Bayesian Inference of Synaptic Quantal Parameters from Correlated Vesicle Release

Abstract

Synaptic transmission is both history-dependent and stochastic, resulting in varying responses to presentations of the same presynaptic stimulus. This complicates attempts to infer synaptic parameters and has led to the proposal of a number of different strategies for their quantification. Recently Bayesian approaches have been applied to make more efficient use of the data collected in paired intracellular recordings. Methods have been developed that either provide a complete model of the distribution of amplitudes for isolated responses or approximate the amplitude distributions of a train of post-synaptic potentials, with correct short-term synaptic dynamics but neglecting correlations. In both cases the methods provided significantly improved inference of model parameters as compared to existing mean-variance fitting approaches. However, for synapses with high release probability, low vesicle number or relatively low restock rate and for data in which only one or few repeats of the same pattern are available, correlations between serial events can allow for the extraction of significantly more information from experiment: a more complete Bayesian approach would take this into account also. This has not been possible previously because of the technical difficulty in calculating the likelihood of amplitudes seen in correlated post-synaptic potential trains; however, recent theoretical advances have now rendered the likelihood calculation tractable for a broad class of synaptic dynamics models. Here we present a compact mathematical form for the likelihood in terms of a matrix product and demonstrate how marginals of the posterior provide information on covariance of parameter distributions. The associated computer code for Bayesian parameter inference for a variety of models of synaptic dynamics is provided in the Supplementary Material allowing for quantal and dynamical parameters to be readily inferred from experimental data sets.

Keywords: Bayesian; EPSP; correlation; plasticity; quantal; stochastic; synapse.

Figure 1

Bayesian inference provides parameter distributions from five sweeps of synthetic data comprising 30 regular spikes at 30 Hz. Marginal posterior distributions (black), maximum a-posteriori estimates (orange crosses) and true parameter values (light blue dots) for the parameters of the synaptic model summarized in Table 1. Posteriors shown after 10⁶ Metropolis-Hastings samples. The true values were n = 7, τ_D = 0.25 s, τ_F = 0.2 s, p₀ = 0.6, p₁ = 0.8, μ_a = 0.25 mV, σ_a = 0.1 mV and σ_b = 0.05 mV.

Figure 2

Joint parameter estimates for the synaptic-dynamics model. (A) Pairwise and individual posterior marginals for release-site number n, depression timescale τ_D, initial release probability p₀, and mean quantal amplitude μ_a. True parameter values and data are the same as Figure 1. Colorbars for the values of the posterior distributions are not shown; the relative differences in value show the shape and sharpness of the pairwise posteriors for each pair of parameters. (B) Pairwise posterior marginal for release site number n and initial release probability p₀ for a case where the true values were n = 35 and p₀ = 0.50 showing a strong anticorrelation. All posteriors shown after 10⁶ Metropolis-Hastings samples.

Figure 3

Bayesian inference captures the shift in synaptic dynamics under application of adenosine. (A) Individual postsynaptic voltage traces under control (top) and adenosine (bottom) conditions. (B) Mean EPSP size for each spike in the stimulation protocol under control (blue) and adenosine (red) conditions. Bars show standard error. (C) Marginal posterior distributions for the parameters of the synaptic model in the control (blue) and adenosine (red) conditions. (D) Pairwise posterior marginals for number of active release sites n and initial release probability p₀ before (left) and after (right) application of adenosine. Posteriors shown after 5 × 10⁶ Metropolis-Hastings samples.

Figure 4

Comparison of likelihood functions that do or do not account for serial correlations in synaptic amplitudes. (A) Posterior distributions for release site number n computed by correlated (solid) and uncorrelated (dashed) likelihood functions for three different values of n (n = 8 dark black; n = 15 middle blue; and n = 35 light blue) for a single sweep of 5 spikes regularly distributed at 30Hz. (B) 95% confidence intervals for correlated (solid) and uncorrelated (dashed) likelihood functions as a function of the number of sweeps for different numbers of spikes per train. Spikes occur at 30Hz, the true value of n is 35, and averages are taken over 10 realizations. Other parameters are the same as for Figure 1 (light-blue dots).