Quantal analysis based on density estimation

Abstract

When direct measurements of the quantal parameters for a synapse cannot be made, these parameters can be extracted from an analysis of the fluctuations in the evoked response at that synapse. In this article, a decision tree is described in which the ability of the data to match simple models of quantal transmission is rigorously compared with its ability to fit progressively more complex models. The Wilks statistic is the basis for this comparison. The procedure commences with optimal transformation of peak amplitude measurements into a probability density function (PDF). It then examines this PDF against various models of transmission, commencing with a multi-modal but non-quantal distribution, then to a multi-modal distribution with quantal peak separation with and without quantal variability, and, finally, the constraints of uniform and non-uniform release probabilities are imposed. These procedures are illustrated by example. A comparison is made between the relative sensitivities of the Wilks statistic and the χ² goodness-of-fit criteria in rejecting inappropriate models at all stages in these procedures.

Introduction

The discovery of quantal transmission at the neuromuscular junction, and the subsequent identification of vesicular packaging of neurotransmitter as the biological correlate of quantisation, was a major landmark in synaptic physiology. This concept has been of enormous value for investigations into modulation of transmission; whether caused by prior impulse activity or by substances that alter transmitter release or post-synaptic responses.

The idealised concept of quantal transmission is one where all vesicles would contain identical amounts of transmitter, which when released would present identical concentration profiles to the post-synaptic receptors. Further, all ligand-gated receptors (of the same type) would have identical binding affinities and identical channel conductances when open. A final requirement would be that in an isopotential cell, ligand-gated channels occur in equal numbers in opposition to each release site at which transmission occurs, or in a dendritic neurone, the numbers are scaled at each release site to provide identical currents at the soma. We know that this combination of idealised requirements is unlikely to hold for real synapses. Given the possible variability intrinsic to transmission at a single active site, as well as the potential variability between sites, especially in dendritic neurones, it is remarkable that quantised responses can be found in recordings from neurones in the CNS.

When confronted with a set of evoked responses collected for the purpose of examining transmission at single active sites of transmission, the first question is whether the responses are quantised. If this can be established with sufficient statistical rigour, then the quantal amplitude, its variance, the number of active sites, and the release probabilities at these sites may be extractable. If not, all is not lost, and lower order information about transmission characteristics can be determined using other approaches (Clamann et al., 1989, Clements, 2003, Clements and Silver, 2000, Dityatev et al., 1992, Robinson et al., 1991).

In this article, we will describe one approach that provides a decision base for accepting (or rejecting) responses as quantised and what the underlying statistics of transmitter release are. The probability density function (PDF) formed by the peak amplitudes of the evoked responses is the target distribution. All putative model distributions to be tested against this PDF are fitted to it using a maximum likelihood criterion and this is estimated using the expectation-maximum (EM) algorithm (Stricker and Redman, 1994). The PDF is fitted first by different mixture models, each consisting of a set of discrete amplitudes occurring with probabilities, which must sum to one but are otherwise arbitrary (unconstrained), and each with variability set by the baseline noise. The optimal number of discrete amplitudes is determined first. This discrete (multi-modal) representation is then compared with various models of continuous (uni-modal) distributions to determine if there is sufficient modality (or ‘peakiness’) to treat the PDF as multi-modal. All statistical comparisons of goodness-of-fit use the log-likelihood ratio obtained from fitting competing models to the data. This ratio, the Wilks statistic, is distributed asymptotically as χ² if the competing models are nested (Wilks, 1938). Practically, nested means that the two competing models have the same number of modes. If they are not nested, the statistic has to be generated from the data using Monte Carlo simulations.

If a PDF emerges from this challenge consisting of discrete amplitudes, further constraints consistent with concepts of quantal transmission can be added. These are quantal separation of discrete amplitudes, with and without quantal variability. If a quantal model cannot be rejected, constraints on the probabilities associated with each discrete amplitude can be added, such as those imposed by uniform or non-uniform release probabilities. The Wilks statistic is employed at each step in this process. When this statistic cannot separate two competing models, the decision is then based on the principle of parsimony. This means that the model with fewer parameters is to be chosen. At the same time, a deliberate bias towards classical transmitter release statistics occurs when choosing the null (H₀) and the alternative hypothesis (H₁).

This approach to quantal analysis was first developed to provide a more rigorous basis for model comparison than had been previously available (Stricker and Redman, 1994, Stricker et al., 1994). It was subsequently applied to the analysis of EPSCs evoked in CA1 pyramidal cells in response to stimulation of Schaffer collaterals before and after induction of long-term potentiation (Stricker et al., 1996a, Stricker et al., 1996b, Stricker et al., 1999). In this article, we have illustrated the methodology by using EPSC data (unpublished data) evoked and recorded in a CA1 pyramidal cell using a similar protocol to that used in Stricker et al., 1996a, Stricker et al., 1996b).

Density estimation methods have been claimed to be unreliable and controversies over the use are still alive. Some of these criticisms have arisen from the use of other methods of parameter estimation and goodness-of-fit criteria. In particular, in contrast to the log-likelihood arguments described above, often χ² goodness-of-fit criteria are used since these are much easier statistics to apply and are more intuitively understandable. In this article we illustrate a systematic evaluation of a hierarchy of statistical models, contrast these results with those obtained using χ² goodness-of-fit criteria and explore some issues associated with using these two methods.