More on myelinated nerve model analysis

An error in a previous analysis of a simple model for myelinated nerve propagation is pointed out and rectified. The model previously chosen does not lend itself to analysis because it has a region ofinfinite negative conductivity which leads to an instability in steady propagation. A model which assumes afinite negative conductivity is examined in detail and a more general results is derived.     

A distributed-parameter model of the myelinated nerve fiber

This paper presents a new model for the characterization of electrical activity in the nodal, paranodal and internodal regions of isolated amphibian and mammalian myelinated nerve fibers. It differs from previous models in the following ways: (1) in its ability to incorporate detailed anatomical and electrophysiological data; (2) in its approach to the myelinated nerve fiber as a multi-axial cable; and (3) in the numerical algorithm used to obtain distributed model equation solutions for potential and current. The morphometric properties are taken from detailed electron microscopic anatomical studies (Berthold & Rydmark, 1983a, Experientia 39, 964-976). The internodal axolemma is characterized as an excitable membrane and model-generated nodal and internodal membrane action potentials are presented. A system of describing equations for the equivalent network model is derived, based on the application of Kirchoff's Current Law, which take the form of multiple cross-coupled parabolic partial differential equations. An implicit numerical integration method is developed and the numerical solution implemented on a parallel processor. Non-uniform spatial step sizes are used, enabling detailed representation of the nodal region while minimizing the number of total segments necessary to represent the overall fiber. Conduction velocities of 20.2 m sec-1 at 20 degrees C for a 15 microns diameter amphibian fiber and 57.6 m sec-1 at 37 degrees C for a 17.5 microns diameter mammalian fiber are achieved, which agrees qualitatively with published experimental data at similar temperatures (Huxley & St?mpfli, 1949, J. Physiol., Lond. 108, 315-339; Rasminsky, 1973, Arch, Neurol. 28, 287-292). The simulation results demonstrate the ability of this model to produce detailed representations of the transaxonal, transmyelin and transfiber potentials and currents, as well as the longitudinal extra-axonal, periaxonal and intra-axonal currents. Also indicated is the potential contribution of the paranodal axolemma to nodal activity as well as the presence of significant longitudinal currents in the periaxonal space adjacent to the node of Ranvier.     

A model of a myelinated nerve axon: threshold behaviour and propagation

A model of a myelinated nerve axon is developed on the basis of FitzHugh-Nagumo dynamics under the assumption that the nodes of Ranvier are of small but finite width. It is shown that a periodic excited state may not exist if the width of the nodes is too small and the leakage across the myelin sheath is too great. The propagation of a super threshold pulse is prevented in the absence of nodes. Global stability of the resting equilibrium state is investigated as well as the propagation of wave front, type solutions.     

Myelin as longitudinal conductor: a multi-layered model of the myelinated human motor nerve fibre

 The myelin sheath is normally regarded as an electrical insulator. Low values of radial conductance and capacitance have been measured, and in electrical models of myelinated axons the contribution of longitudinal conduction within the sheath has been ignored. According to X-ray diffraction studies, however, myelin sheaths comprise alternate lipid and aqueous layers, and the latter may be expected to have a low resistivity. We propose a new model of myelinated axons in which the aqueous layers within the myelin provide appreciable longitudinal and radial conductance, the latter via a spiral pathway. We have investigated the likely contribution of these conductive paths within the myelin to the electrical properties of a human motor nerve fibre by computer simulation, representing the myelin sheath as a series of interconnecting parallel lamellae. With this new model, action potential conduction has been simulated along a 20-node cable, and the electrotonic responses to 100-ms depolarizing and hyperpolarizing current pulses have been simulated for a uniformly polarized fibre. We have found that the hypothesis of a longitudinally conducting myelin sheath improves our previous model in two ways: it is no longer necessary to make implausible assumptions about the resistivity or width of the periaxonal space to simulate realistic electrotonus, and the conduction velocity is appreciably faster (by 8.6%). Received: 19 April 1999 / Accepted in revised form: 11 September 2000     

Threshold conditions for a diffusive model of a myelinated axon

This paper develops and uses comparison principles to study the time evolution of solutions to problems of the form (formula; see text) Such a system models an infinite myelinated axon with discrete, excitable nodes spaced unit distant apart.     

A mathematical model of a myelinated nerve fiber for analysis of impulse conduction mechanisms during the relative refractory period

It was shown by means of a mathematical model of a myelinated nerve fiber (Frankenhaeuser — Huxley) that an increase in threshold and decrease in the amplitude of the action potential (AP) during the relative refractory period are due mainly to sodium inactivation. The contribution of increased potassium permeability to these changes is small, for the chief component of the outgoing ionic current in the node of Ranvier is not the potassium current, but the leak current. Given the ratio between these currents the increase in threshold and graduation of the action potential in the node membrane are less marked than in the membrane of the squid giant axon. At the beginning of the relative refractory period the AP evoked by strong stimulation is conducted only to the next node. Later in the refractory period impulses are conducted incrementally, and the threshold for the spreading impulse is higher than the threshold for spike excitation in the stimulated node. Delay in impulse conduction between refractory nodes leads to the formation of a retrograde depolarization wave. The reasons for differences in the mechanisms of impulse conduction along unmyelinated and myelinated refractory fibers are discussed.Vishnevskii Institute of Surgery, Academy of Medical Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 4, No. 2, pp. 201–207, March–April, 1972.     

On the capacity current in myelinated nerve fibres

Different equivalent circuits are presented to describe the capacity current, Ic, of potential-clamped Ranvier nodes. The corresponding transfer functions are discussed in examining the influence of the involved parameters on size and time course of Ic. The following results were obtained: i. The duration and the amplitude of Ic increased with increasing membrane capacity. ii. The amplitude of Ic increased with increasing gain or decreasing time constant of the clamp amplifier while its duration was shortened. iii. Similar effects were seen with decreasing resistance in series with the nodal membrane. iv. Qualitatively identical but stronger effects were seen with increasing bandwidth of the current measuring internode. Measurements were carried out under experimental conditions close to the calculated ones. They resembled the calculated curves satisfactorily.     

Conduction along myelinated and demyelinated nerve fibres with a reorganized axonal membrane during the recovery cycle: model investigations

The changes in the excitability of the reorganized axonal membrane in myelinated and demyelinated nerve fibres as well as the causes conditioning such changes have been investigated by paired stimulation during the first 30 ms of the recovery cycle. The variations of the action potential parameters (amplitude and velocity) are traced also. The simulation of the conduction along the normal fiber is based on the Frankenhaeuser and Huxley (1964) and Goldman and Albus (1968) equations, while the demyelination is considered to be an elongation of the nodes of Ranvier. The axonal membrane reorganization is achieved by means of potassium channel blocking and increase of the sodium-channel permeability. It is shown that potassium channels block decreases membrane excitability for the myelinated and demyelinated fibres in the cases of initial and paired stimulation. With increasing sodium-channel permeability on the background of the blocked potassium channels, the membrane excitability is increased. For the fibres with a reorganized membrane, a supernormality of the membrane excitability is obtained, the latter remaining unrecovered during the 30 ms cycle under investigation. The supernormality of the excitability grows from the demyelinated fibre without reorganized membrane to the demyelinated fibre with reorganized one. For short interstimulus intervals, the second action potential propagates along the fibres with a reduced velocity and a decreased amplitude. No supernormality of the potential parameters (amplitude, velocity) is observed during the cycle up to 30 ms. The membrane properties of the myelinated and demyelinated fibres with blocked potassium channels recover in the interval from 15 to 20 ms depending on whether the sodium channels' increase of the permeability is added on the background of the blocked potassium channel or not. In the recovery cycle, the axonal membrane reorganization leads to an improvement of the conduction along most severely demyelinated fibres.     

Local anesthetics potently block a potential insensitive potassium channel in myelinated nerve

Effects of some local anesthetics were studied in patch clamp experiments on enzymatically demyelinated peripheral amphibian nerve fibers. Micromolar concentrations of external bupivacaine depolarized the excised membrane considerably. The flicker K+ channel was found to be the most sensitive ion channel to local anesthetics in this preparation. Half-maximum inhibiting concentrations (IC50) for extracellular application of bupivacaine, ropivacaine, etidocaine, mepivacaine, lidocaine, and QX-314 were 0.21, 4.2, 8.6, 56, 220, and > 10,000 microM, respectively. The corresponding concentration-effect curves could be fitted under the assumption of a 1:1 reaction. Application from the axoplasmic side resulted in clearly lower potencies with IC50 values of 2.1, 6.6, 16, 300, 1,200, and 1,250 microM, respectively. The log(IC50)-values of the local anesthetics linearly depended on the logarithm of their octanol:buffer distribution coefficients with two regression lines for the piperidine derivatives and the standard amino-amides indicating an inherently higher potency of the cyclic piperidine series. Amide-linked local anesthetics did not impair the amplitude of the single-channel current but prolonged the time of the channel to be in the closed state derived as time constants tau c from closed-time histograms. With etidocaine and lidocaine tau c was 133 and 7.2 ms, and proved to be independent of concentration. With the most potent bupivacaine time constants of wash in and wash out were 1.8 and 5.2 s for 600 nM bupivacaine. After lowering the extracellular pH from 7.4 to 6.6, externally applied bupivacaine showed a reduced potency, whereas at higher pH of 8.2 the block was slightly enhanced. Intracellular pH of 6.4, 7.2, 8.0 had almost no effect on internal bupivacaine block. It is concluded that local anesthetics block the flicker K+ channel by impeding its gating but not its conductance. The slow blocker bupivacaine and the fast blocker lidocaine compete for the same receptor. Lipophilic interactions are of importance for blockade but besides a hydrophobic pathway, there exists also a hydrophilic pathway to the binding site which could only be reached from the cytoplasmic side of the membrane. Under physiological conditions, blockade of the flicker K+ channel which is more sensitive to bupivacaine than the Na+ channel might lead via membrane depolarization and the resulting sodium channel inactivation to a pronounced block of conduction in thin fibers.     

The extracellular potential of a myelinated nerve fiber in an unbounded medium and in nerve cuff models

A model is presented for the calculation of single myelinated fiber action potentials in an unbounded homogeneous medium and in nerve cuff electrodes. The model consists of a fiber model, used to calculate the action currents at the nodes of Ranvier, and a cylindrically symmetrical volume conductor model in which the fiber's nodes are represented as point current sources. The extracellular action potentials were shown to remain unchanged if the fiber diameter and the volume conductor geometry are scaled by the same factor (principle of corresponding states), both in an unbounded homogeneous medium and in an inhomogeneous volume conductor. The influence of several cuff electrode parameters, among others, cuff length and cuff diameter, were studied, and the results were compared, where possible, with theoretical and experimental results as reported in the literature.     

Continuous conduction of impulses in peripheral myelinated nerve fibers

1. Conduction of impulses in peripheral myelinated fibers of a nerve trunk is a continuous process, since with uninjured nerve fibers: (a) within each internodal segment the conduction time increases continuously and linearly with increasing conduction distance; (b) the presence of nodes of Ranvier does not result in any detectable discontinuity in the conduction of the impulse; (c) the ascending phase of the spike always has an S shape and never presents signs of fractionation; (d) the shape and magnitude of the spike are constant at all points of each internodal segment. 2. Records have been presented of the external logitudinal current that flows during propagation of an impulse in undissected single nerve fiber (Fig. 6). 3. Propagation of impulses across a conduction block occurs with a readily demonstrable discontinuity.     

Distribution of filipin-sterol complexes in the myelinated nerve fiber

Using filipin as a cytochemical probe to reveal the distribution of cholesterol, myelinated peripheral nerve fibers were examined in freeze-fracture replicas. Filipin-sterol complexes were most abundant in the Schwann cell and axonal plasma membranes. In the Schwann cell plasma membrane there was no heterogeneity in complex distribution in relation to the subjacent cytoplasmic network. In myelin lamellae there was a decrease in complexes from outer to inner lamellae and some aggregation of complexes in individual lamellae. The density of complexes in cytoplasmic organelles varied from absent in mitochondria to high in lysosome-like bodies. The results are interpreted in terms of the related biochemical composition and biophysical properties of cell membranes, with particular reference to the myelinated nerve fiber. The influence of diffusion barriers and gradients on the formation of complexes by filipin is considered.     

Extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor

The extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor were studied. The spatial distribution of the transmembrane potential was obtained by integrating the system of partial differential equations characterizing the electric processes in the active myelinated nerve fiber. The spatial distribution of the extracellular potentials at various radial distances in the volume conductor were calculated using the line source model. Up to a certain radial distance (500 m) the discontinuity of the action potential propagation is reflected in the extracellular potentials, while further in the volume conductor the potentials are smooth. The effect of the fiber diameter and the internodal distance on the volume conductor potentials as well as the changes in the magnitude of the extracellular potential (in the time domain) between two adjacent nodes at various radial distances were studied. The radial decline of the peak-to-peak amplitude of the extracellular potential depends on the radial coordinater of the field point and increases with the increase ofr.