A Discontinuous Galerkin Method for the Extracellular Membrane Intracellular Model

TL;DR

This work formulate and analyze interior penalty discontinuous Galerkin methods for coupled elliptic PDEs modeling excitable tissue, represented by intracellular and extracellular domains sharing a common interface and proves the existence and uniqueness of solutions by a reformulation of the problem to one posed on the membrane.

Abstract

We formulate and analyze interior penalty discontinuous Galerkin methods for coupled elliptic PDEs modeling excitable tissue, represented by intracellular and extracellular domains sharing a common interface. The PDEs are coupled through a dynamic boundary condition, posed on the interface, that relates the normal gradients of the solutions to the time derivative of their jump. This system is referred to as the Extracellular Membrane Intracellular model or the cell-by-cell model. Due to the dynamic nature of the interface condition and to the presence of corner singularities, the analysis of discontinuous Galerkin methods is non-standard. We prove the existence and uniqueness of solutions by a reformulation of the problem to one posed on the membrane. Convergence is shown by utilizing face-to-element lifting operators and notions of weak consistency suitable for solutions with low spatial regularity. Further, we present parameter-robust preconditioned iterative solvers. Numerical examples in idealized geometries demonstrate our theoretical findings, and simulations in multiple cells portray the robustness of the method.

Figures (8)

• Figure 1: Intracellular domain $\Omega_i$ (or cell) in blue surrounded by the extracellular domain $\Omega_e$ with interface $\Gamma$ and $\Omega = \Omega_i \cup \Omega_e$. The outer boundary of $\Omega_e$ is denoted by $\Gamma_e$.
• Figure 2: Single cell EMI model \ref{['eq:EMI']}. Problem geometry together with initial mesh (left) and solution at $t=0$ (right) of the test problem in \ref{['ex:onecell']}.
• Figure 3: Problem domain with structured initial mesh (left) and the low regularity (here \ref{['eq:lr']} with $s=0.5$ is shown) solution of the EMI model \ref{['eq:EMI']} at $t=0$ (right).
• Figure 4: EMI model with gap junctions \ref{['eq:EMI_many']}. Problem geometry together with initial mesh (left) and solution at $t=1$ (right) of the two-cell EMI test problem in \ref{['ex:twocell']}. Extracellular interfaces $\Gamma_{(0, 1)}$, $\Gamma_{(0, 2)}$ are highlighted in blue and red. The interface $\Gamma_{(1, 2)}$ between cells $\Omega_1$ and $\Omega_2$ which idealizes a gap junction is shown in green. Brown nodes depict points of intersections of the three interfaces.
• Figure 5: Simplified problem geometries representing long and thin domains encountered in EMI applications. The domain diameter grows due to (left) the elongated shape of a single cell (modeling spatial characteristics of neurons, see buccino2019does) or (right) stacking of cells in one direction (as is common in sheet simulations, e.g. tveito2017cellhuynh2023convergence).

Theorems & Definitions (35)

• Lemma 3.1: Poincaré's inequality over $\tilde{V}_{h0}^k$
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5: Equivalence
• proof