Article Figures & Data
Figures
- 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, σq = 0.03 mV, U = 0.3, τD = 195 ms, τF = 570 ms. D, Same as C for a depressing connection: N = 10, q = 0.15 mV, σq = 0.03 mV, U = 0.25, τD = 670 ms, τF = 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, σq = 0.06 mV, U = 0.27, τD = 202 ms, τF = 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 zout. 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, τD = 179 ms, τF = 279 ms; LSF estimate: A = 1.48 mV, U = 0.27, τD = 236 ms, τF = 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, τD = 95 ms, τF = 28 ms. C, Same as in B but with true parameters: A = 8.1 mV, U = 0.33, τD = 81 ms, τF = 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 τD . Dashed line, Linear regression (R = −0.48, p < 10−4). D, Initial release probability U vs time constant of facilitation τF . 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 τF 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.
Tables
- Table 1.
Parameter estimates obtained with our method (in bold type) compared with published estimates obtained in similar experimental preparations
Parameter Average ± SD Range Preparation Temperature (°C) Method Reference 
2–77 Ferret mPFC, L5 to L5 32–34 MLE 8.1 ± 5.3 3–18 Rat HC, CA1 34 H, QMLE Larkman et al. (1997) 5.1 ± 2.7 2.3–11.1 Rat SC, L4 spiny stellar to L2/3 pyramidal 35–37 MV Silver et al. (2003) N 7.48 ± 7.11 2–22 Rat VC and SC, various layers 35–36 MV, FA Brémaud et al. (2007) 12.21 ± 7.9 2–32 Cat VC, various layers 35–36 MV, FA Brémaud et al. (2007) 53.3 ± 42 7–170 Rat SC, L5 to L5 35 LS Loebel et al. (2009) 3.4 ± 2.2 1–7 Rat VC, L5 to L5 23–26/36 MV, H Hardingham et al. (2010)
0.06–0.32 Ferret mPFC, L5 to L5 32–34 MLE 0.131 ± 0.145 0.084–0.197 Rat HC, CA1 34 H, QMLE Larkman et al. (1991) 0.196 ± 0.062 0.066–0.275 Rat HC, CA1 34 H, QMLE Larkman et al. (1997) q [mV] 0.15 ± 0.09 Rat SC, L4 spiny stellar to L2/3 pyramidal 35–37 MV Silver et al. (2003) 0.37 ± 0.17 0.15–1 Rat VC and SC, various layers 35–36 MV, FA Brémaud et al. (2007) 0.22 ± 0.19 0.07–1 Cat VC, various layers 35–36 MV, FA Brémaud et al. (2007) 0.13 ± 0.04 0.06–0.25 Rat SC, L5 to L5 35 LS Loebel et al. (2009) 0.211 ± 0.065 0.106–0.302 Rat VC, L5 to L5 23–26/36 MV, H Hardingham et al. (2010)
0.00–0.31 Ferret mPFC, L5 to L5 32–34 MLE σq [mV] 0.052 ± 0.085 0.030–0.10 Rat HC, CA1 34 H, QMLE Larkman et al. (1991) 0.065 ± 0.15 Rat SC, L4 spiny stellar to L2/3 pyramidal 35–37 MV Silver et al. (2003)
0.05–0.73 Ferret mPFC, L5 to L5 32–34 MLE 0.53 ± 0.17 0.14–0.81 Rat HC, CA1 34 H, QMLE Larkman et al. (1997)
Rat SC, L5 to L5 32–34 LS Markram et al. (1998)
Rat SC, L5 to L5 32–34 LS Markram et al. (1998) 0.79 ± 0.12 Rat SC, L4 spiny stellar to L2/3 pyramidal 35–37 MV Silver et al. (2003) U 0.46 ± 0.26* 0.15–0.95 Rat SC, L2/3 to L2/3 35 OQA Koester and Johnston (2005) 0.27 ± 0.15 0.03–0.6 Ferret mPFC, L5 to L5 32–34 LS Wang et al. (2006) 0.63 ± 0.6 Rat VC and SC, various layers 35–36 MV, FA Brémaud et al. (2007) 0.69 ± 0.18 Cat VC, various layers 35–36 MV, FA Brémaud et al. (2007) 0.46 ± 0.1 0.25–0.65 Rat SC, L5 to L5 35 LS Loebel et al. (2009) 0.46 ± 0.21 0.15–0.75 Rat VC, L5 to L5 23–26/36 MV, H Hardingham et al. (2010) 0.53 ± 0.05 Rat VC, L5 to L5 32–34 Bayesian Costa et al. (2013)
16–1800 Ferret mPFC, L5 to L5 32–34 MLE
Rat SC, L5 to L5 32–34 LS Markram et al. (1998) τD [ms] 399 ± 295‡ Rat SC, L5 to L5 32–34 LS Markram et al. (1998) 396 ± 163 0–1600 Ferret mPFC, L5 to L5 32–34 LS Wang et al. (2006) 525 ± 134 380–900 Rat SC, L5 to L5 35 LS Loebel et al. (2009)
0–1900 Ferret mPFC, L5 to L5 32–34 MLE τF [ms] 1797 ± 1247‡ Rat SC, L5 to L5 32–34 LS Markram et al. (1998) 292 ± 240 0–1600 Ferret mPFC, L5 to L5 32–34 LS Wang et al. (2006) The last three columns display the temperature at which recordings were taken, the estimation methods, and the reference to the corresponding study, respectively. All synaptic connection are pyramidal to pyramidal, unless stated otherwise. HC, Hippocampus; VC, visual cortex; SC, somato-sensory cortex; L, layer; MV, mean-variance analysis; QMLE, quantal maximum likelihood; LS, least-squares fit; H, histogram; FA, failure analysis; OQA, optical quantal analysis.
*Release probability per connection; †dominantly depressing connections; ‡dominantly facilitating connections.
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