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.
Mots-clés : Multiple traces formulation, semi-implicit time stepping, biological cells, exponential convergence
@article{M2AN_2018__52_2_659_0,
author = {Henr{\'\i}quez, Fernando and Jerez-Hanckes, Carlos},
title = {Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {659--703},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2018019},
mrnumber = {3834439},
zbl = {1404.65148},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2018019/}
}
TY - JOUR AU - Henríquez, Fernando AU - Jerez-Hanckes, Carlos TI - Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 659 EP - 703 VL - 52 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2018019/ DO - 10.1051/m2an/2018019 LA - en ID - M2AN_2018__52_2_659_0 ER -
%0 Journal Article %A Henríquez, Fernando %A Jerez-Hanckes, Carlos %T Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 659-703 %V 52 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2018019/ %R 10.1051/m2an/2018019 %G en %F M2AN_2018__52_2_659_0
Henríquez, Fernando; Jerez-Hanckes, Carlos. Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 659-703. doi : 10.1051/m2an/2018019. http://archive.numdam.org/articles/10.1051/m2an/2018019/
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