Electrical conduction in brain tissue is commonly modeled using classical bidomain models. These models fundamentally assume that the discrete nature of brain tissue can be represented by homogenized equations where the extracellular space, the cell membrane, and the intracellular spare are continuous and exist everywhere. Consequently, they do not allow simulations highlighting the effect of a nonuniform distribution of ion channels along the cell membrane or the complex morphology of the cells. In this talk, we present a more accurate framework for cerebral electrodiffusion with an explicit representation of the geometry of the cell, the cell membrane and the extracellular space. To take full advantage of this framework, a numerical solution scheme capable of efficiently handling three-dimensional, complicated geometries is required. We propose a novel numerical solution scheme using a mortar finite element method, allowing for the coupling of variational problems posed over the non-overlapping intra and extracellular domains by weakly enforcing interface conditions on the cell membrane. This solution algorithm flexibly allows for arbitrary geometries and efficient solution of the separate subproblems. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the presented framework can give new insights in this setting. Simulations of 9 idealized, tightly packed axons show that inducing action potentials in one or more axons yields ephaptic currents that have a pronounced excitatory effect on neighboring axons, but fail to induce action potentials there .
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement 714892 (Waterscales), and from the Research Council of Norway (BIOTEK2021 Digital Life project ‘DigiBrain’, project 248828).
 Ellingsrud A J, Solbrå A, Einevoll G T, et al. Finite element simulation of ionic electrodiffusion in cellular geometries. arXiv.org. 2019.