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.     

Conduction in bundles of demyelinated nerve fibers: computer simulation

This study presents a model of action potential propagation in bundles of myelinated nerve fibers. The model combines the single-cable formulation of Goldman and Albus (1967) with a basic representation of the ephaptic interaction among the fibers. We analyze first the behavior of the conduction velocity (CV) under the change of the various conductance parameters and temperature. The main parameter influencing the CV is the fast sodium conductance, and the dependence of CV on the temperature is linear up to 30 degrees C. The increase of myelin thickness above its normal value (5 microm) gives a slight increase in CV. The CV of the single fiber decreases monotonically with the disruption of myelin, but the breakdown is abrupt. There is always conduction until the thickness is larger than 2% of its original value, at which with at this point a sharp transition of CV to zero occurs. Also, the increase of temperature can block conduction. At 5% of the original thickness there is still spike propagation, but an increase of 2 degrees C causes conduction block. These results are consistent with clinical observations. Computer simulations are performed to show how the CV is affected by local damage to the myelin sheath, temperature alterations, and increased ephaptic coupling (i.e., coupling of electrical origin due to the electric neutrality of all the nerve) in the case of fiber bundles. The ephaptic interaction is included in the model. Synchronous impulse transmission and the formation of "condensed" pulse states are found. Electric impulses with a delay of 0.5 ms are presented to the system, and the numerical results show that, for increasing coupling, the impulses tend to adjust their speed and become synchronized. Other interesting phenomena are that spurious spikes are likely to be generated when ephaptic interaction is raised and that damaged axons suffering conduction block can be brought into conduction by the normal functioning fibers surrounding them. This is seen also in the case of a large number of fibers (N=500). When all the fibers are stimulated simultaneously, the conduction velocity is found to be strongly dependent on the level of ephaptic coupling and a sensible reduction is observed with respect to the propagation along an isolated axon even for low coupling level. As in the case of three fibers, spikes tend to lock and form collective impulses that propagate slowly in the nerve. On the other hand, if only 10% of fibers are stimulated by an external input, the conduction velocity is only 2% less than that along a single axon. We found a threshold value for the ephaptic coupling such that for lower values it is impossible to recruit the damaged fibers into conduction, for values of the coupling equal to this threshold only one fiber can be restored by the nondamaged fibers, and for values larger than the threshold an increasing number of fibers can return to normal functioning. We get values of the ephaptic coupling such that 25% of axons can be damaged without change of the collective conduction.     

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.     

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.     

Digital Computer Solutions for Excitation and Propagation of the Nerve Impulse

The Hodgkin-Huxley model of the nerve axon describes excitation and propagation of the nerve impulse by means of a nonlinear partial differential equation. This equation relates the conservation of the electric current along the cablelike structure of the axon to the active processes represented by a system of three rate equations for the transport of ions through the nerve membrane. These equations have been integrated numerically with respect to both distance and time for boundary conditions corresponding to a finite length of squid axon stimulated intracellularly at its midpoint. Computations were made for the threshold strength-duration curve and for the repetitive firing of propagated impulses in response to a maintained stimulus. These results are compared with previous solutions for the space-clamped axon. The effect of temperature on the threshold intensity for a short stimulus and for rheobase was determined for a series of values of temperature. Other computations show that a highly unstable subthreshold propagating wave is initiated in principle by a just threshold stimulus; that the stability of the subthreshold wave can be enhanced by reducing the excitability of the axon as with an anesthetic agent, perhaps to the point where it might be observed experimentally; but that with a somewhat greater degree of narcotization, the axon gives only decrementally propagated impulses.     

Noise effects on spike propagation in the stochastic Hodgkin-Huxley models

Effects of membrane current noise on spike propagation along a nerve fiber are studied. Additive current noise and channel noise are considered by using stochastic versions of the Hodgkin-Huxley model. The results of computer simulation show that the membrane noise causes considerable variation of the propagation time of a spike (thus changes in interspike intervals) for a small unmyelinated fiber of radius 0.1 approximately 1 micron.     

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.     

A Note on the Local Current Associated with the Rising Phase of a Propagating Impulse in Nonmyelinated Nerve Fibers

To extend our recent paper dealing with the cable properties and the conduction velocity of nonmyelinated nerve fibers (Bull. Math. Biol. 64, 1069; 2002), the behavior of the local current associated with the rising phase of a propagating action potential is discussed. It is shown that the process of charging the membrane capacity by means of the local current plays a crucial role in determining the velocity of nerve conduction. The symmetry of the local current with respect to the boundary between the resting and active regions of the nerve fiber is emphasized. It is noted that there are several simple quantitative rules governing the intensities of the capacitive, resistive and total membrane currents observed during the rising phase of an action potential.     

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.     

Analysis of mechanism of conduction of nerve impulses during the relative refractory phase by a mathematical model of the giant squid axon

By use of the mathematical model of Hodgkin and Huxley we have investigated changes in membrane currents and in ionic permeability of the membrane during conduction of an action potential along a refractory axon. At the start of the relative refractory period, the action potential has a graded character, and is extinguished over a distance of a few millimeters from the site of stimulation (decremental conduction). These changes are brought about not only by the low value of the inactivation variable h and the high potassium conductivity g_K, but also by the inhibitory influence on the recovery process exerted by the spreading wave of depolarization. The graded peak can be propagated along the whole fiber in approximately 0.6 msec at 18.5° after the end of the absolute refractory phase. At this time the wave of the recovery process (increase of h and decrease of g_K) begins to oppose the wave of depolarization, with the result that at each subsequent point along the fiber h increases above and g_K decreases below the preceding value. Under these circumstances, conduction becomes incremental, and in order to evoke a propagated action potential all that is required is to induce a subthreshold local response at the point of stimulation.A. V. Vishnevskii Institute of Surgery, Academy of Medical Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 3, No. 4, pp. 434–441, July–August, 1971.     

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.     

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.     

Optical Coherence Tomography Phase Measurement of Transient Changes in Squid Giant Axons During Activity

Noncontact optical measurements reveal that transient changes in squid giant axons are associated with action potential propagation and altered under different environmental (i.e., temperature) and physiological (i.e., ionic concentrations) conditions. Using a spectral-domain optical coherence tomography system, which produces real-time cross-sectional images of the axon in a nerve chamber, axonal surfaces along a depth profile are monitored. Differential phase analyses show transient changes around the membrane on a millisecond timescale, and the response is coincident with the arrival of the action potential at the optical measurement area. Cooling the axon slows the electrical and optical responses and increases the magnitude of the transient signals. Increasing the NaCl concentration bathing the axon, whose diameter is decreased in the hypertonic solution, results in significantly larger transient signals during action potential propagation. While monophasic and biphasic behaviors are observed, biphasic behavior dominates the results. The initial phase detected was constant for a single location but alternated for different locations; therefore, these transient signals acquired around the membrane appear to have local characteristics.     

Spread of discrete structural changes in synthetic polyanionic gel: a model of propagation of a nerve impulse

Thin fibers of cross-linked polyacrylate gel were prepared by inducing polymerization reaction inside long glass or Tygon tubings. By immersing these gel fibers in salt solutions containing both Ca(2+) and Na(+) at varying ratios, a discontinuous transition from the swollen state to the shrunken was demonstrated. A very sharp boundary was observed between the swollen and shrunken portions of the gel fiber. It was found possible to displace this sharp boundary continuously by application of a weak electric current. Based on the similarity in swelling behavior between nerve fibers and synthetic gel fibers, a non-myelinated nerve fiber carrying an impulse was treated as a cylindrical gel layer consisting of two distinct portions, a swollen (active) portion connected directly to the remaining shrunken (resting) portion. By applying the cable theory to this model of the nerve fiber, mathematical expressions describing the conduction velocity, the maximum rate of potential rise, etc. in terms of the electric parameters of the fiber were derived.     

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.     

Absence of Non-Thermal Microwave Effects on the Function of Giant Nerve Fibers

Functioning of the giant axon of the isolated earthworm ventral nerve cord was examined during exposure to 6.45 GHz microwaves. We used continuous wave and pulsed irradiation, either synchronized with stimuli or asynchronous, lasting for 10–50 min at specific absorption rate from 30 to 230 W/kg. Action potential (AP) conduction velocity and the capability of nerve fiber to answer long-lasting high-frequency stimulation served as indices of microwave effect. Under some experimental conditions the nerve appeared to have extreme sensitivity to subtle temperature changes, induced by irradiation, but no non-thermal microwave effects were detected.     

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.     

Integrative Synaptic Mechanisms in the Caudal Ganglion of the Crayfish

A study of activity recorded with intracellular micropipettes was undertaken in the caudal abdominal ganglion of the crayfish in order to gain information about central fiber to fiber synaptic mechanisms. This synaptic system has well developed integrative properties. Excitatory post-synaptic potentials can be graded, and synaptic potentials from different inputs can sum to initiate spike discharge. In most impaled units, the spike discharge fails to destroy the synaptic potential, thereby allowing sustained depolarization and multiple spike discharge following single pulse stimulation to an afferent input. Some units had characteristics which suggest a graded threshold for spike generation along the post-synaptic fiber membrane. Other impaled units responded to afferent stimulation with spike discharges of two distinct amplitudes. The smaller or "abortive" spikes in such units may represent non-invading activity in branches of the post-synaptic axon. On a few occasions one afferent input was shown to inhibit the spike discharge initiated by another presynaptic input.     

On the cable theory of nerve conduction

Conduction of an impulse in the nonmyelinated nerve fiber is treated quantitatively by considering it as a direct consequence of the coexistence of two structurally distinct regions, resting and active, in the fiber. The profile of the electrical potential change induced in the vicinity of the boundary between the two regions is analyzed by using the cable equations. Simple mathematical formulae relating the conduction velocity to the electrical parameters of the fiber are derived from the symmetry of the potential profile at the boundary. The factors that determine the conduction velocity in the myelinated nerve fiber are reexamined.