  In synaptic molecular communication (MC), the activation of postsynaptic
receptors by neurotransmitters (NTs) is governed by a stochastic
reaction-diffusion process. This randomness of synaptic MC contributes to the
randomness of the electrochemical downstream signal in the postsynaptic cell,
called postsynaptic membrane potential (PSP). Since the randomness of the PSP
is relevant for neural computation and learning, characterizing the statistics
of the PSP is critical. However, the statistical characterization of the
synaptic reaction-diffusion process is difficult because the reversible
bi-molecular reaction of NTs with receptors renders the system nonlinear.
Consequently, there is currently no model available which characterizes the
impact of the statistics of postsynaptic receptor activation on the PSP. In
this work, we propose a novel statistical model for the synaptic
reaction-diffusion process in terms of the chemical master equation (CME). We
further propose a novel numerical method which allows to compute the CME
efficiently and we use this method to characterize the statistics of the PSP.
Finally, we present results from stochastic particle-based computer simulations
which validate the proposed models. We show that the biophysical parameters
governing synaptic transmission shape the autocovariance of the receptor
activation and, ultimately, the statistics of the PSP. Our results suggest that
the processing of the synaptic signal by the postsynaptic cell effectively
mitigates synaptic noise while the statistical characteristics of the synaptic
signal are preserved. The results presented in this paper contribute to a
better understanding of the impact of the randomness of synaptic signal
transmission on neuronal information processing.
