Nonlinear coupling of the Poisson-Nernst Planck system of equations using the Discrete Duality Finite Volumes method, application to ionic and voltage dynamics in neuronal compartments

Description

The Poisson-Nernst Planck (PNP) system of equations is widely recognized as the standard model for characterizing the electrodiffusion of ions in electrolytes, including ionic dynamics in the cellular cytosol. This non-linear system presents challenges for both modeling and simulations, due to the presence of a stiff boundary layer tightly related to the choice of boundary conditions. In this article, we propose a numerical scheme based on the Discrete Duality Finite Volumes method (DDFV) to solve the PNP system of equations, while preserving the positivity of ionic concentrations. The DDFV method presents the main characteristics of modern numerical methods, allowing for the use of unstructured meshes in complex geometries and giving robust and precise approximate solutions. It is particularly attractive for the PNP equations on neuronal geometries, because of its local conservation property, and robustness with respect to mesh distortion. Through several simulations, we illustrate the accuracy of our scheme, achieving secondorder accuracy in space. Furthermore, using a specific test case, we show that our method can resolve steep gradients. Finally, we apply our scheme to investigate the propagation and attenuation of an ionic influx in small neuronal compartments of the dendritic tree: a branch bifurcation and a dendritic spine-the mushroom-like protrusion that receive neuronal inputs. Considering the connection of our neuronal compartments to an ionic reservoir, which could be the dendritic shaft, we observe that the distance to the closest ionic reservoir deeply influences signal propagation. In particular, a spine close to a reservoir acts as an isolated compartment, whereas a spine located farther away is subject to signal invasion. Hence, our numerical results suggest that the local geometry of the dendritic tree has a major influence on spine behavior, which would make plasticity not only at the level of the spine but also at the level of the full dendritic tree.

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