title: Improved Simulation of Electrodiffusion in the Node of Ranvier by Mesh Adaptation.
abstract: In neural structures with complex geometries, numerical resolution of the Poisson-Nernst-Planck (PNP) equations is necessary to accurately model electrodiffusion. This formalism allows one to describe ionic concentrations and the electric field (even away from the membrane) with arbitrary spatial and temporal resolution which is impossible to achieve with models relying on cable theory. However, solving the PNP equations on complex geometries involves handling intricate numerical difficulties related either to the spatial discretization, temporal discretization or the resolution of the linearized systems, often requiring large computational resources which have limited the use of this approach. In the present paper, we investigate the best ways to use the finite elements method (FEM) to solve the PNP equations on domains with discontinuous properties (such as occur at the membrane-cytoplasm interface). 1) Using a simple 2D geometry to allow comparison with analytical solution, we show that mesh adaptation is a very (if not the most) efficient way to obtain accurate solutions while limiting the computational efforts, 2) We use mesh adaptation in a 3D model of a node of Ranvier to reveal details of the solution which are nearly impossible to resolve with other modelling techniques. For instance, we exhibit a non linear distribution of the electric potential within the membrane due to the non uniform width of the myelin and investigate its impact on the spatial profile of the electric field in the Debye layer.
authors.affiliation: Département de mathématiques et de statistique/Groupe Interdisciplinaire de Recherche en Éléments Finis (GIREF), Université Laval, Québec, Québec, Canada.|Neurosciences cellulaires et moléculaires/Institut universitaire en santé mentale de Québec (CRIUSMQ), Université Laval, Québec, Québec, Canada.
authors.forename: Ibrahima
authors.initials: I
authors.lastname: Dione
authors.affiliation: Département de mathématiques et de statistique/Groupe Interdisciplinaire de Recherche en Éléments Finis (GIREF), Université Laval, Québec, Québec, Canada.
authors.forename: Jean
authors.identifier: http://orcid.org/0000-0001-9463-2057
authors.initials: J
authors.lastname: Deteix
authors.affiliation: Département de mathématiques et de statistique/Groupe Interdisciplinaire de Recherche en Éléments Finis (GIREF), Université Laval, Québec, Québec, Canada.
authors.forename: Thomas
authors.initials: T
authors.lastname: Briffard
authors.affiliation: Département de mathématiques et de statistique/Groupe Interdisciplinaire de Recherche en Éléments Finis (GIREF), Université Laval, Québec, Québec, Canada.
authors.forename: Eric
authors.identifier: http://orcid.org/0000-0002-9712-4979
authors.initials: E
authors.lastname: Chamberland
authors.affiliation: Département de mathématiques et de statistique/Groupe Interdisciplinaire de Recherche en Éléments Finis (GIREF), Université Laval, Québec, Québec, Canada.|Neurosciences cellulaires et moléculaires/Institut universitaire en santé mentale de Québec (CRIUSMQ), Université Laval, Québec, Québec, Canada.
authors.forename: Nicolas
authors.initials: N
authors.lastname: Doyon
mesh_terms: D000200:Action Potentials / Q000502:physiology*; D000818:Animals; D003198:Computer Simulation; D004058:Diffusion; D004560:Electricity; D020342:Finite Element Analysis; D006801:Humans; D008959:Models, Neurological; D009415:Nerve Net / Q000502:physiology*; D011901:Ranvier's Nodes / Q000502:physiology*