Exact Distribution of the Quantal Content in Synaptic Transmission

Abstract

During electrochemical signal transmission through synapses, triggered by an action potential (AP), a stochastic number of synaptic vesicles (SVs), called the "quantal content," release neurotransmitters in the synaptic cleft. It is widely accepted that the quantal content probability distribution is a binomial based on the number of ready-release SVs in the presynaptic terminal. But the latter number itself fluctuates due to its stochastic replenishment, hence the actual distribution of quantal content is unknown. We show that exact distribution of quantal content can be derived for general stochastic AP inputs in the steady state. For fixed interval AP train, we prove that the distribution is a binomial, and corroborate our predictions by comparison with electrophysiological recordings from MNTB-LSO synapses of juvenile mice. For a Poisson train, we show that the distribution is nonbinomial. Moreover, we find exact moments of the quantal content in the Poisson and other general cases, which may be used to obtain the model parameters from experiments.

FIG. 1.

(a) Schematic picture of a chemical synapse. The SVs (circular balls) filled with neurotransmitters (black dots) get docked to sites (in green) and remain ready to release at the presynaptic terminal. Each AP triggers release of some SVs to the synaptic cleft, which reach the receptors in postsynaptic terminal to produce evoked postsynaptic current. (b) A schematic input AP train with random intervals $t_s$ (top), and (bottom) the output quantal content varying with time, shown to reach a steady state at long times.

FIG. 2.

The time variation of docked SV number $n\left(t\right)$ is shown as a function of time over two successive ISIs between the $m-2$ and $m-1$ and $m\text{th}$ APs. The respective numbers before an AP $\left(n_{-}\right)$ after an AP $\left(n_{+}\right)$ and quantal content $\left(b\right)$ are shown.

FIG. 3.

Steady state quantal content distributions for (a) fixed AP arrival times $\left(f=1\text{/}T=20\text{Hz}\right)$, and (b) exponentially distributed ISIs (rate $f_0=20\text{Hz}$). The common parameters are $M=50$, $p_r=0.5$, $k=2{\text{s}}^{-1}$. The exact theoretical curves (thick lines) match the KMC data (red symbols). In (a) an approximate (see text) binomial distribution (dashed line) is shown alongside the exact binomial. In (b) the binomial distribution of Eq. (11) with $T\to 1\text{/}{f}_0$ (dashed line) deviates from the exact nonbinomial distribution.

FIG. 4.

The experimental mean quantal content $〈b〉$ is plotted (in red) against time for (a) synapse-1 and (b) synapse-2. The blue curves represent one of the 20 experimental realizations of the quantal content. The yellow box shows the part of data we considered to be in the steady state for plotting the experimental histograms of quantal content in (c) and (d) (in green). In (c) and (d), the parameter values used for the exact theoretical distributions (black curves) for synapse-1 are $M=50$ and $p_{rb}=0.345$ and for the synapse-2 are $M=98$ and $p_{rb}=0.266$.

FIG. 5.

The frequency dependence of the theoretical mean quantal content is matched with the experimental values for the synapse types (a) CA3-CA1, (b) EC-DG, and (c) MNTB-LSO. The parameters (given in Ref. [39], as well as estimated by our fit here) are as follows: (a)$M=688$, $p_r=0.011$, $k=0.0523{\text{s}}^{-1}$ (b)$M=17$, $p_r=0.251$, $k=2.058{\text{s}}^{-1}$ (c)$M=147$, $p_r=0.156$, $k=3.816{\text{s}}^{-1}$.