Mathematical Analysis of Deterministic and Stochastic Models
of Synaptic Transmitter Release

In the brain, information is transmitted from one nerve cell to another by the secretion of neurotransmitters. We have developed mathematical models of neurotransmitter secretion, and the short-term enhancement of secretion that is an important part of the signal processing done at nerve terminals. Although some information can be obtained from simple deterministic models, the biological process is inherently stochastic, so stochastic models are required to capture many important features. Development and analysis of one such model will be described. With this model, short-term enhancement is due to the interaction of two populations of transmitter release sites, each described by probability density functions that evolve according to hyperbolic conservation equations. Equations for the means of these density functions are derived, yielding a system of ordinary differential equations that is much simpler to analyze. Finally, a multiple scale analysis is used to explain several properties of transmitter release observed in experiments.