Tel.:

+49-6221-548261

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+49-6221-548884

Internet:

Email:

peter.bastian@iwr.uni-heidelberg.de

Tel.:

+49-6221-548264

Fax:

+49-6221-548884

Internet:

Email:

stefan.lang@iwr.uni-heidelberg.de

The aim of this project was to understand the generation of local field potential (LFP) from first principles. To that end, we developed a model describing the self-consistent electrical potential and the transport of dissolved ions within an axon and in the extracellular space. Both regions are coupled through a membrane, which is not spatially resolved, and which can be penetrated by the ions. Mathematically, the model consists of the Poisson-Nernst-Planck equations in two regions coupled through a nonstandard interface condition given by an adapted Hodgkin-Huxley model.
Then a numerical model was developed based on the Finite Element method using a fully-coupled and fully-implicit approach. As solver, an inexact Newton method with an inner algebraic multigrid is used. The implementation is based on the DUNE software framework and allows the use of large-scale parallel computers. Another major scientific result was the derivation of previously unknown analytic solutions to the unsteady (i.e. time-dependent) PNP equations for the case where only a single ion species is considered. The resulting crucial equation in terms of the electric field shows similarities to the vector Burgers equation. Several one-dimensional examples and a possible extension to multidimensional problems were extensively described together with the complete derivation (Schönke 2012). The analytical solutions were used directly to test the implementation of the time-dependent PNP system in the DUNE-PDELab software framework. The numerical solutions converge to the analytical ones with the expected order.
We laid focus on the electric potential around a cylindrical axon surrounded by extracellular fluid. Using symmetry assumptions reduction from three to two space dimensions is possible while fully resolving the important Debye layer (Pods et al. 2013). The results of both unmyelinated and myelinated axon were compared to volume conductor (VC) models, which neglect concentration effects on the extracellular potential. We found very good agreements at distances at least 5 microns away from the membrane; for smaller distances, our results show significant deviations from VC models like the line source approximation (LSA) commonly used to calculate the LFP from membrane currents of 1D cable models. The deviations are present in two different regimes: the Debye layer, a few nanometers around the membrane, where deviations to VC models are drastically different due to the presence of strong concentration and potential gradients towards the membrane. At the adjacent diffusion layer, which is on the order of micrometers, we still find notable differences to the potential calculated by LSA. Both models converge for larger distances and show a good agreement at distances larger than about 5 microns. These results formed the core of the Ph.D. thesis of Jurgis Pods (Pods 2014).
Furthermore, we investigated the effects of ephaptic coupling between neighbouring axons under isolated extracellular space conditions. Our results indicate that extracellular potential might be orders of magnitude larger when the extracellular conductivity is severely limited, which might be the case in nerve bundles or regions of high membrane density. Finally, we implemented and compared the \"electroneutral model\", a model that lies between the full PNP system of electrodiffusion and the much simpler VC models (Pods 2015). It provides the major advantage that the Debye layers around the cell membranes do not have to be resolved by the computational grid explicitly. It therefore seems especially promising for full 3D computations, which would allow the inclusion of arbitrary extracellular geometries, such as those stemming from EM imaging. This might be the focus of a future project.

Pods J (2015) A Comparison of Computational Models for the Extracellular Potential of Neurons arXiv:1509.01481 .

Pods, J (2014) Electrodiffusion Models of Neurons and Extracellular Space Using the Poisson-Nernst-Planck Equations - Numerical Simulation of the Intra- and Extracellular Potential for an Axon Model. PhD Thesis. Heidelberg University. http://www.ub.uni-heidelberg.de/archiv/17128 .

(2013) Electrodiffusion Models of Neurons and Extracellular Space Using the Poisson-Nernst-Planck Equations - Numerical Simulation of the Intra- and Extracellular Potential for an Axon Model Biophysical Journal. 105(1):242-254 .

(2012) Unsteady analytical solutions to the Poisson–Nernst–Planck equations Journal of Physics A: Mathematical and Theoretical. 45: 455204. http://stacks.iop.org/JPhysA/45/455204 .