Multiple Traces Formulation and Semi-Implicit Scheme for Modelling Biological Cells under Electrical Stimulation

Abstract

We model the electrical behavior of several biological cells under external stimuli by extending and computationally improving the semi-implicit multiple traces formulation presented in (Henr´ıquez et al., Numerische Mathematik, 2016). Therein, the electric potential and current for a single cell are retrieved through the coupling of boundary integral operators and non-linear ordinary differential systems of equations.
Yet, the low-order discretization scheme presented becomes impractical when accounting for interactions among multiple cells. In this note, we consider multi-cellular systems and show existence and uniqueness
of the resulting non-linear evolution problem in finite time. Our main tools are analytic semigroup theory along with mapping properties of boundary integral operators in Sobolev spaces. Thanks to the smoothness
of cellular shapes, solutions are highly regular at a given time. Hence, spectral spatial discretization can be employed, thereby largely reducing the number of unknowns. Time-space coupling is achieved via a
semi-implicit time-stepping scheme shown to be stable and convergent. Numerical results in two dimensions validate our claims and match observed biological behavior for the Hodgkin-Huxley dynamical model.