On biomembrane electrodiffusive models

Two models are used in the literature, to study the electric behaviour of cellular membranes such as in protein aggregates, excitable media or ionic currents for examples. The first one is the Electroneutral Model based on Nernst-Planck and Poisson equations with a specific condition of microscopic electroneutrality. The second one is the Cable Model valid for long wavelengths based on an analogy between an electric cable and a cell. Convincing experiments have justified the Cable equation. First, we show that these two models are in contradiction. More precisely the assumption of electroneutrality is not considered in the Cable Model. The main difference between the two models is highlighted by the analysis of the well known voltage instability due to a negative differential conductance. Then, we derive a new semi-microscopic model (the Biomembrane Electrodiffusive Model, called BEM) valid for phenomena at any wavelength. The BEM is based on Nernst-Planck and Poisson equations but, doesn't imply microscopic electroneutrality. It reveals the capacitive behaviour of the membrane. In the limit of long wavelengths, one recovers the behaviour described within the Cable framework, as shown precisely in the study of the negative differential conductance analysis. Finally, we demonstrate the intimate link between the last models: the Cable Model appears as the limit of the BEM for large wavelengths with some prerequisites which are discussed. The effects of geometry and asymmetrical media are introduced.