Solving the EMI Equations using Finite Element Methods

This chapter discusses 2 X 2 symmetric variational formulations and associated finite element methods for the EMI equations. We demonstrate that the presented methods converge at expected rates, and compare the approaches in terms of approximation of the transmembrane potential. Overall, the choice of which formulation to employ for solving EMI models becomes largely a matter of desired accuracy and available computational resources.

Iterative Solvers for EMI Models

This chapter deals with iterative solution algorithms for the four EMI formulations derived in (17, Chapter 10.1007/978-3-030-61157-6_5). Order optimal monolithic solvers robust with respect to material parameters, the number of degrees of freedom of discretization as well as the time-stepping parameter are presented and compared in terms of computational cost. Domain decomposition solver for the single-dimensional primal formulation is discussed.

Modeling Cardiac Mechanics on a Sub-Cellular Scale

We aim to extend existing models of single-cell mechanics to the EMI framework, to define spatially resolved mechanical models of cardiac myocytes embedded in a passive extracellular space. The models introduced here will be pure mechanics models employing fairly simple constitutive laws for active and passive mechanics. Future extensions of the models may include a coupling to the electrophysiology and electro-diffusion models described in the other chapters, to study the impact of spatially heterogeneous ion concentrations on the cell and tissue mechanics.

Operator Splitting and Finite Difference Schemes for Solving the EMI Model

We want to be able to perform accurate simulations of a large number of cardiac cells based on mathematical models where each individual cell is represented in the model. This implies that the computational mesh has to have a typical resolution of a few µm leading to huge computational challenges. In this paper we use a certain operator splitting of the coupled equations and showthat this leads to systems that can be solved in parallel. This opens up for the possibility of simulating large numbers of coupled cardiac cells.

A Cell-Based Model for Ionic Electrodiffusion in Excitable Tissue

This chapter presents the KNP-EMI model describing ion concentrations and electrodiffusion in excitable tissue. The KNP-EMI model extends on the EMI model by removing the assumption that ion concentrations are constant in time and space, and may as such be more appropriate in connection with modelling e.g. spreading depression, stroke and epilepsy. The KNP-EMI model defines a system of time-dependent, nonlinear, mixed dimensional partial differential equations. We here detail the derivation of the system and present a numerical example illustrating how ion concentrations evolve during neuronal activity.

Derivation of a Cell-Based Mathematical Model of Excitable Cells

Excitable cells are of vital importance in biology, and mathematical models have contributed significantly to understand their basic mechanisms. However, classical models of excitable cells are based on severe assumptions that may limit the accuracy of the simulation results. Here, we derive a more detailed approach to modeling that has recently been applied to study the electrical properties of both neurons and cardiomyocytes. The model is derived from first principles and opens up possibilities for studying detailed properties of excitable cells.We refer to the model as the EMI model because both the extracellular space (E), the cell membrane (M) and the intracellular space (I) are explicitly represented in the model, in contrast to classical spatial models of excitable cells. Later chapters of the present text will focus on numerical methods and software for solving the model. Also, in the next chapter, the model will be extended to account for ionic concentrations in the intracellular and extracellular spaces.

Improving Neural Simulations with the EMI Model

Mathematical modeling of neurons is an essential tool to investigate neuronal activity alongside with experimental approaches. However, the conventional modeling framework to simulate neuronal dynamics and extracellular potentials makes several assumptions that might need to be revisited for some applications. In this chapter we apply the EMI model to investigate the ephaptic effect and the effect of the extracellular probes on the measured potential. Finally, we introduce reduced EMI models, which provide a more computationally efficient framework for simulating neurons with complex morphologies.