include variability in quantal size

Variations in quantal size can occur at an individual release site (the so-called intrasite, type I or temporal quantal variance) or as a result of different quantal sizes at different release sites (intersite, type II or spatial quantal variance). These two types of quantal variance affect the relationship between variance (or *CV*) and synaptic conductance in different ways (Wahl, Stratford, Larkman & Jack, 1995; Frerking & Wilson 1996; see also Sigworth, 1980, for equivalent situation with ion channel subconductance levels), and can complicate the interpretation of quantal parameters estimated with the binomial model. At high release probabilities (*P*¯_{r}→ 1), only variance arising from intrasite quantal variation remains. An upper limit can therefore be estimated for intrasite quantal variability from the *CV* of the EPSC (0.0103 ± 0.0009, *n* = 6, in 4 mm Ca^{2+}/0.5 mm Mg^{2+}), the quantal size (recorded independently, see below) and the number of release sites. This gave an upper limit of *CV* = 0.19 indicating that quantal variability at an individual site is relatively low.

The total quantal variance (intra- and intersite) can, in principle, be estimated from the distribution of miniature EPSCs. However, except during early development, Purkinje cells are innervated by both parallel and climbing fibre inputs, so it is not possible to measure CF miniature currents in isolation. We have overcome this problem by evoking CF EPSCs in the presence of Sr^{2+} (which causes desynchronized transmitter release from activated terminals; Abdul-Ghani, Valiante & Pennefather, 1996). This gave a frequency of CF quanta approximately 10-fold higher than the background frequency in the absence of CF stimulation. Figure 3*A* shows a cell in which CF EPSCs were evoked in 5 mm Sr^{2+} and 1 mm Mg^{2+} at 0.2 Hz. We analysed synaptic events in a 520 ms window starting 200 ms after the evoked event, in which there was no correlation of quantal amplitude with time (*P* > 0.05, Spearman rank test), indicating that the level of postsynaptic receptor desensitization was constant. The distribution of quantal amplitudes was skewed (see Fig. 3*B*, □) and had a *CV* of 0.37 ± 0.01 (background noise corrected; *n* = 4). Since the intrasite variability was relatively small, this measure of total quantal variability can be used as an estimate for intersite quantal variability. It is possible that this could still be an underestimate of intersite variability for MPF analysis because any voltage drop down the dendrite during CF-evoked currents would increase variability seen at the soma (see Methods).

If the simple binomial model is used under conditions where *q* is non-uniform across sites, the estimate of *q*_{b} will be weighted towards the larger events, and *N*_{b} may be underestimated. Under such conditions, a simple multinomial model is required to estimate the true mean quantal size. However, this can be reduced to a simple binomial expression multiplied by a constant that includes the intersite variability (Markram *et al.* 1997; see Methods). A simple multinomial model (eqn (4)), with a fixed intersite quantal variability of *CV*_{qII} = 0.37, gave an estimate of 580 ± 80 (*n* = 6) for the number of functional release sites (*N*_{m}; subscript denotes multinomial model), 13.7% greater than that estimated from the simple binomial (*N*_{b}). The mean quantal size (*q*_{m}) was 0.44 ± 0.03 nS, 88% of that estimated from the simple binomial (*q*_{b}). However, since the product of these two parameters was unchanged, estimates of the maximal response and release probability are similar for both models. Both the binomial and multinomial models assume intrasite quantal variance is negligible. If this is not the case, and the CF intrasite quantal variability is close to our estimate of its the upper limit, *P*¯_{r} could be underestimated by ≤ 4%.

### Non-uniform release probability

A potential problem with the application of simple binomial and simple multinomial models to central synapses is that they assume that the *P*_{r} is uniform across release sites. Non-uniformity in *P*_{r} has been observed at several central synapses (Walmsley *et al.* 1988; Rosenmund *et al.* 1993; Stricker *et al.* 1996; Isaacson & Hille, 1997; Murthy *et al.* 1997) and would complicate our analysis because the EPSC variance will be less than for the uniform *P*_{r} case and, thus, estimates of quantal parameters will be incorrect (Quastel, 1997). Since we were unable to measure *P*_{r} at each release site, we investigated the effects of non-uniformity in *P*_{r} on MPF analysis with simulations. We have modelled spatial non-uniformity in *P*_{r} by using sets of beta distributions to describe both the *P*_{r} distribution (Fig. 4*A*) and how it changes as a function of *P*¯_{r} (Fig. 4*B*). These models indicate that large dispersions in *P*_{r} can result in substantial distortion of the EPSC variance-conductance relationship (Fig. 4*C*). However, they also show that estimates of *q* made from low *P*¯_{r} regions are likely to be little affected by dispersion in *P*_{r} when the variance-conductance relationship is skewed towards larger values. Although the pooled variance-conductance data (normalized to the maximum conductance from each cell) had a symmetrical parabolic shape (Fig. 5*B*), indicating that there was little systematic dispersion in release probability, some plots from individual cells (Fig. 5*A*) were skewed towards larger values, consistent with dispersion in *P*_{r} at some climbing fibre connections (see below).

### A model that incorporates both non-uniform quantal size and non-uniform release probability

Since the level of spatial non-uniformity in *P*_{r} is likely to vary from cell to cell, we developed a multinomial model that includes both non-uniform quantal size and non-uniform release probability (eqn (9)). This compound multinomial model incorporates spatial non-uniformity in *P*_{r}, yet remains relatively simple, by assuming that the dispersion in *P*_{r} as a function of mean release probability can be approximated by a family of beta distributions with a particular α value (see Methods). Application of the compound multinomial model gave estimates of *q*_{m} = 0.50 ± 0.03 nS, with a range of 0.40-0.58 nS, and *N*_{m} = 510 ± 50, with a range of 313-652 (*n* = 6). The dispersion in *P*_{r}, as indicated from the α value, ranged from 1.2 to 5.1 × 10^{4} for the six cells. In half of the cells α < 10, indicating significant dispersion in *P*_{r} (Fig. 5*A* this can be calculated from α using eqn (5) - but see Methods for interpretation). However, when the compound multinomial model was fitted to the pooled data (Fig. 5*B*), the large α value obtained (10^{4}) confirmed that there was no systematic distortion of the variance-conductance relationships by dispersion in *P*_{r} at the CF synapse.

### Testing the method of multiple-probability fluctuation analysis

We have tested our method by comparing results from MPF analysis with several independent measures. One quantal parameter that could be measured directly was the quantal size. As illustrated in Fig. 3*B*, direct measurement of the quantal size in Sr^{2+} gave a peak conductance of 0.55 ± 0.01 nS (*n* = 4), similar to *q*_{m} estimated from the MPF analysis using the compound multinomial model (*P* = 0.26; Student's *t* test). We also estimated the quantal size in the presence of 2 mm Ca^{2+}. This was done by using Cd^{2+} (a non-specific blocker of calcium channels) to reduce the release probability to very low levels such that it was possible to resolve quantal events (Issacson & Walmsley, 1995; data not shown). The signal-to-noise ratio was relatively low in these Cd^{2+} recordings, so we estimated the quantal size from the ratio of the variance and mean current (simplification of eqn (2) at low *P*¯_{r}), as this did not rely on separating events from failures. The quantal size was 0.64 ± 0.09 nS (*n* = 5), not significantly different from *q*_{b} estimated from MPF analysis using the compound binomial model (*P* = 0.47; see Table 2 for comparison of quantal size obtained with different methods).

In a second independent test we reduced the postsynaptic responsiveness to 14 ± 1% of control (*n* = 5) with a submaximal concentration (1 μm) of the non-NMDA receptor antagonist CNQX. Figure 6*A* shows the averaged EPSCs, while Fig. 6*B* shows the effect of CNQX on the *CV*-conductance plot. Estimates for *N*_{m} were unaffected by CNQX (*P* = 0.94, Student's *t* test) but, as expected, the quantal size was substantially reduced (to 13 ± 2%). Indeed, the ratio of the control quantal size to that in CNQX, estimated from the fit, was not different from the ratio of the EPSC amplitudes (*P* = 0.09, Student's *t* test). This demonstrates that the MPF analysis method is capable of correctly assigning a postsynaptic locus to the actions of CNQX. These control experiments confirmed that MPF analysis, when used with an appropriate statistical model, gave quantitative information of quantal parameters at this synaptic connection.

### Quantal parameters measured during frequency-dependent depression

One of the most striking features of the CF input is the frequency dependence of the EPSC amplitude during sustained stimulation. This is illustrated in Fig. 7*A*, which shows averaged CF EPSCs obtained at different stimulation frequencies (0.033-10 Hz) under steady-state conditions. The relationship between EPSC peak amplitude and stimulus frequency is shown in Fig. 7*B* At 10 Hz stimulation the mean peak current was only 2 ± 1% (*n* = 4) of that at 0.033 Hz. In general, such depression could arise presynaptically, as has been long established at the NMJ (del Castillo & Katz, 1954*b*). But, at central synapses the situation is more controversial since both presynaptic (Larkman *et al.* 1991; Mennerick & Zorumski, 1995; Borst & Sakmann, 1996; von Gersdorff *et al.* 1997) and postsynaptic (Trussell *et al.* 1993; Zhang & Trussell, 1994) mechanisms have been proposed. We have investigated the locus of depression at the climbing fibre-Purkinje cell connection by establishing how the underlying quantal parameters were affected by stimulation frequency.

In the MPF analysis described above (Fig. 2), EPSCs were recorded at a fixed low frequency of stimulation, and different release probabilities were set by varying the [Ca^{2+}]/[Mg^{2+}] ratio of the external solution. We repeated this protocol at a higher stimulation frequency to examine which of the quantal parameters changed during the depression of the synaptic conductance. Figure 8*A* shows that changing the stimulation frequency from 0.2 to 3 Hz resulted in a significant depression in the EPSC amplitude in 2 mm Ca^{2+} and 1 mm Mg^{2+}, as expected. The plots in Fig. 8*B* and *C* show the *CV*-conductance and variance-conductance relationships recorded at 3 Hz stimulation. In both cases the data collected at 3 Hz fall on the same theoretical relationships estimated from the low-frequency data illustrated in Fig. 2 (continuous lines) but are shifted to smaller amplitudes, indicating that *N* and *q* remain the same for these two stimulation frequencies. When the variance-conductance plots obtained at 3 Hz were fitted (with α values constrained to the 0.2 Hz control value) they gave a value of *q*_{m} that was not different (*q*_{m} = 0.50 ± 0.04 nS; *P* = 0.86, Student's *t* test) from that obtained at 0.2 Hz. This demonstrates that the EPSC depression at 3 Hz stimulation is caused by a reduction in release probability rather than by a change in the postsynaptic responsiveness.

We next examined whether depression of the EPSC was still presynaptic at frequencies up to 10 Hz. However, instead of changing the [Ca^{2+}]/[Mg^{2+}] ratio and stimulating at a single frequency, we recorded synaptic currents at several different frequencies in the same external solution. As shown in Fig. 9, simply changing the stimulation frequency over the range 0.033-10 Hz gave a full variance-conductance relationship where each data point corresponds to a single frequency of stimulation. Fitting the compound multinomial model to variance-synaptic conductance plots gave values of 420 ± 50 for *N*_{m} and 0.76 ± 0.1 nS for *q*_{m} (*n* = 5). Two factors complicate comparison of these estimates with those where the [Ca^{2+}]/[Mg^{2+}] ratio was altered. There was a significant positive correlation between *q*_{m} and *G* (recorded in 2 mm Ca^{2+} at 0.2 Hz; *r* = 0.7363, *P* = 0.01, Spearman rank test, slope = 1.49 × 10^{−3}) and a difference in the synaptic conductance between the two groups of cells (group mean at 0.2 Hz; 271 and 199 nS). When these factors were taken into account by normalizing to a synaptic conductance of 200 nS there was no difference between the two *q*_{m} values (*P* = 0.148; see Table 2 for values). The α values (mean α = 4.4; range, 0.5-18; α and synaptic conductance were not correlated, *P* = 0.4) obtained from variance-conductance plots where frequency was varied indicate that dispersion in *P*_{r} became greater at higher stimulation frequencies. If a beta distribution is a good approximation to underlying *P*_{r} distribution, the α values we observe at the CF correspond to a range of coefficient of variation of release probabilities that overlap with those estimated directly at hippocampal synapses (Issacson & Hille, 1997; Murthy *et al.* 1997). These results strongly suggest that frequency-dependent synaptic depression at this connection is entirely presynaptic over the range 0.033-10 Hz. Calculation of *P*¯_{r} for the frequency range over which depression was presynaptic showed that there was a 50-fold reduction in release probability between the lowest and highest stimulation frequencies (see Fig. 10*A*). The mean release probability ranged from 0.90 ± 0.03 (*n* = 5) during stimulation at 0.033 Hz to 0.02 ± 0.01 at 10 Hz (*n* = 4; recorded in 2 mm Ca^{2+}, 1 mm Mg^{2+}).

### Testing for the presence of presynaptic metabotropic receptors

We investigated whether metabotropic glutamate receptors (mGluRs) mediate frequency-dependent depression at the CF terminal by applying antagonists to group I, II and III mGluRs at different stimulation frequencies. EPSC amplitude remained unchanged in the presence of 500 μm MCPG, 200 μm MTPG or 250 μm MPPG at both low (0.2 Hz) and high (3 or 5 Hz) stimulation frequencies (*n* = 3). Furthermore, the mGluR agonist L-AP4 (50 μm), had no effect on the climbing fibre EPSC at 0.2 or 3 Hz (*n* = 4, *P* > 0.05, Student's *t* test). These results indicate that frequency-dependent depression at the CF is not mediated by activation of mGluRs.

### Temporal characteristics of transmitter release probability

To understand the temporal characteristics of presynaptic depression in more detail, we used two experimental approaches to gather information about release probability kinetics. First, we used paired-pulse stimulation repeated at low frequency (0.033 Hz), which induced presynaptic depression.

Interpreting EPSC recovery from depression in terms of release probability assumes that the depression is presynaptic, as we have shown for sustained stimulation up to 10 Hz. However, in the paired-pulse experiments, depression of a second EPSC after a 100 ms interval is not directly comparable with the depression by sustained stimulation at 10 Hz because *P*¯_{r} for the first EPSC in the pair (EPSC1, Fig. 10*B*) was close to one. We tested whether a paired-pulse depression at a 100 ms interval was presynaptic by estimating the quantal size from variance-conductance relationships constructed from the first and second EPSCs in the pair. The mean quantal size estimated from a simple multinomial fit was 0.39 ± 0.05 nS (*n* = 5) in these experiments, not significantly different from the quantal size estimated from MPF analysis (0.44 nS; *P* = 0.46, Student's *t* test). However, we cannot rule out the possibility that a small contribution of desensitization remains but is not detectable with our method.

Figure 10*B* shows several superimposed paired-pulse trials; as the interval between the two EPSCs was increased from 100 ms, the amplitude of the second EPSC recovered. In five out of six cells, a single exponential function gave a poor fit to the recovery data. In these cells a fit to the sum of two exponential functions gave time constants of 350 ± 90 ms (37 ± 7%) and 3.2 ± 0.4 s (*n* = 5) as illustrated in Fig. 10*C*. Since *P*¯_{r} for EPSC1 was close to one, this biphasic recovery represents the time course of recovery of *P*¯_{r} following transmitter release at practically all release sites.

In order to establish whether processes with similar kinetics were involved during sustained stimulation, we used a second approach, illustrated in Fig. 11*A* This shows an example of the onset time course of EPSC depression during a 180 s train at 2.8 Hz stimulation. The amplitude of the second EPSC in the conditioning train dropped to 57 ± 4% (*n* = 5) of the first EPSC. The subsequent depression in EPSC amplitude (small circles) could not adequately be described by a single exponential function in three out of five cells, so a dual exponential function was used. In the remaining two cells the onset was well fitted with a single exponential, and this had a the time constant similar to the slow component of the dual exponential fit in the other cells (τ_{1} = 0.9 ± 0.4 s, 32 ± 5%, *n* = 3; τ_{2} = 34 ± 3 s, *n* = 5). Each conditioning train was followed by a single EPSC of variable latency (EPSC_{recovery}, large circles in Fig. 11*A*), and trials were separated by an interval of 200 s to allow full recovery of *P*¯_{r}. The pooled recovery time course data (*n* = 5), shown in Fig. 11*B*, were fitted with the sum of two exponentials with time constants of 1.7 s (amplitude, 61%) and 24 s. These results indicate that following release the recovery of *P*¯_{r} is slow at the CF synapses and that the kinetics of recovery of the presynaptic terminal depends on the stimulation protocol and therefore on the history of the synapse.

## DISCUSSION

In this paper we describe a new quantal analysis method which is applicable when quantal size and release probability are both non-uniform. We have used this method to investigate transmission at the cerebellar climbing fibre-Purkinje cell synaptic connection. We show that this distributed synaptic input has a large number of functional release sites and that the mean release probability across all sites is high at low stimulation frequencies. As the frequency of stimulation increases, the release probability falls off dramatically due to a slow recovery of release probability at each site. These properties appear specialized to act as a low-pass filter, promoting reliable transmission at low frequencies but inhibiting transmission at sustained high frequencies.

### Multiple-probability fluctuation analysis

The major experimental difference between MPF analysis and previous quantal analysis methods (del Castillo & Katz, 1954*a*,Clamann *et al.* 1989; Malinow & Tsien, 1990; Bekkers & Stevens, 1990; Frerking & Wilson, 1996) is that quantal parameters are estimated from multiple measurements made over a wide range of experimentally imposed release probabilities at a single input (but see Miyamoto, 1975). This approach has several advantages. Estimating *q* and *N* from a number of measurements made under different probability conditions is likely to provide more reliable quantal estimates than if only a single experimental condition is analysed. Unlike methods that involve fitting amplitude histograms, MPF analysis allows quantitative estimates of *N* and *q* at synaptic connections where quantal size is non-uniform. This property may be widespread in the CNS neurons, as many miniature distributions are highly skewed. This could result from intra- or intersite non-uniformity in *q* or from spatial non-uniformity in *q* arising from electrotonic attenuation of signals from more distal synapses. If it is not possible to estimate quantal variability independently from the mini distribution, application of the binomial model will give a weighted mean quantal size that deviates from the true mean by a factor of $1+C{V}_{q}^{2}$.

The other main advantage of MPF analysis is that the shape of the relationship between variance and mean synaptic conductance contains information about the dispersion in *P*_{r}. This allows application of models that do not assume that release probability is uniform or that dispersion in *P*_{r} occurs at any fixed level across cells. Recent direct measurements of *CV*_{Pr} at hippocampal synapses (0.3-0.7; Isaacson & Hille 1997; Murthy *et al.* 1997), and the substantial effect of non-uniform *P*_{r} on uniform statistical models (see Methods), highlight the importance of methods that incorporate non-uniform release probability (Jack *et al.* 1981; Walmsley *et al.* 1988; Stricker *et al.* 1996). It should be noted, however, that although our method incorporates dispersion in *P*_{r}, it is possible that errors in estimating *q* and *N* could arise if *q* and *P*_{r} were strongly correlated (see Methods for extreme case). Since errors in estimated *N* and *q* occur in opposite directions and therefore cancel when multiplied together, estimates of mean release probability are relatively insensitive to these complicating synaptic characteristics. At the CF synapse such correlations, if present, must be weak given the similarity between the quantal size estimated from MPF analysis and that measured independently.

MPF analysis is applicable to other types of synaptic connections, even those where peaks in amplitude histograms cannot be resolved. It may therefore be useful in identifying the quantal parameters that change during other forms of synaptic plasticity such as long-term potentiation and long-term depression. To extract the full quantal description of the synapse with MPF analysis, it is necessary to manipulate *P*¯_{r} over a wide range, as the shape of the relationship is necessary for models that incorporate non-uniform *P*_{r}. Fortunately, synapses with a low *P*¯_{r} in control conditions often exhibit presynaptic facilitation, so inducing such facilitation, or increasing calcium, could be used to raise *P*¯_{r} to levels suitable for MPF analysis. If variance-conductance data are skewed towards smaller values, that is, in a direction opposite to that observed at the climbing fibre, a different compound multinomial model is appropriate (eqn (11)) because the steeper slope of the high *P*¯_{r} region will give the best estimate of *q*_{m} (see Methods). Synapses with few release sites would require a greater number of observations to describe the variance accurately, and the underlying *P*_{r} distribution may not be well approximated by a continuous beta distribution. In such cases, when the variance-conductance plot is not symmetrical, *q*_{m} should be estimated from the high or low *P*¯_{r} region with the steepest slope. This is because the region with the steepest slope will be least affected by non-uniform *P*_{r}, which tends to reduce variance, and will therefore give a better estimate of *q*_{m}.

### Number of functional transmitter release sites

Our results show that a single climbing fibre input has the largest number of functional release sites so far described for a central synaptic connection (mean value, ~500). This is 1-2 orders of magnitude greater than typical values for other central excitatory synapses (1-30 sites; Jack *et al.* 1981; Walmsley *et al.* 1988; Gulyás, Miles, Sík, Tóth, Tamamakl & Freund, 1993; Jonas, Major & Sakmann, 1993; Stevens & Wang, 1995; Silver *et al.* 1996; Stricker *et al.* 1996; Buhl, Tamas, Szilagyi, Stricker, Paulsen & Somogyi, 1997), and exceeds the number of release sites at giant synapses in the auditory pathway (100-200 sites; Forsythe & Barnes-Davies, 1993; Trussell *et al.* 1993; Zhang & Trussell, 1994; Isaacson & Walmsley, 1995; Borst & Sakmann, 1996). Indeed, one has to look outside the brain to the NMJ (Heuser, Reese & Landis, 1974) for a synaptic connection with a comparable number of transmitter release sites. We are confident that our estimates of the number of functional release sites are reasonably accurate; we obtained similar values of *N* with three different experimental protocols (when *P*¯_{r} was changed with Ca^{2+} and Mg^{2+}, with frequency, and in the presence of CNQX) and when data obtained at different release probabilities were used for the estimate. Consistent with our estimates from MPF analysis, the quantal content of the maximal EPSC amplitude (430; and therefore a lower limit for *N*) can be calculated using the quantal size measured directly in Sr^{2+}. Table 2 shows *q* and *N* values obtained under different conditions.

It is interesting to consider the physical basis of *N*. In our analysis, *N* represents the number of functional release sites that can each generate a quantal EPSC. If any ‘silent synapses’ are present (where *P*_{r} remains at zero or where there are no postsynaptic receptors) they will not be detected with our method and *N* will be less than the number of anatomical sites. In the adult rat, the CF has about 300 varicosities, a number that appears to be relatively conserved across species (Llinas, Bloedel & Hillman, 1969; Larramendi, 1969; Rossi, Borsello, Vaudano & Strata, 1993). We might expect a similar situation in our preparation as, in the mouse at least, the number of varicosities reaches the adult level by about P10 (Larramendi, 1969). The mean number of 500 functional release sites revealed by our analysis is therefore greater than the number of varicosities and may correspond to the number of synaptic densities, since electron microscopy studies show that CF varicosities generally have more than one synaptic density (Palay & Chan-Palay, 1974).

### Quantal size

Quantitative comparisons of quantal size across cell types are complicated by differences in the quality of voltage clamp in each preparation and the different methods of measurement. Our values for the CF quantal size (Table 2) are almost certainly underestimates due to filtering by the electrode-Purkinje cell circuit, but are still large compared with CF miniatures in neonatal (P2-P3) Purkinje cells (Momiyama, Silver & Cull-Candy, 1996), or with miniatures at the cerebellar mossy fibre-granule cell synaptic connection (Silver, Traynelis & Cull-Candy, 1992). During the evoked EPSC, transmitter release is asynchronous (Isaacson & Walmsley, 1995; Diamond & Jahr, 1995; Silver *et al.* 1996), causing some temporal dispersion of quantal events. Because such events have a monotonic rise and decay, the peak of the evoked EPSC need not be an integer multiple of the quantal amplitude. Thus, quantal size estimated from methods that rely on measurements of fluctuations in peak amplitude of evoked EPSCs might be expected to be slightly smaller than those measured directly.

Quantal size may vary with *P*¯_{r} if release sites are physically close and the postsynaptic receptors are not saturated by transmitter (Silver *et al.* 1996), since transmitter from neighbouring sites may overlap or ‘spillover’ (reviewed by Barbour & Häusser, 1997). Overlap is most likely to occur at higher release probabilities because the mean distance between sites that release transmitter is at a minimum. Although we observed a slowing in the EPSC decay with increasing *P*¯_{r} (Methods), estimates of *q*_{m} from high and low *P*¯_{r} were similar (Fig. 5*B*), suggesting that any spillover has little effect on quantal amplitude.

### Synaptic depression and frequency dependence of release probability

Our results indicate that at low stimulation frequencies the probability of transmitter release is close to one at each CF release site. This high *P*¯_{r}, together with the large number of sites, underlies the invariant nature of CF EPSCs under these conditions (*CV* ~1%). Increasing the frequency of CF stimulation from 0.033 to 10 Hz caused a profound depression of the EPSC, resulting from a 50-fold reduction in release probability rather than a change in the postsynaptic responsiveness or a change in the number of release sites. What mechanism could underlie such a wide dynamic range in the probability of release at each site? We first considered the possibility that the synaptic depression was mediated by presynaptic autoreceptors. However, our results indicate that metabotropic glutamate receptors (mGluRs) have little or no effect on the regulation of transmitter release at the CF synaptic connection, since depression was not altered by mGluR antagonists, and the EPSC was unaffected by the mGluR agonist L-AP4. These results agree with those of Hashimoto & Kano (1998), who have recently proposed that paired-pulse depression at the CF is presynaptic and that the depression is not mediated by mGluRs. This contrasts with the situation at the giant synapses of the calyx of Held, where activation of mGluRs substantially reduces the EPSC amplitude (Barnes-Davies & Forsythe, 1995). Adenosine is also thought to be released from CF terminals and adenosine receptors are known to be present (Takahashi, Kovalchuk & Attwell, 1995). However, the maximal depression produced by applied adenosine (30%; Takahashi *et al.* 1995) is too small to account for the frequency-dependent depression that we observe.

Rather than arising from the effects of autoreceptor activation, the depression we observed appears to result from some intrinsic property of the release sites themselves. Low-frequency paired-pulse stimulation revealed the time course of recovery of *P*¯_{r} across all sites. Two exponential components were necessary to describe this recovery, suggesting that multiple processes are involved at each release site. Alternatively, the two presynaptic components of recovery we observe could reflect different populations of release sites present on a single climbing fibre, each with a different rate of recovery of *P*_{r}. Although we feel this possibility is less likely, it is consistent with the apparently greater dispersion in *P*_{r} observed at higher stimulation frequencies (Fig. 9). Further insights came from examining the onset of and subsequent recovery from depression during sustained stimulation. Both onset and recovery exhibited a similarly slow component that was not apparent in the paired-pulse data, possibly reflecting a process uniquely associated with sustained release. The different components of recovery we observe might include recovery from calcium channel inactivation (Klien, Shapiro & Kandel, 1980) or kinetically distinct steps in vesicle recycling (Ryan *et al.* 1993) and/or a refractory period caused by the transient loss of release site competence (Katz, 1996), all of which have kinetics in this range. The time course of the slowest component is comparable with that of vesicle endocytosis at other excitatory synapses (Ryan *et al.* 1993), raising the possibility that this process limits the availability of quanta at 3 Hz.

Our experiments indicate that synaptic depression arises when the interstimulus interval becomes shorter than the mean time necessary for a release site to recover fully. They also show that recovery kinetics differ between paired-pulse and sustained stimulation and are thus dependent on the pattern of synaptic activity, possibly changing with time during the stimulus train. The particularly slow second component of recovery observed following sustained stimulation is likely to account, at least in part, for the greater depression observed during sustained rather than paired-pulse stimulation. Overall, our results suggest that the high initial release probability and the remarkably slow kinetics of recovery of release probability following release underlie the profound frequency-dependent depression of the EPSC. Comparing the time course of presynaptic recovery at different synapses suggests that the time taken for a release site to recover can vary widely at different central connections (Stevens & Wang, 1995; Mennerick & Zorumski, 1995; von Gersdorff *et al.* 1997). This raises the possibility that frequency-dependent depression occurs at those synapses where facilitating effects of an increase in residual calcium are masked by the slower kinetics of recovery of release probability (Zucker, 1989; Quastel, 1997), and that it is the temporal relationship between these two processes, together with the absolute *P*_{r}, that determines whether a synapse facilitates or depresses.

### Transmission at the climbing fibre-Purkinje cell synaptic connection

Climbing fibres exert a powerful excitatory influence on Purkinje cells and represent an important determinant of cerebellar output. The large number of release sites, the large quantal size and the high release probability all appear specialized to ensure that transmission at the CF synaptic connection is highly reliable at low frequencies. So, at typical *in vivo* firing rates of around 0.5-3 Hz (Armstrong & Rawson, 1979), the CF input is likely to act as a relay. Frequency-dependent depression at giant synapses in the auditory pathway is caused, in part, by desensitization of the postsynaptic receptors (Trussell *et al.* 1993; Zhang & Trussell, 1994). Desensitization could also play a role in CF depression (as suggested by Takahashi *et al.* 1995), but our data show this is only likely at frequencies > 10 Hz. Consistent with this idea, fast application of glutamate to patches has shown that Purkinje cell non-NMDA receptors recover from desensitization with a time constant of about 30 ms (Häusser & Roth, 1997), which would limit the effect of desensitization to frequencies above 10 Hz if it was determined by the frequency at which quanta are released at a particular site rather than by the ambient level of glutamate. Although the characteristic Purkinje cell complex spike is usually activated by a single climbing fibre action potential (Eccles *et al.* 1966), postsynaptic depression may occur during transmission when the olivary neuron fires, not one, but a rapid burst of action potentials.

Although our results were obtained at room temperature, it should be noted that olivary neurons can fire at frequencies of around 10 Hz (Llinas & Yarom, 1986), and sustained stimulation at these frequencies results in failure to excite the Purkinje cell *in vivo* (Eccles, Provini, Strata & Táboriková, 1968). Thus, it is possible that presynaptic depression at the CF synaptic connection may be acting as a low-pass filter. Recent modelling studies indicate that temporal changes in release probability can shape the frequency-dependent transmission properties at central connections (Sen *et al.* 1996; Tsodyks & Markram, 1997). The unusual temporal characteristics of release probability at the CF terminal may therefore have an important influence on output coding of the cerebellum.

## Acknowledgments

This work was suppported by a Wellcome Trust Programme Grant, a Wellcome Trust Collaborative Award and an International Research Scholars Award from the Howard Hughes Medical Institute to S. G. C.-C. A. M. was funded by a Wellcome Trust Travelling Research Fellowship. R. A. S. was funded by a Wellcome Trust Research Career Development Fellowship during part of this work. We thank Michael Häusser and Arnd Roth for simulations, Steve Traynelis for providing software, and David Attwell, Mark Farrant, Michael Häusser, Bernard Katz, Zoltan Nusser, Arnd Roth and Tomoyuki Takahashi for discussion and comments on the manuscript.

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- Non-NMDA glutamate receptor occupancy and open probability at a rat cerebellar synapse with single and multiple release sites.[J Physiol. 1996]
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