Table 1.
Parameters used in the stochastic model for synaptic transmission.
Fig 1.
Schematic of a chemical synapse and sample realizations of its corresponding stochastic model.
A) The process of SV docking and undocking in the active zone of the axon terminal, and the evoked release of neurotransmitter molecules. The lower panel shows the different time-varying probabilities pd,i, pu,i, pr,i related to the AP, where
, that govern the reversible transitions between an empty site (ES) and an occupied site (OS) upon SV docking/undocking, and SV fusion (see text for details). B) A sample stochastic realization of the model showing a buildup in the number of docked SVs between successive APs, and a reduction in docked SV numbers from fusion and neurotransmitter release in response to APs (top). The corresponding quantal content (QC) – the number of SV fusion events per AP – is shown in the bottom plot. The number of docking sites is assumed to be M = 100, and all sites are occupied with SVs at the start of the AP train. Each docked SV has a constant release probability pr = 0.5, and drawing from a binomial distribution results in the first QC to be 45. This drops the number of docked SVs to 55 after the first AP, and docked SVs replenish till the arrival of the second AP. The docked SV dynamics in the inter-stimulus interval is governed by kinetic rates that are chosen so as to result in refilling and undocking probabilities pd = 0.4 and pu = 0.1, respectively. (Appendix A in S1 File).
Fig 2.
Transient reduction in quantal content (QC) is associated with an increased QC Fano factor.
A) The average QC, i.e., the average number of synaptic SVs fusing per AP as predicted by Eqs 5 and (10) for a constant release probability pr = 0.15 (blue, orange, and gray dots), and undocking probability pu = 0. To capture synaptic facilitation (yellow dots) we also consider a time-varying release probability: pr,1 = 0.15 (first stimulus), pr,2 = 0.2 (second stimulus), pr,3 = 0.25 (third stimulus) and pr = 0.3 for the fourth and all remaining stimuli. The number of docking sites M = 200 are all assumed to be filled upon arrival of the 1st AP. The per-site SV refilling probability is pd = 0.3 (blue dots), 0.15 (orange dots), 0.05 (gray dots), or 0.02 (yellow dots). B) The corresponding QC Fano factor FFi over time as predicted by (7).
Fig 3.
Alternative parameter regimes with identical mean transient QC yield contrasting QC fluctuation statistics.
Different predictions for the QC Fano factor, each resulting in the same mean QC corresponding to pr = 0.15 and pd = 0.05 in Fig 2 (the bottom-most, yellow line). The top line corresponds to the Fano factor FFi predicted for pr = 0.15 and pd = 0.05 from (7). The middle curve is obtained from (7) with a high release probability (pr = 0.9) and a corresponding time-varying refilling probability pd,i to get the same mean synaptic depression. The bottom curve corresponds to (7) with parameters , and here the same mean synaptic depression occurs due to a reduction in the number of docking sites M. In all cases, the undocking probability is assumed to be zero (pu = 0) and each docking site is occupied at the beginning of the AP train (p1 = 1).
Fig 4.
Normalized synaptic depression and QC fluctuation statistics as a function of release and refilling probabilities.
A) Plots of the steady-state QC Fano factor as given by Eq 16. B) Steady-state correlation between successive QCs as given by Eq 20. C) Normalized synaptic depression assumed to be equal to
in Eq 13 as a function of the release and refilling probabilities. From panel B, one can see that if both probabilities pr and pd are simultaneously high or simultaneously low, this leads to uncorrelated QCs. However, the two scenarios make contrasting predictions on the Fano factor in panel A. In particular, high probability values lead to a Fano factor close to zero, whereas low probability values result in a Fano factor close to one.
Fig 5.
Fluctuation statistics of the quantal content (QC) for the auditory MNTB-LSO synapses.
A) Results of a whole-cell patch-clamp recording from a single LSO neuron at a 50-Hz challenge for 1 min (3000 stimuli) as obtained from [69]. Each point represents the QC after a single stimulus pulse. B) Normalized QC values to the first 20 stimuli show the initial depression behavior and the subsequent average synaptic depression level. Values are normalized to the first stimulus QC. C) Steady-state QC distribution as obtained using QCs from stimulus numbers 10 to 3000 together with a fit to a binomial distribution. The steady-state QC Fano factor is obtained as , where
denotes the 95% confidence intervals as obtained by bootstrapping. D) The scatter plot between successive QCs from AP number 10 to 3000 shows a weak negative correlation with a Pearson correlation coefficient
.
Fig 6.
The MNTB-LSO synapses are characterized by high refilling and release probabilities.
A) A quantitative fit of Eq 24 with pr = 0.23 & pd = 0.2 to the transient QC dynamics (orange line). B) Model-predicted QC dynamics as per Eq 24 with pr = 0.93 & pd = 0.53 (gray line), and as predicted by solving (2) for a constant release probability pr = 0.93, zero undocking probability pu,i = 0 and a time-varying refilling probability as shown in the inset (green line). C) Model-predicted steady-state QC Fano factors from (16) for constant low probabilities (pr = 0.23 & pd = 0.2 in orange) or high probabilities (pr = 0.93 & pd = 0.53 in green). Only the latter scenario is consistent with fluctuation statistics from the electrophysiological data (shown in blue). D) Model-predicted steady-state correlations between successive QCs from (20) for constant low (orange) and high probabilities (green), with only the latter fitting the experimentally obtained QC correlations (blue). Error bars on the data are the 95% confidence interval on the steady-state statistics as obtained from bootstrapping QCs from stimulus numbers 10 to 3000.
Fig 7.
Identification of release and refilling probabilities for MNTB-LSO synapses using QC fluctuation statistics.
The left-most-plot marks the region of parameter space (in terms of the refilling and release probabilities pd and pr, respectively) consistent with a steady-state QC Fano factor as predicted by the formula (16) to be less than . The FF upper bound is based on a
range around the experimentally-observed
in Fig 5C. The other plots mark the parameter space consistent with QC correlations as given by (20) to be
and the normalized synaptic depression as given by (13) in the range
based on the electrophysiological data in Fig 5. The right-most-plot shows the intersection of all three consistent regions narrowing the parameter space to a region with a high release probability (
) and
.