BidomainBEM: multiscale and multiphysics modeling of neurons in electric fields

Background. Understanding how device-generated electric fields activate neurons is essential for explaining experimental observations and improving brain stimulation therapies. The cable equation models commonly used for this purpose assume that the neuron does not perturb its own local electric field, which rules out phenomena such as transverse stimulation, ephaptic coupling between neighboring cells, and entrainment by weak fields. Bidomain (or "whole") finite element models capture the full coupling but require volume meshes that resolve microscale membranes inside a macroscale head model, and therefore do not scale to realistic morphologies.

Objective. Create bidomain solvers that scale, so that the full electromagnetic coupling among stimulation devices, tissue, and the intracellular, membrane, and extracellular regions of neurons can be analyzed for morphologically realistic cells and, eventually, for populations of ephaptically coupled neurons.

Approach. BidomainBEM is the first integral-equation bidomain formulation. A boundary element method solves an integral equation that couples the device-induced field, tissue inhomogeneity, and the membrane-induced field using surface meshes only, and it is coupled to nonlinear membrane dynamics and stepped in time. Because only cell and tissue surfaces are meshed, the relative placement of devices and cells can be changed without regenerating a mesh.

Main results. Comparison studies show that the boundary element approach produces accurate results for both electric and magnetic stimulation. Unlike bidomain finite element methods, it does not require multiscale volume meshes, so modeling cells, or tightly packed populations of cells, with microscale features embedded in a macroscale head model becomes computationally tractable. The work received the first-place student paper award at the 2023 Applied Computational Electromagnetics Society Symposium and has been presented in invited talks at the URSI International Symposium on Electromagnetic Theory and the IEEE International Conference on Numerical Electromagnetic and Multiphysics Modeling and Optimization.

Significance. Bidomain solvers allow realistic cell geometries, device fields, and neuron populations to be treated together in a single model. BidomainBEM is the foundation for our scalable bidomain solvers (hierarchical-matrix fast direct solvers and boundary element–cable hybrids), and for the long-term goal of simulating cortical-column networks of thousands of realistic neurons with full electromagnetic coupling.

Publications.

D. M. Czerwonky (G), A. S. Aberra, and L. J. Gomez, "A boundary element method of bidomain modeling for predicting cellular responses to electromagnetic fields," Journal of Neural Engineering, vol. 21, no. 3, pp. 036050, 2024. link

D. M. Czerwonky (G) and L. J. Gomez, "Integral Equation for Analyzing Neuron Response to Non-invasive Electromagnetic Brain Stimulation," International Applied Computational Electromagnetics Society Symposium, March 2023 (first place, student paper competition).

D. M. Czerwonky (G) and L. J. Gomez, "Integral equation for analyzing cell’s response to device E-fields," IEEE MTT-S International Conference on Numerical Electromagnetic and Multiphysics Modeling and Optimization, June 2023 (invited).

L. J. Gomez and D. M. Czerwonky (G), "Integral Equation for Analyzing Neuron Cell Response to Transcranial Magnetic Stimulation," URSI International Symposium on Electromagnetic Theory, May 2023 (invited).

D. M. Czerwonky (G) and L. J. Gomez, "Bidomain Boundary Element Modeling of Pseudo-realistic Neurons with Advanced Membrane Mechanisms," International Applied Computational Electromagnetics Society Symposium, May 2024.

D. Czerwonky (G) and L. J. Gomez, "Bidomain Boundary Integral Equation for Analyzing a Neuron Cell’s Response to Non-invasive Brain Stimulation," Progress In Electromagnetics Research Symposium, April 2024.