The impact of internodal segmentation in biophysical nerve fiber models

Abstract

Implementation of double cable models to simulate the behavior of myelinated peripheral nerve fibers requires defining a segmentation of the internode between successive nodes of Ranvier. The number of internodal segments is a model parameter that is not well agreed on, with values in the literature ranging from 1 to more than 500. Moreover, a lot of studies also lack a sensitivity study or a rationale behind the implementation used. In a model of a myelinated nerve fiber developed in our group, the segmentation scheme (i.e., the number of segments and their individual morphology) strongly influenced model outcomes such as action potential shape and velocity, stimulation threshold and absolute refractory period. In the present study these influences were investigated systematically in homogeneous neurons with different diameters. Uniformly segmented internodes were found to require several hundreds of segments (and associated computational power) to reach model outcomes differing by less than 1 % from the asymptotic value. In fact, in the majority of segmentation schemes the main determinant is not the number of segments, but the length λ of the internodal segments directly adjacent to the nodes of Ranvier. If λ is larger than approximately 10 μm, model outcomes for the tested fibers are almost independent of the total number of segments. Furthermore, λ can be optimized to enable models using just three segments per internode, to reach physiologically relevant model outcomes with limited computational resources. However, to study anatomical or physiological details of the internode itself, an appropriately detailed segmentation scheme is crucial.

Appendix

In order to be able to apply a single formula for the transneural potential in different neural compartments (i.e., in both nodes of Ranvier and internodal segments), all neural compartments were labeled by a single index n. This way, if for example 5 internodal segments were modeled per internode, the first node of Ranvier would be labeled n = 1, the 5 consecutive internodal segments n = 2, 3,...6, the second node of Ranvier n = 7, etc. Application of Kirchhoff’s law in a neural compartment (e.g., the point labeled ‘n’ in Figure 1) yields the following equation for the transneural potential V _n :

$\begin{array}{l}\frac{d{V}_n}{ dt}=\frac{d{\displaystyle {V}_n^{\prime }}}{ dt}+\\ {}\frac{G_{n-1}^a}{C_n^{my}}\left[\varDelta {V}_{n-1}+\varDelta {V}_{n-1}^e\right]+\frac{G_n^a}{C_n^{my}}\left[\varDelta {V}_{n+1}+\varDelta {V}_{n+1}^e\right]+\\ {}\frac{G_{n-1}^p}{C_n^{my}}\left[\varDelta {V}_{n-1}-\varDelta {\displaystyle {V}_{n-1}^{\prime }}+\varDelta {V}_{n-1}^e\right]+\\ {}\frac{G_n^p}{C_n^{my}}\left[\varDelta {V}_{n+1}-\varDelta {\displaystyle {V}_{n+1}^{\prime }}+\varDelta {V}_{n+1}^e\right]-\frac{G_n^{my}}{C_n^{my}}\left[{V}_n-{\displaystyle {V}_n^{\prime }}\right]\end{array}$A.1

with the transmembrane voltage V ^′ _n defined by

$\begin{array}{l}\frac{d{\displaystyle {V}_n^{\prime }}}{ dt}=-\frac{G_n^L}{C_n^m}\left[{\displaystyle {V}_n^{\prime }}-{V}^L\right]-\frac{1}{C_n^m}{I}_n^{act}\left({\displaystyle {V}_n^{\prime }}\right)+\\ {}\frac{G_{n-1}^a}{C_n^m}\left[\varDelta {V}_{n-1}+\varDelta {V}_{n-1}^e\right]+\frac{G_n^a}{C_n^m}\left[\varDelta {V}_{n+1}+\varDelta {V}_{n+1}^e\right]\end{array}$A.2

In equations A.1 and A.2, G ^a _n , G ^p _n ,

G ^my _n and G ^L _n are the axonal conductance, the conductance of the peri-axonal space and the conductance of the myelin layer at node n and the passive leakage conductance.

C ^a _n , C ^p _n , and C ^my _n resemble the corresponding capacitances. G ^my _n and C ^my _n are calculated by:

${G}_n^{my}=2\pi \left[{R}_n+{\delta}^p+\frac{N_n{\delta}^{my}}{2}\right]{L}_n\frac{g_{my}}{N_n}$A.3

${C}_n^{my}=2\pi \left[{R}_n+{\delta}^p+\frac{N_n{\delta}^{my}}{2}\right]{L}_n\frac{c_{my}}{N_n}$A.4

in which R _n is the axonal radius, δ ^p is the thickness of the periaxonal layer, N _n is the number of myelin layers, δ ^my is the myelin thickness, L _n is the axonal length of the nth neural compartment and c ^my and g ^my are the capacitance and conductance of the myelin per unit area. I ^act _n is the sum of currents arising from the ion flow through the nodal voltage gated ion channels, the equations for which were taken from Schwarz et al. (1995), from which also the electrical and kinetics parameters were used, together with the correction by Wesselink et al. (1999). ΔV _n − 1 = V _n − 1 − V _n , ΔV _n + 1 = V _n + 1 − V _n and V ^e _n is the extracellular potential. The latter voltages all are relative to a resting potential E _r , which was calculated by the Goldman equation. As already mentioned in the text, the equations were solved using a backward Euler integration algorithm with a standard step size of 0.001 μs, as it turned out that all results were stable for time step sizes of 0.001 μs and smaller.