Today, the classical homogenized models for simulating excitable tissue are challenged by new mathematical frameworks that explicitly represent and resolve the geometry of extracellular and intracellular spaces and cellular membranes (EMI models). These mixed-dimensional models crucially enable abstract representation of heterogeneous and physiologically realistic membrane mechanisms, such as the heterogeneous distribution of the astrocytic water channel AQP4. EMI models typically predict the electrical properties of the tissue, with the underlying assumption that the ion concentrations are constant. Although this assumption is only an approximation, the resulting models still give accurate predictions of neuronal electrodynamics in many scenarios. They do however fail in capturing the numerous phenomena related to shifts in the extracellular ion concentrations.
Here, we discuss an alternative approach to detailed modelling of excitable tissue that also accounts for ion concentrations and electrodiffusion. In particular, we consider a mathematical model describing the distribution and evolution of ion concentrations and potentials in a geometrically-explicit representation of the intra and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the EMI framework, we refer to this model as the KNP-EMI model. 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