The Narrow Escape problem in a flat cylindrical microdomain with application to diffusion in the synaptic cleft

The mean first passage time (MFPT) for a Brownian particle to reach a small target in cellular microdomains is a key parameter for chemical activation. Although asymptotic estimations of the MFPT are available for various geometries, these formula cannot be applied to degenerated structures where one dimension of is much smaller compared to the others. Here we study the narrow escape time (NET) problem for a Brownian particle to reach a small target located on the surface of a flat cylinder, where the cylinder height is comparable to the target size, and much smaller than the cylinder radius. When the cylinder is sealed, we estimate the MFPT for a Brownian particle to hit a small disk located centrally on the lower surface. For a laterally open cylinder, we estimate the conditional probability and the conditional MFPT to reach the small disk before exiting through the lateral opening. We apply our results to diffusion in the narrow synaptic cleft, and compute the fraction and the mean time for neurotransmitters to find their specific receptors located on the postsynaptic terminal. Finally, we confirm our formulas with Brownian simulations.