Excellent models have been presented in the literature which relate membrane potential to transverse membrane current and which describe the propagation of action potentials along the axon, for both myelinated and nonmyelinated fibers. There is not, however, an adequate model for nerve excitation which allows one to compute the threshold of a nerve fiber for pulses of finite duration using electrodes that are not in direct contact with the fiber. This paper considers this problem and presents a model of the electrical properties of myelinated nerve which describes the time course of events following stimulus application up to the initiation of the action potential. The time-varying current and potential at all nodes can be computed from the model, and the strength-duration curve can be determined for arbitrary electrode geometries, although only the case of a monopolar electrode is considered in this paper. It is shown that even when the stimulus is a constant-current pulse, the membrane current at the nodes varies considerably with time. The strength-duration curve calculated from the model is consistent with previously published experimental data, and the model provides a quantitative relationship between threshold and fiber diameter which shows there is less selectivity among fibers of large diameter than those of small diameter.

Analysis of a Model for Excitation of Myelinated Nerve

The sections in this article are:


1
Introduction
1.1
Core Conductor Concept

1.2
Perspective

1.3
Comment

1.4
Reviews and Monographs




2
Brief Historical Notes
2.1
Early Electrophysiology

2.2
Electrotonus

2.3
Passive Membrane Electrotonus

2.4
Passive Versus Active Membrane

2.5
Cable Theory

2.6
Core Conductor Concept

2.7
Core Conductor Theory

2.8
Estimation of Membrane Capacitance

2.9
Resting Membrane Resistivitiy

2.10
Passive Cable Parameters of Invertebrate Axons

2.11
Importance of Single Axon Preparations

2.12
Estimation of Parameters for Myelinated Axons

2.13
Space and Voltage Clamp




3
Dendritic Aspects of Neurons
3.1
Axon-Dendrite Contrast

3.2
Microelectrodes in Motoneurons

3.3
Theoretical Neuron Models and Parameters

3.4
Class of Trees Equivalent to Cylinders

3.5
Motoneuron Membrane Resistivity and Dendritic Dominance

3.6
Dendritic Electrotonic Length

3.7
Membrane Potential Transients and Time Constants

3.8
Spatiotemporal Effects with Dendritic Synapses

3.9
Excitatory Postsynaptic Potential Shape Index Loci

3.10
Comments on Extracellular Potentials

3.11
Additional Comments and References




4
Cable Equations Defined
4.1
Usual Cable Equation

4.2
Steady-state Cable Equations

4.3
Augmented Cable Equations

4.4
Comment: Cable Versus Wave Equation

4.5
Modified Cable Equation for Tapering Core

4.6
General Solution of Steady-state Cable Equation

4.7
Basic Transient Solutions of Cable Equation

4.8
Solutions Using Separation of Variables

4.9
Fundamental Solution for Instantaneous Point Charge




5
List of Symbols

6
Assumptions and Derivation of Cable Theory
6.1
One Dimensional in Space

6.2
Intracellular Core Resistance

6.3
Ohm's Law for Core Current

6.4
Conservation of Current

6.5
Relation of Membrane Current to Vi

6.6
Effect of Assuming Extracellular Isopotentiality

6.7
Passive Membrane Model

6.8
Resulting Cable Equation for Simple Case

6.9
Physical Meaning of Cable Equation Terms

6.10
Physical Meaning of τ

6.11
Physical Meaning of λ

6.12
Electrotonic Distance, Length, and Decrement

6.13
Effect of Placing Axon in Oil

6.14
Effect of Applied Current

6.15
Comment on Sign Conventions

6.16
Effect of Synaptic Membrane Conductance

6.17
Effect of Active Membrane Properties




7
Input Resistance and Steady Decrement with Distance
7.1
Note on Correspondence with Experiment

7.2
Cable of Semi-infinite Length

7.3
Comments about R∞, G∞, Core Current, and Input Current

7.4
Doubly Infinite Length

7.5
Case of Voltage Clamps at X1 and X2

7.6
Relations Between Axon Parameters

7.7
Finite Length: Effect of Boundary Condition at X= X1

7.8
Sealed End at X= X1: Case of B1 = 0

7.9
Voltage Clamp(V1 = 0) at X = X1: Case of B1 = ∞

7.10
Semi-infinite Extension at X = X1: Case of B1 = 1

7.11
Input Conductance for Finite Length General Case

7.12
Branches at X = X1

7.13
Comment on Branching Equivalent to a Cylinder

7.14
Comment on Membrane Injury at X = X1

7.15
Comment on Steady Synaptic Input at X= X1

7.16
Case of Input to One Branch of Dendritic Neuron Model




8
Passive Membrane Potential Transients and Time Constants
8.1
Passive Decay Transients

8.2
Time Constant Ratios and Electrotonic Length

8.3
Effect of Large L and Infinite L

8.4
Transient Response to Applied Current Step, for Finite Length

8.5
Applied Current Step with L Large or Infinite

8.6
Voltage Clamp at X = 0, with Infinite L

8.7
Voltage Clamp with Finite Length

8.8
Transient Response to Current Injected at One Branch of Model




9
Relations Between Neuron Model Parameters
9.1
Input Resistance and Membrane Resistivity

9.2
Dendritic Tree Input Resistance and Membrane Resistivity

9.3
Results for Trees Equivalent to Cylinders

9.4
Result for Neuron Equivalent to Cylinder

9.5
Estimation of Motoneuron Parameters

Core Conductor Theory and Cable Properties of Neurons

This article reviews the determinants of conduction velocity in myelinated nerve fibers. For fibers exhibiting geometric similarity, conduction velocity is nearly proportional to diameter. However, in classes of geometrically similar fibers, myelin thickness and internode distance are linearly related to diameter. Therefore, it is not possible to assess the relative contributions of each of these factors. The relative effects of changes in each of these parameters on conduction velocity can be assessed from studies on fibers representing different similarity classes or from theoretical parametric studies. These studies show that for a fixed axon diameter, conduction velocity increases with myelin thickness. For a fixed total fiber diameter, there is an optimal ratio of axon diameter to total fiber diameter (and therefore to myelin thickness) at which conduction velocity is maximized. Conduction velocity is dependent on internode distance, with a broad maximum centered around the value observed in normal peripheral fibers. Conduction velocity is also dependent on temperature and the properties of the axonal milieu. Thus, the conduction speed of any given fiber reflects a number of parameters and is not determined by any single structural characteristic.

Determinants of conduction velocity in myelinated nerve fibers.

Conventional electrophysiological tests of nerve function focus on the number of conducting fibers and their conduction velocity. These tests are sensitive to the integrity of the myelin sheath, but provide little information about the axonal membrane. Threshold tracking techniques, in contrast, test nerve excitability, which depends on the membrane properties of the axons at the site of stimulation. These methods are sensitive to membrane potential, and to changes in membrane potential caused by activation of ion channels and electrogenic ion pumps, including those under the myelin sheath. This review describes the range of threshold tracking techniques that have been developed for the study of human nerves in vivo: resting threshold is compared with the threshold altered by a change in environment (e.g., ischemia), by a preceding single impulse (e.g., refractoriness, superexcitability) or impulse train, or by a subthreshold current (e.g., threshold electrotonus). Few clinical studies have been reported so far, mainly in diabetic neuropathy and motor neuron disease. Threshold measurements seem well suited for studies of metabolic and toxic neuropathies but insensitive to demyelination. Until suitable equipment becomes more widely available, their full potential is unlikely to be realized.

Threshold tracking techniques in the study of human peripheral nerve.

The permeability of K channels to various cations is studied in myelinated nerve. Ionic currents under voltage clamp are measured in Ringer solution containing tetrodotoxin and a high concentration of the test ion. Reversal potentials for current in K channels are determined and used with the Goldman-Hodgkin-Katz equation to calculate relative permeabilities. The ratios PTl:PK:PRb:PNHNH4 are 2.3:1.00:0.92:0.13. No other ions are found to be measurably permeant including Li+, Na+, Cs+, methylamine, guanidine, hydrazine, or hydroxylamine. The ratio PNa/PK is less than 0.01. Potassium conductance is depressed at pH values below 5.0. Leakage conductance is higher in K, Rb, Cs, NH4, and Tl Ringer than in Na Ringer, but the selectivity sequence probably is not the same as for K channels. The hypothesis is offered that the narrowest part of the K channel is a circle of oxygen atoms about 3 A in diameter with low electrostatic field strength.

/pdf/potassium-channels-in-myelinated-nerve-selective-4il4xnglk5.pdf

L. Goldman

Papers

Computation of Impulse Conduction in Myelinated Fibers; Theoretical Basis of the Velocity-Diameter Relation