  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.
