# Geometrical effects on nonlinear electrodiffusion in cell physiology

## TL;DR

This paper develops new electrical laws based on nonlinear electro-diffusion theory to understand how local geometrical features like curvature influence the electrical properties of cells, with implications for nano-pipette design and neural signal analysis.

## Contribution

It introduces an asymptotic approximation for electro-diffusion in complex geometries, revealing how charge concentration depends on local curvature and shape.

## Key findings

- Charge concentrates at cusp-shaped features as charge increases
- Derived asymptotic solutions for steady-state electro-diffusion in complex geometries
- Insights into voltage variations in neurons with heterogeneous structures

## Abstract

We report here new electrical laws, derived from nonlinear electro-diffusion theory, about the effect of the local geometrical structure, such as curvature, on the electrical properties of a cell. We adopt the Poisson-Nernst-Planck (PNP) equations for charge concentration and electric potential as a model of electro-diffusion. In the case at hand, the entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson's equation. We construct an asymptotic approximation for certain singular limits to the steady-state solution in a ball with an attached cusp-shaped funnel on its surface. As the number of charge increases, they concentrate at the end of cusp-shaped funnel. These results can be used in the design of nano-pipettes and help to understand the local voltage changes inside dendrites and axons with heterogenous local geometry.