Search Time for a Small Ribbon and Application to Vesicular Release at Neuronal Synapses

The arrival of a Brownian particle at a narrow cusp located underneath a ball is a model of vesicular release at neuronal synapses, triggered by calcium ions. The asymptotic computation of the arrival time presents several difficulties that can be overcome using conformal mappings and asymptotic analysis of the model equations. Using a regular expansion of the solution of the Laplace equation in the mapped domain, we compute the solution involving both small and large spatial scales. We derive novel asymptotic formulas for Brownian escape through cusps in both two and three dimensions. The range of validity of the asymptotic formulas is challenged by stochastic simulations. Finally, we apply the analysis to estimate the vesicular release probability at presynaptic terminals and, in particular, we suggest that vesicular organization imposes a severe constraint on calcium channel localization: diffusing calcium ions can trigger vesicular release only in a specific range of positions that we provide.