A boundary element method of bidomain modeling for predicting cellular responses to electromagnetic fields.

Commonly used cable equation approaches for simulating the
effects of electromagnetic fields on excitable cells make several simplifying assumptions
that could limit their predictive power. Bidomain or "whole" finite element methods
have been developed to fully couple cells and electric fields for more realistic neuron
modeling. Here, we introduce a novel bidomain integral equation designed for
determining the full electromagnetic coupling between stimulation devices and the
intracellular, membrane, and extracellular regions of neurons.
Methods: Our proposed boundary element formulation offers a solution to an
integral equation that connects the device, tissue inhomogeneity, and cell membrane-
induced E-fields. We solve this integral equation using first-order nodal elements
and an unconditionally stable Crank-Nicholson time-stepping scheme. To validate
and demonstrate our approach, we simulated cylindrical Hodgkin-Huxley axons and
spherical cells in multiple brain stimulation scenarios.
Main Results: Comparison studies show that a boundary element approach
produces accurate results for both electric and magnetic stimulation. Unlike bidomain
finite element methods, the bidomain boundary element method does not require
volume meshes containing features at multiple scales. As a result, modeling cells, or
tightly packed populations of cells, with microscale features embedded in a macroscale
head model, is simplified, and the relative placement of devices and cells can be varied
without the need to generate a new mesh.
Significance: Device-induced electromagnetic fields are commonly used to modulate
brain activity for research and therapeutic applications. Bidomain solvers allow
for the full incorporation of realistic cell geometries, device E-fields, and neuron
populations. Thus, multi-cell studies of advanced neuronal mechanisms would greatly
benefit from the development of fast-bidomain solvers to ensure scalability and the
practical execution of neural network simulations with realistic neuron morphologies.