Eccentric multipole representations of current generators in a spherical volume conductor

In Yeh, Martinek and de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958), a method is presented for determining successively better central multipole representations of the current generators in a homogeneous conducting sphere by measuring surface potentials at a successively increasing number of points. This paper generalizes the method such that the multipoles may be located at any chosen point in the conductor. The spherical harmonic expansion is advantageously used and the “interior sphere theorem” of Ludford, Martinek and Yeh (Proc. Cambridge Philos. Soc.,51, 389–93, 1955) makes possible disturbance potential expressions in closed forms. A method for approximate determination of the eccentricity is also presented. In the theory of electrocardiography, the eccentric multipoles can more accurately represent the heart as a current generator with fewer surface potential measurements than the central multipoles. This investigation was supported by The National Heart Institute under Research Grant H-2263(c-4).     

Multipole representation of an eccentric dipole and an eccentric double-layer

Recently a theorem for representing current generators in a volume conductor by the superposition of a central dipole, quadrupole, octopole, etc., has been established by G. C. K. Yeh, J. Martinek, and H. de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958). This theorem makes possible the representation of any discrete or line, surface- or volume-distributed current source by a unique model which can be determined for each given case by surface potential measurements and closed form analysis. In this paper the multipole representations of an eccentric dipole and an eccentric double-layer are obtained in terms of the various parameters of the assumed singularities, and the contributions to surface potentials due to each of the multipoles are compared. Certain numerical results corresponding to those of E. Frank (Amer. Heart J.,46, 364–78, 1953) are carried out and compared. Furthermore, the multipole representation of a partially damaged double-layer is also determined and compared with that of an undamaged one. It is concluded that within the range of parameters corresponding to human subjects the higher-order multipoles can contribute significantly to the surface potentials compared with the dipole. This investigation was supported by the National Heart Institute under a research grant H-2263(c).     

The inconsistencies of present-day mathematical-physical methods pertaining to current generators in volume conductors used in electrocardiography

The present-day practices of electrocardiography and vectorardiography are based upon the theory that the surface potential differences can be assumed to be due to a single dipole inside the body. It is shown in this paper that a dipole cannot account for all the surface potentials due to realistic current generators, and hence the determination of the current generator from surface potential measurements based upon such a theory will lead to inconsistent representations of the heart for one and the same subject. To demonstrate this point two eccentric dipoles of different strengths and locations representing two muscle fibers are taken to be the current generator in a homogeneous spherical conductor. The exact surface potentials are then expressed by means of the “interior sphere theorem” of the authors. With these expressions the magnitude, direction, and location of the resultant dipole are determined by the method of D. Gabor and C. V. Nelson (J. App. Physics,25, 413–16, 1954). The surface potentials due to this resultant dipole are again exactly expressed by means of the “interior sphere theorem” and compared with those due to the eccentric dipoles assumed. It can be seen that the differences can be considerable. It is suggested that the multipole model of the authors (Bull. Math. Biophysics,20, 203–16, 1958) be used as a more accurate and the only unique representation of the heart. This investigation was supported by the National Heart Institute under a research grant H-2263(c).     

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.     

Comparison of surface potentials due to several singularity representations of the human heart

In electrocardiography the electrical potentials due to the heart actions can be analyzed by assuming the human body to be a conductor of homogeneous medium and the heart to be a combination of singularities within it. For a spherical conductor the “interior sphere theorem” of G. Ludford, J. Martinek, and G. Yeh (Proc. Cambridge Phil. Soc.,51, 389–93, 1955) renders potential expressions due to any singularity. For a conductor of prolate spheroidal shape the potential expressions due to a source-sink pair and a general dipole have been given by J. R. Wait (Jour. App. Physics,24, 496–97, 1953) and the authors (paper at the Conference on the Electrophysiology of the Heart, Feb. 16–17, 1956, in New York, to appear in theAnn. N. Y. Acad. Sciences) respectively. (A theorem which applies to any singularity inside a prolate or oblate spheroid will be published by the authors soon). This paper presents numerical and graphical results of potentials on the surfaces of a prolate spheroid and a sphere due to source-sink pairs and dipoles of several locations and directions and compares the various representations. A discussion on the judicious choice of heart models concludes the paper. This investigation was supported by The National Heart Institute under a research grant H-2263.     

On Bioelectric Potentials in an Inhomogeneous Volume Conductor

Green's theorem is used to derive two sets of expressions for the quasi-static potential distribution in an inhomogeneous volume conductor. The current density in passive regions is assumed to be linearly related instantaneously to the electric field. Two equations are derived relating potentials to an arbitrary distribution of impressed currents. In one, surfaces of discontinuity in electrical conductivity are replaced by double layers and in the other, by surface charges. A multipole equivalent generator is defined and related both to the potential distribution on the outer surface of the volume conductor and to the current sources. An alternative result involves the electric field at the outer surface rather than the potential. Finally, the impressed currents are related to electrical activity at the membranes of active cells. The normal component of membrane current density is assumed to be equal at both membrane surfaces. One expression is obtained involving the potentials at the inner and outer surfaces of the membrane. A second expression involves the transmembrane potential and the normal component of membrane current.     

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.     

On multipole representation of current generators

In Yeh, Martinek, and de Beaumont (Bull. Math. Biophysics,20, 203–216, 1958), multipole representations of current generators in a volume conductor are used, based upon the Taylor series expansion of the potential function. In Yeh, (Bull. Math. Biophysics,23, 263–276, 1961) multipole representations of current generators in a spherical volume conductor are used, based upon the spherical harmonic expansion. This paper correlates these two systems of multipole representations so that formulations in terms of one system of the representations may be readily transformed into formulations in terms of the other system. Since the Taylor series representation is more graphic, whereas the spherical harmonic representation is more compact, such a transformation between these two systems of formulations can serve useful purposes in the application of the theory of electrocardiography. This investigation was supported by the National Heart Institute under Research Grant H-2263 (c-4).     

Physiology and physiopathology of somatic sensations studied in man by the method of evoked potentials

The physiology of somatic sensation can be investigated noninvasively in man by recording the electric activity of peripheral nerves, spinal cord and brain. Since these responses have a small voltage, it is necessary to use electronic averaging methods for improving the signal-to-noise ratio. These methods are described and discussed, as well as principles of interpretation of somatosensory evoked potentials. It is agreed that the traces thus obtained involve a series of components (extracellular potentials) which reflect distinct neural generators. These generators have been identified and localized at different levels of the subcortical somatosensory pathway and in different cortical areas. Several components reflect generators located under the recording electrodes (nearfield potentials), while other reflect extracellular potentials diffusing at a distance in the volume conductor of the neck and head (farfield potentials). The analysis of these components provides a wealth of new data for the physiology and pathophysiology of the somatic sensory system in man. Besides so-called "obligatory" components that are present irrespective of the attention of the subject, the studies have uncovered "cognitive" components which reflect neural mechanisms involved in the intellectual processus of perception and decision.     

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.     

Disentanglement of local field potential sources by independent component analysis

The spontaneous activity of working neurons yields synaptic currents that mix up in the volume conductor. This activity is picked up by intracerebral recording electrodes as local field potentials (LFPs), but their separation into original informative sources is an unresolved problem. Assuming that synaptic currents have stationary placing we implemented independent component model for blind source separation of LFPs in the hippocampal CA1 region. After suppressing contaminating sources from adjacent regions we obtained three main local LFP generators. The specificity of the information contained in isolated generators is much higher than in raw potentials as revealed by stronger phase-spike correlation with local putative interneurons. The spatial distribution of the population synaptic input corresponding to each isolated generator was disclosed by current-source density analysis of spatial weights. The found generators match with axonal terminal fields from subtypes of local interneurons and associational fibers from nearby subfields. The found distributions of synaptic currents were employed in a computational model to reconstruct spontaneous LFPs. The phase-spike correlations of simulated units and LFPs show laminar dependency that reflects the nature and magnitude of the synaptic currents in the targeted pyramidal cells. We propose that each isolated generator captures the synaptic activity driven by a different neuron subpopulation. This offers experimentally justified model of local circuits creating extracellular potential, which involves distinct neuron subtypes.     

Characteristics of spinal cord-evoked responses in man

The averaged electrical potentials evoked by the stimulation of the peripheral nerves were recorded with surface electrodes over the lumbosacral, lower thoracic and cervical spine and with epidurally placed electrodes in the cervical area. The waveforms of the lumbosacral and cervical spinal cord potentials show similar complexity reflecting peripheral and central generators. The larger negative wave with at least two components is followed by a slower positive deflection. Evoked potentials recorded over the cervical segments of the spinal cord with epidural electrodes are of much higher amplitude and more complex waveform than those recorded with surface electrodes.     

Simultaneous analysis of the cardiac electric and magnetic fields using the scalar multipole expansion

A theoretical analysis of electric and magnetic fields of the heart, based solely upon the scalar multipole expansion, is carried out in order to gain an insight into the interrelation of the data contained in electro- and magnetocardiological measurements. The usual multipole expansion is applied for the electric field, however corresp[nding equivalent multipoles are formulated as idealized generators, having not only flow sources, but also vortex sources of the field. Furthermore, the magnetic field in a homogeneous infinite volume conductor is expressed as a sum of two series, the first being the usual multipole expansion of the nonvortex component of the magnetic field, and the second being a sequence of magnetic fields set up by the aforementioned electric multipole generators reduced to axial form. The former term is uniquely defined by the electric multipole components, but the latter reflects properties of the cardiogenerator that can be revealed only by means of magnetic measurements. Features of the electric and magnetic multipole components as integral characteristics of the cardiogenerator are discussed and concepts of the magnetic centre and magnetic axis of the cardiogenerator are proposed. The analysis is illustrated by examples of simple generator configurations.     

Separate generators with distinct orientations for N20 and P22 somatosensory evoked potentials to finger stimulation?

Sequential spatial maps of scalp potentials, obtained with a 16-channel montage, were used in 12 healthy subjects in order to assess the temporal and spatial distribution of early cortical SEPs to single finger stimulation. It was found that when the contralateral parietal N20 negativity peaks there is a synchronous frontal P20 positivity, supporting the view of a tangentially orientated dipolar generator for this couple of scalp SEPs components. It was not possible to show a distribution of N20 peak on the scalp that would parallel the somatotopic finger representations in area S1; however, the orientation of the putative dipolar source of the N20/P20 complex was found to change according to the finger stimulated. A central P22 component was also constantly obtained without any synchronous negativity on the scalp surface corresponding to the electrode array; a clear somatotopic organisation was found for P22. These features favour the hypothesis that this latter component has a radially orientated generator situated in the prerolandic motor cortex, close to the scalp surface. Because of overlapping between the P20 and P22 components, the determination of P22 onset latency was hazardous in some cases, and spatial mapping was then essential to identify this component. The conclusion that the contralateral parietal N20 and central P22 could be generated by separate dipolar generators with distinct orientations is supported by recent data from combined electrical and magnetic field recording.     

Spherical classification of wavelet transformed EMG intensity patterns

Electromyograms of different muscles can be submitted to a wavelet-transform and arranged in a Multi-Muscle Pattern (MMP). The MMP represents the intensity of the EMG signals of a number of muscles simultaneously in time/frequency space. As previously shown, the MMPs can be represented by points in an Euclidian vector space that was called pattern space. The variability of the MMPs is represented by the distribution of the scattered points in pattern space. The purpose of this study was to investigate the distribution of the points and use the properties of the distribution to classify MMPs. The first task was to test whether the points representing a group of MMPs were located between the inner and outer boundary of a sphere-like domain in whitened pattern space as theoretically predicted. The mean of these points and thus of the MMPs is represented by a point at the center of the sphere. The hypothesis was that the spheres representing points of the MMPs of barefoot and shod runners were sufficiently separated and distinguishable in pattern space to allow classification of the runners according to their shod condition. The results confirmed the hypothesis and revealed that the recognition rate was over 80%. One can conclude and generalize that the points representing MMPs recorded for a certain condition reside between the inner and outer boundary of the sphere. The classification based on the spherical feature represents a much better discrimination than one based on the distance from the mean.     

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     

Sheet conductor model of brain slices for stimulation and recording with planar electronic contacts

Current and voltage in a brain slice are considered, taking into account the boundary conditions at the surface to an electrolyte bath and at the substrate of an electron conductor. A sheet conductor model is introduced with ohmic leak conductance to the bath and capacitive coupling to the substrate. It assigns a current-source density of neuronal activity to extracellular field potentials recorded by planar contacts, and it relates the current of planar capacitive contacts to the field potential that elicits neuronal activity. Two examples are analytically solved: the recording across a layered brain slice and the stimulation by a circular electrode. The study forms the basis for neurophysical experiments with brain slices or retinae on microelectronic chips.     

Three-dimensional organization of uninfected tissue in soybean root nodules and its relation to cell specialization in the central region

Summary The three-dimensional structure of uninfected tissue in the central infected region of soybean nodules has been studied using several methods. Transverse and longitudinal sections have been examined. A view of the surface of the central region has been obtained by partially clearing nodules and removing their cortex. A three-dimensional reconstruction of the central region has been assembled using transverse sections. It was found that some of the uninfected cells participate in forming characteristic structural aggregates. A centrally located spherical region with a diameter half that of the entire infected region is filled with interconnected aggregates of uninfected cells. A set of branching interconnected planes of uninfected cells emanate from the sphere and extend out to the surface of the infected region. The planes are oriented generally in a longitudinal direction but are sometimes at other angles. These planes divide the nodule into compartments of various sizes and shapes. The planes are perforated by irregularly shaped groups of infected cells. Uninfected cells also often occur arranged in lines oriented approximately radially in the central region. Irregularly shaped fine aggregates of uninfected cells occur outside the central sphere formation and are interconnected by narrow lines of uninfected cells. All of these types of formations could provide uninfected paths all or part of the way from the center of the central region to the cortex. The planes and lines often contain uninfected cells elongated perpendicular to the surface of the central region, suggesting that the routes may serve in the transport of substances from the inside to the surface of the central region. The distribution of plasmodesmata also appears to favor transport from one uninfected cell to another.Abbreviation TER tubular endoplasmic reticulum     

Extracellular potentials of single active muscle fibres: Effects of finite fibre length

The extracellular action potentials (ECAPs) of single active muscle fibres immersed an isotropic volume conductor were investigated. The origination of excitation in the motor end-plate and its reaching the fibre end were taken into consideration. It was explained why at short radial distances the ECAPs over the fibre at points close to the end were similar in shape to the first time derivative and at points close to the motor end-plate-to the first time derivative of the intracellular action potential (ICAP) taken with minus sign. The fibre end changed the ECAP which would be recorded if the fibre was infinite and this change called pure termination potential (PTP) was a biphase positive-negative potential, proportional to the first time derivative of the ICAP at points close to the membrane and over the very end. With increasing the radial and axial distances PTP decreases in amplitude. Taking into account the PTP, the genesis of the terminal positive phase of the ECAPs (Gydikov and Kosarov 1972a, b) can be explained. The onset of the fibre or the motor end-plate also changed the potential which would be recorded if the fibre was infinite. This change was given the term of pure onset potential (POP)-a biphase negative-positive potential, proportional to the first time derivative of the ICAP taken with minus sign at a point close to the membrane and over the motor end-plate. With increasing the radial and the axial distance POP decreased in amplitude.Close to the membrane PTP and POP were commensurable with the potential of an infinite fibre only at points close to the ends or to the motor end-plate. At long radial distances they were commensurable with the potential of an infinite fibre for all axial distances.     

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.