Our knowledge of the mechanism of nerve action is still insufficient to predict what happens in terms of electrical and chemical structure. If our theoretical knowledge were as weak in every aspect as it is upon the behaviour of the excitable membrane, we should not be able to predict confidently any relation but this-that two nerves which were exactly the same in structure, composition and properties, must behave in exactly the same way. However, both Ringer's fluid and axoplasm appear to be homogeneous conductors, so the current distribution in those regions is exactly defined by electrical theory. If, then, two nerves are not identical but have membranes with the same specific properties, it might still be possible to compare them exactly. For, though the properties of the membrane are unknown, they are the same in the two nerves, and though the distribution of current in Ringer and axoplasm is different it is precisely understood. It turns out that there is only a very narrow range of conditions which will permit this argument being applied. It is when the two fibres are dimensionally similar, which is nearly the same as their being geometrically similar. Now the interesting thing is that the nerve fibres do, in fact, exhibit the structural similarity demanded by the theory, and hence it is worth while to examine how far the variation in properties also accords with theoretical predictions.

A theory of the effects of fibre size in medullated nerve

SUMMARY

1
Four methods of determining the potential difference across the surface membrane of living cells are described.

2
In a wide range of excitable tissues the resting membrane potential is of the order of 50–100 mV. and the action potential of the order of 80–130 mV.

3
At the height of activity the potential difference across the membrane is reversed by 30–50 mV.

4
The potassium concentration inside most excitable cells is 20–50 times greater than that in the external medium; sodium is 3–15 times more concentrated outside than it is inside, while chloride is 5–50 times more concentrated outside than inside. Isolated fibres lose potassium and gain sodium and chloride ions.

5
Potassium appears to exist as a free ion inside nerve and muscle fibres.

6
The nature of the organic anions which balance the high concentration of potassium inside excitable cells is still largely unknown. In certain cases amino-acids such as aspartic acid are present in high concentrations.

7
The resting membrane behaves as though it were moderately permeable to K+ and Cl- but sparingly permeable to Na+. The absolute magnitude of the resting potential is similar to that calculated from the potassium concentrations if allowance is made for the contributions of chloride and other ions. Movements of K+ and Cl- as determined by radioactive tracers or by chemical methods agree with a quantitative formulation of this hypothesis.

8
It is necessary to suppose that sodium is continuously pumped out of excitable cells by a process which depends on metabolism.

9
Electrical activity is due to a large and specific increase in the permeability to sodium. The reversed potential difference across the active membrane arises from the concentration difference of sodium and varies with the external concentration of sodium in the same manner as the theoretical potential of a sodium electrode.

10
In many cells, conduction of impulses is impossible if the external medium does not contain sodium or lithium ions.

11
The rate of rise of the action potential varies with the concentration of sodium ions in the external medium.

12
Sodium enters a nerve fibre when it is active. The quantity entering 1 cm.2 of membrane during one impulse is of the order of 3 μμmol.

13
Entry of sodium is approximately balanced by the leakage of a corresponding quantity of potassium.

14
It is suggested that sodium enters the nerve fibre during the rising phase of the action potential and that potassium leaves during the falling phase.

15
The permeability changes during the action potential probably consist of a rapid but transient increase in the permeability to sodium and a delayed increase in the permeability to potassium. It is suggested that both permeability changes vary with membrane potential in a graded but reversible manner. This hypothesis is applied to the phenomena of subthreshold activity, accomodation and oscillatory behaviour.

16
In vertebrate myelinated fibres there is much evidence to show that conduction is saltatory; this suggests that sodium entry is confined to the nodes of Ranvier, and that the internodes are depolarized by local circuit action.

17
Provided that nerves are not stimulated at a high rate, recovery heat production is sufficient to account for the metabolic extrusion of sodium after activity.

The ionic basis of electrical activity in nerve and muscle

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

Human nerve fibers exhibit a distinct pattern of threshold fluctuation following a single action potential known as the recovery cycle. We developed geometrically and electrically accurate models o...

Modeling the excitability of mammalian nerve fibers: influence of afterpotentials on the recovery cycle

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.

https://physoc.onlinelibrary.wiley.com/doi/pdf/10.1113/jphysiol.1949.sp004335

Evidence for saltatory conduction in peripheral myelinated nerve fibres

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Evidence for saltatory conduction in peripheral myelinated nerve fibres