The “Lillie Transition”: models of the onset of saltatory conduction in myelinating axons

Almost 90 years ago, Lillie reported that rapid saltatory conduction arose in an iron wire model of nerve impulse propagation when he covered the wire with insulating sections of glass tubing equivalent to myelinated internodes. This led to his suggestion of a similar mechanism explaining rapid conduction in myelinated nerve. In both their evolution and their development, myelinating axons must make a similar transition between continuous and saltatory conduction. Achieving a smooth transition is a potential challenge that we examined in computer models simulating a segmented insulating sheath surrounding an axon having Hodgkin-Huxley squid parameters. With a wide gap under the sheath, conduction was continuous. As the gap was reduced, conduction initially slowed, owing to the increased extra-axonal resistance, then increased (the “rise”) up to several times that of the unmyelinated fiber, as saltatory conduction set in. The conduction velocity slowdown was little affected by the number of myelin layers or modest changes in the size of the “node,” but strongly affected by the size of the “internode” and axon diameter. The steepness of the rise of rapid conduction was greatly affected by the number of myelin layers and axon diameter, variably affected by internode length and little affected by node length. The transition to saltatory conduction occurred at surprisingly wide gaps and the improvement in conduction speed persisted to surprisingly small gaps. The study demonstrates that the specialized paranodal seals between myelin and axon, and indeed even the clustering of sodium channels at the nodes, are not necessary for saltatory conduction.     

Nerve conduction theory: Some mathematical consequences of Bernstein's model

The generally accepted permeability theory of nerve conduction is presented in mathematical form. The resulting velocity formula is found to agree well with data on squid giant axon, but predicts velocities considerably too high in the case ofNitella. The dependence of velocity on fiber diameter is discussed for both medullated and non-medullated nerve, it being shown theoretically that velocity is proportional to the square root of diameter for non-medullated and to the diameter for medullated nerve. The equations relating the shape of the action spike to the observed permeability changes are given but are not solved.     

Action currents, internodal potentials, and extracellular records of myelinated mammalian nerve fibers derived from node potentials.

The potential distribution within the internodal axon of mammalian nerve fibers is derived by applying known node potential waveforms to the ends of an equivalent circuit model of the internode. The complete spatial/temporal profile of action potentials synthesized from the internodal profiles is used to compute the node current waveforn, and the extracellular action potential around fibers captured within a tubular electrode. For amphibia, the results agreed with empirical values. For mammals, the amplitude of the node currents plotted against conduction velocity was fitted by a straight line. The extracellular potential waveform depended on the location of the nodes within the tube. For tubes of length from 2 to 8 internodes, extracellular wave amplitude (mammals) was about one-third of the product of peak node current and tube resistance (center to ends). The extracellular potentials developed by longitudinal and radial currents in an anisotropic medium (fiber bundle) are compared.     

Effect of diameter on membrane capacity and conductance of sheep cardiac Purkinje fibers

Membrane electrical properties were measured in sheep cardiac Purkinje fibers, having diameters ranging from 50 to 300 mum. Both membrane capacitance and conductance per unit area of apparent fiber surface varied fourfold over this range. Membrane time constant, and capacitance per unit apparent surface area calculated from the foot of the action potential were independent of fiber diameter, having average values of 18.8 +/- 0.7 ms, and 3.4 +/- 0.25 muF/cm2, respectively (mean +/- SEM). The conduction velocity and time constant of the foot of the action potential also appeared independent of diameter, having values of 3.0 +/- 0.1 m/s and 0.10 +/- 0.007 ms. These findings are consistent with earlier suggestions that in addition to membrane on the surface of the fiber, there exists a large fraction of membrane in continuity with the extracellular space but not directly on the surface of the fiber. Combining the electrical and morphological information, it was possible to predict a passive length constant for the internal membranes of about 100 mum and a time constant for chaning these membranes in a passive 100-mum fiber of 1.7 ms.     

A study of conduction velocity in nonmyelinated nerve fibers.

By treating a nonmyelinated nerve fiber as a continuous cable consisting of three distinct zones (Resting, transitional, and excited), the following mathematical expression was derived: (formula: see text) where v is the conduction velocity, d the diameter of the fiber, R the resistance of the membrane of unit area at the peak of excitation, rho the resistivity of the medium inside the fiber, and C the capacity of membrane per unit area. The validity of this expression was demonstrated by using squid giant nerve fibers intracellularly perfused with dilute salt solutions. The relationship between these results and previous theories and experiments on conduction velocity is discussed.     

Anatomical basis for an apparent paradox concerning conduction velocities of two identified axons in Aplysia.

Larger axons usually have faster conduction velocities, lower thresholds, and larger extracellular action potentials than smaller axons. However, it has been shown that the largest fiber, R2, in the right pleurovisceral connective of the marine mollusc, Aplysia, has a higher threshold and a slower conduction velocity than does the smaller axon of cell RI, even though the amplitude of R2's spike is larger than R1's spike. One explanation of this apparent parodox is that the two axons have different "intrinsic membrane and axoplasmic constants" (Goldman, L. (1961), J. Cell Comp. Physiol. 57: 185-191). However, the deep infolding of R2's axonal membrane suggested that differences in the shape of the two axons might also account for the paradox. Accordingly, we measured the conduction velocities of the two axons and then examined the same axons in the electron microscope in order to measure their volumes and surface areas. Our morphological observations indicate that the extensive infolding of surface membrane causes R2 to have a smaller volume to surface area ratio than R1. Thus, since conduction velocity is proportional to the square root of the volume to surface area ratio (Hodgkin, A.L. (1954), J. Physiol. 125: 221-224), it is predictable that the smaller axon would have a faster conduction velocity. The results suggest that the paradoxical conduction velocities can be explained largely as resulting from differences in the shapes of the two axons. However, certain discrepancies between the measured and the predicted values suggest that other factors are contributing as well.     

Computation of Impulse Initiation and Saltatory Conduction in a Myelinated Nerve Fiber

A mathematical model of the electrical properties of a myelinated nerve fiber is given, consisting of the Hodgkin-Huxley ordinary differential equations to represent the membrane at the nodes of Ranvier, and a partial differential cable equation to represent the internodes. Digital computer solutions of these equations show an impulse arising at a stimulating electrode and being propagated away, approaching a constant velocity. Action potential curves plotted against distance show discontinuities in slope, proportional to the nodal action currents, at the nodes. Action potential curves plotted against time, at the nodes and in the internodes, show a marked difference in steepness of the rising phase, but little difference in peak height. These results and computed action current curves agree fairly accurately with published experimental data from frog and toad fibers.     

Axons and myelinating glia: An intimate contact

The coordination of the vertebrate nervous system requires high velocity signal transmission between different brain areas. High speed nerve conduction is achieved in the myelinated fibers of both the central and the peripheral nervous system where the myelin sheath acts as an insulator of the axon. The interactions between the glial cell and the adjacent axon, namely axo-glial interactions, segregate the fiber in distinct molecular and functional domains that ensure the rapid propagation of action potentials. These domains are the node of Ranvier, the paranode, the juxtaparanode and the internode and are characterized by multiprotein complexes between voltage-gated ion channels, cell adhesion molecules, members of the Neurexin family and cytoskeletal proteins. In the present review, we outline recent evidence on the key players of axo-glial interactions, depicting their importance in myelinated fiber physiology and disease.     

Conduction velocities and fiber diameters in a cutaneous nerve of the pigeon

Summary The characteristics of fibers of a cutaneous nerve supplying the wing skin of the pigeon have been investigated with electrophysiological and electron microscopic techniques.Recordings of the compound action potential showed four distinct peaks with conduction velocities of about 30 m/s, 12 m/s, 4 m/s and 0.5 m/s.From electron micrographs both fiber diameters and thickness of myelin sheath were assessed and used as criteria for segregating various fiber populations. Altogether four groups could be discerned: large thickly myelinated fibers, small thickly myelinated fibers, small thinly myelinated fibers, and unmyelinated or C-fibers. The subdivision of the thickly myelinated fibers into two populations is evidenced mainly by corresponding peaks in the compound action potential. The thinly myelinated fibers with a mean diameter of 2 m contributed about 90% of all myelinated fibers in this nerve.When comparing fiber dimensions and conduction velocities of this avian nerve with those of mammalian cutaneous nerves, the lower CV's of avian nerve fibers can be explained by smaller diameters and thinner myelin sheaths.The results of this investigation are a prerequisite for latency considerations in central somatosensory pathways in birds.Abbreviations CAP compound action potential - CV conduction velocity - D fiber diameter - d axon diameter - g ratio d/D - m thickness of myelin sheath     

Theoretical consideration of a possible mechanism in the conduction process of thin-sheathed nerve fibers

The propagation of a transverse disturbance along a tubular membrane enclosing a fluid medium and embedded in another is considered. It is shown that the velocity of propagation of such a disturbance can be identified with the velocity of the conduction process of thin-sheathed nerve fibers. The required values of the associated parameters, tension and pressure, appear not unreasonable. The results obtained indicate that experimental observations on the relation between the conduction velocity and the fiber diameter, as well as the effects of longitudinal stretching and transverse squeezing on the velocity of the conduction process in nerve, may be correlated on such a basis.     

Anatomical basis for an apparent paradox concerning conduction velocities of two identified axons in Aplysia

Larger axons usually have faster conduction velocities, lower thresholds, and larger extracellular action potentials than smaller axons. However, it has been shown that the largest fiber, R2, in the right pleurovisceral connective of the marine mollusc, Aplysia, has a higher threshold and a slower conduction velocity than does the smaller axon of cell R1, even though the amplitude of R2's spike is larger than R1's spike. One explanation of this apparent paradox is that the two axons have different “intrinsic membrane and axoplasmic constants” (Goldman, L. (1961), J. Cell Comp. Physiol. 57: 185–191). However, the deep infolding of R2's axonal membrane suggested that differences in the shape of the two axons might also account for the paradox. Accordingly, we measured the conduction velocities of the two axons and then examined the same axons in the electron microscope in order to measure their volumes and surface areas. Our morphological observations indicate that the extensive infolding of surface membrane causes R2 to have a smaller volume to surface area ratio than R1. Thus, since conduction velocity is proportional to the square root of the volume to surface area ratio (Hodgkin, A. L. (1954), J. Physiol. 125: 221–224), it is predictable that the smaller axon would have a faster conduction velocity. The results suggest that the paradoxical conduction velocities can be explained largely as resulting from differences in the shapes of the two axons. However, certain discrepancies between the measured and the predicted values suggest that other factors are contributing as well.     

Hedysari Extract Improves Regeneration after Peripheral Nerve Injury by Enhancing the Amplification Effect

Radix Hedysari is an herbal preparation frequently used in traditional Chinese medicine. It can promote regeneration after peripheral nerve injury, but its effect on the amplification ratio (the ratio of distal to proximal fibers) during peripheral nerve regeneration has not yet been examined. In this study, we explored the effect of Hedysari extract on the amplification ratio in the peripheral nerve. Male Sprague-Dawley rats were separated into three groups at random: normal group (without surgery), model group (given sleeve nerve bridging surgery, but without adjuvant treatment) and treatment group (given sleeve nerve bridging surgery and then given Hedysari extract as adjuvant treatment). Twelve weeks after surgery, general observations, electrophysiological examination, histological analysis, morphometric measurements, and amplification ratio calculations were made. The results showed that nerve conduction velocity, the fiber and axon diameter, the g-ratio, the number of regenerating nerve fibers and the amplification ratio were better in the treatment group than in the model group, suggesting that Hedysari extract can effectively promote the growth of lateral buds in the proximal nerve stump and substantially improve the amplification effect during peripheral nerve regeneration.     

An analysis of the relationships between subthreshold electrical properties and excitability in skeletal muscle

Skeletal muscle activation requires action potential (AP) initiation followed by its sarcolemmal propagation and tubular excitation to trigger Ca(2+) release and contraction. Recent studies demonstrate that ion channels underlying the resting membrane conductance (G(M)) of fast-twitch mammalian muscle fibers are highly regulated during muscle activity. Thus, onset of activity reduces G(M), whereas prolonged activity can markedly elevate G(M). Although these observations implicate G(M) regulation in control of muscle excitability, classical theoretical studies in un-myelinated axons predict little influence of G(M) on membrane excitability. However, surface membrane morphologies differ markedly between un-myelinated axons and muscle fibers, predominantly because of the tubular (t)-system of muscle fibers. This study develops a linear circuit model of mammalian muscle fiber and uses this to assess the role of subthreshold electrical properties, including G(M) changes during muscle activity, for AP initiation, AP propagation, and t-system excitation. Experimental observations of frequency-dependent length constant and membrane-phase properties in fast-twitch rat fibers could only be replicated by models that included t-system luminal resistances. Having quantified these resistances, the resulting models showed enhanced conduction velocity of passive current flow also implicating elevated AP propagation velocity. Furthermore, the resistances filter passive currents such that higher frequency current components would determine sarcolemma AP conduction velocity, whereas lower frequency components excite t-system APs. Because G(M) modulation affects only the low-frequency membrane impedance, the G(M) changes in active muscle would predominantly affect neuromuscular transmission and low-frequency t-system excitation while exerting little influence on the high-frequency process of sarcolemmal AP propagation. This physiological role of G(M) regulation was increased by high Cl(-) permeability, as in muscle endplate regions, and by increased extracellular [K(+)], as observed in working muscle. Thus, reduced G(M) at the onset of exercise would enhance t-system excitation and neuromuscular transmission, whereas elevated G(M) after sustained activity would inhibit these processes and thereby accentuate muscle fatigue.     

The velocity of conduction in nerve fiber and its electric characteristics

The theory developed in this paper shows that the propagation of spike potential along a nerve fiber and the conduction of an electric wave along an inert inorganic conductor follow a common quantitative relationship. This result gives further support to the belief that propagation of excitation is an electrical process. The basic idea of the theory is derived from the consideration that velocity has, by its mathematical definition, a local meaning; conduction in a nerve is completely determined by the local characteristics of the latter, as well as those of the wave. The final formula derived does not make use of any other field of science beyond the fundamental principles of electricity. It gives the conduction velocity in terms of the electric characteristics of the fiber and of the duration of the spike potential. The formula is in agreement with the known dependence of the conduction velocity on various parameters characterizing the axon. The computed velocity agrees with the measured ones on the squid giant axon, crab nerve axon, frog muscle fiber and Nitella cell. The membrane inductance appears as a velocity controling agent which prevents also a possible distortion of the spike potential during conduction. The structural meaning of the electric characteristics of the axon membrane is discussed from the viewpoint of the diffusion theory. A formula for the velocity of spread of the electrotonus is also derived.     

Regeneration of spinal electrocyte fibers in Sternarchus albifrons: Development of axon-schwann cell relationships and nodes of Ranvier

Summary The electrocyte fibers in the gymnotid Sternarchus albifrons are highly differentiated myelinated axons which exhibit several types of nodes of Ranvier and characteristically short internode lengths. In the present study, regeneration of the electrocyte fibers following removal of the tail was examined by electron microscopy. By 36 days following extirpation, the regenerating electrocyte axons exhibit Type I nodes of Ranvier, with a normal morphology, and Type II nodes of Ranvier with a large nonmyelinated gap and a polypoid elaboration of the axon surface. Moreover, in the regenerating axons the internode length diameter ratios are quite small. Thus, relatively normal axon-Schwann cell relations and a relatively normal differentiation of the axon surface are achieved during regeneration of the Sternarchus electrocyte fibers.Supported in part by the Medical Research Service, Veterans Administration and by grants from the National Institutes of Health and the Paralyzed Veterans of America     

Intraretinal axon diameter: a single cell analysis in the marmoset (Callithrix jacchus)

We have labelled individual retinal ganglion cells of a New World primate, the common marmoset (Callithrix jacchus) with neurobiotin and then measured axon, soma and dendritic field diameter. A total of 111 cells were analysed (62 parasol cells, 22 midget cells, 16 hedge cells and 11 small bistratified cells). When all retinal ganglion cells were grouped together axon diameter was positively correlated to soma diameter. When analysed according to cell class only midget cells showed a positive correlation between soma size and mean axon diameter. Dendritic field diameter and mean axon diameter of both parasol and midget cells showed significant correlations. Axon diameter is not constant along the intraretinal length of the axon and the rate of change in diameter appears to be related to the cell class and the initial size of the axon. Midget cell axons showed a rapid increase of up to 20% over the first 200 μm in contrast to parasol cell axons which increased more slowly over this distance but then showed a marked increase in diameter of up to 40% over the next 450 μm. However, axon diameter did not remain at these increased diameters but decreased at greater distances from the soma. The degree to which an axon changes its diameter is related to retinal ganglion cell class and the initial size of the axon. We postulate that these variations in intraretinal axon diameter may have a direct influence on conduction velocity and reflect a compensatory mechanism to minimise spatiotemporal dispersion along the visual pathway.     

Schmidt-Lanterman clefts: a morphometric study in human sural nerve

In the human sural nerve, large myelinated fibers contained 35 Schmidt-Lanterman (SL) clefts per mm, and small myelinated fibers contained only eight SL clefts per mm. The incidence of SL clefts is linearly related to myelin thickness. The SL clefts extended over 13 micron in large and over 9 micron in small fibers, the total extent of the SL region amounting to nearly 50% of internodal length in large and to 6% in small fibers. In the SL region, the fiber diameter was 6% larger than outside this region, and the axon was 17% smaller in large and 28% smaller in small fibers. The paranodal-nodal region occupied less than 2% of internodal length in large fibers and 6.5% in small fibers; in the nodal region the axon diameter was reduced by 40-50%.     

The diameters of frozen-hydrated chromatin fibers increase with DNA linker length: evidence in support of variable diameter models for chromatin

The diameters of chromatin fibers from Thyone briareus (sea cucumber) sperm (DNA linker length, n = 87 bp) and Necturus maculosus (mudpuppy) erythrocytes (n = 48 bp) were investigated. Soluble fibers were frozen into vitrified aqueous solutions of physiological ionic strength (124 mM), imaged by cryo-EM, and measured interactively using quantitative computer image-processing techniques. Frozen-hydrated Thyone and Necturus fibers had significantly different mean diameters of 43.5 nm (SD = 4.2 nm; SEM = 0.61 nm) and 32.0 nm (SD = 3.0 nm; SEM = 0.36 nm), respectively. Evaluation of previously published EM data shows that the diameters of chromatin from a large number of sources are proportional to linker length. In addition, the inherent variability in fiber diameter suggests a relationship between fiber structure and the heterogeneity of linker length. The cryo-EM data were in quantitative agreement with space-filling double-helical crossed-linker models of Thyone and Necturus chromatin. The data, however, do not support solenoid or twisted-ribbon models for chromatin that specify a constant 30 nm diameter. To reconcile the concept of solenoidal packing with the data, we propose a variable-diameter solid-solenoid model with a fiber diameter that increases with linker length. In principle, each of the variable diameter models for chromatin can be reconciled with local variations in linker length.     

Electrical activity and structural correlates of giant nerve fibers in Kuruma shrimp (Penaeus japonicus)

Morphology and recordings of electrical activity of Kuruma shrimp (Penaeus japonicus) giant medullated nerve fibers were carried out. A pair of giant fibers with external diameter of about 120 μ and 10 μ in myelin thickness were found in the ventral nerve cord. The diameter of the axon is about 10 μ. Thus there is a wide gap between the axon and the external myelin sheath. Each axon is doubly coated directly by Schwann cells and indirectly by the myelin sheath layer which is produced by those Schwann cells. Impulse conduction velocities of these giant fibers showed a range between 90–210 m/sec at about 22°C. Large action potentials (up to 113 mV, rise time of 0.16–0.3 msec, maximum rate of rise of 650–1250 V/sec, half decay time of 0.2–0.3 msec, maximum rate of fall of 250–450 V/sec and total duration of less than 1.5 msec) could be obtained by inserting microelectrodes or by longitudinal insertion of 25 μ diameter capillary electrodes into the gap but no DC-potential difference was observed across the myelin sheath. Transmyelin electrical parameters were very favorable for fast impulse conduction: myelin resistance of 3 × 10⁴ Ω cm²; time constant of 0.38 msec; myelin capacitance of 1.35 × 10^?8 F/cm²; gap fluid resistivity of 23 Ω cm. The existence of nodes of Ranvier could not be demonstrated morphologically, but electrophysiological evidence suggests that a type of saltatory conduction occurs in these giant fibers.     

Variation of muscle stiffness with force at increasing speeds of shortening

Single frog skeletal muscle fibers were attached to a servo motor and force transducer by knotting the tendons to pieces of wire at the fiber insertions. Small amplitude, high frequency sinusoidal length changes were then applied during tetani while fibers contracted both isometrically and isotonically at various constant velocities. The amplitude of the resulting force oscillation provides a relative measure of muscle stiffness. It is shown from an analysis of the transient force responses observed after sudden changes in muscle length applied both at full and reduced overlap and during the rising phase of short tetani that these responses can be explained on the basis of varying numbers of cross bridges attached at the time of the length step. Therefore, the stiffness measured by the high frequency length oscillation method is taken to be directly proportional to the number of cross bridges attached to thin filament sites. It is found that muscle stiffness measured in this way falls with increasing shortening velocity, but not as rapidly as the force. The results suggest that at the maximum velocity of shortening, when the external force is zero, muscle stiffness is still substantial. The findings are interpreted in terms of a specific model for muscle contraction in which the maximum velocity of shortening under zero external load arises when a force balance is attained between attached cross bridges some of which are aiding and others opposing shortening. Other interpretations of these results are also discussed.