The effect of subthreshold prepulses on the recruitment order in a nerve trunk analyzed in a simple and a realistic volume conductor model

 The influence of subthreshold depolarizing prepulses on the threshold current-to-distance and the threshold current-to-diameter relationship of myelinated nerve fibers has been investigated. A nerve fiber model was used in combination with both a simple, homogeneous volume conductor model with a point source and a realistic, inhomogeneous volume conductor model of a monofascicular nerve trunk surrounded by a cuff electrode. The models predict that a subthreshold depolarizing prepulse will desensitize Ranvier nodes of fibers in the vicinity of the cathode and thus cause an increase in the threshold current of a subsequent pulse to activate these fibers. If the increase in threshold current of the excited node is large enough, the excitation will be accompanied by a strong hyperpolarization of adjacent nodes, preventing the propagation of action potentials in these fibers. As fibers close to the electrode are more desensitized by prepulses than more distant ones, it is possible to stimulate distant fibers without stimulating such fibers close to the electrode. Moreover, as larger fibers are more desensitized than smaller ones, smaller fibers have lower threshold currents than larger fibers up to a certain distance from the electrode. The realistic model has provided an additional condition for the application of this method to invert nerve fiber recruitment, i.e., real or virtual anodes should be close to the cathode. When using a cuff electrode for this purpose, in the case of monopolar stimulation the cuff length (determining the position of the virtual anodes) should not exceed twice the internodal length of the fibers to be blocked. Similarly, the distance between cathode and anodes should not exceed the internodal length of these fibers when stimulation is to be applied tripolarly. Received: 15 May 2000 / Accepted in revised form: 9 February 2001     

Extracellular currents and potentials of the active myelinated nerve fiber.

This paper is concerned with the accurate and rapid calculation of extracellular potentials and currents from an active myelinated nerve fiber in a volume conductor, under conditions of normal and abnormal conduction. The neuroelectric source for the problem is characterized mathematically by using a modified version of the distributed parameter model of L. Goldman and J. S. Albus (1968, Biophys. J., 8:596-607) for the myelinated nerve fiber. Solution of the partial differential equation associated with the model provides a waveform for the spatial distribution of the transmembrane potential V(z). This model-generated waveform is then used as input to a second model that is based on the principles of electromagnetic field theory, and allows one to calculate easily the spatial distribution for the potential everywhere in the surrounding volume conductor for the nerve fiber. In addition, the field theoretic model may be used to calculate the total longitudinal current in the extracellular medium (I0L(z)) and the transmembrane current per unit length (im(z)); both of these quantities are defined in connection with the well-known core conductor model and associated cable equations in electrophysiology. These potential and current quantities may also be calculated as functions of time and as such, are useful in interpreting measured I0L(t) and im(t) data waveforms. An analysis of the accuracy of conventionally used measurement techniques to determine I0L(t) and im(t) is performed, particularly with regard to the effect of electrode separation distance and size of the volume conductor on these measurements. Also, a simulation of paranodal demyelination at a single node of Ranvier is made and its effects on potential and current waveforms as well as on the conduction process are determined. In particular, our field theoretic model is used to predict the temporal waveshape of the field potentials from the active, non-uniformly conducting nerve fiber in a finite volume conductor.     

Analytical theory for extracellular electrical stimulation of nerve with focal electrodes. I. Passive unmyelinated axon.

The cable model of a passive, unmyelinated fiber in an applied extracellular field is derived. The solution is valid for an arbitrary, time-varying, applied field, which may be determined analytically or numerically. Simple analytical computations are presented. They explain a variety of known phenomena and predict some previously undescribed properties of extracellular electrical stimulation. The polarization of a fiber in an applied field behaves like the output of a spatial high-pass and temporal low-pass filter of the stimulus. High-frequency stimulation results in a more spatially restricted region of fiber excitation, effectively reducing current spread relative to that produced by low-frequency stimulation. Chronaxie measured extracellularly is a function of electrode position relative to the stimulated fiber, and its value may differ substantially from that obtained intracellularly. Frequency dependence of psychophysical threshold obtained by electrical stimulation of the macaque cochlea closely follows the frequency dependence of single-fiber passive response.     

Potential and current distributions in a cylindrical bundle of cardiac tissue.

The intracellular and interstitial potentials associated with each cell or fiber in multicellular preparations carrying a uniformly propagating wave are important for characterizing the electrophysiological behavior of the preparation and in particular, for evaluating the source contributed by each fiber. The aforementioned potentials depend on a number of factors including the conductivities characterizing the intracellular, interstitial, and extracellular domains, the thickness of the tissue, and the distance (depth) of the field point from the surface of the tissue. A model study is presented describing the extracellular and interstitial potential distribution and current flow in a cylindrical bundle of cardiac muscle arising from a planar wavefront. For simplicity, the bundle is considered as a bidomain. Using typical values of conductivity, the results show that the intracellular and interstitial potential of fibers near the center of a very large bundle (greater than 10 mm) may be approximated by the potentials of a single fiber surrounded by a limited extracellular space (a fiber in oil), hence justifying a core-conductor model. For smaller bundles, the peak interstitial potential is less than that predicted by the core-conductor model but still large enough to affect the overall source strength. The magnitude of the source strength is greatest for fibers lying near the center of the bundle and diminishes sharply for fibers within 50 microns of the surface.     

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.     

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.     

A fiber-reinforced composite model of the viscoelastic behavior of the brainstem in shear.

Brainstem trauma occurs frequently in severe head injury, often resulting in fatal lesions due to importance of brainstem in crucial neural functions. Structurally, the brainstem is composed of bundles of axonal fibers distinctly oriented in a longitudinal direction surrounded by an extracellular matrix. We hypothesize that the oriented structure and architecture of the brainstem dictates this mechanical response and results in its selective vulnerability in rotational loading. In order to understand the relationship between the biologic architecture and the mechanical response and provide further insight into the high vulnerability of this region, a structural and mathematical model was created. A fiber-reinforced composite model composed of viscoelastic fibers surrounded by a viscoelastic matrix was used to relate the biological architecture of the brainstem to its anisotropic mechanical response. Relevant model parameters measured include the brainstem's composite complex moduli and relative fraction of matrix and fiber. The model predicted that the fiber component is three times stiffer and more viscous than the matrix. The fiber modulus predictions were compared with experimental tissue measurements. The optic nerve, a bundle of tightly packed longitudinally arranged myelinated fibers with little matrix, served as a surrogate for the brainstem fiber component. Model predictions agreed with experimental measures, offering a validation of the model. This approach provided an understanding of the relationship between the specific biologic architecture of the brainstem and the anisotropic mechanical response and allowed insight into reasons for the selective vulnerability of this region in rotational head injury.     

Manifestations of dynamic coding of the amplitude-modulated sounds on the level of auditory nerve fibres

Average firing rate of the auditory nerve fiber as function of the level of the tone with the frequency equal to characteristic frequency of the fibers, can be defined as an input-output characteristic. It is known that the steepening of the input-output characteristic of the real auditory nerve fiber is more, and the width is less than the spontaneous activity of the fiber. The latter characterizes fiber's ability to generate spikes, if the stimulus is absent. However it is known, that the real auditory nerve fibers with low spontaneous activity reproduce amplitude modulation of the signals much better, than the fibers with high spontaneous activity. From the results of simulation experiments, it follows that the dynamic properties of the auditory nerve fibers, providing fine tuning or adaptation of a fiber threshold under the stimulus level but not the static input-output characteristics, are the reason of fibers reproduction of stimuli amplitude modulations. However the auditory nerve fibers with high spontaneous activity due to abrupt input-output characteristic are capable to reproduce modulations of sounds whose levels are lower than a threshold of the fiber, if a weak signal adds to a weak broadband noise. This is a phenomenon of stochastic resonance found in the reactions of auditory nerve fibers.     

Computer simulation of action potential propagation in septated nerve fibers.

The nonlinear, core-conductor model of action potential propagation down axisymmetric nerve fibers is adapted for an implicit, numerical simulation by computer solution of the differential equations. The calculation allows a septum to be inserted in the model fiber; the thin, passive septum is characterized by series resistance Rsz and shunt resistance Rss to the grounded bath. If Rsz is too large or Rss too small, the signal fails to propagate through the septum. Plots of the action potential profiles for various axial positions are obtained and show distortions due to the presence of the septum. A simple linear model, developed from these simulations, relates propagation delay through the septum and the preseptal risetime to Rsz and Rss. This model agrees with the simulations for a wide range of parameters and allows estimation of Rsz and Rss from measured propagation delays at the septum. Plots of the axial current as a function of both time and position demonstrate how the presence of the septum can cause prominent local reversals of the current. This result, not previously described, suggests that extracellular magnetic measurements of cellular action currents could be useful in the biophysical study of septated fibers.     

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.     

Chloride conductance of denervated gastrocnemius fibers from normal goats.

Membrane potentials, cable parameters, and component resting ionic conductances of gastrocnemius fibers from normal goats were measured in vitro at six to 32 days following denervation by section of the tibial nerve. Denervated fibers were depolarized an average of 11.6 +/- 1.5 mV (six preparations) from the control mean of 62.1 +/- 1.0 mV (124 fibers) over the period studied. Fibrillation, tetrodotoxin-resistant action potentials, and anode-break excitation were present in the denervated preparations after 13 days. The control cable parameters from 124 fibers (13 preparations) were membrane resistance, 1052 +/- 70 omega-cm2 and membrane capacitance, 6.2 muF/cm2. In denervated fibers membrane resistance increased two to three times in the 13 to 32 day period; membrane capacitance increased about 50% in normal solution at eight to nine, 27-28, and 32 days. Myoplasmic resistivity was assumed to be 112 omega-cm. Measurements were made at 38 degrees C. Component resting conductances were determined from the cable parameters in normal and chloride-free solution. Mean chloride conducantance GC1 and mean potassium conductance GK of control fibers were 776 +/- 49 mumhos/cm2 and 175 +/- 15 mumhos/cm2 (92 fibers), respectively. Following denervation GC1 increased slightly at six to nine days then fell to low values at 16 to 32 days that were close to or indistinguishable from zero. GK increased significantly to 372 +/- 40 mumhos/cm2 and 499 +/- 90 mumhos/cm2 at 16 to 20 and 32 days, respectively. It was concluded from these findings that GC1 and GK of mammalian skeletal muscle are controlled by factors from the nerve and/or muscle action potentials. Goat muscle is different from frog muscle in which GC1 does not change and GK decreases during denervation.     

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.     

A multi-modular tensegrity model of an actin stress fiber

Stress fibers are contractile bundles in the cytoskeleton that stabilize cell structure by exerting traction forces on the extracellular matrix. Individual stress fibers are molecular bundles composed of parallel actin and myosin filaments linked by various actin-binding proteins, which are organized end-on-end in a sarcomere-like pattern within an elongated three-dimensional network. While measurements of single stress fibers in living cells show that they behave like tensed viscoelastic fibers, precisely how this mechanical behavior arises from this complex supramolecular arrangement of protein components remains unclear. Here we show that computationally modeling a stress fiber as a multi-modular tensegrity network can predict several key behaviors of stress fibers measured in living cells, including viscoelastic retraction, fiber splaying after severing, non-uniform contraction, and elliptical strain of a puncture wound within the fiber. The tensegrity model can also explain how they simultaneously experience passive tension and generate active contraction forces; in contrast, a tensed cable net model predicts some, but not all, of these properties. Thus, tensegrity models may provide a useful link between molecular and cellular scale mechanical behaviors and represent a new handle on multi-scale modeling of living materials.     

The correction factors for sucrose gap measurements and their practical applications.

The distribution of extracellular and intracellular potential in the sucrose gap apparatus, previously established for a single fiber using the cable equations for a core conductor model (Jirounek and Straub, Biophys. J., 11:1, 1971), is obtained for a multifiber preparation. The exact equation is derived relating the true membrane potential change to the measured potential differences across the sucrose gap, the junction potentials between sucrose and physiological solution, the membrane potential in the sucrose region, and the electrical parameters of the preparation in each region of the sucrose gap. The extracellular potential distribution has been measured using a modified sucrose gap apparatus for the frog sciatic nerve and the rabbit vagus nerve. The results indicate a hyperpolarization of the preparations in the sucrose region, of 60--75 mV. The hyperpolarization is independent of the presence of junction potentials. The calculation of the correction terms in the equation relating the actual to the measured potential change is illustrated for the case of complete depolarization by KC1 on one side of the sucrose gap. The correction terms in the equation are given for various experimental conditions, and a number of nomographic charts are presented, by means of which the correction factors can be rapidly evaluated.     

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.     

Histological study of motor innervation to long nuclear chain intrafusal fibers in the muscle spindle of the cat

Several muscle spindles of the cat tenuissimus muscle were cut in serial, 1-micron thick transverse sections and stained with toluidine blue in search for long nuclear chain intrafusal muscle fibers. Five complete poles of the long chain fibers were located. Each fiber pole displayed one plate-type motor ending situated in the extracapsular fiber region. The endings were supplied by myelinated motor axons that originated from intramuscular nerve fascicles containing motor axons to extrafusal muscle fibers. One of the endings was innervated by a collateral from a motor axon that supplied an extrafusal end-plate. Ultrastructurally, the long chain endings resembled extrafusal end-plates. They were more complex, in terms of prominence of sole-plate and degree of post-junctional folding, than any other intrafusal ending present in the spindles. The motor endings of the long chain fibers were assumed to be the terminals of static (fast) skeletofusimotor axons, which preferentially innervate the longest nuclear chain fibers of cat muscle spindles.     

Innervation of the cat pineal gland by neuropeptide Y-immunoreactive nerve fibers: an experimental immunohistochemical study

An immunohistochemical study of the cat pineal gland was performed using a rabbit polyclonal antibody directed against neuropeptide Y (NPY) and an antibody directed against the C-terminal flanking peptide of neuropeptide Y (CPON). Numerous NPY- and CPON-immunoreactive (IR) nerve fibers were demonstrated throughout the gland and in the pineal capsule. The number of IR nerve fibers in the capsule was high and from this location fibers were observed to penetrate into the gland proper via the pineal connective tissue septa, often following the blood vessels. From the connective tissue septa IR fibers intruded into the parenchyma between the pinealocytes. Many IR nerve fibers were observed in the pineal stalk and in the habenular as well as the posterior commissural areas. The number of NPY/CPON-IR nerve fibers in pineal glands from animals bilaterally ganglionectomized two weeks before sacrifice was low. The source of most of the extrasympathetic NPY/CPONergic nerve fibers is probably the brain from where they enter the pineal via the pineal stalk. However, an origin of some of the fibers from parasympathetic ganglia cannot be excluded due to the presence of a few IR fibers in the pineal capsule of ganglionectomized animals. It is concluded that the cat pineal is richly innervated with NPYergic nerve fibers mostly of sympathetic origin. The posttranslational processing of the NPY promolecule results in the presence of both NPY and CPON in intrapineal nerve fibers.     

A new analytical method of studying post-synaptic currents.

A powerful methodology for analyzing post-synaptic currents recorded from central neurons is presented. An unknown quantity of transmitter molecules released from presynaptic terminals by electrical stimulation of nerve fibers generates a post-synaptic response at the synaptic site. The current induced at the synaptic junction is assumed to rise rapidly and decay slowly with its peak amplitude being proportional to the number of released transmitter molecules. The signal so generated is then distorted by the cable properties of the dendrite, modeled as a time-invariant, linear filter with unknown parameters. The response recorded from the cell body of the neuron following the electrical stimulation is contaminated by zero-mean, white, Gaussian noise. The parameters of the signal are then evaluated from the observation sequence using a quasi-profile likelihood estimation procedure. These parameter values are then employed to deconvolve each measured post-synaptic response to produce an optimal estimate of the transmembrane current flux. From these estimates we derive the amplitude of the synaptic current and the relative amount of transmitter molecules that elicited each response. The underlying amplitude fluctuations in the entire data sequence are investigated using a non-parametric technique based on kernel smoothing procedures. The effectiveness of the new methodology is illustrated in various simulation examples.     

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.     

Nonselective motor innervation of nuclear bag1 intrafusal muscle fibers in the cat

The motor nerve supply to cat nuclear bag1 intrafusal muscle fibers was reconstructed from light and electron microscopy of serial transverse sections of spindles in the tenuissimus muscle. Twenty-six of thirty poles of bag1 fibers that were examined received motor innervation. Every innervated bag1 pole received at least one (range 1-3) selective motor axon that supplied this fiber type only. Four of the innervated bag1 poles (15%) received additional motor supply from a nonselective motor axon that also innervated one nuclear chain fiber in the same spindle pole. The chain fibers co-innervated with bag1 fibers were among the longest chain fibers although they were shorter than two long chain fibers also present in the spindle poles. In cross-sections stained with toluidine blue they displayed 1-3 equatorial nuclei side by side, and there were fewer intermyofibrillar granules in their polar regions than in most of the other chain fibers. The endings of nonselective motor axons on the bag1 and chain fibers were morphologically and ultrastructurally dissimilar. It is suggested that instances of common innervation of the (dynamic) bag1 fiber and a (static?) chain fiber represent an integral and, presumably, functionally meaningful part of the motor pattern in some cat spindles.