From Three-Dimensional Electrophysiology to the Cable Model: an Asymptotic Study

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.