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.