  The flow of ions through permeable channels causes voltage drop in
physiological nanodomains such as synapses, dendrites and dendritic spines, and
other protrusions. How the voltage changes around channels in these nanodomains
has remained poorly studied. We focus this book chapter on summarizing recent
efforts in computing the steady-state current, voltage and ionic concentration
distributions based on the Poisson-Nernst-Planck equations as a model of
electro-diffusion. We first consider the spatial distribution of an uncharged
particle density and derive asymptotic formulas for the concentration
difference by solving the Laplace's equation with mixed boundary conditions. We
study a constant particles injection rate modeled by a Neumann flux condition
at a channel represented by a small boundary target, while the injected
particles can exit at one or several narrow patches. We then discuss the case
of two species (positive and negative charges) and take into account motions
due to both concentration and electrochemical gradients. The voltage resulting
from charge interactions is calculated by solving the Poisson's equation. We
show how deep an influx diffusion propagates inside a nanodomain, for
populations of both uncharged and charged particles. We estimate the
concentration and voltage changes in relations with geometrical parameters and
quantify the impact of membrane curvature.
