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

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.

In this appendix, we will provide theoretical arguments indicating the asymptotic nature of the errors introduced by removing the ephaptic current. These arguments are founded on strong assumptions on the analytical properties of the solutions and rigorous mathematical arguments would require a priori proofs of these properties. We therefore emphasize that the arguments provided here just indicate relations and it is an open problem to rigorously prove these relations mathematically.
As discussed above, the key step in deriving the classical Cable model is to remove the ephaptic current, I eph . We have given computational evidence indicating that This relation can also be derived from the classical summation formula (see Equation (27) in the paper). If we assume that (27) holds and assume that ∂ 2 ue ∂x 2 is uniformly bounded, we find that where we have used that η = hσ i 4 ; recall that h = l y = l z (the width of the neuron). Next, our aim in this appendix is to provide a rough estimate of the error introduced in the membrane potential by removing the ephaptic current given by (12) in the paper. The theoretical bound will be based on the assumption that the term ∂ 2 ue ∂x 2 in (12) is bounded independently of the parameter η. In order to derive the bound, we compare the two models given by and Here subscript t represents the derivative with respect to t and subscript xx represents the double derivative with respect to x. For simplicity, we assume that both models are equipped with the boundary condition v = v rest at x = 0 and x = l x . By subtracting (1) from (2), we find that the error with the boundary condition e(0, t) = e(l x , t) = 0, initial condition e(·, 0) = 0, and where g(x, t) = g L + g s (x)e − t−t 0 α .
By multiplying (3) by e and integrating over the length of the neuron, we get First, we note that and secondly, we use the Poincaré inequality (see e.g. [1]) to find that In order to estimate the last term of (4), we note that, for any a, b and ε = 0, we have By using this inequality with a = e, and b = u e xx , we find that We define and note that, by (4, 5, 6, 7), we have where we have introduced Again, if we assume that the extracellular potential is faithfully represented by the classical summation formula (27), we have

Equation (8) can be written as
with A = η/2, B = 2η/l 2 x + g L and C = ηF 0 /2. Provided that B > Aε 2 this ODE will be bounded by the steady state .
Choosing ε 2 = B/2A in order to minize this upper bound, it follows that Since, F 0 = O(1/σ 2 e ), we find that for small values of η we have where we recall that h = l y = l z represents the width of the neuron. Finally, we conclude that This estimate indicates that the error introduced by removing the ephaptic current is reduced as h or σ i are reduced, and it is reduced if σ e or g L is increased.