INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS THEORY AND RESULTS DISCUSSION APPENDIX REFERENCES

In fluctuation analysis of synaptic transmission, an alternative approach to the classical histogram method emerged over the past decade, which is referred to as ensemble noise analysis (Clamann et al., 1989), multiple probability fluctuation analysis (Silver et al., 1998), or variance-mean analysis (Reid and Clements, 1999; Oleskevich et al., 2000). It was adapted to synaptic transmission from ion channel noise analysis (Sigworth, 1980) and is based on the model- independent determination of the mean and the variance of synaptic responses under a set of conditions that cover a suitable range of transmitter release probabilities. The obtained relationship between variance and mean is then compared to the relationship predicted by a statistical model of synaptic transmission to estimate synaptic parameters. The classical, simple binomial model predicts a parabolic variance-mean relationship. A detailed description and discussion of the method was provided by Clements and Silver (2000). So far, the variance-mean analysis has mainly been restricted to steady-state sequences recorded under a variety of conditions, resulting in a range of mean response sizes (Silver et al., 1998; Reid and Clements, 1999; Oleskevich et al., 2000). In addition, sequences of double pulses (Oleskevich et al., 2000) and long repetitive trains of stimuli have also been used (Clamann et al., 1989), but no analysis dedicated to such nonstationary cases was presented.

Here we describe methods for applying the variance-mean analysis to short trains of synaptic responses for studying the mechanisms underlying synaptic short-term plasticity. In addition, we introduce the analysis of the covariance between successive synaptic responses in practice, which was already discussed theoretically by Vere-Jones (1966) and Quastel (1997). The covariance approach has some advantages over the variance-mean analysis alone and provides additional information about synaptic parameters during short-term plastic changes.

Fluctuation or noise analysis of synaptic transmission dates back to Del Castillo and Katz (1954). They introduced the quantal theory and quantal analysis of synaptic transmission based on the observation that evoked postsynaptic responses in a muscle fiber vary randomly between integer multiples of the spontaneous miniature response. Their analysis was based on binomial statistics, including Poisson statistics as a limiting case, with three parameters determining the size of a stimulus-evoked response: the average response size of the quantal unit q, and the binomial parameters p and N (e.g., McLachlan, 1978). Although p is generally associated with the release probability of one quantal unit, the interpretation of N is still controversial. It is suggested to be the number of docking sites, the available number of docked vesicles, or the number of morphologically defined active zones (Korn et al., 1982; Redman, 1990; Walmsley, 1993; Oleskevich et al., 2000).

Quantal analysis is applied to determine the functional parameters of a given synapse and to correlate any modification of synaptic efficacy with a change in one or more of the three parameters. In the classical histogram approach, as many synaptic responses as possible are collected in an amplitude histogram, which is treated as a multimodal distribution, with each mode representing a different number of quanta released. Ideally, this requires the identification of peaks in the histogram at a spacing of one quantal unit. The latter can be measured independently by recording spontaneously occurring miniature events (for reviews, see Redman, 1990; Walmsley, 1993). Especially in the CNS, the histogram approach is often compromised due to the presence of factors that obscure the amplitude quantization, such as high quantal contents, strong quantal-size variability, and heterogeneity in the release probability.

Frerking and Wilson (1996) summarize the coefficients of variation (CV_q) of quantal size distributions from miniature EPSC recorded in different preparations to be in the range of 44-90%. Heterogeneity in the release probability has been reported for a number of synapses (Walmsley et al., 1988; Rosenmund et al., 1993; Dobrunz and Stevens, 1997; Murthy et al., 1997; Sakaba and Neher, 2001). The degree of heterogeneity expressed in terms of coefficient of variation (CV_pp) is in the range of 22-71% in synapses of group 1 muscle afferents onto spinocerebellar tract neurons (Walmsley et al., 1988) and >50% in hippocampal synapses (Murthy et al., 1997).

These findings can be accounted for by applying compound binomial, multinomial, and compound multinomial models in the quantal analysis (Redman, 1990; Walmsley, 1993; Silver et al., 1998). However, in the case of histograms simply lacking peaks, quantization is contentious in spite of sophisticated fitting or deconvolution algorithms. The advantage of the variance-mean analysis is that the fluctuations of synaptic responses under different conditions are first quantified in a model-independent way by the variance and mean. The simple binomial model of synaptic transmission predicts a parabolic variance-mean relationship. Extensions of the theory accounting for quantal size variability or heterogeneity in the release probability introduce certain distortions to the simple variance-mean parabola (Silver et al., 1998). Thus, by fitting the respective relationship, the data can be interpreted in a very comprehensible way. Furthermore, the variance-mean approach is less noise sensitive and provides more and more reliable information, because it integrates or combines the data of different recording conditions with different mean response size.

When the variance-mean analysis is restricted to steady-state data sequences recorded at various conditions, e.g., by varying the external Ca²⁺ concentration or by application of long stimulus trains at different frequencies (which leads to various steady states of depression), the data necessarily yield information about synaptic parameters in steady state only. Here we present how the variance-mean approach can be applied to short, repetitive trains of non-stationary responses to study transient changes in the synaptic parameters during synaptic short-term plasticity. In short trains of stimuli the mean response amplitude is usually different for each stimulus due to short-term synaptic plasticity, such as paired-pulse facilitation (PPF) and short-term depression (STD). Applying such trains of stimuli repetitively and allowing sufficient time for recovery in between trains, one can assume that corresponding responses in different trains represent identical conditions.

For quantitative analysis of nonstationary data, we explicitly derive equations for estimating synaptic parameters on the basis of the classical binomial model of synaptic transmission. We assume N independent release sites, which are either empty or occupied by a release-competent vesicle. Thus, a release site is not necessarily equivalent to an active zone, which is rather considered to represent a small group of release sites. We adopt the concept suggested by Zucker (1989) and Quastel (1997) that the release probability consists of the product of the probability p_A that a vesicle is available at a release site and the output probability p₀ in case a vesicle is available for release. We allow for heterogeneity by considering a nonuniform output probability among release sites. Intra- and intersite quantal variability is taken into account. Distinguishing p_A and p₀ provides an interpretation of synaptic depression by vesicle depletion. Furthermore, it allows an interpretation of the covariance between successive synaptic responses by depletion as shown theoretically by Vere-Jones (1966) and Quastel (1997). Quastel (1997) also suggested postsynaptic effects to contribute to the covariance between successive synaptic responses in addition to depletion. We allow for such postsynaptic effects, e.g., due to postsynaptic receptor saturation or desensitization, in our interpretation of the covariance.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS THEORY AND RESULTS DISCUSSION APPENDIX REFERENCES

All the presented equations are derived assuming a binomial model of synaptic transmission as discussed in the introduction, applying basic principles of statistics. For verifying the hypotheses and studying situations, which cannot be solved analytically, we performed Monte Carlo simulations. The routines for simulation and analysis were programmed in IGORPRO (Wavemetrix, Lake Oswego, OR) and carried out on PC computers.

Release from N = 500 release sites in response to trains of five stimuli was simulated. Any release site could be either in the occupied, release competent state or in the empty state. Spontaneous transitions between the states, i.e., refilling, undocking, or spontaneous release, were computed according to a first-order kinetic scheme assuming an infinite reserve pool, as shown in Fig. 1 A. In such a scheme, the transition rate constants are determined by the recovery-time constant and the fraction of occupied sites at dynamic equilibrium, for which we used the parameters reported and proposed for the calyx of Held synapse. These were 4 s (von Gersdorff et al., 1997) and 80% (Meyer, 1999), respectively. The transition probabilities and the simulation time-step size were chosen such that the probability for a forward and backward transition within the same simulation time step was less than 0.001. Between the five evoked responses of the train, a number of simulation steps equivalent to 10 ms was computed, and between trains, a larger number equivalent to 10 s. Trains were repeated 10,000 times.

View larger version (33K): [in this window] [in a new window]   FIGURE 1   Simulation and analysis of synaptic transmission. (A) Kinetic scheme applied for the spontaneous release site transitions between the empty and the occupied, release competent state. (B) Histogram of the miniature EPSC data from the calyx of Held, which was used to generate ranked quantal-size data sets for the simulations. The CVq is 0.5. (C) Ranked quantal-size data sets of different size (250, 469, 500, 1000), the bold line represents the ranked original data set. (D) Simulated quantal size reduction (Eq. 1) in terms of quantal size ratio qm/q0 depending on the number m of preceding release events if the quantal size ratio q1/q0 after a single release event is 90, 80, or 50% as indicated on the right. (E) SNR for the segment-wise variance estimation as described in Methods, plotted as a function of the segment size for a data set containing 100 samples: Independent, nonoverlapping segments (broken line); half-overlapping segments (dotted line); and completely overlapping segments (line). The dots mark the segment sizes, which are eligible in the respective approach.

For evoked responses, an output probability was assigned to each release site. In the homogeneous case, this was the same for all sites. Heterogeneity was introduced by dividing the sites into two groups of 250 each, and assigning different output probabilities to the two groups. Evoked release was simulated by generating a random number distributed evenly in the interval [0, 1] for each release site in the occupied state and comparing this number to the output probability assigned to that site. If the random number was smaller, a release event took place, the site was set to the empty state, and the size of the quantal response was determined as described below.

The quantal size assigned to individual release events was derived from miniature excitatory postsynaptic current (mEPSC) amplitude data recorded in the calyx of Held synapse, a histogram of which is shown in Fig. 1 B. The amplitude data was sorted according to increasing size and scaled such that the mean quantal size was equal to 1 in arbitrary units. The ranking numbers of the sorted data were used for assigning a quantal size to release events. Ranking number sets of a certain size were generated by extending or compressing the sorted amplitude data versus ranking number by interpolation, as shown in Fig. 1 C. In the case of intrasite quantal variability, a random number between 0 and 999 was generated and the corresponding quantal size chosen from a data set with 1000 ranking numbers. For intersite quantal variability, each of the 500 release sites was assigned a quantal size from a data set with 500 ranking numbers. In the case of two groups of 250 sites, each of the 250 release sites was assigned a quantal size from a separate data set with 250 ranking numbers in order not to introduce correlation between release probability and quantal size.

A quantal size reduction during a train depending on the number of previous release events was implemented by a simplified empirical desensitization scheme. The quantal size is proportional to the number of open channels, which is reduced in case of desensitization. We assume a kinetic scheme such that channels enter the desensitized state from a transmitter-bound state, which does not necessarily have to be an open state, and further that equilibration of transmitter and receptor channels is faster than the transition to desensitization. Then one simply gets for the ratio of the reduced quantal size over the unperturbed quantal size in dependence on the number of previous release events m from the law of mass action

(1)

Here D is defined by the quantal size ratio in the case of a single prior release event via D  q₀/q₁  1, and m is limited by the number of postsynaptically interacting sites, denoted M in the simulations presented in Fig. 4. The relationship between the quantal size ratio and the number of preceding release events is shown in Fig. 1 D for a range of q₁/q₀ ratios. Assuming that stimuli occur faster than recovery from desensitisation (_recov = 19 ms, Trussell et al., 1993), all preceding events were treated to contribute equally.

For the practical analysis contamination of the variance and covariance estimates by drifts or trends in the recording due to rundown or any other instability has to be minimized. Clamann et al. (1989) and Quastel (1997) suggested determination of mean, variance, and covariance by calculating these parameters over short segments or groups of sequential records and averaging the obtained values to give the overall or grand values of the parameter. There are several possibilities for dividing data into segments, such as nonoverlapping independent segments, half-overlapping and maximally overlapping segments. We calculated the accuracy of the variance estimates using these three approaches as a function of segment and data set size to determine an optimal segment size (see the Appendix). The accuracy of all three approaches is shown in Fig. 1 E as signal-to-noise ratio (SNR) for N = 100 and segment sizes between 2 and 10. It is seen that SNR is higher for larger segment sizes. This is particularly apparent for nonoverlapping segments. Therefore, one would prefer large segment sizes for ideal data. However, long segments can be expected to be contaminated by long-term trends more severely than short segments. Thus, there is a trade-off between potentially better SNR with larger signals and sensitivity to nonstationarities. We decided to use maximally overlapping segments of size two (Clamann et al., 1989, used independent segments of size 5 and 6), which yields the maximal suppression of contamination by long-term trends and drifts, but suffers only relatively little deficit in SNR compared to larger segments. For the covariance estimate, we calculated the accuracy for the condition of maximally overlapping segments of size 2, as presented in the Appendix.

Fits were performed with the built-in procedure of IGORPRO, based on minimization of ². Fits to variance-mean plots were always weighted with the reciprocal of the standard deviation and constrained to pass the origin. If not otherwise stated results are reported as mean ± SEM.

	THEORY AND RESULTS

TOP ABSTRACT INTRODUCTION METHODS THEORY AND RESULTS DISCUSSION APPENDIX REFERENCES

Binomial model of synaptic transmission

Assume N release sites, which can be occupied by no more than one vesicle. Indexes x and y refer to release sites, indexes i and j to stimulus or response numbers. The model distinguishes between the all-or-none release process and the generation of the quantal postsynaptic response (considered here as a current; in the case of membrane potential, nonlinear summation might have to be considered). The all-or-none release process is associated with the parameter r with

(2)

Any given site is assumed to obey binomial statistics. The probability that a release event occurs at site x in response to stimulus i is the product of the probability p_Aix, that a vesicle is available at site x immediately before stimulus i, and the probability p_0ix, that an available vesicle is released from site x at stimulus i (Vere-Jones, 1966; Zucker, 1989; Quastel, 1997).

(3)

p_Aix and p_0ix are not assumed to be independent, see Eq. 19, Eq. 34 and following. Taking into account that the release probability is heterogeneous among release sites (Walmsley et al., 1988; Rosenmund et al., 1993; Murthy et al., 1997; Sakaba & Neher, 2001), the mean release probability over all release sites is p_Aip_0i with standard deviation _pp and coefficient of variation CV_pp ( /mean), such that

(4a)

(4b)

In case an all-or-none release event occurs, the size of the quantal postsynaptic response q has some statistics associated to it, too. Intra- and intersite quantal variability can be distinguished (Frerking and Wilson, 1996). At a single release site x, the mean quantal size is q_Intra, with standard deviation _qIntra and coefficient of variation CV_qIntra, such that

(5a)

(5b)

assuming that the intrasite quantal variability is the same at all sites, but not necessarily for all stimuli. Intersite quantal variability arises from the intrasite quantal size having different means among release sites. Intersite quantal variability with mean q_Inter, standard deviation _qInter, and coefficient of variation CV_qInter gives

(6a)

(6b)

(6c)

We follow the general assumption that release probability and mean quantal size are not correlated among release sites (but see Silver et al. 1998).

The EPSC recorded in response to stimulus i is the sum of the outputs over all release sites

(7)

The mean EPSC amplitude I_i in response to stimulus i is the expectation of the sum in Eq. 7. With substitution of Eqs. 3, 5a and 6a this yields

(8)

Variance and covariance are determined by the second moment. The second moment, M2, of the EPSC amplitude is

(9)

Substitution of Eq. 7 and 8 yields

(10)

Estimates for N and q from the variance-mean plot

The variance in the response amplitude at stimulus i is obtained by evaluating Eq. 10 for i = j. For the interpretation of the first expectation term E(q_ixr_ixq_jyr_jy) in Eq. 10, different cases have to be distinguished. For the variance these are 1) Case one (release events occurring at the same site): x = y, i = j. Again with the assumption that release process and quantal size are independent, substitution of Eqs. 3 and 5b leads to

(11)

2) Case two (release events occurring at separate sites): x  y; i = j. Again with the assumption of release site independence regarding the release process, independence of the release process and quantal size, and no interaction regarding the quantal size, (e.g., due to persistence of neurotransmitter in the synaptic cleft opposing an active zone or spill-over on the time scale of release events), this is with substitution of Eqs. 3 and 5a,

(12)

Substituting Eqs. 11 and 12 into Eq. 10 yields the variance,

(13)

Based on the assumption that release probability and quantal size are not correlated, after insertion of Eq. 6b, it follows that

(14)

Combining Eq. 8 and Eq. 14 yields the classical parabolic variance mean relationship in case that the quantal size is independent of the stimulus number, i.e. q_iInter = q = const. (Sigworth, 1980; Silver et al., 1998)

(15)

and in the linear form (Heinemann and Conti, 1992),

(16)

where the fitting parameters q* and N_Var are related to the true parameters by

(17)

(18)

Note that CV_pp in Eq. 18 is not necessarily constant, but may be a function of the average release probability p_Ap₀ (Quastel, 1997; Silver et al., 1998). The linear form in Eq. 16 has the advantage that procedures can be applied for weighted fitting, which consider uncertainties in both variables (Orear, 1982), to determine q* and N_var from the y-axis intercept and the slope, respectively.

Estimating N and q from the covariance

The covariance in the response amplitudes at stimulus i and stimulus j is obtained by evaluating Eq. 10 for i  j. Regarding the first expectation term E(q_ixr_ixq_jyr_jy) in Eq. 10, three cases have to be distinguished.

1)  x = y. This considers previous and subsequent release from the same release site. Release occurring at both stimulus i and stimulus j requires that the release site is reoccupied in between. Defining the probability of this reoccupation as p_Ai|jx it follows, that

(19)

In the framework of a vesicle pool model p_Ai|jx can be expressed by the solution of the differential equations for vesicle recycling. For very simple models of recycling, it is given by p_Ai|jx = p_A1x(1  exp( t/)), where t is the inter-stimulus interval and  the time constant of recovery (e.g., Weis et al., 1999). Furthermore, the quantal size of the subsequent release event might be affected by the previous release, e.g., due to desensitization or saturation. This is denoted by q_jx(r_ix) below. More accurately, this should also depend on how much was released previously, i.e. the quantal size q_ix. However, we assume that the variability in the neurotransmitter amount due to quantal size variability is negligible compared to the variability due to the fluctuation in the number of released vesicles. Then,

(20)

Substitution of Eqs. 3, 5a, and 19 yields

(21)

2)  y  [x  M/2, x + M/2]. This considers release from any site to interact with previous release of its M neighbors, due to desensitization or saturation. The quantal size of the subsequent response depends on the previous release from the neighboring sites with E(r_ixq_jy) = cov(r_ix, q_jy) + E(r_ix)E(q_jy). Thus,

(22)

and substitution of Eqs. 3 and 5a yields

(23)

3)  y / [x  M/2, x + M/2]. In this case, there is no interaction between the sites. The expectation of the product is the product of the expections. With Eqs. 3 and 5a, this gives

(24)

Inserting Eqs. 21, 23 and 24 into Eq. 10 and rearranging, the covariance between successive overall response amplitudes is

(25)

This equation is more complex and has not been discussed in the literature as much as the equation for the variance. Therefore the different aspects of the covariance analysis are considered separately in turn.

First it is assumed that there is effectively no refilling, i.e., p_Ai|jx = 0, which holds if the interstimulus intervals are brief enough. Furthermore, any effects of preceding release events on the subsequent quantal size are neglected, i.e., cov(r_ix, q_jx) = 0, and the output probability is assumed to be homogenous, i.e., CV_pp = 0. In this case of substitution of Eqs. 3, 5a, and 6c into Eq. 25 gives the covariance caused by vesicle depletion alone

(26)

From this, N and quantal size q can be calculated as follows. Combining Eqs. 8 and 26 yields

(27)

with N = N_cov (1 + CV). The expression for N_cov including heterogeneity in the output probability is given in Eq. 32 for comparison to the estimate from the variance in Eq. 18. Combining Eqs. 8, 14, 17, and 26 gives

(28)

with q* as defined in Eq. 17. This allows the calculation of the quantal size for each response in a train separately, which is more specific than q*, determined as a common parameter of all responses in the variance-mean parabola. (Note that there are two ways to calculate q. In practice, we take the average.)

The covariance gives a better estimate for N in case of p heterogeneity than does the variance-mean plot

Our simulations suggested (see below) that the covariance analysis gives a better N estimate than does the variance-mean parabola for mean output probability of p₀ = 0.5 as documented in Table 1 and discussed below (see particularly Table 1, last line of the middle section in all cases the correct N value is 500). So the question arose whether it is a general feature that the covariance analysis is superior to the variance-mean parabola in the estimation of N, and why this might be the case. In the following analysis, it is again assumed that there is effectively no refilling, i.e., p_Ai|jx = 0. Furthermore, effects of preceding release events on the subsequent quantal size are neglected for simplicity, i.e., cov(r_ix, q_jx) = 0. In this case, Eq. 25 simplifies to

(29)

Combination of Eq. 8 with Eq. 29 and insertion of Eq. 4a yields

(30)

with

(31)

such that, because of Eq. 27,

(32)

Comparing Eq. 32 to Eq. 18 yields an explanation for the question posed above by considering that   cov(p_Aip_0i, p_Ajp_0j), see Eq. 4b for _pp. Thus one can argue that

(33)

This means that the covariance can yield a better estimate for N than the variance-mean parabola. To analyze this further, the case j = i + 1 is considered. Because refilling is assumed to be negligible, p_Ai+1 can be replaced by what remains after the previous response

(34)

and the evaluation of C_i,i+1 yields after some rearrangement

(35)

Assuming an initially uniform occupancy, i.e. p = p_A1², Eq. 35 yields

(36)

As a special case of heterogeneity in the output probability, we consider the situation (as assumed for the Monte Carlo simulations) with no facilitation (p₀₁ = p₀₂ = p₀) and two groups of equal numbers of release sites, one having an output probability of p₀ = p₀  a, and the other p₀ = p₀ + a, with some parameter a (p₀  a), such that that p = p₀² (1 + CV) and p = p₀³(1 + 3CV). This yields

(37)

Thus the optimum condition, i.e., C_1,2 = 0, for determining N from the covariance can readily be calculated in terms of mean output probability, which is p₀ = 0.5. Furthermore, it holds that (1 + C_1,2) < (1 + CV) unless CV_p approaches 1, as shown in Fig. 2 A (the case for two groups of release sites).

To evaluate C_1,2 for a more realistic p₀ distribution, we applied the beta distribution suggested by Silver et al. (1998), again assuming the case where p₀₁ = p₀₂ = p₀, and modeling the heterogeneity in p₀ according to a beta distribution. Note that Silver et al. (1998) used the beta distribution for the release probability, which we consider here as the product p_A · p₀, i.e., of a probability of availability and an output probability. However, assuming that the availability is homogenous at the first stimulus, p_A can be considered as a scaling factor. p₀ is then distributed according to the beta distribution. The probability density function is

(38)

where B(, ) designates the beta function. Evaluation of the nth moment yields

(39)

Such that the parameters  and  are related to the mean and coefficient of variation of the p₀ distribution in the following way:

(40a)

(40b)

Insertion of Eqs. 39 and 40 into Eq. 36 yields a complicated analytical expression, which is not explicitly given here. The calculated result is shown in Fig. 2 B, where (1 + C_1,2) is compared to (1 + CV) for the beta distribution. Again, it is seen that the correction factor for N_cov is quite close to 1, even for large heterogeneity, if p₀ is close to 0.5. 

View larger version (14K): [in this window] [in a new window]   FIGURE 2   Correction factor (1 + C1,2) for the estimate of N from the covariance in comparison to the correction factor (1 + CV), which applies to the variance-mean plot (heavy solid line). (1 + C1,2) is shown for mean output probabilities of p0 = 0.3, 0.5, and 0.7 versus the coefficient of variation CVp ranging from 0 to 1. (A) (1 + C1,2) in the case of two groups of release sites with different p0 (lines, for details refer to text) and as obtained from the simulation shown in Fig. 3 and Table 1 regarding intrasite quantal variability (symbols). (B) (1 + C1,2) in the event that p0 is distributed according to a beta distribution.

Simulations with heterogeneity in the output probability

To compare the N estimates from the variance-mean plot (Eq. 15) with the estimate from the covariance (Eq. 27), simulations were carried out as described in the methods for N = 500 release sites with mean output probabilities p₀ of 0.3, 0.5, and 0.7 and different degrees of heterogeneity in the output probability. We also compared the effects of intra- and intersite quantal variability. The results are summarized in Figs. 2 and 3, and Table 1. In each case, a parabolic fit to the simulated data points is shown (Fig. 3). The degree of curvature in the variance-mean parabolas, and, consequently, the N_var estimate is effectively determined by the first (largest!) response in the train. Later responses in the train lie on the linearly rising part of the parabolas, see Fig. 3. Intersite quantal variability generally leads to an underestimation of N, see Fig. 3, B, D, and F, and Table 1, as expected from Eqs. 18 and 32 with CV_qInter = 0.5 (note, that N/N_var-values and N/N_cov-values of Table 1 are systematically larger by a factor of (1 + CV)  1.25 for the case of intersite quantal variability as compared to intrasite variability).

View larger version (27K): [in this window] [in a new window]   FIGURE 3   Variance-mean plots of simulated data. Simulations with N = 500 release sites, mean output probability of p0 = 0.3 (A, B), 0.5 (C, D), and 0.7 (E, F), and different degrees of heterogeneity are shown. The legend in panel B applies to the whole figure. Quantal size variability had a CVq of 0.5 in each case. In the left column (A, C, E), the quantal size variability was assumed to be of the intrasite type. In the right column (B, D, F), the case intersite variability is shown. The theoretical correction factors for the N estimates from the variance-mean plot (1 + CV) are given on the right-hand side to indicate the degree of heterogeneity. The N estimates obtained from the parabolic fits (Nvar) are compared to the estimates from the covariance (Ncov) in Table 1. All data are shown in arbitrary units with respect to a quantal size of 1 unit.

At low mean output probability, i.e., p₀ = 0.3, the variance-mean parabola (Fig. 3, A and B) and the covariance approach (Table 1, top) both underestimate N with increasing p₀ heterogeneity to an extent accurately predicted by Eqs. 18 and 32 with the correction factors (1 + CV) and (1 + C_1,2) as plotted in Fig. 2 A (solid and thin lines), respectively. The estimates from the covariance are acceptable for small degrees of heterogeneity. At medium mean output probability, i.e., p₀ = 0.5, N_cov slightly overestimates N, but the estimates appear not to be affected by any degree of p₀ heterogeneity, because (1 + C_1,2)  1, as expected from Eq. 37 (Fig. 2 A, dotted line). The variance-mean parabola, however, underestimates N with increasing heterogeneity (Fig. 3, C and D, Table 1, middle). At high mean output probability, i.e., p₀ = 0.7, the variance-mean estimate is affected by increasing heterogeneity (Fig. 3, E and F), but less compared to the case of small mean output probability of p₀ = 0.3 (Fig. 3, A and B). The reason for this is that the same degree of heterogeneity yields a lower CV_p at higher mean. The covariance approach considerably overestimates N at high p₀ (Table 1, bottom), as expected from Eq. 37 (Fig. 2 A, broken line). The analytically derived dependence of the correction factor (1 + C_i,i+1) for the N_cov estimates on the mean and degree of heterogeneity in the output probability in case of two groups of release sites (Eq. 37), is confirmed by our simulations. Figure 2 A shows good agreement between the analytical relationship (lines) and the values obtained for (1 + C_i,i+1) by dividing N = 500 by N_cov from the simulations (symbols; Eq. 32).

This shows that the covariance approach yields a good N estimate, which is unaffected by heterogeneity in the output probability under conditions with intermediate mean output probability, ideally p₀ = 0.5. In the range between p₀ = 0.3 and p₀ = 0.7, the N_cov estimate is accurate within ±40% as long as CV_p < 75%. The variance-mean plot provides reasonable N estimates for small degrees of heterogeneity under conditions of high mean output probability. However, both approaches suffer the same from the presence of intersite quantal variability as expected from Eqs. 18 and 32.

Effect of saturation and desensitization on the estimates from the covariance

It must be considered that correlation might not only arise from depletion of vesicles, but also from desensitization or saturation, in case of persistence of neurotransmitter in the synaptic cleft or spillover as a consequence of repetitive activity (Trussell et al., 1993; Otis et al., 1996a; Barbour and Häusser, 1997).

Here, it is again assumed that there is effectively no refilling, i.e., p_Ai|jx = 0. Effects of preceding release events on the subsequent quantal size are considered, i.e., cov(r_ix, q_jx)  0, but quantal size variability and heterogeneity in the release probability are now neglected for simplicity. In this case, substitution of Eqs. 3 and 4a into Eq. 25 gives for the total covariance caused by depletion and postsynaptic effects,

(41)

where M is the number of postynaptically interacting release sites, as detailed in the methods section. Comparing Eq. 41 and Eq. 26 (in the absence of quantal size variability, i.e., CV_qInter = 0), it is seen that the total covariance caused by depletion and postsynaptic effects can be expressed as the covariance caused by depletion alone (Cov) increased by a factor depending on the covariance caused by postsynaptic effects

(42)

with

(43)

Below we will show that D_i,i+1 is always negative, if release leads to a decrease in quantal size, such that the correction factor (1  D_i,i+1) represents, indeed, the sum of the covariance of presynaptic origin and that caused by postsynaptic effects.

Thus, for estimating N and q from the total covariance (Eq. 42) according to Eq. 27 and Eq.