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The transfer of electro-chemical signals from the pre-synaptic to the post-synaptic terminal of a neuronal or neuro-muscular synapse is the basic building block of neuronal communication. When triggered by an action potential, the pre-synaptic terminal releases neurotransmitters into the synaptic cleft through exocytotic vesicle fusion. The number of synaptic vesicles that fuse, i.e., the quantal content, is stochastic, and widely assumed to be binomially distributed. However, the quantal content depends on the number of release-ready vesicles, a random variable that depends upon a stochastic replenishment process, as well as stochastic interspike intervals of action potentials. The quantal content distribution suitably accounting for these two stochastic processes was not known explicitly in the literature. Here we analytically obtain the exact probability distribution of the number of vesicles released into the synaptic cleft, in the steady state reached during stimulation by a train of action potentials. We show that this distribution is binomial, with modified parameters, only when stimulated by constant frequency signals. Other forms of stochastic stimulation, e.g. Poissonian action potential train, lead to distributions that are non-binomial. The general formula valid for arbitrary distributions of the input inter-spike interval may be employed to study neuronal transmission under diverse experimental conditions. We provide exact theoretical moments which may be compared with experiments to estimate the model parameters. We corroborate our theoretical predictions through comparison with the quantal content distributions obtained from electrophysiological recordings from MNTB-LSO synapses of juvenile mice. We also confirm our theoretically predicted frequency dependence of mean quantal content by comparing with experimental data from hippocampal and auditory neurons.Author summaryThe synapse is a specialized junction between a pre-synaptic neuron and another neuron or muscle cell for neuronal communication. Action Potentials (APs) in the pre-synaptic neuron reach the synapse and trigger neurotransmitter-loaded vesicles to release their cargo in the nanometer-sized gap between the two cells. The pool of vesicles that are ready to be released is then replenished. The number of vesicles that release neurotransmitters, the quantal content and the number that are replenished are both random numbers. However, the probability distribution that describes the quantal content while taking the stochastic replenishment process into account was not known analytically, although a binomial distribution was often assumed. Exact mathematical results are not frequent in biological processes, but we could find an exact analytical solution of the problem in the long time (steady state) limit of AP stimulation. We show that for constant frequency stimulation the distribution of quantal content is indeed a binomial, but for stochastic stimulations (like Poisson train) generically it is not a binomial. The exact distributions and moments that we derive would greatly simplify estimation of model parameters, as we demonstrate by comparing our theory with data from electrophysiological recordings of auditory neurons. |
