Minimizing the caliber of myelinated axons by means of nodal constrictions

Abstract

In myelinated axons, most of the voltage-gated ion channels are concentrated at the nodes of Ranvier, which are short gaps in the myelin sheath. This arrangement leads to saltatory conduction and a larger conduction velocity than in nonmyelinated axons. Intriguingly, axons in the peripheral nervous system that exceed about 2 μm in diameter exhibit a characteristic narrowing of the axon at nodes that results in a local reduction of the axonal cross-sectional area. The extent of constriction increases with increasing internodal axonal caliber, reaching a threefold reduction in diameter for the largest axons. In this paper, we use computational modeling to investigate the effect of nodal constrictions on axonal conduction velocity. For a fixed number of ion channels, we find that there is an optimal extent of nodal constriction which minimizes the internodal axon caliber that is required to achieve a given target conduction velocity, and we show that this is sensitive to the precise geometry of the axon and myelin sheath in the flanking paranodal regions. Thus axonal constrictions at nodes of Ranvier appear to be a biological adaptation to minimize axonal volume, thereby maximizing the spatial and metabolic efficiency of these processes, which can be a significant evolutionary constraint. We show that the optimal nodal morphologies are relatively insensitive to changes in the number of nodal sodium channels.

Keywords: axon morphology; computational modeling; conduction velocity; nodal constrictions; node of Ranvier.

Fig. 1.

The single-cable model. We simulated a generic axon fiber, consisting of 30 repeating INTER-JUXTA-PARA-NODE-PARA-JUXTA-INTER sections. Nodes contained fixed numbers of sodium channels (5 × 10³, 1.5 × 10⁴, or 2.5 × 10⁴), which have fast Na⁺ membrane dynamics and a linear leak conductance. JUXTA sections contained fast-gated potassium channels. While all sections contained a membrane capacitance in parallel with their respective conductance, nonnodal regions contained two capacitors in series, one representing the myelin capacitance C_myelin, and the other representing the membrane capacitance C_mem. The figure is not to scale. INTER, internode; JUXTA, juxtaparanode; PARA, paranode; and NODE, node itself.

Fig. 2.

Representation of the node and internode regions. A: the three tapering designs of the axons and the myelin around the nodes of Ranvier used in this study. B: the cylindrical capacitor model for the internode. The axon fiber consists of the axon (the inner core) with diameter D_axon, the membrane with thickness Δ_mem, and a layer of myelin of thickness Δ_myelin generating the outer cylinder. The fiber diameter D_fiber is then determined by D_fiber = D_axon + 2 Δ_mem + 2 Δ_myelin ≈ D_axon + 2 Δ_myelin.

Fig. 3.

Analysis of velocity gain using a linear paranodal taper. The conduction velocity is plotted vs. nodal diameter for values of fiber diameter (internode plus twice the myelin thickness) ranging from 4 μm to 20 μm (indicated below each line) and 5 × 10³ nodal sodium channels (A), 1.5 × 10⁴ nodal sodium channels (B), and 2.5 × 10⁴ nodal sodium channels (C). The numbers below the curves indicate the fiber diameters. The dashed lines indicate velocities for unconstricted axons, i.e., where the nodal and internodal axon diameters are identical. The thick solid lines indicate the maximum conduction velocity for each fiber diameter. The insets in A–C indicate the percentage gain of conduction velocity compared with an unconstricted axon. In D, we show the conduction velocity as a function of the nodal diameter for three values of the fiber diameter with a constant specific sodium conductance of g_Na = 3.0 S/cm², i.e., when the numbers of nodal sodium channels increase linearly with the nodal diameter. The inset displays the linear tapering of the axon and myelin sheath in the PARA region.

Fig. 4.

Analysis of space cost using a linear paranodal taper. We show the contour lines of constant conduction velocity on the Cartesian plane of internodal axon diameter vs. nodal diameter for a range of conduction velocities and 5 × 10³ nodal sodium channels (A), 1.5 × 10⁴ nodal sodium channels (B), or 2.5 × 10⁴ nodal sodium channels (C). The numbers below the lines indicate the target conduction velocities. The nodal and internodal diameters are identical on the dashed line. The thick solid lines indicate fiber morphologies with the minimal internodal axon diameter for the given target conduction velocities. The insets indicate the percentage increase in volume (i.e., spatial cost) for an unconstricted axon (internodal diameter = nodal diameter) compared with an axon with optimal nodal constrictions.

Fig. 5.

Analysis of velocity gain and space cost for a nonlinear paranodal taper. In A, the conduction velocity is plotted against the nodal diameters for values of fiber diameter ranging from 4 μm to 20 μm (indicated below each line) and 5 × 10³ nodal sodium channels. The dashed lines in A indicate velocities for an unconstricted fiber, i.e., an axon with identical nodal and internodal diameters. The thick solid line indicates the velocities of fibers optimized for maximum conduction velocities. The inset in A indicates the percentage gain in conduction velocity for a range of fiber diameters compared with a fiber with an unconstricted axon. In B, we show contour lines of constant conduction velocity on the Cartesian plane of nodal diameter vs. internodal diameter for conduction velocities ranging from 10 m/s to 40 m/s (indicated below each line) and 5 × 10³ nodal sodium channels. The nodal and internodal axon diameters are identical on the dashed line. The thick solid lines indicate fiber morphologies with the minimal internodal axon diameter for the given target conduction velocities. The insets indicate the percentage increase in volume (i.e., spatial cost) for an unconstricted axon (internodal diameter = nodal diameter) compared with an axon with optimal nodal constrictions. In C, we show the conduction velocity as a function of the nodal diameter for three values of the fiber diameter when the sodium channel density is constant, i.e., when the numbers of nodal sodium channels increase linearly with the nodal diameter. The inset displays the nonlinear tapering of the axon and myelin in the PARA region.

Fig. 6.

Analysis of velocity gain and space cost for a steplike taper. In A, the conduction velocity is plotted against the nodal diameters for values of fiber diameter ranging from 4 μm to 20 μm (indicated below each line) and 5 × 10³ nodal sodium channels. The dashed lines in A indicate velocities for an unconstricted fiber, i.e., an axon with identical nodal and internodal diameters. The thick solid line indicates the velocities of fibers optimized for maximum conduction velocities. The inset in A indicates the percent gain in conduction velocity for a range of fiber diameters compared with a fiber with an unconstricted axon. In B, we show contour lines of constant conduction velocity on the Cartesian plane of nodal diameter vs. internodal diameter for conduction velocities ranging from 10 m/s to 40 m/s (indicated below each line) and 5 × 10³ nodal sodium channels. The nodal and internodal axon diameters are identical on the dashed line. The thick solid line indicates fiber morphologies with the minimal internodal axon diameter for the given target conduction velocities. The inset indicates the percentage increase in volume (i.e., spatial cost) for an unconstricted axon (internodal diameter = nodal diameter) compared with an axon with optimal nodal constrictions. In C, we show the conduction velocity as a function of the nodal diameter for three values of the fiber diameter when the sodium channel density is constant, i.e., when the numbers of nodal sodium channels increase linearly with the nodal diameter. The inset displays the steplike tapering of the axon and myelin in the PARA region.

Fig. 7.

Analysis of the effects of an extended PARA length or a nodal bulge. We show the contour lines of constant conduction velocity on the Cartesian plane of internodal diameter vs. nodal diameter for conduction velocities of 20 m/s, 30 m/s, and 50 m/s (indicated below each line) and 1.5 × 10⁴ nodal sodium channels. The nodal and internodal diameters are identical on the dotted line. The solid lines represent simulations in the presence of an extended PARA length (A) or a nodal bulge (B), while the dashed lines indicate simulations in their absence.

Fig. 8.

Analysis of the effect of sodium channel number on the optimal fiber morphology. We show the computed lines of optimal fiber morphologies for a nonlinear tapering of the axon and myelin sheath in PARA (see inset Fig. 5C) and 5 × 10³ (dot-dashed line), 1.5 × 10⁴ (dashed line), and 2.5 × 10⁴ (solid line) nodal sodium channels.

Fig. 9.

Comparison of actual and predicted fiber morphologies. In A, we show morphological data for feline dorsal spinal nerve root (squares) and ventral spinal nerve root (circles) extracted from Figs. 1 and 2 of Rydmark (1981). In B, we show the computed lines of optimal fiber morphologies for a linear tapering of the axon and myelin sheath in the PARA region (left-most solid/dashed lines), for a nonlinear tapering (middle solid/dashed lines), and for an abrupt steplike narrowing of the axon and myelin sheath at the JUXTA:PARA interface (right-most solid/dashed lines). The dashed lines represent fibers having a 4-μm PARA region length, and the solid lines represent fibers having an extended 8-μm PARA region length. The arrows indicate the corresponding tapering method used.

Fig. 10.

Analysis of the number of additional nodal sodium channels required to match the conduction velocity gains resulting from optimal nodal constrictions. A: the conduction velocity of a 20 μm fiber (12.9 μm internodal diameter) is plotted vs. the number of nodal sodium channels for an optimal (constricted) cable and an unconstricted (uniform) cable. For a specified conduction velocity (vertical axis), the horizontal distance between the two curves represents the number of additional sodium channels needed for the uniform cable. B: the necessary internodal diameter for a conduction speed of 25 m/s is shown as a function of the number of nodal sodium channels for the optimal cable and the uniform cable. For a specified internodal diameter (vertical axis), the horizontal distance between the curves represents the number of additional sodium channels needed for the uniform cable to conduct with the same speed of 25 m/s.