Microscopic and Macroscopic Response of a Cortical Neuron to an External Electric Field Computed with the Boundary Element Fast Multipole Method

The goal of this study is to demonstrate how one can compute the activating function and surface charge density resulting from application of an external electric field to a high-resolution realistic neuronal morphology. We use the boundary element fast multipole method (BEM-FMM) on an ordinary computer to accurately perform these computations in under 2-10 minutes for a dense surface mesh of a single neuron with approximately 1.4 million triangles. Prior work used commercial finite element method (FEM) software which required creation of a volumetric tetrahedral mesh between fine neuronal arbor, potentially resulting in prohibitively large volume sizes and long mesh generation times. We used the example of a human pyramidal neuron with an externally applied E-field to show how our approach can quickly and accurately compute the induced surface charge density on the cell surface and the activating function of the cable equation. We found that the induced surface charge density perturbs the macroscopically applied E-field on a microscopic spatial scale. The strength of the perturbation depends on the conductivity contrast; the stronger the contrast, the larger the perturbation. In our example, the induced surface charge density may change the average activating function by up to 75%. We also embedded this neuron model into a detailed macroscopic human head model and simulated a realistic TMS excitation using the BEM-FMM method for the combined model. The solution obtained in this case predicted a smaller activating function error. The difference between the microscopic and the macroscopic effect of the externally applied electric field is of much interest to users of extracellular stimulation techniques and merits further study.