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

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

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.     

Active muscle fiber as an equivalent cardiac current generator

Equations are derived describing potentials due to an active muscle fiber in an infinite medium in terms of two surface integrals—one of the propagated action potential and the other of the membrane current density, both integrals being taken over the surface of the muscle. These equations are incorporated into an equivalent cardiac current generator in which the left ventricle (i.e. the current source) is represented by a three-dimensional wedge and the thorax (i.e. the volume conductor), by a homogeneous circular cylinder. Since this current generator expresses the body surface potentials in terms of the membrane current density and the membrane potential at any point on the surface of the electrically active muscle fiber, the calculated ECG can be correlated with theactual sources within the heart. This equivalent cardiac generator possesses many of the physical and physiological properties of cardiac muscle. The equations were evaluated numerically on a digital computer. The results indicate that equivalent cardiac current generators of this type can yield clinically significant results and that further research is necessary to investigate their properties fully.     

Electrical behavior of a skeletal muscle fiber in a volume conductor of finite extent

A model study of the spatial distribution of the extracellular potentials and current densities arising from an active single skeletal muscle fiber in a cylindrical volume conductor of finite radial extent is presented. The paper examines the influence of the radius of the volume conductor,b, on the extracellular potentials and currents at different field points. The equivalent sources with respect to the extracellular potential are investigated as well. The axial source density associated with the primary and secondary sources is calculated using the expressions for the intracellular and extracellular potentials. The density of the secondary sources is a decreasing function of the radius of the conducting medium and approaches zero whenb becomes infinitely big.     

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.     

Mathematical modelling of intra- and extracellular potentials generated by active structures: effects of a step change in structure diameter

A mathematical model developed in our laboratory is used to estimate and analyse extracellular potentials generated in a volume conductor by a geometrically inhomogeneous structure with a step increase or a step decrease in its diameter. The transmembrane potentials were calculated using the model of Hodgkin and Huxley (1952) and the method of Joyner et al. (1978). Variations in waveforms of the transmembrane and extracellular potentials were described and discussed. Differences in waveforms of the extracellular potentials and in declines of their components are due to changes in the source which generates these potentials. In case of a propagation block the peak-to-peak amplitude of the extracellular potentials calculated over the area of the block may be higher than that over the area of propagation of action potentials. The possible applications of the results to the analysis of extracellular potentials recorded around actual motoneurons during their orthodromic or antidromic activation are discussed.     

Modeling Magnetomyograms of Uterine Contractions during Pregnancy Using a Multiscale Forward Electromagnetic Approach

Understanding the mechanisms of uterine contractions during pregnancy is especially important in predicting the onset of labor and thus in forecasting preterm deliveries. Preterm birth can cause serious health problems in newborns, as well as large financial burdens to society. Various techniques such as electromyography (EMG) and magnetomyography (MMG) have been developed to quantify uterine contractions. However, no widely accepted method to predict labor based on electromagnetic measurement is available. Therefore, developing a biophysical model of EMG and MMG could help better understand uterine contractions, interpret real measurements, and detect labor. In this work, we propose a multiscale realistic model of uterine contractions during pregnancy. At the cellular level, building on bifurcation theory, we apply generalized FitzHugh-Nagumo (FHN) equations that produces both plateau-type and bursting-type action potentials. At the tissue level, we introduce a random fiber orientation model applicable to an arbitrary uterine shape. We also develop an analytical expression for the propagation speed of transmembrane potential. At the organ level, a realistic volume conductor geometry model is provided based on magnetic resonance images of a pregnant woman. To simulate the measurements from the SQUID Array for Reproductive Assessment (SARA) device, we propose a sensor array model. Our model is able to reproduce the characteristics of action potentials. Additionally, we investigate the sensitivity of MMG to model configuration aspects such as volume geometry, fiber orientation, and pacemaker location. Our numerical results show that fiber orientation and pacemaker location are the key aspects that greatly affect the MMG as measured by the SARA device. We conclude that sphere is appropriate as an approximation of the volume geometry. The initial step towards validating the model against real MMG measurement is also presented. Our results show that the model is flexible to mimic the limited-propagation magnetic signature during the emergence and decay of a uterine contraction.     

Interstitial potentials and their change with depth into cardiac tissue.

The electrical source strength for an isolated, active, excitable fiber can be taken to be its transmembrane current as an excellent approximation. The transmembrane current can be determined from intracellular potentials only. But for multicellular preparations, particularly cardiac ventricular muscle, the electrical source strength may be changed significantly by the presence of the interstitial potential field. This report examines the size of the interstitial potential field as a function of depth into a semi-infinite tissue structure of cardiac muscle regarded as syncytial. A uniform propagating plane wave of excitation is assumed and the interstitial potential field is found based on consideration of the medium as a continuum (bidomain model). As a whole, the results are inconsistent with any of the limiting cases normally used to represent the volume conductor, and suggest that in only the thinnest of tissue (less than 200 micron) can the interstitial potentials be ignored.     

A distributed-parameter model of the myelinated nerve fiber

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

Modeling the interactions between stimulation and physiologically induced APs in a mammalian nerve fiber: dependence on frequency and fiber diameter

Electrical stimulation of nerve fibers is used as a therapeutic tool to treat neurophysiological disorders. Despite efforts to model the effects of stimulation, its underlying mechanisms remain unclear. Current mechanistic models quantify the effects that the electrical field produces near the fiber but do not capture interactions between action potentials (APs) initiated by stimulus and APs initiated by underlying physiological activity. In this study, we aim to quantify the effects of stimulation frequency and fiber diameter on AP interactions involving collisions and loss of excitability. We constructed a mechanistic model of a myelinated nerve fiber receiving two inputs: the underlying physiological activity at the terminal end of the fiber, and an external stimulus applied to the middle of the fiber. We define conduction reliability as the percentage of physiological APs that make it to the somatic end of the nerve fiber. At low input frequencies, conduction reliability is greater than 95% and decreases with increasing frequency due to an increase in AP interactions. Conduction reliability is less sensitive to fiber diameter and only decreases slightly with increasing fiber diameter. Finally, both the number and type of AP interactions significantly vary with both input frequencies and fiber diameter. Modeling the interactions between APs initiated by stimulus and APs initiated by underlying physiological activity in a nerve fiber opens opportunities towards understanding mechanisms of electrical stimulation therapies.     

A discrete formalism for the computation of extracellular potentials

To facilitate the computation of field potentials generated by a population of neurons a discrete formalism of the physical laws governing propagation of current in a conductive medium (volume conductor theory) is proposed. The formalism is used in combination with the compartmental model of Rall, which models the membrane activity of a neuron taking into account the electrical as well as the geometrical properties of a neuronal membrane. The direct objective is to use computer programs based on this combination in order to simulate field potentials caused by synchronous excitation of the granule cells of the fascia dentata of the hippocampus stimulated by means of the afferent fibers (perforant path) from the entorhinal cortex. To demonstrate the validity of the formalism the extracellular field of an action potential propagating in an axon has been modelled. The extracellular action potential shows a two or threephasic character which is dependent on the direction of propagation of the membrane activity.     

A Mathematical Evaluation of the Core Conductor Model

This paper is a mathematical evaluation of the core conductor model where its three dimensionality is taken into account. The problem considered is that of a single, active, unmyelinated nerve fiber situated in an extensive, homogeneous, conducting medium. Expressions for the various core conductor parameters have been derived in a mathematically rigorous manner according to the principles of electromagnetic theory. The purpose of employing mathematical rigor in this study is to bring to light the inherent assumptions of the one dimensional core conductor model, providing a method of evaluating the accuracy of this linear model. Based on the use of synthetic squid axon data, the conclusion of this study is that the linear core conductor model is a good approximation for internal but not external parameters.     

Electrical Interactions via the Extracellular Potential Near Cell Bodies

Ephaptic interactions between a neuron and axons or dendrites passing by its cell body can be, in principle, more significant than ephaptic interactions among axons in a fiber tract. Extracellular action potentials outside axons are small in amplitude and spatially spread out, while they are larger in amplitude and much more spatially confined near cell bodies. We estimated the extracellular potentials associated with an action potential in a cortical pyramidal cell using standard one-dimensional cable theory and volume conductor theory. Their spatial and temporal pattern reveal much about the location and timing of currents in the cell, especially in combination with a known morphology, and simple experiments could resolve questions about spike initiation. From the extracellular potential we compute the ephaptically induced polarization in a nearby passive cable. The magnitude of this induced voltage can be several mV, does not spread electrotonically, and depends only weakly on the passive properties of the cable. We discuss their possible functional relevance.     

Relation between inter- and intracellular action potentials of frog isolates muscle fiber at various temperatures

Extra- and intracellular action potentials (AP) of isolated muscle fibres plunged into volume conductor were studied at different temperatures. Changes of the first and second AP derivatives and their temperature dependence were described. The changes are explained by temperature effect on the density of input and output ion currents. Changes of the shape of extracellular potential with temperature increase were described. They were concerned with the changes of the first and second AP derivatives and are due to the peculiar distribution of the potential field in the volume conductor around an excitable fibre of the finite length.     

Electrical phenomena in nerve; crab nerve

The resting and action potentials of the leg nerves of the spider crab are reduced by procaine, cocaine, iodoacetate, KCl, and veratrine. The first three agents depress the sensitivity of the resting potential to anoxia, while the last can be shown to augment it. Glucose sustains activity and the polarized state in the absence of oxygen, an effect blocked by iodoacetate; corresponding concentrations of lactate and pyruvate are inert under most experimental conditions. DDT and veratrine both induce repetitive activity following an impulse, but only the latter does so with a marked increase in negative after-potential. The negative after-potential induced by veratrine is decreased by KCl relatively more than the spike or the resting potential. Elevation of the calcium content of the medium increases this after-potential. Neither ion appreciably alters the time constant of repolarization. The recovery is more rapid than that obtained following prolonged activity of both veratrinized and unveratrinized nerves. Repolarization following a tetanus is accelerated by an increase in the volume of solution in contact with the fibers; associated with this is an augmentation of the positive after-potential which normally follows a bout of activity. Yohimbine induces a positive after-potential following individual spikes which is depressed by an elevation of the potassium or calcium content of the medium. These observations are discussed from the standpoint of the available evidence for the involvement of potassium at the surface of the fibers as regulated by a labile permeability and metabolism. The potassium liberated by the action potential, calculated from the polarization changes, agrees closely with an available analytical figure; less direct observations are also found to be consistent with this view.     

Multielectrode spiral cuff for ordered and reversed activation of nerve fibres

In this paper we present the modelling, design, and experimental testing of a nerve cuff multielectrode system for selective activation of fibres in superficial peripheral nerve trunk regions which is capable of activating fibres in physiological order. The multielectrode system consists of 45 platinum electrodes embedded within a self-curling spiral silicone sheet organized in fifteen longitudinal groups consisting of three electrodes spaced equidistally around the circumference of the cuff. Electrodes in the centre band acted as stimulating cathodes while the two electrodes of the same group in the two outer bands were connected together and corresponded to the position of a particular cathode, serving as anodes to block the nascent action potentials by membrane hyperpolarization. The interpolar distance was 6 mm on both sides, resulting in a total cuff length of about 20 mm. The cuff was constructed with a diameter to fit the size of the dog sciatic nerve. Preliminary animal testing of the nerve cuff was performed on the sciatic nerve of a Bigley female dog. In the 45-electrode stimulation system, biphasic cathodic first pulses with quasitrapezoidal-shaped cathodic and square anodic parts were delivered through the particular group of tripolar electrodes to effect both selective stimulation of motor axons within the gastrocnemius muscle fascicle, and differential block by membrane hyperpolarization. The test was repeated using rectangular cathodic first biphasic current pulses delivered monopolarly on the central electrode of the same group while connected anodes were replaced by a common anode situated elsewhere in the surrounding tissue. In both experiments an isometric torque in the ankle joint elicited by the gastrocnemius muscle was measured and compared. It was shown that tripolar activation with quasitrapezoidal stimulation pulses elicited an isometric torque with a peak value of 0.83 Nm in 65 ms after onset of delivering stimulating pulses in comparison to the monopolar activation with rectangular biphasic pulses where the peak of the same value was observed in 45 ms after onset. Thus, the multipolar cuff stimulating monopolarly provided an effective means of activating motor axons selectively within the gastrocnemius muscle fascicle, while more physiological recruitment of the muscle fibres was evident when stimulating tripolarly.     

The origin of the human auditory brain-stem response wave II

Auditory brain-stem responses (ABRs) were recorded from human subjects undergoing neurosurgical procedures which exposed the auditory nerve. Scalp recordings indicated that the latency of the negativity between waves (I_n) and II (I_n) and the latency of positive peak II (II_p) were shorter when the nerve was suspended in air than when the nerve was submerged in cerebrospinal fluid or saline, while earlier and later waves remained unaffected. These results could not be attributed to changes in stimulus or recording parameters or conduction velocity. Computational and somatosensory experimental evidence of stationary potentials generated by physical properties of the volume conductor, including changes in conductivity or geometry, are presented to develop a model of wave II_p generation. The results of this study suggest that wave II_p (and probably I_n) are manifestations of current flux asymmetries across conductivity boundaries created by the temporal bone-cerebrospinal fluid intradural space-brain-stem interfaces. The current flux asymmetries are generated as the propagating auditory nerve action potential crosses the conductivity boundaries. These results also indicate that the physical characteristics of the volume conductor and neural pathways must be considered when interpreting surface recorded evoked potentials.