# Asymptotic Analysis of the Narrow Escape Problem in Dendritic Spine Shaped Domain: Three Dimension

## TL;DR

This paper derives high-order asymptotic formulas for the mean first passage time of Brownian particles in a 3D dendritic spine-shaped domain, simplifying the complex boundary problem for better analytical understanding.

## Contribution

It extends previous 2D analyses to 3D, providing a rigorous asymptotic expansion for the narrow escape problem in dendritic spines.

## Key findings

- Derived high-order asymptotic expansion for mean first passage time
- Simplified the boundary value problem from Dirichlet-Neumann to Robin-Neumann
- Generalized previous 2D results to three-dimensional domains

## Abstract

This paper deals with the three-dimensional narrow escape problem in dendritic spine shaped domain, which is composed of a relatively big head and a thin neck. The narrow escape problem is to compute the mean first passage time of Brownian particles traveling from inside the head to the end of the neck. The original model is to solve a mixed Dirichlet-Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper we seek to transfer the original problem to a mixed Robin-Neumann boundary value problem by dropping the thin neck part, and rigorously derive the asymptotic expansion of the mean first passage time with high order terms. This study is a nontrivial generalization of the work in \cite{Li}, where a two-dimensional analogue domain is considered.