Errors in the estimation of the variance: Implications for multiple-probability fluctuation analysis

Abstract

Synapses play a crucial role in information processing in the brain. Amplitude fluctuations of synaptic responses can be used to extract information about the mechanisms underlying synaptic transmission and its modulation. In particular, multiple-probability fluctuation analysis can be used to estimate the number of functional release sites, the mean probability of release and the amplitude of the mean quantal response from fits of the relationship between the variance and mean amplitude of postsynaptic responses, recorded at different probabilities. To determine these quantal parameters, calculate their uncertainties and the goodness-of-fit of the model, it is important to weight the contribution of each data point in the fitting procedure. We therefore investigated the errors associated with measuring the variance by determining the best estimators of the variance of the variance and have used simulations of synaptic transmission to test their accuracy and reliability under different experimental conditions. For central synapses, which generally have a low number of release sites, the amplitude distribution of synaptic responses is not normal, thus the use of a theoretical variance of the variance based on the normal assumption is not a good approximation. However, appropriate estimators can be derived for the population and for limited sample sizes using a more general expression that involves higher moments and introducing unbiased estimators based on the h-statistics. Our results are likely to be relevant for various applications of fluctuation analysis when few channels or release sites are present.

Introduction

Neuronal communication in the brain is mediated predominantly by chemical synapses, which operate through the release, from the presynaptic neuron, of a neurotransmitter that diffuses in the synaptic cleft and induces electrical changes in the postsynaptic cell. According to the quantal hypothesis of transmitter release (del Castillo and Katz, 1954, Katz, 1969), neurotransmitter is packaged in discrete quantities (quanta) that are released in an all-or-none fashion at specific sites. Assuming that each site acts independently, binomial and Poisson statistics can be used to model the stochastic behavior of synaptic transmission. Thus, synaptic efficacy can be described using three quantal parameters (Katz, 1969, Vere-Jones, 1966): the number of functional release sites (N), the mean probability of release (P) and the amplitude of the response to a single quantum (Q). Changes in one or more of these parameters account for modifications in synaptic strength.

Several “quantal analysis” methods have been developed to estimate quantal parameters and/or to identify the site of changes in synaptic efficacy, for example by examining the failure rate or the amplitude histograms of synaptic responses (Katz, 1969). However, these methods can be difficult to apply at central synapses when the signal-to-noise ratio is low due to the small quantal size and non-uniform Q compromises the discrimination of peaks in the amplitude histograms (Redman, 1990, Walmsley, 1995). To overcome some of these difficulties, new approaches inspired by stationary and non-stationary noise analysis of ion channels (Heinemann and Conti, 1992, Sigworth, 1980) have been developed as an extension of previous methods used in the 1970s and 1980s (Clamann et al., 1989, Miyamoto, 1975). Multiple-probability fluctuation analysis (MPFA: Silver et al., 1998; also known as variance–mean analysis: Reid and Clements, 1999) is a recent method that allows estimation of quantal parameters from the relationship between the variance and mean of synaptic responses recorded under different release probability conditions (Clements and Silver, 2000, Silver, 2003). Quantal parameters are estimated by fitting the variance–mean relationship with a binomial or multinomial model using the least squares optimization method. MPFA has been applied at a number of central synapses (Foster and Regehr, 2004, Lawrence et al., 2004, Oleskevich et al., 2000, Sargent et al., 2005, Silver et al., 1998, Silver et al., 2003, Sims and Hartell, 2005). These studies have shown that the number of functional release sites per connection can vary from one to several hundred and the release probability per site varies widely at central synapses.

The binomial model is more appropriate than the Poisson for describing neurotransmitter release at central synapses (Kuno, 1964), since some of these connections are characterized by few release sites and a release probability that can be quite high under physiological conditions (Silver et al., 2003). According to this model, the mean peak amplitude of the synaptic current I and the associated variance σ² are given by

$$I=NPQ,$$

$${\sigma }^2=N{Q}^{2}P(1-P),$$

so that a parabolic relationship holds between them:

$${\sigma }^2=IQ-\frac{I^2}{N}.$$

Fig. 1A shows Monte Carlo simulations of postsynaptic currents at three different probabilities for a connection with five release sites (a value typical for central synapses). The mean synaptic current increases linearly with P, while trial-to-trial fluctuations arise presynaptically from the variability in the number of quanta released. The theoretical relationship between variance and mean of synaptic current is clearly parabolic (Eq. (3)) and the black dots show the tested probabilities (Fig. 1A). The empty squares show the relationship between variance and mean measured from samples of 200 simulated currents for each P.

Several studies have shown that the assumptions required for applying a simple binomial model often do not hold for central synapses. A multinomial model has therefore been developed in order to take into account both asynchronous release and non-uniform Q (Frerking and Wilson, 1996, Silver, 2003). Quantal variability can be present at the single site level (intrasite variability) and across sites (intersite variability). Intrasite variability arises from fluctuations in the size of quantal events from trial-to-trial and from asynchrony in their latency, and can be quantified in terms of the associated coefficients of variation (CV_QS and CV_QL, respectively). Intersite variability arises from differences in the mean quantal size from site to site and the coefficient of variation of this component is defined as CV_QII. Both intrasite and intersite variability can be quite easily implemented in a multinomial model, where the mean current and the variance are given by

$$I=NP{\overline{Q}}_{\text{p}},$$

$${\sigma }^2=N{\overline{Q}}_{\text{p}}^{2}P(1-P)(1+\text{C}{\text{V}}_{\text{QII}}^2)+N{\overline{Q}}_{\text{p}}^{2}P\text{C}{\text{V}}_{\text{QI}}^2,$$

where ${\overline{Q}}_{\text{p}}$ represents the mean quantal size at the time of the peak of the mean synaptic current and is therefore affected by asynchronous release. CV_QI includes all the variability observed at the single site level, being defined by

$$\text{C}{\text{V}}_{\text{QI}}^2=\text{C}{\text{V}}_{\text{QL}}^2+\text{C}{\text{V}}_{\text{QS}}^2.$$

Substituting Eq. (4), the variance can be expressed in terms of I:

$${\sigma }^2=\left({\overline{Q}}_{\text{p}}I-\frac{I^2}{N}\right)(1+\text{C}{\text{V}}_{\text{QII}}^2)+{\overline{Q}}_{\text{p}}I\text{C}{\text{V}}_{\text{QI}}^2.$$

Fig. 1B shows simulated synaptic currents with intrasite and intersite quantal variability. Because of the asynchrony in release, the mean currents are slightly reduced compared to those shown in Fig. 1A for the same probabilities. The main difference in the shape of the variance–mean relationship (black line), compared with the binomial case, is due to the introduction of intrasite variability, which increases linearly with release probability (dashed line). As in Fig. 1A, black dots correspond to the tested probabilities, while black empty squares represent measured values from sets of 200 synaptic currents.

Different groups have shown that the probability of release can also be non-uniform across release sites at central synapses (Murthy et al., 1997, Walmsley et al., 1988). In this case, assuming that there are no correlations between release probability and quantal size across sites (Frerking and Wilson, 1996, Silver et al., 1998), the mean current and the variance are given by (Silver, 2003):

$$I=N\overline{P}{\overline{Q}}_{\text{p}},$$

$${\sigma }^2=N{\overline{Q}}_{\text{p}}^2\overline{P}[1-\overline{P}(1+\text{C}{\text{V}}_{\text{P}}^2)](1+\text{C}{\text{V}}_{\text{QII}}^2)+N{\overline{Q}}_{\text{p}}^2\overline{P}\text{C}{\text{V}}_{\text{QI}}^2,$$

where $\overline{P}$ is the mean release probability across sites at the time of the peak of the current, while CV_P represents the coefficient of variation of release probability across sites and changes with $\overline{P}$. By substitution, the variance can be expressed in terms of the mean current:

$${\sigma }^2=\left[{\overline{Q}}_{\text{p}}I-\frac{I^2}{N}(1+\text{C}{\text{V}}_{\text{P}}^2)\right](1+\text{C}{\text{V}}_{\text{QII}}^2)+{\overline{Q}}_{\text{p}}I\text{C}{\text{V}}_{\text{QI}}^2.$$

CV_P and how it changes with $\overline{P}$ has been modeled using families of beta functions β(α,β) so that (Silver, 2003):

$$\text{C}{\text{V}}_{\text{P}}=\sqrt{\frac{1-\overline{P}}{\overline{P}+\alpha }}$$

and the variance–mean relationship depends on only one additional parameter α:

$${\sigma }^2=\left[{\overline{Q}}_{\text{p}}I-\frac{{\overline{Q}}_{\text{p}}{I}^2(1+\alpha )}{I+N{\overline{Q}}_{\text{p}}\alpha }\right](1+\text{C}{\text{V}}_{\text{QII}}^2)+{\overline{Q}}_{\text{p}}I\text{C}{\text{V}}_{\text{QI}}^2,$$

It follows that low α values indicate non-uniform release probability and the model reduces to the uniform one for α → ∞. The grey line in Fig. 1B shows the theoretical relationship between variance and mean of synaptic currents for a multinomial model with non-uniform P (Eq. (12); α = 1). The grey dots and empty squares represent the theoretical and simulated values. Although the variance is rather insensitive to non-uniformity for low release probabilities, it is significantly reduced at high $\overline{P}$ producing a skew in the variance–mean relationship.

Quantal parameters can be estimated by fitting the variance–mean relationship using the standard least squares optimization method when the error in the measurement of the current is negligible compared to the error in the variance. However, one of the assumptions underlying the use of this fitting procedure for estimating the coefficients and their uncertainties is that the variance of the dependent variable is constant over all values (Ryan, 1997), a statement that clearly does not hold in many experimental conditions. Moreover, such uniform weights do not permit the goodness-of-fit of the model, and therefore the significance of the fit, to be determined. For these reasons, data points should be weighted in the optimization routine. This is usually done by weighting the contribution that each data point makes by the reciprocal of its variance (Ryan, 1997), so that more precisely defined points will have more influence over the parameter estimates and vice versa. In this way, the goodness-of-fit can be established using the χ²-test (Taylor, 1997) and the uncertainties in the estimates of the coefficients can be calculated by propagating the error on each data point (Press et al., 2002). Since the use of the weighted least squares method assumes that the weights are known exactly and this is almost never the case in real applications, it is important to use their proper estimators. In order to perform weighted fit of the variance–mean relationship, we determined the theoretical estimators of the variance of the variance under different experimental conditions. The accuracy with which the variance can be estimated depends on the variance itself, the shape of the distribution and the number of observations acquired (n). We derived estimators of the variance of the variance for a general and a normal parent distribution and for different sample sizes. We used Monte Carlo simulations of synaptic transmission to test the accuracy and reliability of the different estimators for various experimental conditions.