Core Conductor Model Research Articles

Positive sharp wave and fibrillation potential modeling.

A finite muscle fiber simulation program which calculates the extracellular potential for any given intracellular action potential (IAP) was used to model a fibrillation potential and a positive sharp wave. This computer model employs the core conductor model assumptions for an active muscle fiber and allows two distinct types of end effects: a cut or a crush. A "cut end" is defined as a membrane segment with the termination of both active and passive ion channels. The "crush end" is simulated as a focal membrane segment which blocks action potential propagation, and is connected to a region of normal membrane on either side of it so that a normal transmembrane potential is maintained beyond the crush zone. A prototypical positive sharp wave of appropriate amplitude and duration could only be detected extracellularly by using an IAP of the configuration found in denervated rat muscle recorded from a muscle fiber terminating in a crush segment of membrane.

Origin of the apparent tissue conductivity in the molecular and granular layers of the in vitro turtle cerebellum and the interpretation of current source-density analysis.

1. We determined the origin of the apparent tissue conductivity (sigma 2) of the turtle cerebellum in vitro. 2. Application of a current with a known current density (J) along the longitudinal axis of a conductivity cell produced an electric field in the cerebellum suspended in the cell. The measured electric field (E) perpendicular to the cerebellar surface indicated a significant inhomogeneity in sigma a (= J/E) with a major discontinuity between the molecular layer (0.25 +/- 0.05 S/m, mean +/- SD) and granular layers (0.15 +/- 0.03 S/m) (n = 39). 3. This inhomogeneity was more pronounced after anoxic depolarization. The value of sigma a decreased to 0.11 +/- 0.03 and 0.040 +/- 0.008 S/m in the molecular and granular layers, respectively. The ratio of sigma a S in the two layers increased from 1.67 in the normoxic condition to 2.75 after anoxic depolarization. 4. This difference in sigma a across the two layers was present within the range of frequencies (DC to 10 kHz) studied where the phase of sigma a was small (less than +/- 2 degrees) and therefore sigma a was ohmic. 5. The inhomogeneity in sigma a was in part due to an inhomogeneity in the extracellular conductivity (sigma e) as determined from the extracellular diffusion of ionophoresed tetramethylammonium. Like sigma a, the value of sigma e was also higher in the molecular layer (0.165 S/m) than in the granular layer (0.097 S/m). The inhomogeneity in sigma e was due to a smaller tortuosity and a larger extracellular volume fraction in the molecular layer compared with the granular layer. 6. sigma a was, however, consistently higher, by approximately 50%, than sigma e. A core conductor model of the cerebellum indicated that these discrepancies between sigma a and sigma e were attributable to additional conductivity produced by a passage of the longitudinal applied current through the intracellular space of Purkinje cells and ependymal glial cells, with the glial compartment playing the dominant role. Cells with a long process and a short space constant such as the ependymal glia evidently enhance the effective "extracellular" conductivity by serving as intracellular conduits for the applied current. The result implies that the effective sigma e may be larger than sigma e for neuronally generated currents in the turtle cerebellum because the space constant for Purkinje cells is several times greater than that for the ependymal glia and consequently Purkinje cell-generated currents travel over a long distance relative to the space constant of glial cells.(ABSTRACT TRUNCATED AT 400 WORDS)

TUTORIAL ON NEUROBIOLOGY: FROM SINGLE NEURONS TO BRAIN CHAOS

Those classical models are reviewed that are most widely used by neurobiologists to explain the dynamics of neurons and neuron populations, and by modelers to implement artificial neural networks. Each neuron has input fibers called dendrites that integrate and an axon that transmits the output. The differing fiber architectures reflect these dissimilar dynamic operations. The basic tools to describe them are the RC model of the membrane, the core conductor model of the fibers, the Hodgkin–Huxley model of the trigger zone, and the modifiable synapse. Populations additionally require description of macroscopic state variables, the types of nonlinearity (most importantly the sigmoid curve and the dynamic range compression at the input to the cortex), and the types and strengths of connections. The properties of these neural masses can be characterized with the tools of nonlinear dynamics. These include description of point, limit cycle, and chaotic attractors for the cerebal cortex, as well as the types and mechanisms for the state transitions between basins of attraction during learning and perception.

On the relation between axial resistance and conductivity in linear cable models

In the classical core conductor model the intra- and extracellular media of an infinitely long nerve or muscle fiber in a volume conductor are represented by parallel axial resistances. Thus, volume conductor properties are neglected. In this paper a study is presented that relates the conductivity of the intra- or extracellular medium of the fiber to the axial resistance used in the core conductor model. Expressions for the axial resistances that are dependent on the spatial frequency of the membrane current source density are derived from a volume conductor approach.

Magnetic field of a single muscle fiber. First measurements and a core conductor model

We present the first measurements of the magnetic field from a single muscle fiber of the frog gastrocnemius, obtained by using a toroidal pickup coil coupled to a room-temperature, low-noise amplifier. The axial currents associated with the magnetic fields of single fibers were biphasic and had peak-to-peak amplitudes ranging between 50 and 100 nA, depending primarily on the fiber radius. With an intracellular microelectrode, we measured the action potential of the same fiber, which allowed us to determine that the intracellular conductivity of the muscle fiber in the core conductor approximation was 0.20 +/- 0.09 S/m. Similarly, we found that the effective membrane capacitance was 0.030 +/- 0.011 F/m2. These results were not significantly affected by the anisotropic conductivity of the muscle bundle. We demonstrate how our magnetic technique can be used to determine the transmembrane action potential without penetrating the membrane with a microelectrode, thereby offering a reliable, stable, and atraumatic method for studying contracting muscle fibers.

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.

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.

The Closed Forn Solution to the Periodic Core-Conductor Model Using Asymptotic Analysis

The distribution of the transmembrane potential along an infinite strand of cardiac cells generated by a point source under steady-state conditions has been calculated using the asymptotic analysis method. With the intracellular conductivity changing periodically in space, the problem can be treated as dependent upon two variables: the large scale variable x covering the whole strand, and the small scale variable y defined on the unit cell. The solution is given as a two-scale expansion in powers of the period length. Each term of the expansion can be determined by solving the differential equations derived by decomposing the original problem. These equations do not have to be solved simultaneously; moreover, the linearity of the problem allows the separation of the x and y dependence in the higher order terms. The series converges quickly, and for all practical purposes, the solution containing zero-, first-, and second-order terms has a negligible truncation error. The subsequent terms of the solution have the following physiological interpretation: The zero-order term is the solution to the classical core-conductor model obtained by the homogenization of the periodic model, the first-order term acts as the dipole sources located at junctions, and finally, the second-order term resembles the monopole sources arising at junctions.

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.

A Two-Part Model for Determining the Electromagnetic and Physiologic Behavior of Cuff Electrode Nerve Stimulators

A two-part model for determining the electrophysiological behavior of a cuff electrode designed for unidirectional nerve stimulation is presented. In the first part, a model is described which evaluates the potential field of the cuff electrode system in the absence of a nerve. The model recognizes boundary conditions through the specification of secondary sources on the cuff; the unknown source strengths are found through the solution of a Fredholm integral equation. In the second part of the study, the potential field derived for the cuff is taken to be the applied field of a myelinated nerve fiber considered through a linear core-conductor model. The simulation confirms the existence of a window of the unidirectional stimulation, a condition that must exist to implement a collision block. The result corresponds to that obtained experimentally by van den Honert and Mortimer [11], [12], and suggests the applicability of such models to determine optimum cuff electrode designs.

Magnetic measurements of action currents in a single nerve axon: a core-conductor model.

Measurements have been performed on the medial giant axon of the crayfish in which both microelectrode recordings of the transmembrane action potential and magnetic recordings of the axial, intracellular action current were obtained at a single location along the nerve. This approach is unique because the combination of electric and magnetic techniques allows for independent measurements of transmembrane potential and intracellular current without assumptions regarding membrane capacitance or axonal resistances. The availability of both types of data recorded at a single location on the axon allows the core-conductor model to be solved explicitly for the internal axial resistance of the axon, the membrane capacitance, and the membrane conduction current density without the need to make a series of subthreshold measurements of passive cable properties. The extracellular magnetic measurements can be used both to determine the transmembrane action potential without the need to penetrate the nerve membrane with a microelectrode, and to obtain approximate values for the sodium and potassium peak conductances.

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.

A physico-mathematical analysis of elliptical nerve and muscle fibres

In the framework of the tridimensional core conductor model, the current flow field of an elliptical nerve or muscle fibre in a volume conductor is studied. As the quasi-static conditions are valid, the Laplace equation applies. Expressions for the intracellular and extra cellular potential fields and the membrane current are exactly derived. As a limit the solutions for the circular case are recovered. Finally a sketch of an approximate method of calculation is outlined and the first elliptical correction to the usual membrane current is evaluated.

A core-conductor model of the cardiac Purkinje fibre based on structural analysis.

1. Structural analysis of voltage clamp preparations of sheep cardiac Purkinje fibres was carried out using methods based on light and electron microscopic observations. Results demonstrate the marked structural variability in preparations that appear, outwardly, simple.2. The length of the cell aggregate was measured in vitro, and the mean area of cross-section, by light microscopic methods in serially sampled transverse sections. The number of intercellular clefts and path lengths of cell profiles distributed at the lateral surface and along clefts were determined from photomicrographs. Accuracy of these estimates was improved by obtaining, electron microscopically, values representing the degree of membrane folding in longitudinal and transverse planes.3. Cleft width was evaluated from electron micrographs. Measurements made on sections that there tilted through wide arcs with a goniometer indicate that cleft width is quite variable and, on average, somewhat greater than 400 A.4. A three-dimensional core-conductor model is presented to aid in quantitative interpretation of electrophysiological experiments. Application of the model to individual preparations requires evaluation of the length and perimeter of cross-section of cell aggregates and of the mean number, width and depth of intercellular clefts, with appropriate corrections for fine sarcolemmal folding. Methods are given for estimating these structural parameter values from measurements of external dimensions of cell aggregates.5. The core-conductor model is applied in an accompanying analysis of the linear electrical characteristics of Purkinje membrane.

Linear analysis of membrane conductance and capacitance in cardiac Purkinje fibres.

1. The membrane capacitance and slope conductance of sheep cardiac Purkinje tissue were examined under conditions of constant and slowly changing voltage within the potential range: -70 to -100 mV. Data were secured by applying constant current pulses, or by imposing ramp commands or small changes in holding potential under voltage clamp control.2. The electrical measurements, combined with light and electron microscopic data on preparation structure, were interpreted with the aid of a core-conductor model that permits quantitative representation of cleft- and lateral surface membranes for individual preparations (Hellam & Studt, 1974).3. The analysis led to the following conclusions. The slope conductance of unit membrane area (G(m)) is quite low in the normal resting potential range; at -80 mV the value is 0.09 mmho/cm(2). G(m) exhibits a marked decline with depolarization in the region positive to -88 mV, but only slight voltage-dependence at more negative potentials within the range of present observations. The capacitance of unit membrane area (C(m)), from voltage ramp data collected near -90 mV, is 0.9 muF/cm(2).4. Comparison with results of earlier studies indicates that the differences between present and previous experimental estimates of these membrane characteristics result from differences in modelling and structural analysis of tissue preparations.

Muscle spindle receptors

The depolarization of the sensory terminals of muscle spindle primary endings is studied in terms of a simplified core conductor model of the system. The terminals branching from a group Ia afferent fibre have different geometrical structures, depending on whether they innervate the nuclear bag or the nuclear chain intrafusal fibres of a muscle spindle.

Passive Membrane Potentials: A Generalization of the Theory of Electrotonus

THE THEORY OF ELECTROTONUS, WHICH HAS BEEN WELL DEVELOPED FOR SMALL CYLINDERS, IS EXTENDED: the fundamental potential equations for a membrane of arbitrary shape are derived, and solutions are found for cylindrical and spherical geometries. If two purely conductive media are separated by a resistance-capacitance membrane, then Laplace's equation describes the potential in either medium, and two boundary equations relate the transmembrane potential to applied currents and to currents flowing into the membrane from each medium. The core conductor model, on which most previous work on cylindrical electrotonus has been based, gives rise to a one dimensional diffusion equation, the cable equation, for the transmembrane potential in a small cylinder. Under the assumptions of the core conductor model the more general equations developed here are shown to reduce to the cable equation. The two theories agree well in predicting the transmembrane potential in a small cylinder owing to an applied current step, and the extracellular potential for this cylinder is estimated numerically from the general theory. A detailed proof is given for the isopotentiality of a spherical soma membrane.

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.

Analysis of Certain Errors in Squid Axon Voltage Clamp Measurements

Localized membrane current and potential measurements were made on the squid giant axon in voltage clamp experiments. Spatial control of potential was impaired by the use of axial current supplying electrodes with surface resistance greater than 20 ohms for a centimeter length of axon. No region of membrane which was indeed subjected to a potential step showed more than one inward current peak. Other patterns were results of space clamp failure. Membrane current and potential patterns during space clamp failure were approximately reproduced in computations on a model containing two membrane patches obeying the equations of Hodgkin and Huxley. Non-uniformities in the axon or electrodes are not necessary for non-uniform electrical behavior. An extension of the core conductor model which includes the axial wire and external solution has been analyzed. The space constant of electrotonic spread is less than 0.5 mm with a usable electrode. Errors of about 5 per cent are introduced by ignoring the external solution. Resistance between the membrane and the control electrodes reduces the control and a few ohm cm(2) could lead to serious errors in interpretation.

The distribution of membrane current in nerve with longitudinal linearly increasing applied current

The distribution of membrane current in three models of nerve, when a longitudinal, linearly increasing current is applied, is derived. For the simple core conductor model it is shown that, if the region over which such a current is applied is large compared to the space constant of the model, the membrane current in the mid-portion of the region is a constant, independent of the distance, the time following the application of the current, and the impedance of the membrane. The effect of nonlinear membrane electrical properties is discussed. It is further shown that these conclusions apply equally to the case in which the simple model is surrounded by another concentric sheath (the double cable model). In this case the impedance of the sheath does not influence the membrane current in the mid-polar region. Finally, it is shown that the form of the solution for the saltatory model, for this type of applied current, is identical with that for the simple model.