In this paper a methodology of mathematical description of the synthesis, storage and release of the neurotransmitter during the fast synaptic transport is presented. The proposed model is based on the initial and boundary value problem for a parabolic nonlinear partial differential equation (PDE). Presented approach enables to express space and time dependences in the process: rate of vesicular replenishment, gradients of vesicular concentration and, through the boundary conditions, the location of docking and release sites. The model should be a good starting point for future numerical simulations since it is based on thoroughly studied parabolic equation. In the article classical and variational formulation of the problem is presented and the unique solution is shown to exist. The model is referred to the model based on ordinary differential equations (ODEs), created by Aristizabal and Glavinovic (AG model). It is shown that, under some assumptions, AG model is a special case of the introduced one.