Quantifying Repetitive Transmission at Chemical Synapses: A Generative-Model Approach

Abstract

The dependence of the synaptic responses on the history of activation and their large variability are both distinctive features of repetitive transmission at chemical synapses. Quantitative investigations have mostly focused on trial-averaged responses to characterize dynamic aspects of the transmission--thus disregarding variability--or on the fluctuations of the responses in steady conditions to characterize variability--thus disregarding dynamics. We present a statistically principled framework to quantify the dynamics of the probability distribution of synaptic responses under arbitrary patterns of activation. This is achieved by constructing a generative model of repetitive transmission, which includes an explicit description of the sources of stochasticity present in the process. The underlying parameters are then selected via an expectation-maximization algorithm that is exact for a large class of models of synaptic transmission, so as to maximize the likelihood of the observed responses. The method exploits the information contained in the correlation between responses to produce highly accurate estimates of both quantal and dynamic parameters from the same recordings. The method also provides important conceptual and technical advances over existing state-of-the-art techniques. In particular, the repetition of the same stimulation in identical conditions becomes unnecessary. This paves the way to the design of optimal protocols to estimate synaptic parameters, to the quantitative comparison of synaptic models over benchmark datasets, and, most importantly, to the study of repetitive transmission under physiologically relevant patterns of synaptic activation.

Keywords: expectation-maximization; generative modeling; quantal analysis; repetitive transmission; short-term plasticity.

Figure 1.

The generative model and sample synthetic traces. A, Schematics of the synaptic model. Upon spike (blue), only the docked vesicles (yellow) can be released. The postsynaptic response (gray) is proportional on average to the number of vesicles released. In between spikes, vesicles dock to noncompetent release sites (black arrow) with constant probability per unit time. B, Graphical model of the statistical dependencies among the number of docked vesicles before spike (S⁻) and after spike (S ⁺) and observable responses (R). The number of docked vesicles is not observable. C, Sample synthetic traces and average trace for a facilitating connection: N = 10, q = 0.15 mV, σ_q = 0.03 mV, U = 0.3, τ_D = 195 ms, τ_F = 570 ms. D, Same as C for a depressing connection: N = 10, q = 0.15 mV, σ_q = 0.03 mV, U = 0.25, τ_D = 670 ms, τ_F = 15 ms.

Figure 2.

Stimulation protocol and variability of synaptic responses (experimental data). A, Five sample single-trial trains of experimentally measured postsynaptic responses (gray traces) illustrate the large trial-to-trial variability. The trial-averaged trace (black trace) reveals facilitating transmission (compare first and second response). Stimulation consists of a regular train of eight spikes followed by a recovery spike (top blue trace). B, Same as in A for a depressing connection (compare first and second responses). C, Histograms of the CVs of the synaptic responses in the train and upon the recovery spike (gray-shaded panel) across the dataset. The red numbers denote the respective average CVs. The size of the bins is 0.25, apart from the last one, which includes all CVs >0.75.

Figure 3.

Maximum-likelihood estimation of the synaptic parameters from experimental data. A, Log-likelihood (left) and associated model parameters (right) as a function of N for a sample connection. The maximum is attained at N = 17 with q = 0.18 mV, σ_q = 0.06 mV, U = 0.27, τ_D = 202 ms, τ_F = 449 ms. B, Top, Average experimental responses (black line) vs average model responses (red line) for the same connection as in A. Bottom, Same as in the top panel for the coefficients of variation. Error bars indicate the 95% confidence interval of the model prediction. C, Cumulative distribution function of the log-likelihood for four instances of the leave-one-out procedure (blue curves). The red dots indicate the log-likelihood of the left-out trials. Same connection as in A. D, Distribution over the dataset of the average z_out. E, Distribution over the dataset of the average coefficients of variation of the estimates obtained in the leave-one-out procedure.

Figure 4.

Estimating uncertainty and correlations between parameter estimates with parametric bootstrap. A, Distributions of the relative errors obtained by re-estimating the parameters of the sample connection in Fig. 3A from synthetically generated responses. The dots on the x-axes indicate the corresponding averages. B, Standard error vs bias of the relative error for all the connections. C, Pearson correlation coefficients between all pairs of parameter estimates averaged over all the connections.

Figure 5.

Maximum-likelihood estimation (MLE) vs least-squares fitting (LSF). A, Estimates obtained with MLE vs those obtained with LSF for all the connections in the dataset. B, Average experimental responses (black line) for the connection corresponding to the cyan dot in A together with the result of LSF (blue line) and the MLE prediction with 95% confidence interval (red line + bars). MLE estimate: A = 0.91 mV, U = 0.38, τ_D = 179 ms, τ_F = 279 ms; LSF estimate: A = 1.48 mV, U = 0.27, τ_D = 236 ms, τ_F = 28 ms. C, Cumulative distribution functions of the relative errors of MLE (red), MLE with shuffled responses (green) and LSF (blue) estimates from synthetic experiments. D, Average and SD (bars) of relative errors for MLE (red), MLE with shuffled responses (green), and LSF (blue) estimates as a function of the number of trials (synthetic experiments).

Figure 6.

The condition number and the accuracy of the least-squares estimates. A, The range of relative errors vs the condition number of the estimates obtained by least-squares fitting (synthetic experiments). B, Sample synthetic average responses (black) together with average model responses (blue) resulting from the least-squares fitting. The parameters reported in the panels are the least-squares estimates. True parameters were as follows: A = 4.8 mV, U = 0.07, τ_D = 95 ms, τ_F = 28 ms. C, Same as in B but with true parameters: A = 8.1 mV, U = 0.33, τ_D = 81 ms, τ_F = 100 ms.

Figure 7.

Distributions and correlations of the parameters (experimental data). A, Distributions over the dataset of the different synaptic parameters. B, Number of release sites N vs average first response in the train. Dashed line, Linear regression (R = 0.76, p < 10⁻¹³). C, Initial release probability U vs time constant of the docking process τ_D. Dashed line, Linear regression (R = −0.48, p < 10⁻⁴). D, Initial release probability U vs time constant of facilitation τ_F. Dashed line, Linear regression (R = −0.34, p < 10⁻³).

Figure 8.

Comparison of different stimulation protocols (synthetic experiments). A, Schematics of the protocols. B, Box plots of the distributions of the relative errors of the estimates for all the parameters for sample synthetic connection. Stimulation frequency was 5 Hz. C, Cumulative distribution function of the minimal relative errors on the estimate of τ_F at varying stimulation frequencies for the regular (blue lines), the Poisson (red lines) and the single-sweep (magenta lines) protocols. Cumulative distribution functions at the same stimulation frequency are all statistically different (Kolmogorov–Smirnov test, p < 0.01). This is true for all stimulation frequencies.