Evaluation of a Resistor Network for Solving Electrical Problems on Ohmic Media

Abstract

Computations of fields from electrodes in extracellular ohmic media in fiber bundles with complex cross-sectional geometries are nowadays aided by the use of Finite Element Methods. However, when endogenous fields from the activity of fibers and ephaptic coupling are taken into account, coupling FEM with a neural model is a formidable task. We present an alternative to this approach, consisting on a Resistor Network (RN) fully implemented in NEURON. This results in a stable EMI model with which we avoid coupling NEURON to an external FEM or Poisson solver and dealing with the corresponding computational burden and risk of numerical instability. The RN is designed to model bundles of parallel fibers in nerves. It embeds the fibers within and uses Voronoi tessellations on their naturally random locations to calculate the values of the resistances interconnecting fibers and regions of the extracellular space. This approach provides a numerical solver for electrical fields, but also with a nearest-neighbour ephaptic coupling model. In this work, we assess the capability of the RN to solve electrical problems when using meshes built from such tessellation techniques on continuous ohmic media. Results show that the RN solves simple problems with good accuracy when compared to FEM and analytic results. Therefore, it is safe to use this method to compute electrical fields in the applications we pursue. This opens a new possibility for effectively studying ephaptic coupling in somewhat complex nerve trunks.