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 multiple traces formulation introduced in Henríquez et al. [Numer. Math. 136 (2016) 101–145]. 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 second order convergent. Numerical results in two dimensions validate our claims and match observed biological behavior for the Hodgkin–Huxley dynamical model.

Derivation of cable equation by multiscale analysis for a model of myelinated axons

“…The existence of a mild solution follows from the classical semigroup theory (see, for example, [5]). For the existence of more regular solutions see [4], [3]. We present just an idea of the proof.…”

“…On the Ranvier nodes we assume the continuity of currents (2), and the Hodgkin-Huxley dynamics for the transmembrane potential (3). Following the Hodgkin-Huxley model, the applied current through the membrane is a sum of the capacitive current c m ∂ t [u ε ], where c m is the membrane capacitance per unit area, and the ionic current I ion ([u ε ], g ε ) through the ion channels.…”

Time-Domain Multiple Traces Boundary Integral Formulation for Acoustic Wave Scattering in 2D

“…Yet, to our knowledge, no time-domain CQ MTF has been described in the literature. In fact, only semi-implicit time schemes have been coupled to local MTF to describe non-linear dynamics of neural transmission [22,21].…”