[Paper Review] Finite element simulation of ionic electrodiffusion in cellular geometries

Ada J. Ellingsrud|arXiv (Cornell University)|Nov 8, 2019

Neuroscience and Neural Engineering48 references17 citations

TL;DR

This paper introduces the KNP-EMI model, a finite element-based numerical framework that simulates ionic electrodiffusion in detailed cellular geometries by coupling the electroneutral Kirchhoff-Nernst-Planck (KNP) equations with the Extracellular-Membrane-Intracellular (EMI) framework. The method enables accurate, stable, and flexible simulations of ion concentration dynamics and ephaptic coupling in complex 3D neural tissue, revealing that ionic ephaptic coupling—driven by Nernst potential shifts—can be stronger than electric ephaptic coupling in tightly packed unmyelinated axons.

ABSTRACT

Mathematical models for excitable cells are commonly based on cable theory, which considers a homogenized domain and spatially constant ionic concentrations. Although such models provide valuable insight, the effect of altered ion concentrations or detailed cell morphology on the electrical potentials cannot be captured. In this paper, we discuss an alternative approach to detailed modelling of electrodiffusion in neural tissue. The mathematical model describes the distribution and evolution of ion concentrations in a geometrically-explicit representation of the intra- and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the Extracellular-Membrane-Intracellular (EMI) framework, we refer to this model as the KNP-EMI model. Here, we introduce and numerically evaluate a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree. Moreover, we compare the electrical potentials predicted by the KNP-EMI and EMI models. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the KNP-EMI framework can give new insights in this setting.

Motivation & Objective

• To develop a numerically stable and flexible framework for simulating ionic electrodiffusion in complex, geometrically explicit neural tissue.
• To overcome limitations of homogenized models like cable theory, which assume constant ion concentrations and extracellular potentials.
• To enable the study of ephaptic coupling arising from ion concentration changes and resulting diffusion potentials in unmyelinated axon bundles.
• To provide a computationally feasible alternative to the full Poisson-Nernst-Planck (PNP) framework by assuming electroneutrality in bulk tissue.
• To compare predictions of the KNP-EMI model with those of the EMI and cable theory models, particularly regarding extracellular potential and ion concentration dynamics.

Proposed method

• Develops a mortar-based finite element formulation for solving the KNP-EMI model, which couples the electroneutral KNP equations with the EMI framework.
• Uses mixed finite element methods to handle the coupled system of partial differential equations describing ion transport and membrane potential across intra- and extracellular domains.
• Applies the method to 3D geometric representations of neurons and extracellular space, enabling simulations on arbitrary-dimensional domains with variable polynomial degrees.
• Implements a consistent treatment of bulk conductivities σi and σe based on local ion concentrations, differing from tissue-averaged values in homogenized models.
• Employs a time-implicit scheme to allow stable simulations at coarser spatial and temporal resolutions than required by PNP models.
• Validates the method through comparisons with EMI and cable theory models, focusing on extracellular potential and ion concentration evolution.

Experimental results

Research questions

• RQ1 How does the KNP-EMI model predict ephaptic coupling in unmyelinated axon bundles compared to cable theory?
• RQ2 What is the relative contribution of ionic ephaptic coupling (via Nernst potential shifts) versus electric ephaptic coupling (via extracellular potential) in tightly packed axons?
• RQ3 How do ion concentration changes in the extracellular space affect membrane potential and neuronal activity in morphologically detailed tissue?
• RQ4 To what extent do the KNP-EMI predictions differ from those of the EMI model when ion concentration dynamics are explicitly included?
• RQ5 Can the KNP-EMI framework simulate long-duration phenomena such as spreading depression with realistic ion pump dynamics?

Key findings

• The KNP-EMI model predicts stronger ephaptic coupling via ionic concentration changes (ionic ephaptic coupling) than via extracellular potential (electric ephaptic coupling), especially in tightly packed unmyelinated axons.
• The model reveals that extracellular K+ concentration can rise to tens of mM during pathological states such as spreading depression, significantly altering reversal potentials.
• Simulations show that the KNP-EMI framework produces different ephaptic coupling predictions than cable theory due to a more accurate representation of bulk conductivities based on local ion concentrations.
• The finite element formulation enables stable and flexible simulations on complex 3D geometries with arbitrary polynomial degrees and spatial dimensions.
• The KNP-EMI model is computationally more efficient than the PNP framework while retaining physical accuracy by assuming electroneutrality in bulk tissue.
• The model highlights the importance of including ion pumps and co-transporters in membrane mechanisms to prevent overestimation of concentration changes over short timescales.