  We report here new electrical laws, derived from nonlinear electro-diffusion
theory, about the effect of the local geometrical structure, such as curvature,
on the electrical properties of a cell. We adopt the Poisson-Nernst-Planck
(PNP) equations for charge concentration and electric potential as a model of
electro-diffusion. In the case at hand, the entire boundary is impermeable to
ions and the electric field satisfies the compatibility condition of Poisson's
equation. We construct an asymptotic approximation for certain singular limits
to the steady-state solution in a ball with an attached cusp-shaped funnel on
its surface. As the number of charge increases, they concentrate at the end of
cusp-shaped funnel. These results can be used in the design of nano-pipettes
and help to understand the local voltage changes inside dendrites and axons
with heterogenous local geometry.
