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.     

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

For myelinated fibers, it is experimentally well established that spike conduction velocity is proportional to fiber diameter. However no really satisfactory theoretical treatment has been proposed. To treat this problem a theoretical axon was described consisting of lengths of passive leaky cable (internode) regularly interrupted by short isopotential patches of excitable membrane (node). The nodal membrane was assumed to obey the Frankenhaeuser-Huxley equations. The explicit diameter dependencies of the various parameters were incorporated into the equations. The fiber diameter to axon diameter ratio was taken to be constant, and the internode length was taken to be proportional to the fiber diameter. Both these conditions reflect the situation that exists in real, experimental fibers. Dimensional analysis shows that these anatomical conditions are equivalent to Rushton's (1951) assumption of corresponding states. Hence, conduction velocity will be proportional to fiber diameter, in complete agreement with the experimental findings. Digital computer solutions of these equations were made in order to compute a set of actual velocities. Computations made with constant internode length or constant myelin thickness (i.e., nonconstant fiber diameter to axon diameter ratio) did not show linearity of the velocity-diameter relation.     

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.     

Spontaneous firing and myelination of very small axons

In this article the question of what evolutionary factors guided acquisition of myelin in the nervous system is addressed. The conclusion that conduction velocity of action potentials along the axon has been the only motive force needs reformulation, as other factors may have played a central role as well. In particular, protection against firing of spontaneous action potentials which may result from the simultaneous opening of only few (less than 10) sodium channels at the nodes of small (less than 1 micron diameter) myelinated axons, may have greatly contributed to discouraging myelination of axons smaller than 1 micron.     

Propagation of action potentials in squid giant axons. Repetitive firing at regions of membrane inhomogeneities

Effects of reduction in potassium conductance on impulse conduction were studied in squid giant axons. Internal perfusion of axons with tetraethylammonium (TEA) ions reduces G K and causes the duration of action potential to be increased up to 300 ms. This prolongation of action potentials does not change their conduction velocity. The shape of these propagating action potentials is similar to membrane action potentials in TEA. Axons with regions of differing membrane potassium conductances are obtained by perfusing the axon trunk and one of its two main branches with TEA after the second branch has been filled with normal perfusing solution. Although the latter is initially free of TEA, this ion diffuses in slowly. Up until a large amount of TEA has diffused into the second branch, action potentials in the two branches have very different durations. During this period, membrane regions with prolonged action potentials are a source of depolarizing current for the other, and repetitive activity may be initiated at transitional regions. After a single stimulus in either axon region, interactions between action potentials of different durations usually led to rebound, or a short burst, of action potentials. Complex interactions between two axon regions whose action potentials have different durations resembles electric activity recorded during some cardiac arrhythmias.     

Rapid conduction and the evolution of giant axons and myelinated fibers

Nervous systems have evolved two basic mechanisms for increasing the conduction speed of the electrical impulse. The first is through axon gigantism: using axons several times larger in diameter than the norm for other large axons, as for example in the well-known case of the squid giant axon. The second is through encasing axons in helical or concentrically wrapped multilamellar sheets of insulating plasma membrane--the myelin sheath. Each mechanism, alone or in combination, is employed in nervous systems of many taxa, both vertebrate and invertebrate. Myelin is a unique way to increase conduction speeds along axons of relatively small caliber. It seems to have arisen independently in evolution several times in vertebrates, annelids and crustacea. Myelinated nerves, regardless of their source, have in common a multilamellar membrane wrapping, and long myelinated segments interspersed with 'nodal' loci where the myelin terminates and the nerve impulse propagates along the axon by 'saltatory' conduction. For all of the differences in detail among the morphologies and biochemistries of the sheath in the different myelinated animal classes, the function is remarkably universal.     

A model of motor and sensory axon activation in the median nerve using surface electrical stimulation

Surface electrical stimulation has the potential to be a powerful and non-invasive treatment for a variety of medical conditions but currently it is difficult to obtain consistent evoked responses. A viable clinical system must be able to adapt to variations in individuals to produce repeatable results. To more fully study the effect of these variations without performing exhaustive testing on human subjects, a system of computer models was created to predict motor and sensory axon activation in the median nerve due to surface electrical stimulation at the elbow. An anatomically-based finite element model of the arm was built to accurately predict voltages resulting from surface electrical stimulation. In addition, two axon models were developed based on previously published models to incorporate physiological differences between sensory and motor axons. This resulted in axon models that could reproduce experimental results for conduction velocity, strength-duration curves and activation threshold. Differences in experimentally obtained action potential shape between the motor and sensory axons were reflected in the models. The models predicted a lower threshold for sensory axons than motor axons of the same diameter, allowing a range of sensory axons to be activated before any motor axons. This system of models will be a useful tool for development of surface electrical stimulation as a method to target specific neural functions.     

Neuregulin-1 type III determines the ensheathment fate of axons

The signals that determine whether axons are ensheathed or myelinated by Schwann cells have long been elusive. We now report that threshold levels of neuregulin-1 (NRG1) type III on axons determine their ensheathment fate. Ensheathed axons express low levels whereas myelinated fibers express high levels of NRG1 type III. Sensory neurons from NRG1 type III deficient mice are poorly ensheathed and fail to myelinate; lentiviral-mediated expression of NRG1 type III rescues these defects. Expression also converts the normally unmyelinated axons of sympathetic neurons to myelination. Nerve fibers of mice haploinsufficient for NRG1 type III are disproportionately unmyelinated, aberrantly ensheathed, and hypomyelinated, with reduced conduction velocities. Type III is the sole NRG1 isoform retained at the axon surface and activates PI 3-kinase, which is required for Schwann cell myelination. These results indicate that levels of NRG1 type III, independent of axon diameter, provide a key instructive signal that determines the ensheathment fate of axons.     

Relative Conduction Velocities of Small Myelinated and Non-myelinated Fibres in the Central Nervous System

IN peripheral nerve, most axons with diameters of less than 1 µm do not have myelin sheaths, while most fibres more than 1 µm in diameter are myelinated^1,2. In the central nervous system, axons as small as 0.2 µm in diameter may be myelinated^2–5. In his paper on the effects of myelin on conduction velocity, Rushton⁶ concluded that 1 µm is the “critical diameter” above which “myelin increases conduction velocity” and below which “conduction is faster without myelination”. This conclusion is referred to widely (see, for example, refs. 7–9). In this communication we demonstrate that the analysis leading to this conclusion is based on morphological data¹⁰ which do not apply either to central or to peripheral fibres, so that myelinated fibres considerably smaller than 1 µm might be expected to conduct more rapidly than non-myelinated fibres of similar size.     

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.     

Seasonal variation in conduction velocity of action potentials in squid giant axons

To determine whether the electrical properties of the squid giant axon are seasonally acclimated, action potentials, recorded at different temperatures, were compared between giant axons isolated from Loligo pealei caught in May, from relatively cold waters (approximately 10 degrees-12 degrees C), and in August, from relatively warm waters (approximately 20 degrees C). Parameters relating to the duration of the action potential (e.g., maximum rate of rise, maximum rate of fall, and duration at half-peak) did not change seasonally. The relationship between conduction velocity and temperature remained constant between seasons as well, in spite of the fact that May axons were significantly larger than August axons. When normalized to the fiber diameter, mean May conduction velocities were 83% of the August values at all temperatures tested, and analysis of the rise time of the action potential foot suggested that a change in the axoplasmic resistivity was responsible for this difference. Direct measurements of axoplasmic resistance further supported this hypothesis. Thus seasonal changes in the giant axon's size and resistivity are not consistent with compensatory thermal acclimation, but instead serve to maintain a constant relationship between conduction velocity and temperature.     

The corpus callosum in primates: processing speed of axons and the evolution of hemispheric asymmetry

Interhemispheric communication may be constrained as brain size increases because of transmission delays in action potentials over the length of axons. Although one might expect larger brains to have progressively thicker axons to compensate, spatial packing is a limiting factor. Axon size distributions within the primate corpus callosum (CC) may provide insights into how these demands affect conduction velocity. We used electron microscopy to explore phylogenetic variation in myelinated axon density and diameter of the CC from 14 different anthropoid primate species, including humans. The majority of axons were less than 1 µm in diameter across all species, indicating that conduction velocity for most interhemispheric communication is relatively constant regardless of brain size. The largest axons within the upper 95th percentile scaled with a progressively higher exponent than the median axons towards the posterior region of the CC. While brain mass among the primates in our analysis varied by 97-fold, estimates of the fastest cross-brain conduction times, as conveyed by axons at the 95th percentile, varied within a relatively narrow range between 3 and 9 ms across species, whereas cross-brain conduction times for the median axon diameters differed more substantially between 11 and 38 ms. Nonetheless, for both size classes of axons, an increase in diameter does not entirely compensate for the delay in interhemispheric transmission time that accompanies larger brain size. Such biophysical constraints on the processing speed of axons conveyed by the CC may play an important role in the evolution of hemispheric asymmetry.     

Axonal excitability revisited

The original papers of Hodgkin and Huxley (J. Physiol. 116 (1952a) 449, J. Physiol. 116 (1952b) 473, J. Physiol. 116 (1952c) 497, J. Physiol. 117 (1952d) 500) have provided a benchmark in our understanding of cellular excitability. Not surprisingly, their model of the membrane action potential (AP) requires revisions even for the squid giant axon, the preparation for which it was originally formulated. The mechanisms they proposed for the voltage-gated potassium and sodium ion currents, IK, and INa, respectively, have been superceded by more recent formulations that more accurately describe voltage-clamp measurements of these components. Moreover, the current-voltage relation for IK has a non-linear dependence upon driving force that is well described by the Goldman-Hodgkin-Katz (GHK) relation, rather than the linear dependence on driving force found by Hodgkin and Huxley. Furthermore, accumulation of potassium ions in the extracellular space adjacent to the axolemma appears to be significant even during a single AP. This paper describes the influence of these various modifications in their model on the mathematically reconstructed AP. The GHK and K+ accumulation results alter the shape of the AP, whereas the modifications in IK and INa gating have surprisingly little effect. Perhaps the most significant change in their model concerns the amplitude of INa, which they appear to have overestimated by a factor of two. This modification together with the GHK and the K+ accumulation results largely remove the discrepancies between membrane excitability of the squid giant axon and the Hodgkin and Huxley (J. Physiol. 117 (1952d) 500) model previously described (Clay, J. Neurophysiol. 80 (1998) 903).     

The role of different size vestibulospinal neurons in the static control of posture

1. In addition to giant cells, originally described by Deiters, the lateral vestibular nucleus contains also medium- and small-size cells. The role that these neurons exert in the static control of posture has been investigated in precollicular decerebrate cats in which the resting discharge of spontaneously active vestibulospinal neurons projecting to lumbosacral segments of the spinal cord (IVS neurons) has been related to the cell size inferred on the basis of the conduction velocity of their axons. 2. In control experiments, the IVS neurons with slower axonal conduction velocity and, by inference, having thinner axons and smaller cell bodies differed from those having faster conduction velocity by displaying a higher resting discharge rate and a relatively regular interspike interval distribution, i.e. a lower coefficient of variation (CV). 3. The resting discharge of the IVS neurons, which corresponded on the average to 24.5 +/- 15.7, S.D. imp./sec, in control experiments, increased significantly to 44.1 +/- 23.8, S.D. imp./sec after ablation of the cerebellar vermis and the fastigial nuclei, leading to a great increase in postural activity, while the proportion of regularly discharging units (with the lowest CV) increased. Moreover, the negative correlation between resting discharge of all the recorded IVS neurons and the conduction velocity of the corresponding axons, which was quite slight in the experiments with the cerebellum intact, greatly increased after partial cerebellectomy. This finding was due to a prominent increase in resting discharge of the small-size IVS neurons, while the discharge of the large-size IVS neurons was, on the average, comparable to that obtained in the controls. It appears, therefore, that the cerebellum exerts a prominent tonic inhibitory influence on the small-size IVS neurons, which are thus responsible for the great increase in decerebrate rigidity after cerebellectomy. 4. The resting discharge rate of the IVS neurons was not, on the average, greatly modified after ipsilateral acute (aVN) and chronic vestibular neurectomy (cVN) with respect to the controls. However, the proportion of regularly discharging units (with the lowest CV) decreased after aVN, but increased after cVN. The relation found in control experiments, i.e. the faster the conduction velocity of VS axon the lower was the unit discharge at rest, was lost after aVN, due to a decrease in resting discharge rate of the slow neurons. The mean discharge rate of these units, however, recovered after cVN, so that the negative correlation between resting discharge rate and axonal conduction velocity was reestablished.(ABSTRACT TRUNCATED AT 400 WORDS)     

In vivo responses of paired giant mechanoreceptor neurons in Aplysia abdominal ganglion.

Two neurons with cell bodies symmetrically located in the abdominal ganglion and giant axons in the left (L1) and right (R1) pleurovisceral connectives of Aplysia californica were examined in vivo and in vitro. Direct stimulation of R1 and L1 in the intact animal does not elicit any observable behavior, suggesting that they are neither motoneurons nor command neurons. These cells respond in vivo to sudden onset mechanical stimulation of widespread regions of the body. R1 and L1 spikes are initiated in at least three different loci: (1) the peripheral axon in the foot, (2) the neuropil of the pleural and/or pedal ganglion, and (3) the neuropil of the abdominal ganglion. Furthermore, R1 and L1 probably have two different mechanisms for spike initiation: (1) sensory (foot), and (2) synaptic (abdominal and/or head ganglia). The different loci for spike initiation account for the bidirectional conduction of R1 and L1 spikes. As sensory (mechanoreceptor) neurons, R1 and L1 have peripheral axons in the ipsilateral posterior pedal nerve, show low threshold responses to stimulation of the ipsilateral posterior foot, they are rapidly adapting their responses do not decrease with repetition, and they are not blocked by high Mg++/low Ca++ solutions. As synaptically-driven neurons, R1 and L1 have widespread bilateral responsiveness, their responses decrease with repetition and their inputs are blocked with high Mg++/low Ca++ solutions. These neurons integrate sensory and synaptic inputs and conduct bidirectionally, however, their output connections must be specified before their behavioral function can be understood.     

Optical measurement of conduction in single demyelinated axons

Demyelination was initiated in Xenopus sciatic nerves by an intraneural injection of lysolecithin over a 2-3-mm region. During the next week macrophages and Schwann cells removed all remaining damaged myelin by phagocytosis. Proliferating Schwann cells then began to remyelinate the axons, with the first few lamellae appearing 13 d after surgery. Action potentials were recorded optically through the use of a potential-sensitive dye. Signals could be detected both at normal nodes of Ranvier and within demyelinated segments. Before remyelination, conduction through the lesion occurred in only a small fraction of the fibers. However, in these particular cases we could demonstrate continuous (nonsaltatory) conduction at very low velocities over long (greater than one internode) lengths of demyelinated axons. We have previously found through loose patch clamp experiments that the internodal axolemma contains voltage-dependent Na+ channels at a density approximately 4% of that at the nodes. These channels alone, however, are insufficient for successful conduction past the transition point between myelinated and demyelinated regions. Small improvements in the passive cable properties of the axon, adequate for propagation at this site, can be realized through the close apposition of macrophages and Schwann cells. As the initial lamellae of myelin appear, the probability of success at the transition zone increases rapidly, though the conduction velocity through the demyelinated segment is not appreciably changed. A detailed computational model is used to test the relative roles of the internodal Na+ channels and the new extracellular layer. The results suggest a possible mechanism that may contribute to the spontaneous recovery of function often seen in demyelinating disease.     

The incidence and properties of beta axons to muscle spindles in the cat hind limb.

The distribution of beta axons to muscle spindles in the tenuissimus and abductor digiti quinti medius (A.D.Q.M.) muscles of the hind limb of the cat was determined by testing the action of single motor axons, capable of producing extrafusal contraction, isolated in the ventral spinal roots on the discharges of individual muscle spindle primary sensory endings recorded in the dorsal spinal roots. The proportion of spindles with beta innervation was 41% in A.D.Q.M. and 30% in tenuissimus. The proportion of fast motor axons that were beta axons was 28% in the A.D.Q.M. and 11% in tenuissimus; usually each beta axon innervated a single spindle while no spindle received more than two beta axons. The beta axons were dynamic in nature and those to any one muscle tended to have slightly lower conduction velocities than the alpha axons though some overlap did occur. The extent to which beta axons can account for the fact that in isolated spindles axons selective to either nuclear bag or nuclear chain fibres are found in about equal proportions whereas a ratio of three static to one dynamic gamma axons is found electrophysiologically is discussed. An explanation for the low incidence of beta innervation previously found electrophysiologically and the considerably higher incidence found histologically is given.     

Impulse Propagation at the Septal and Commissural Junctions of Crayfish Lateral Giant Axons

Transmission across the septal junctions of the segmented giant axons of crayfish is accounted for quantitatively by a simple equivalent circuit. The septal membranes are passive, resistive components and transmission is ephaptic, by the electrotonic spread of the action current of the pre-septal spike. The electrotonic spread appears as a septal potential, considerably smaller than the pre-septal spike, but usually still large enough to initiate a new spike in the post-septal segments. The septal membranes do not exhibit rectification, at least over a range of ± 25 mv polarization and this accounts for their capacity for bidirectional transmission. The commissural branches, which are put forth by each lateral axon, make functional connections between the two axons. Transmission across these junctions can also be bidirectional and is probably also ephaptic. Under various conditions, the ladder-like network of cross-connections formed by the commissural junctions can give rise to circus propagation of impulses from one axon to the other. This can give rise to reverberatory activity of both axons at frequencies as high as 400/sec.     

Membrane Capacitive Memory Alters Spiking in Neurons Described by the Fractional-Order Hodgkin-Huxley Model

Excitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.     

Synaptic potential in the motor giant axon of the crayfish

Some electrical properties of the synapses between central giant axons (presynaptic) and the motor giant axon (postsynaptic) of the crayfish abdominal nerve cord have been investigated. Postsynaptic potential change in response to presynaptic volleys contains two components: a spike potential and a synaptic potential of very long time course. Amplitude of the synaptic potential is graded according to the number of active presynaptic axons. Conductance increase in the synaptic membrane endures over most of the period of potential change, and it is this rather than the "electrical time constant" of the membrane that in large measure determines the form of the synaptic potential. Temporal summation of synaptic potential occurs during repetitive presynaptic stimulation, and after such stimulation the rate of decay of synaptic potential is greatly slowed.