An Evaluation of the Accuracy of Classical Models for Computing the Membrane Potential and Extracellular Potential for Neurons

Abstract

Two mathematical models are part of the foundation of Computational neurophysiology; (a) the Cable equation is used to compute the membrane potential of neurons, and, (b) volume-conductor theory describes the extracellular potential around neurons. In the standard procedure for computing extracellular potentials, the transmembrane currents are computed by means of (a) and the extracellular potentials are computed using an explicit sum over analytical point-current source solutions as prescribed by volume conductor theory. Both models are extremely useful as they allow huge simplifications of the computational efforts involved in computing extracellular potentials. However, there are more accurate, though computationally very expensive, models available where the potentials inside and outside the neurons are computed simultaneously in a self-consistent scheme. In the present work we explore the accuracy of the classical models (a) and (b) by comparing them to these more accurate schemes. The main assumption of (a) is that the ephaptic current can be ignored in the derivation of the Cable equation. We find, however, for our examples with stylized neurons, that the ephaptic current is comparable in magnitude to other currents involved in the computations, suggesting that it may be significant—at least in parts of the simulation. The magnitude of the error introduced in the membrane potential is several millivolts, and this error also translates into errors in the predicted extracellular potentials. While the error becomes negligible if we assume the extracellular conductivity to be very large, this assumption is, unfortunately, not easy to justify a priori for all situations of interest.

A Cell-Based Framework for Numerical Modeling of Electrical Conduction in Cardiac Tissue

“…To simplify the notation, we describe here the 2D case only; the extension to three dimensions is immediate. The spatial discretization employed here is taken from Tveito et al [8]. Figure 2 shows the four different types of nodes used in the computations.…”

“…The first equation is given directly by (8), the second is a first-order finite difference discretization of (7), and the third is an implicit discretization of (9) of the form…”

“…For the EMI system, the Poisson problems (1) and (2) are coupled by the conditions (4) and (5) and the conditions (7) and (8).…”

“…For this reason, we consider an operator splitting approach to solve the EMI model defined by (1)(2)(3)(4)(5)(6)(7)(8)(9). The system (1-9) is solved by first applying given initial conditions for v and w. Then, for each time step n, we assume that the solutions v n−1 and w n−1 are known for t = t n−1 on Ŵ and Ŵ 1,2 respectively.…”

“…Indeed, the presentation here is very much motivated by the formulation presented by Stinstra et al [5] and by Agudelo-Toro and Neef [4]. Furthermore, the EMI approach was used to study the effect of the ephaptic coupling of neurons in Tveito et al [8].…”