Electrodiffusion Models of Axon and Extracellular Space Using the Poisson-Nernst-Planck Equations
In studies of the brain and the nervous system, extracellular signals - as measured by local field potentials (LFPs) or electroencephalography (EEG) - are of capital importance, as they allow to simultaneously obtain data from multiple neurons. The exact biophysical basis of these signals is, however, still not fully understood. Most models for the extracellular potential today are based on volume conductor theory, which assumes that the extracellular fluid is electroneutral and that the only contributions to the electric field are given by membrane currents, which can be imposed as boundary conditions in the mathematical model. This neglects a second, possibly important contributor to the extracellular field: the time- and position-dependent concentrations of ions in the intra- and extracellular fluids. In this thesis, a 3D model of a single axon in extracellular fluid is presented based on the Poisson-Nernst-Planck (PNP) equations of electrodiffusion. This fundamental model includes not only the potential, but also the concentrations of all participating ion concentrations in a self-consistent way. This enables us to study the propagation of an action potential (AP) along the axonal membrane based on first principles by means of numerical simulations. By exploiting the cylinder symmetry of this geometry, the problem can be reduced to two dimensions. The numerical solution is implemented in a flexible and efficient way, using the DUNE framework. A suitable mesh generation strategy and a parallelization of the algorithm allow to solve the problem in reasonable time, with a high spatial and temporal resolution. The methods and programming techniques used to deal with the numerical challenges of this multi-scale problem are presented in detail.