Voltage laws for three-dimensional microdomains with cusp-shaped funnels derived from Poisson-Nernst-Planck equations

TL;DR

This paper derives new three-dimensional electrostatic laws for cusp-shaped domains in electrolytes, showing how geometry influences voltage profiles, with applications to neuroscience dendritic spines.

Contribution

It introduces asymptotic approximations for the Poisson-Nernst-Planck equations in cusp-shaped domains, revealing geometry-dependent voltage laws in non-electroneutral electrolytes.

Findings

Cusp geometry significantly affects voltage distribution.

Derived electrostatic laws match numerical simulations.

Applications to neuronal dendritic spines are discussed.

Abstract

We study the electro-diffusion properties of a domain containing a cusp-shaped structure in three dimensions when one ionic specie is dominant. The mathematical problem consists in solving the steady-state Poisson-Nernst-Planck (PNP) equation with an integral constraint for the number of charges. A non-homogeneous Neumann boundary condition is imposed on the boundary. We construct an asymptotic approximation for certain singular limits that agree with numerical simulations. Finally, we analyse the consequences of non-homogeneous surface charge density. We conclude that the geometry of cusp-shaped domains influences the voltage profile, specifically inside the cusp structure. The main results are summarized in the form of new three-dimensional electrostatic laws for non-electroneutral electrolytes. We discuss applications to dendritic spines in neuroscience.