Finite Element Simulation of Ionic Electrodiffusion in Cellular Geometries.

Ada J Ellingsrud,Marie E Rognes,Geir Halnes,Gaute T Einevoll,Andreas Solbrå

doi:10.3389/fninf.2020.00011

Abstract

Mathematical models for excitable cells are commonly based on cable theory, which considers a homogenized domain and spatially constant ionic concentrations. Although such models provide valuable insight, the effect of altered ion concentrations or detailed cell morphology on the electrical potentials cannot be captured. In this paper, we discuss an alternative approach to detailed modeling of electrodiffusion in neural tissue. The mathematical model describes the distribution and evolution of ion concentrations in a geometrically-explicit representation of the intra- and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the Extracellular-Membrane-Intracellular (EMI) framework, we refer to this model as the KNP-EMI model. Here, we introduce and numerically evaluate a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree. Moreover, we compare the electrical potentials predicted by the KNP-EMI and EMI models. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the KNP-EMI framework can give new insights in this setting.

Highlights

The most common computational models for excitable cells are those based on cable theory (Rall, 1977; Koch, 1999)
We present the governing equations for ionic electrodiffusion in neural tissue with a geometrically explicit representation of the cellular membranes in section 2.1 below
Summary of Governing Equations In summary, the mathematical framework for electrodiffusion with explicit geometrical representation of the cell membranes is comprised of the bulk equations (1), (2) with (3), the interface conditions (7), (11) with (6) and (9), and (15) with (14), the initial conditions (24) and (26), and additional boundary conditions

Summary

Introduction

The most common computational models for excitable cells are those based on cable theory (Rall, 1977; Koch, 1999). Multi-compartmental neuron models based on cable theory are widely used within the field of neuroscience to simulate large network of interacting neurons (see e.g., Markram et al, 2015). In such models, only synaptic interactions between neurons are considered, whereas changes in the extracellular field and extracellular ion concentrations associated with a neuron’s activity are assumed to be too small to have any influence on its neighboring neurons (or itself). Concentration changes are often limited by neuronal and glial uptake mechanisms that strive toward maintaining concentrations close to basal levels

Methods

Results

Conclusion