Model Studies of the Magnetocardiogram

A general expression is developed for the quasi-static magnetic field outside an inhomogeneous nonmagnetic volume conductor containing internal electromotive forces. Multipole expansions for both the electric and magnetic fields are derived. It is shown that the external magnetic field vanishes under conditions of axial symmetry. The magnetic field for a dipole current source in a sphere is derived, and the effect of an eccentric spherical inhomogeneity is analyzed. Finally the magnetic dipole moment is calculated for a current dipole in a conducting prolate spheroid.     

Brain stimulation using electromagnetic sources: theoretical aspects.

We prove that, at the frequencies generally proposed for extracranial stimulation of the brain, it is not possible, using any superposition of external current sources, to produce a three-dimensional local maximum of the electric field strength inside the brain. The maximum always occurs on a boundary where the conductivity jumps in value. Nevertheless, it may be possible to achieve greater two-dimensional focusing and shaping of the electric field than is currently available. Towards this goal we have used the reciprocity theorem to present a uniform treatment of the electric field inside a conducting medium produced by a variety of sources: an external magnetic dipole (current loop), an external electric dipole (linear antenna), and surface and depth electrodes. This formulation makes use of the lead fields from magneto- and electroencephalography. For the special case of a system with spherically symmetric conductivity, we derive a simple analytic formula for the electric field due to an external magnetic dipole. This formula is independent of the conductivity profile and therefore embraces spherical models with any number of shells. This explains the "insensitivity" to the skull's conductivity that has been described in numerical studies. We also present analytic formulas for the electric field due to an electric dipole, and also surface and depth electrodes, for the case of a sphere of constant conductivity.     

A note on dipole sources in conducting biological tissues

The study of current flow fields in biologicaltissue requires finding the potential field from dipole sources such that the normal gradient vanishes at the exterior surface. A convenient way to determine the dipole field is by taking the gradient of the potential field set up by a point source. However, the point source problem is ill-posed when the normal gradient is required to vanish at the outer surface. In the paper the nature of this problem is discussed and several methods for overcoming the difficulty are presented.     

EEG and implanted sources in the brain

Localisation procedures are based on models of the EEG that are relatively simple. The models are based on assumptions and choices of parameters that can be mistaken. Thus, it is crucial to validate the localisation procedures used in EEG. One of the options is to use the data obtained with electrodes that are implanted within the brain of an epileptic patient as part of the pre-surgical evaluation. When one of two neighbouring electrodes is used as a current source and the other as a current sink this can be regarded as a current dipole. The current injected has to be below the threshold for activation of cells. The position of this dipole can be deduced from magnetic resonance or X-ray images. The current dipole gives rise to a potential distribution at the scalp that can be measured by EEG. The measurements can be compared with the potential distribution that is calculated in a forward computation. Another method is to use the measured potential at the scalp to localize the source and to compare the result with the actual position of the dipole. In this paper the measured potential distributions at the scalp due to implanted dipoles were used to evaluate different volume conductor models. Since intracerebral and subdural electrodes were introduced through trephine holes over the fronto-central areas, and the diameter of the holes was rather large, approximately 23 mm, special effort was put into modelling the skull. Two important assumptions could be validated in this study: the electric currents within the head are Ohmic and a dipole can be used to model the induced electric activity of pairs of contacts on subdural electrodes or intra cerebral electrodes.     

Dipole characterization of single neurons from their extracellular action potentials

The spatial variation of the extracellular action potentials (EAP) of a single neuron contains information about the size and location of the dominant current source of its action potential generator, which is typically in the vicinity of the soma. Using this dependence in reverse in a three-component realistic probe + brain + source model, we solved the inverse problem of characterizing the equivalent current source of an isolated neuron from the EAP data sampled by an extracellular probe at multiple independent recording locations. We used a dipole for the model source because there is extensive evidence it accurately captures the spatial roll-off of the EAP amplitude, and because, as we show, dipole localization, beyond a minimum cell-probe distance, is a more accurate alternative to approaches based on monopole source models. Dipole characterization is separable into a linear dipole moment optimization where the dipole location is fixed, and a second, nonlinear, global optimization of the source location. We solved the linear optimization on a discrete grid via the lead fields of the probe, which can be calculated for any realistic probe + brain model by the finite element method. The global source location was optimized by means of Tikhonov regularization that jointly minimizes model error and dipole size. The particular strategy chosen reflects the fact that the dipole model is used in the near field, in contrast to the typical prior applications of dipole models to EKG and EEG source analysis. We applied dipole localization to data collected with stepped tetrodes whose detailed geometry was measured via scanning electron microscopy. The optimal dipole could account for 96% of the power in the spatial variation of the EAP amplitude. Among various model error contributions to the residual, we address especially the error in probe geometry, and the extent to which it biases estimates of dipole parameters. This dipole characterization method can be applied to any recording technique that has the capabilities of taking multiple independent measurements of the same single units.     

基于模拟退火法由脑磁图推测电流偶极子参数

利用模拟退火（Ｓｉｍｕｌａｔｅｄ Ａｎｎｅａｌｉｎｇ） 算法,由脑磁图（ ＭＥＧ） 数据反演脑内作为磁源的单电流偶极子参数,可以得到理想的结果。在上述工作的基础上,对脑内多电流偶极子参数的反演,则呈现如下状况：即以少于实际源数目的偶极子为源假设反演,目标函数得不到极小优化。反之,目标函数可以得到极小优化, 但出现多余的伪偶极子, 且这些伪偶极子在多次不同条件的反演结果中,处于不稳定状态。若将多次反演结果中处于不稳定状态的偶极子作为伪偶极子的判据而将其排除,则可以得到一种判断磁源偶极子数目的方法     

Real-Time Localization of Moving Dipole Sources for Tracking Multiple Free-Swimming Weakly Electric Fish

In order to survive, animals must quickly and accurately locate prey, predators, and conspecifics using the signals they generate. The signal source location can be estimated using multiple detectors and the inverse relationship between the received signal intensity (RSI) and the distance, but difficulty of the source localization increases if there is an additional dependence on the orientation of a signal source. In such cases, the signal source could be approximated as an ideal dipole for simplification. Based on a theoretical model, the RSI can be directly predicted from a known dipole location; but estimating a dipole location from RSIs has no direct analytical solution. Here, we propose an efficient solution to the dipole localization problem by using a lookup table (LUT) to store RSIs predicted by our theoretically derived dipole model at many possible dipole positions and orientations. For a given set of RSIs measured at multiple detectors, our algorithm found a dipole location having the closest matching normalized RSIs from the LUT, and further refined the location at higher resolution. Studying the natural behavior of weakly electric fish (WEF) requires efficiently computing their location and the temporal pattern of their electric signals over extended periods. Our dipole localization method was successfully applied to track single or multiple freely swimming WEF in shallow water in real-time, as each fish could be closely approximated by an ideal current dipole in two dimensions. Our optimized search algorithm found the animal’s positions, orientations, and tail-bending angles quickly and accurately under various conditions, without the need for calibrating individual-specific parameters. Our dipole localization method is directly applicable to studying the role of active sensing during spatial navigation, or social interactions between multiple WEF. Furthermore, our method could be extended to other application areas involving dipole source localization.     

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 extrastriate generators of the EP to checkerboard onset. A source localization approach

The cortical origin of the pattern onset EP has been investigated over a time window which covers the entire positive-negative-positive complex of the pattern onset EP. On the basis of a dipole source localization approach, the position, orientation and strength of the underlying sources of the pattern onset EP were estimated. For large check stimuli, chosen to have a weak edge specific component in the response, still two components are needed to account for the variance of the responses. Each component corresponds to a single dipole source, and both originate in the extrastriate cortex. These components dominate, respectively, the initial and the late positive peaks of the pattern onset EP.The equivalent dipole sources of the two components show different behaviors with respect to the position of the stimulus in the visual field. The topography and behavior of the equivalent dipole source underlying the early positive component suggest an origin in area 18. The invariance with stimulus location of the dipole source underlying the late positive component suggests an origin beyond area 18. The different topographies of the components also account for the differences in surface distribution of the pattern onset EP to large check stimulation of the upper and lower sectors of the visual field.     

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.     

Dipole Theory of Heat Production and Absorption in Nerve Axon

Exact formulas are derived for the energy change of a dipole system with two energy states (or bands) in a changing field in two cases: (a) no dipole flip-flop and (b) dipole flip-flop caused by stimulation. Based on these formulas, the positive and negative heats are calculated. The results are in good agreement with experiment in case b but are 60-180% larger in case a. Furthermore, the theory shows that the negative heat cannot be less than the positive heat in case a but can be either way in case b, the latter result being found prevalent in experiment. It is concluded that nerve excitation is most likely to involve dipole flip-flop at the membrane surface. The theory is consistent in the interpretations and correlations of the electrical, optical, and thermal effects observed in nerve axon.     

A Mathematical-Physical Model of the Genesis of the Electrocardiogram

A fundamental problem of cardiac electrophysiology is that of relating quantitatively the electrical activity within the heart to the complete timevarying potential distribution at the body surface. A new numerical method is described for the calculation of the surface potential on an irregularly shaped closed external surface due to an arbitrary source distribution in a medium containing regions of different conductivity, subject to the appropriate boundary conditions. The method is intended to provide an exact theoretical analysis of the experimental data acquired by A. M. Scher and others who have been mapping the pathways of ventricular depolarization in dogs and other animals. In anticipation of the above research program, a number of exploratory computations are reported. For example, the surface potential distribution has been calculated for a cylinder of human torso cross-section with a hemispherical dipole layer current source in approximate heart position and orientation and containing “lungs” of conductivity different from that of the surrounding medium. Under certain conditions, when lung-like inhomogeneities are introduced, a simple dipole source can generate a potential distribution having the multiple maxima and minima characteristic of higher multipole sources.     

Effects of dipole position,orientation and noise on the accuracy of EEG source localization

Background  

The electroencephalogram (EEG) reflects the electrical activity in the brain on the surface of scalp. A major challenge in this field is the localization of sources in the brain responsible for eliciting the EEG signal measured at the scalp. In order to estimate the location of these sources, one must correctly model the sources, i.e., dipoles, as well as the volume conductor in which the resulting currents flow. In this study, we investigate the effects of dipole depth and orientation on source localization with varying sets of simulated random noise in 4 realistic head models.     

Laminar distribution of current sources in the rabbit superior colliculus during optic nerve stimulation

The electric field of evoked potentials developing in the rabbit superior colliculus in response to single electrical stimulation of the optic nerve was investigated. Regions of localization of sources and sinks of synaptic current creating the evoked potential were discovered by the current source density analysis method. Sinks with the shortest latency appeared in the depth of str. griseum superficiale, after which activity created by two dipoles appeared: The sink of one dipole was located in the upper part of str. griseum superficiale, the sink of the other in the lower part. The sinks thus found corresponded in their temporal characteristics and location to the principal components of the combined evoked potential. On the basis of these data the order of activation of the various systems of fibers and synapses responsible for evoked potential formation can be traced.     

Cellular spin resonance

Suspended cells can at times be seen to spin when in an ac electric field. The phenomenon is observed to be linked to cell colony age. We can now understand this in terms of the applied field acting synchronously upon a natural oscillating electric dipole associated with cell division. The dipole field strength thus estimated from spin drag agrees with the value earlier determined by the dielectrophoretic attraction (a non-uniform field effect on neutral bodies) of dividing cells for polarizable powders.Neither the source nor the role of the demonstrated rf electrical oscillations (about 10 000 Hz) of the dividing cells is as yet known. Whether the oscillations are necessary or incidental, is to be determined.     

Uncertainty in source partitioning using stable isotopes

Stable isotope analyses are often used to quantify the contribution of multiple sources to a mixture, such as proportions of food sources in an animal's diet, or C₃ and C₄ plant inputs to soil organic carbon. Linear mixing models can be used to partition two sources with a single isotopic signature (e.g., '¹³C) or three sources with a second isotopic signature (e.g., '¹⁵N). Although variability of source and mixture signatures is often reported, confidence interval calculations for source proportions typically use only the mixture variability. We provide examples showing that omission of source variability can lead to underestimation of the variability of source proportion estimates. For both two- and three-source mixing models, we present formulas for calculating variances, standard errors (SE), and confidence intervals for source proportion estimates that account for the observed variability in the isotopic signatures for the sources as well as the mixture. We then performed sensitivity analyses to assess the relative importance of: (1) the isotopic signature difference between the sources, (2) isotopic signature standard deviations (SD) in the source and mixture populations, (3) sample size, (4) analytical SD, and (5) the evenness of the source proportions, for determining the variability (SE) of source proportion estimates. The proportion SEs varied inversely with the signature difference between sources, so doubling the source difference from 2‰ to 4‰ reduced the SEs by half. Source and mixture signature SDs had a substantial linear effect on source proportion SEs. However, the population variability of the sources and the mixture are fixed and the sampling error component can be changed only by increasing sample size. Source proportion SEs varied inversely with the square root of sample size, so an increase from 1 to 4 samples per population cut the SE in half. Analytical SD had little effect over the range examined since it was generally substantially smaller than the population SDs. Proportion SEs were minimized when sources were evenly divided, but increased only slightly as the proportions varied. The variance formulas provided will enable quantification of the precision of source proportion estimates. Graphs are provided to allow rapid assessment of possible combinations of source differences and source and mixture population SDs that will allow source proportion estimates with desired precision. In addition, an Excel spreadsheet to perform the calculations for the source proportions and their variances, SEs, and 95% confidence intervals for the two-source and three-source mixing models can be accessed at http://www.epa.gov/wed/pages/models.htm.     

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.     

Spatial resolution of neuromagnetic records: theoretical calculations in a spherical model

Spatial resolution of magnetoencephalography (MEG) was studied by computer simulations using a spherical conductor model for the head. The accuracy obtainable in the absolute location of a dipole was found by calculating the confidence limits for source location in 3 dimensions. The accuracy in determining the relative locations of two sources was estimated by calculating the smallest shift in source location that could be detected with statistical significance. The results were used to illustrate the dependence of spatial resolution on several factors including noise, source depth, source strength, flux transformer configuration and the choice of the measurement locations. Under optimal conditions, separations of a couple of millimeters in superficial non-simultaneous sources can be detected, whereas for deeper sources the resolution is worse.     

Prey detection mechanism of elasmobranchs

Elasmobranchs can detect a little amount of electric fields and they have characteristic approach strategies to find an electric dipole source generated by prey or conspecifics. They appear to align the body at a constant angle with the current flow line of the electric field while swimming towards prey. However, it has not been studied how they process the perception of electric fields for the approach behaviour or what kind of neural mechanism is used. We use a model of electrosensory perception with electrodynamics and explore a possible approach mechanism based on the sensory landscape distributed on electroreceptors. This paper presents that elasmobranchs can estimate the direction of the electric field by swaying their head, which will be a basis information for their particular approach behaviour. A velocity profile of voltage gradients and intensity difference among the ampullary clusters can be another cues to detect a prey source.     

Multipole representations of current generators in a volume conductor

As far as the potential distribution outside the current generators is concerned, any current source distribution may be replaced by a suitable collection of multipoles. If these current generators lie close to the geometrical center of the volume conductor, a central dipole is a good approximation for potentials at surface points which are at considerable distances from the center. For better accuracy and for points close to the center, additional singularities such as a central quadrupole, a central octopole, etc., should be included. Potential expressions due to such multipoles in a spherical conductor can be obtained in closed forms by means of the “interior sphere theorem”. This paper presents a method for determining successively better multipole representations of the current generators in a homogeneous conducting sphere by measuring surface potentials at a successively increasing number of points. It is shown that Einthoven's triangle and Wilson's tetrahedron in the theory of electrocardiography are first and second approximations of this method. This concept also applies to conductors of other shapes. This investigation was supported by The National Heart Institute under a research grant H-2263(c).