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…
Objective: A novel hybrid method called Cable-BEM has been developed to simulate neuronal responses to electric fields. This method aims to bridge the gap between conventional cable models, which excel at modeling membrane dynamics but fall short in capturing bidirectional, field-mediated interactions, and fully coupled volumetric methods, which are computationally intensive due to the need for extensive 3D meshing.
Approach: Cable-BEM employs a bidomain boundary element method combined with a wire-kernel approach. By analytically integrating boundary integral kernels around cylindrical neuronal compartments, the method maintains the efficiency of traditional cable equations' 1D degrees of freedom while fully coupling intracellular, extracellular, and membrane dynamics. The system is solved using a semi-implicit Crank-Nicolson scheme, which enhances stability and convergence properties.
To address the dense nature of the resulting integral operators, Cable-BEM incorporates an Adaptive Cross Approximation (ACA) and Hierarchical Off-Diagonal Low-Rank (HODLR) compression scheme. These techniques significantly reduce the memory footprint, achieving up to a 4.6-fold reduction for large 225-cell networks, without compromising numerical accuracy.
Validation: The Cable-BEM solver has been thoroughly validated against reference implementations of the full-surface bidomain boundary element method (BEM). The results show excellent agreement in activation thresholds, with a relative error of only 1.3% across a variety of stimulation geometries. This validation confirms the method's accuracy and reliability in simulating complex field-mediated phenomena.
Main Result: The ability of Cable-BEM to resolve subtle, distance-dependent ephaptic interactions has been demonstrated through simulations involving biophysically realistic, multi-compartment Purkinje cells. The solver successfully captured the progressive phase synchronization of these cells, highlighting the method's capability to investigate subtle and critical aspects of neuronal interactions mediated by electric fields.
Significance: Cable-BEM offers a computationally scalable and mesh-free framework that addresses the limitations of conventional cable models and fully coupled volumetric methods. By providing a powerful and practical foundation, this hybrid approach enables researchers to delve into the intricate dynamics of large-scale, multicellular neuronal networks influenced by electric fields, paving the way for more sophisticated studies in neuroscience and neurophysiology.
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