Subtle paranodal injury slows impulse conduction in a mathematical model of myelinated axons

Abstract

This study explores in detail the functional consequences of subtle retraction and detachment of myelin around the nodes of Ranvier following mild-to-moderate crush or stretch mediated injury. An equivalent electrical circuit model for a series of equally spaced nodes of Ranvier was created incorporating extracellular and axonal resistances, paranodal resistances, nodal capacitances, time varying sodium and potassium currents, and realistic resting and threshold membrane potentials in a myelinated axon segment of 21 successive nodes. Differential equations describing membrane potentials at each nodal region were solved numerically. Subtle injury was simulated by increasing the width of exposed nodal membrane in nodes 8 through 20 of the model. Such injury diminishes action potential amplitude and slows conduction velocity from 19.1 m/sec in the normal region to 7.8 m/sec in the crushed region. Detachment of paranodal myelin, exposing juxtaparanodal potassium channels, decreases conduction velocity further to 6.6 m/sec, an effect that is partially reversible with potassium ion channel blockade. Conduction velocity decreases as node width increases or as paranodal resistance falls. The calculated changes in conduction velocity with subtle paranodal injury agree with experimental observations. Nodes of Ranvier are highly effective but somewhat fragile devices for increasing nerve conduction velocity and decreasing reaction time in vertebrate animals. Their fundamental design limitation is that even small mechanical retractions of myelin from very narrow nodes or slight loosening of paranodal myelin, which are difficult to notice at the light microscopic level of observation, can cause large changes in myelinated nerve conduction velocity.

Figure 1. Relevant anatomy.

(a) Schematic longitudinal section of a myelinated axon. The width of each node of Ranvier is denoted s. The distance between nodes is denoted L. Arrows indicate flow of positive ionic current during depolarization of Node 1 as conduction of the action potential moves toward Node 2. This sketch is foreshortened in the axial dimension. Anatomically L/s ∼ 1000. (b) Schematic cross section of a myelinated nerve or fiber tract. Each axon (solid black) is surrounded by a sheath of myelin (white) and in turn surrounded by a sheath of non-myelinated tissue (shaded) known as endoneurium or neuropil. The mean radius of the sheath of endoneurium is denoted r_e and the thickness of the sheath is denoted h. The cross section of endoneurium, 2πr_eh, is much greater than that of the axon.

Figure 2. Electrical model for multiple nodes of Ranvier.

Arrows indicate directions of positive ionic current. Shading indicates foreshortened myelinated regions. In life the actual distance between nodes (∼1000 µm) is much greater than the width of a single node (∼1 µm). Current is denoted by i, capacitance by C, resistance by R, and voltage by V. Lumped ionic currents from sodium and potassium channels in and around each node are shown as a single current source.

Figure 3. Modeled action potentials at successive nodes of Ranvier in a normal myelinated axon.

The dotted horizontal line represents threshold potential. Average axon diameter 1.0 micrometer; number of nodes of Ranvier 21; node width 0.65 micrometer; width of juxtaparanodal region on both sides of a node 5.0 micrometers. The resistivity of intracellular fluid 200 ohm-cm; normal paranodal resistance 3.2×10¹⁰ ohms; specific membrane capacitance of axonal membrane 1 mircofarad per square centimeter; resting axonal membrane potential –85 mV; threshold potential –50 mV; sodium equilibrium potential +67 mV; potassium equilibrium potential –95 mV. The time step for numerical integration was 0.1 microsecond.

Figure 4. Normal ionic currents for node 6 in the myelinated axon model of Figure 3.

Figure 5. Normal myelinated nerve conduction velocity as a function of the node width.

Figure 6. Modeled action potentials at successive nodes of Ranvier in a model myelinated axon.

Simulated crush injury to nodes on the right. The dotted horizontal line represents threshold potential. Average axon diameter 1.0 micrometer; number of nodes of Ranvier 21; normal node width 0.65 micrometer; injured node width 1.95 micrometer; width of juxtaparanodal region on both sides of a node 5.0 micrometers. The resistivity of intracellular fluid 200 ohm-cm; normal paranodal resistance 3.2×10¹⁰ ohms throughout the model; specific membrane capacitance of axonal membrane 1 mircofarad per square centimeter; resting axonal membrane potential –85 mV; threshold potential –50 mV; sodium equilibrium potential +67 mV; potassium equilibrium potential –95 mV. The time step for numerical integration was 0.1 microsecond.

Figure 7. Modeled action potentials at successive nodes of Ranvier in a model myelinated axon.

Simulated crush injury to nodes on the right. The dotted horizontal line represents threshold potential. In this simulation node width was increased three-fold and paranodal resistance was decreased to one tenth the normal value. Average axon diameter 1.0 micrometer; number of nodes of Ranvier 21; normal node width 0.65 micrometer; injured node width 1.95 micrometer; width of juxtaparanodal region on both sides of a node 5.0 micrometers. The resistivity of intracellular fluid 200 ohm-cm; normal paranodal resistance 3.2×10¹⁰; injured paranodal resistance 3.2×10⁹ ohms; specific membrane capacitance of axonal membrane 1 mircofarad per square centimeter; resting axonal membrane potential –85 mV; threshold potential –50 mV; sodium equilibrium potential +67 mV; potassium equilibrium potential –95 mV. The time step for numerical integration was 0.1 microsecond.

Figure 8. Modeled action potentials at successive nodes of Ranvier in a model myelinated axon.

Simulated crush injury to nodes on the right with 300 percent increase in bare nodal width and a 99% decrease in paranodal resistance. The dotted horizontal line represents threshold potential. Average axon diameter 1.0 micrometer; number of nodes of Ranvier 21; normal node width 0.65 micrometer; injured node width 1.95 micrometer; width of juxtaparanodal region on both sides of a node 5.0 micrometers. The resistivity of intracellular fluid 200 ohm-cm; normal paranodal resistance 3.2×10¹⁰; injured paranodal resistance 3.2×10⁸ ohms; specific membrane capacitance of axonal membrane 1 mircofarad per square centimeter; resting axonal membrane potential –85 mV; threshold potential –50 mV; sodium equilibrium potential +67 mV; potassium equilibrium potential –95 mV. The time step for numerical integration was 0.1 microsecond.

Figure 9. Treatment of simulated crush injury to nodes on the right with a potassium channel blocker that reduces peak potassium conductance in all nodes by 80%.

Parameters of injury: 300 percent increase in bare nodal width and a 99% decrease in paranodal resistance. The dotted horizontal line represents threshold potential. Average axon diameter 1.0 micrometer; number of nodes of Ranvier 21; normal node width 0.65 micrometer; injured node width 1.95 micrometer; width of juxtaparanodal region on both sides of a node 5.0 micrometers. The resistivity of intracellular fluid 200 ohm-cm; normal paranodal resistance 3.2×10¹⁰; injured paranodal resistance 3.2×10⁸ ohms; specific membrane capacitance of axonal membrane 1 mircofarad per square centimeter; resting axonal membrane potential –85 mV; threshold potential –50 mV; sodium equilibrium potential +67 mV; potassium equilibrium potential –95 mV. The time step for numerical integration was 0.1 microsecond.