  Cellular electrophysiology is often modeled using the cable equations. The
cable model can only be used when ionic concentration effects and three
dimensional geometry effects are negligible. The Poisson model, in which the
electrostatic potential satisfies the Poisson equation and the ionic
concentrations satisfy the drift-diffusion equation, is a system of equations
that can incorporate such effects. The Poisson model is unfortunately
prohibitively expensive for numerical computation because of the presence of
thin space charge layers at internal membrane boundaries. As a computationally
efficient and biophysically natural alternative, we introduce the
electroneutral model in which the Poisson equation is replaced by the
electroneutrality condition and the presence of the space charge layer is
incorporated in boundary conditions at the membrane interfaces. We use matched
asymptotics and numerical computations to show that the electroneutral model
provides an excellent approximation to the Poisson model. Further asymptotic
calculations illuminate the relationship of the electroneutral or Poisson
models with the cable model, and reveal the presence of a hierarchy of
electrophysiology models.
