  The narrow escape problem arises in deriving the asymptotic expansion of the
solution of an inhomogeneous mixed Dirichlet-Neumann boundary value problem. In
this paper, we mainly deal with narrow escape problem in a smooth domain
connected to a long neck-Dendritic spine shape domain, which has a certain
significance in biology. Since the special geometry of dendritic spine, we
develop a new model for solving this narrow escape problem which is
Neumann-Robin Boundary Model. This model transform spine singular domain to
smooth spine head domain by inserting Robin boundary condition to the
connection part between spine head and neck. We rigorously find the high-order
asymptotic expansion of Neumann-Robin Boundary Model and apply it to the
solution of narrow escape problem in a dendritic spine shape domain. Our
results show that the asymptotic expansion of the Neumann-Robin Boundary Model
can be easily applied to the narrow escape problem for any smooth spine head
domain with straight spine neck. By numerical simulations, we show that there
is great agreement between the results of our Neumann-Robin Boundary Model and
the original escape problem. In this paper, we also get some results for
non-straight long spine neck case by considering curvature of spine neck.
