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# QUANTITATIVE BIOLOGY SEMINAR

Stabilization of the Electroneutral Model

The electroneutral model for neuronal electrical activity generalizes the popular cable model in electrophysiology. It accounts for electrodiffusion of ions without fully resolving the very fine space-charge layers that typically render Poisson-Nernst-Plank models numerically intractable. The electroneutral model is also applicable to semiconductors, electro-kinetic fluids, and ion channels in cell membranes.

Within the electroneutral model, the concentrations of each ion species are tracked in regions of space separated by capacitive cell membranes with Hodgkin-Huxely ion channels. The evolution of the ion concentrations are governed by the drift-diffusion equations with a flux that is the sum of a term from Fick's first law and a drift term to account for the effects of the electrostatic field. As an approximation, we assume electroneutrality at each point in space. This has two important consequences. The first is that the equation that determines the electrostatic potential, Poisson's equation, must be modified to be consistent with electroneutrality. Second, a boundary layer analysis determines new boundary conditions by accounting for the charge in the Debye layers as a surface charge. I will present this derivation of the electroneutral model.

In the original and most straightforward formulation of the electroneutral model, small perturbations to ion concentrations grow exponentially in time. This instability can be removed by introducing a temporal low pass filtering of the boundary condition, but this has no physical basis. I will present two new and physically motivated modifications to the electroneutral model that result in well-posed problems. These stabilizations will be demonstrated for a specific electrodiffusion problem using an eigenfunction approach. Additionally, I will present comparisons of the various versions of the electroneutral model, solved using a finite volume method, to some exact solutions of the detailed physics.