  The distribution of voltage in sub-micron cellular domains remains poorly
understood. In neurons, the voltage results from the difference in ionic
concentrations which are continuously maintained by pumps and exchangers.
However, it not clear how electro-neutrality could be maintained by an excess
of fast moving positive ions that should be counter balanced by slow diffusing
negatively charged proteins. Using the theory of electro-diffusion, we study
here the voltage distribution in a generic domain, which consists of two
concentric disks (resp. ball) in two (resp. three) dimensions, where a negative
charge is fixed in the inner domain. When global but not local
electro-neutrality is maintained, we solve the Poisson-Nernst-Planck equation
both analytically and numerically in dimension 1 (flat) and 2 (cylindrical) and
found that the voltage changes considerably on a spatial scale which is much
larger than the Debye screening length, which assumes electro-neutrality. The
present result suggests that long-range voltage drop changes are expected in
neuronal microcompartments, probably relevant to explain the activation of far
away voltage-gated channels located on the surface.
