Electrodiffusion Models of Axon and Extracellular Space Using the Poisson-Nernst-Planck Equations

Abstract

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. Special attention is paid to the Debye layer, the region with strong concentration gradients close to the membrane, which is explicitly resolved by the computational mesh. The focus lies on the evolution of the extracellular electric potential at different membrane distances. Roughly, the extracellular space can be divided into three distinct regions: first, the distant farfield, which exhibits a characteristic triphasic waveform in response to an action potential traveling along the membrane. This is consistent with previous modeling efforts and experiments. Secondly, the Debye layer close to the membrane, which shows a completely different extracellular response in the form of an “AP echo”, which is also observed in juxtacellular recordings. Finally, there is the intermediate or diffusion layer located in between, which shows a gradual transition from the Debye layer potential towards the farfield potential. Both of these potentialregions show marked deviations from volume conductor models, which can be attributed to the redistribution of concentrations and associated ion fluxes. These differences are explained by analyzing the capacitive and ionic components of the potential. In an extension, we also include myelination into the model, which has a significant impact on the extracellular field. Again, the numerical results are compared to volume conductor models. Finally, a model study is carried out to assess the magnitude of ephaptic effects, i.e. the influence of the electric field of one cell on a neighboring cell, in a somewhat artificial geometry. While the results probably can not be interpreted quantitatively in the majority of physiological situations, the qualitative behavior shows interesting effects. An axon can elicit an action potential in a surrounding bundle of axons, given that the distance is small enough and the resistivity of the extracellular medium is significantly increased. Further results of this study are extremely large extracellular potentials with amplitudes up to 100 mV and an unusual neuronal firing mode in which the cell is not depolarized by an increase in the intracellular potential, but by a decrease in the extracellular potential. Some literature references are given that show that these observations are consistent with previous studies.