Boundary integral formulation and semi-implicit scheme coupling for modeling cells under electrical stimulation

Abstract

We model the electrical activity of biological cells under external stimuli via a novel boundary integral (BI) formulation together with a suitable time-space numerical discretization scheme. Ionic channels follow a non-linear dynamic behavior commonly described by systems of ordinary differential equations dependent on the electric potential jump across the membrane. Since potentials in both intra--- and extracellular domains satisfy an electrostatic approximation, we represent them using solely Dirichlet and Neumann traces over the membrane via boundary potentials. Hence, the volume problem is condensed to one posed over the cell boundary. A second-order time-stepping semi-implicit numerical Galerkin scheme is proposed and analyzed wherein BI operators are approximated by low-order basis functions, with stability independent of space discretization.