Spatio-temporal modeling of saltatory conduction in neurons using Poisson–Nernst–Planck treatment and estimation of conduction velocity

Action potential propagation along the axons and across the dendrites is the foundation of the electrical activity observed in the brain and the rest of the nervous system. Theoretical and numerical modeling of this action potential activity has long been a key focus area of electro-chemical neuronal modeling, and over the years, electrical network models of varying complexity have been proposed. Specifically, considering the presence of nodes of Ranvier along the myelinated axon, single-cable models of the propagation of action potential have been popular. Building on these models, and considering a secondary electrical conduction pathway below the myelin sheath, the double-cable model has been proposed. Such cable theory based treatments, including the classical Hodgkin–Huxley model, single-cable model, and double-cable model have been extensively studied in the literature. But these have inherent limitations in their lack of a representation of the spatio-temporal evolution of the neuronal electro-chemistry. In contrast, a Poisson–Nernst–Planck (PNP) based electro-diffusive framework accounts for the underlying spatio-temporal ionic concentration dynamics and is a more general and comprehensive treatment. In this work, a high-fidelity implementation of the PNP model is demonstrated. This electro-diffusive model is shown to produce results similar to the cable theory based electrical network models, and in addition, the rich spatio-temporal evolution of the underlying ionic transport is captured. Novel to this work is the extension of PNP model to axonal geometries with multiple nodes of Ranvier, its correlation with cable theory based models, and multiple variants of the electro-diffusive model — PNP without myelin, PNP with myelin, and PNP with the myelin sheath and peri-axonal space. Further, we apply this spatio-temporal model to numerically estimate conduction velocity in a rat axon using the three model variants. Specifically, spatial saltatory conduction due to the presence of myelin sheath and the peri-axonal space is investigated.

Statement of Significance: In this work, we present a comprehensive PDE based treatment for modeling neuronal action potential generation and propagation and provide a first-of-its-kind framework for computationally estimating action potential conduction velocities. This electro-diffusive model, based on a Poisson-Nernst-Planck (PNP) formulation, is shown to produce results similar to the cable theory based electrical network models, and in addition, the rich spatio-temporal evolution of the underlying ionic transport is captured. Novel to this work is the extension of PNP model to axonal geometries with multiple nodes of Ranvier, its correlation with cable theory based models, and multiple variants of the electro-diffusive model - PNP without myelin, PNP with myelin, and PNP with the myelin sheath and periaxonal space. Further, we apply this spatio-temporal model to numerically estimate conduction velocity in a rat axon using the three model variants. Specifically, spatial saltatory conduction due to the presence of myelin sheath and the peri-axonal space is investigated.