Modeling calcium dynamics in neurons with endoplasmic reticulum: existence, uniqueness and an implicit-explicit finite element scheme

TL;DR

This paper models calcium dynamics in neurons with endoplasmic reticulum using diffusion-reaction equations, proving mathematical properties and developing an efficient finite element scheme with proven stability and convergence.

Contribution

It provides the first rigorous proof of existence, uniqueness, and boundedness for this calcium model, and introduces a novel implicit-explicit finite element scheme with stability and optimal convergence.

Findings

Mathematically proved existence, uniqueness, and boundedness of solutions.

Developed an implicit-explicit finite element scheme with mesh-independent stability.

Numerical experiments confirm theoretical stability and convergence results.

Abstract

Like many other biological processes, calcium dynamics in neurons containing an endoplasmic reticulum are governed by diffusion-reaction equations on interface-separated domains. Interface conditions are typically described by systems of ordinary differential equations that provide fluxes across the interfaces. Using the calcium model as an example of this class of ODE-flux boundary interface problems, we prove the existence, uniqueness and boundedness of the solution by applying comparison theorem, fundamental solution of the parabolic operator and a strategy used in Picard's existence theorem. Then we propose and analyze an efficient implicit-explicit finite element scheme which is implicit for the parabolic operator and explicit for the nonlinear terms. We show that the stability does not depend on the spatial mesh size. Also the optimal convergence rate in $H^1$ norm is obtained. Numerical experiments illustrate the theoretical results.