A Bidomain Boundary Element–Cable Method for Modeling Neuronal Responses to Electric Fields

Objective

Extracellular electric fields critically influence neural activity through both exogenous neuromodulation and endogenous ephaptic coupling. While conventional cable models efficiently simulate membrane dynamics, they fail to capture bidirectional, field-mediated interactions self-consistently, and fully coupled volumetric methods require computationally prohibitive 3D meshing. We present Cable-BEM, a hybrid wire-kernel bidomain boundary element method designed to resolve these limitations.

Approach

By analytically integrating boundary integral kernels around cylindrical neuronal compartments, Cable-BEM fully couples intracellular, extracellular, and membrane dynamics while strictly retaining the highly efficient 1D degrees of freedom of traditional cable equations. The system is advanced using a semi-implicit Crank–Nicolson scheme. To overcome the dense nature of the resulting integral operators, we implement an Adaptive Cross Approximation (ACA) and Hierarchical Off-Diagonal Low-Rank (HODLR) compression scheme.

Main result

The solver was rigorously validated against full-surface bidomain boundary element method (BEM) reference implementations, demonstrating tight agreement in activation thresholds (within 1.3% relative error) across diverse stimulation geometries.

The ACA-HODLR compression scheme achieved substantial memory footprint reductions—by a factor of up to 4.6 for large 225-cell networks—without sacrificing numerical accuracy. Furthermore, we utilized the framework to resolve subtle, distance-dependent ephaptic interactions, successfully demonstrating the progressive phase synchronization of biophysically realistic, multi-compartment Purkinje cells.

Significance

Cable-BEM provides a computationally scalable, mesh-free framework that establishes a powerful and practical foundation for investigating complex field-mediated phenomena in large-scale, multicellular neuronal networks.