Neural bidomain model for multidimensional ephaptic coupling of neural spike propagation along myelinated fiber bundles with the nodes of Ranvier

A novel mathematical model is proposed to investigate the distributions and functions of ephaptic coupling in generally-shaped neural fiber bundles in a multidimensional space, corresponding to the spatiotemporal interactions between propagating fiber bundles via the extracellular space. The proposed macroscopic model is derived from the classic Frankenhaeuser-Huxley model for neural spike propagation along the general neural fiber bundles in two domains (bidomain) of the nonoverlapping intracellular and extracellular space, except the common nodes of Ranvier. A high-order continuous Galerkin scheme provides an efficient two-dimensional (2D) computational simulation with moving frames (orthonormal basis vectors) representing intracellular and extracellular conductivity and nonoverlapping domains. The governing equation is mathematically and computationally validated against the existing 1D models of propagations along neural fiber bundles with the aligned nodes of Ranvier. The proposed model reveals the critical role of neural fiber bundle configurations by simulating their 2D propagations. For sufficiently large fiber bundles, synchronizing neural spike propagation along two fiber bundles with misaligned nodes of Ranvier is almost impossible. Moreover, the magnitude of interference of propagations passing-by along the fiber bundles depends on the fiber-bundle configuration, significantly depending on the location of the closest distance of two propagations, either in the myelin sheath or nodes of Ranvier. Finally, the curved fiber bundles demonstrate drastically different ephaptic coupling compared to straight fiber bundles due to the unique extracellular potential distribution by the curved fiber bundles. The new simulations highlight the importance of intrinsic fiber-bundle structures in neural spike propagation and brain functions. Author summaryFiber bundles communicate via chemical interactions across synaptic junctions and electrical interactions across gap junctions. Ephaptic coupling is another communication method between fiber bundles, which is of a small magnitude compared with other communication methods. The effect of ephaptic coupling is known on a relatively large scale and is mostly affected by the intrinsic structure of neural fiber bundles. The study of ephaptic coupling has attracted considerable attention since the 1940s, when the term ephaptic coupling was first coined. However, the lack of a multidimensional model of neural spike propagation with various configurations of myelinated fiber bundles has prohibited further investigations of ephaptic coupling in brain subregions. This paper proposes a multidimensional model of neural spike simulation in the context of partial differential equations. The proposed model aims to simulate neural spike propagations in complex configurations of fiber bundles to study the functions of ephaptic coupling in such brain structures. In this paper, the new simulations of the misaligned nodes of Ranvier, interfering opposite-traveling fiber bundles, and curved fiber bundles demonstrate the critical roles of intrinsic fiber-bundle structures, including the location and alignment of the nodes of Ranvier, in neural spike propagation and brain functions.