Estimation of nonuniform quantal parameters with multiple-probability fluctuation analysis: theory, application and limitations
Introduction
Fluctuations in the amplitude of the end plate potential (del Castillo and Katz, 1954) and the discovery of spontaneous miniature potentials (Fatt and Katz, 1952) lead to the idea that transmitter was released in discrete all-or-none units (quanta) (Katz, 1969). These observations suggested that models based on Poisson and binomial statistics could be used to describe the stochastic behaviour of synaptic transmission if the underlying quantal events were independent. Three ‘quantal’ parameters are now commonly used to describe the properties of transmission at a synaptic connection: the amplitude of the postsynaptic response to a single quantum of transmitter (quantal size, Q), the number of independent release sites (N) and the probability of release at each site (P). The values of these quantal parameters determine the size of the postsynaptic response and how variable it is from trial-to-trial. During synaptic plasticity changes in synaptic efficacy can be described in terms of changes in quantal parameters: presynaptic modulation is associated with changes in P, while postsynaptic alterations are associated with changes in Q. Plasticity that involves the formation of new contacts (or increases in P at existing contacts where P is initially zero) will be reflected in a change in N. Comparison of quantal parameters before and after synaptic plasticity can therefore be used to identify whether presynaptic or postsynaptic mechanisms are involved. Moreover, the initial value of P, recorded at low frequency, is important for understanding short-term plasticity during high frequency trains, since high P values are associated with paired-pulse depression while synapses with low P often facilitate (Magleby, 1979, Oleskevich et al., 2000, Zucker and Regehr, 2002). Knowledge of these properties are important for understanding synaptic physiology because they determine both the reliability of synaptic transmission and the range of frequencies (bandwidth) over which signal transmission can be maintained (Brenowitz and Trussell, 2001).
Amplitude fluctuations in synaptic responses from trial-to-trial contain information about Q, P and N and these parameters can be extracted using statistical analysis—so called ‘quantal analysis’ methods. Early studies used quantal analysis to test the quantal nature of transmission at the neuromuscular junction (NMJ) (Katz, 1969). These studies were carried out under low probability conditions taking advantage of the large number of release sites at the NMJ to apply a simple release model based on Poisson statistics. Lowering release probability minimises possible interactions between quanta that can arise presynaptically due to interaction of calcium entering at neighbouring release sites and postsynaptically due to spillover of transmitter. It therefore ensures that the underlying quantal events are independent and thus that they can be described using the independent variables N, P and Q—an important assumption underlying the application of conventional statistical models. Moreover, the potentially complicating effects of nonuniform release probability (del Castillo and Katz, 1954) are minimised at low P. Although similar approaches have been applied to a giant central synapses (Isaacson and Walmsley, 1995, Sahara and Takahashi, 2001), the large number of release sites required for Poisson statistics makes application of a Poisson model inappropriate at most central synapses. Moreover, the questions presently being addressed have moved on from establishing the quantal nature of transmission to understanding the mechanisms underlying transmission under physiological conditions when release probability can be high (Silver et al., 1998). Under these conditions binomial or multinomial models are more appropriate. Until recently most studies of central synapses have estimated quantal properties from amplitude histograms of synaptic responses, where Q corresponds to the interval between peaks and P is determined from the amplitude distribution (Jack et al., 1990, Redman, 1990). Compound binomial models have been used to account for nonuniform release probability (Jack et al., 1981, Walmsley et al., 1988) but interpretation of peaks in the amplitude distribution is complicated at synapses where quantal size (and release probability) is not uniform across release sites (Stricker et al., 1996, Walmsley, 1995). Moreover, unlike the NMJ where thousands of channels underlie the quantal event, quanta at central synapses are small being mediated by as few as 10–20 channel openings (Silver et al., 1996). Signal-to-noise limitations therefore make resolution of individual quantal events and quantal peaks in amplitude histograms difficult in many central neurons.
Difficulties in applying quantal analysis methods, based on analysis of amplitude histograms, to central synapses have spurred several groups to develop new approaches inspired by noise analysis of ion channels (Anderson and Stevens, 1973, Katz and Miledi, 1970, Sigworth, 1980). These new methods extend previous methods developed in the 1970s and 1980s (Clamann et al., 1989, Miyamoto, 1975, Segal et al., 1985). They include stationary analysis of the mean and variance of synaptic responses recorded at several different release probabilities (Clements and Silver, 2000, Reid and Clements, 1999, Silver et al., 1998) as illustrated in Fig. 1 (also see Frerking and Wilson, 1996) and the analysis of mean, variance (Humeau et al., 2002, Meyer et al., 2001) and covariance (Scheuss and Neher, 2001) of nonstationary trains of postsynaptic currents (PSCs). Analysis of higher moments (skew and kurtosis) have also been used to estimate release rates during prolonged tonic release from voltage clamped presynaptic terminals of the Calyx of Held (Neher and Sakaba, 2001). In this paper, I will describe the theoretical basis of multiple-probability fluctuation analysis (MPFA) (Silver et al., 1998), also known as variance mean analysis (Clements and Silver, 2000), and explain how the properties of synapses with nonuniform quantal parameters can be described with a multinomial model for transmission. I will also examine the practical application and limitations of MPFA and discuss the utility of this approach with reference to related methods and recent optical approaches for determining quantal parameters (Oertner et al., 2002, Wang et al., 1999, Yuste et al., 1999).
Section snippets
The binomial model
The binomial model is one of the simplest models that can describe release over a wide range of release probabilities and was first applied to central synapses by Kuno (1964). The mean amplitude of the synaptic response (Ī; see Fig. 1) isand for a binomial model the variance (σ2) isThe relationship between variance and Ī is thereforeFitting this parabolic function to the relationship between the variance and mean current, which can be determined from the peak of
What type of synaptic response?
The first question that should be asked when carrying out MPFA is what synaptic response should be measured. In most cases it is best to analyse synaptic currents under voltage-clamp because it alleviates driving force issues and the synaptic response should be independent of voltage-gated channels. However, if synapses are far from the soma, and voltage clamp quality is poor, it may be best to record PSPs instead. The main advantage of EPSP recording is that it is much less sensitive to
Comparison with other methods
MPFA (also known as variance–mean analysis (Clements and Silver, 2000, Reid and Clements, 1999, Clements, 2003) is a simple method for estimating the mean quantal parameters at synaptic connections when release probability and quantal size are not uniform. It extends previous quantal analysis methods by circumventing difficulties in identifying quantal peaks in amplitude histograms (Redman, 1990, Stricker et al., 1996), allowing estimation of quantal parameters when quantal variance is large
Acknowledgements
This work was supported by the Wellcome trust (R.A.S. is in receipt of a Wellcome Trust Senior Research Fellowship), E.C. and MRC. I would like to thank David Digregorio and Tom Nielsen for providing software, Chiara Saviane for help with the expressions for sampling errors and John Clements, D.D., T.N. Peter Sargent and C.S. for comments on the manuscript.
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