Stanley Rush

Current distribution in the brain from surface electrodes.

N CONTEMPORARY medical research, many I examples are found, particularly in the fields of electroanesthesia, electrosleep, electrotherapy, and brain hemodynamics, of the application of electric currents to the surface of the head. In each of these examples, the objectives can be achieved only if the current passes through the skull into the brain. It is apparent that the amount and distribution of current entering the brain is of great consequence. Yet the literature discloses no systematic mathematical or experimental method for predicting current flow.

EEG Electrode Sensitivity-An Application of Reciprocity

In this paper, the reciprocity theorem is used to determine the sensitivity of EEG leads to the location and orientation of sources in the brain. Quantitative information used in determining the sensitivity is derived from constant potential plots of a three-concentric-sphere mathematical model of the head with current applied through surface leads (the reciprocal problem), and from an electrolytic tank employing a human skull. Advantages of the reciprocal or lead field approach are outlined and the following conclusions are drawn. 1) Leads placed at the end of a diameter through the center of the brain have a range of sensitivity due to source location of only 3 to 1. 2) For the same electrode placement, sensitivity is maximum to sources oriented parallel to the line of the electrodes regardless of source location. 3) Electrodes spaced 5 cm apart are about ten times more sensitive to proximal cortical sources (by virtue of position) than to sources near the center of the brain.

A Practical Algorithm for Solving Dynamic Membrane Equations

Many investigators work with the Hodgkin-Huxley model of membrane behavior or extensions thereof. In these models action potentials are found as solutions of simultaneous non-linear differential equations which must be solved using numerical techniques on a digital computer. Recent membrane models showing pacemaker activity, such as that of McAllister, Noble, and Tsien, involve solutions covering long periods of time, up to fisve seconds, and many ionic currents. Those added requirements make it desirable to have an efficient algorithm to minimize computer costs, and a systematic and simple solution method to keep the program writing and debugging to manageable levels.

Trunk extensor EMG-torque relationship

The integrated surface electromyogram (IEMG) of the lumbar erectores spinae and the torque generated were simultaneously recorded from 27 healthy subjects in the standing posture while they pulled isometrically against resistance provided by a harness around the shoulders. The IEMG – torque ratio (efficiency of electrical activity, or EEA) was used to characterize each subject. Individual recordings showed evidence of nonlinearity of the IEMG–torque relationship in that a statistically better fit to experimental recordings was obtained by using two straight lines with a breakpoint between them. However, with repeated testing, the gradients of these two lines were more variable than the slope of the single straight line fitted to the entire recording. The slope of the best fit line (EEA) was less for recordings made during torque decrease than for increasing-torque recordings. This also showed as a “hysteresis” pattern in the recordings. The coefficient of variability (within subjects) of the EEA was greater in day-to-day testing (24%) than with repeated pulls at the same testing session (14%). This was similar to variability of the maximum generated torque. About 25% of the variability between subjects was found to be due to anthropometric differences. The residual variability of the relationship would limit the accuracy of IEMG as a measure of muscular effort under changing torque conditions. However, the EEA may be useful for characterizing muscle performance, especially when maximum effort cannot be achieved.

Qualitative effects of thoracic resistivity variations on the interpretation of electrocardiograms: The low resistance surface layer

A lthough the torso is often assumed in electrocardiographic studies to be homogeneous, it is not. In a previous paper’ we have considered the resistivity changes which produce the most drastic modification of surface electrocardiograms (ECG’s). These are those variations which occur in and around the heart itself, i.e., the differences in conductivity between heart muscle, blood, and lung, as well as the lower resistivity of heart muscle in the direction of its fibers compared to that in a transverse direction. In this article, we discuss what we consider to be the second most important resistivity change, namely, that occurring in the vicinity of the surface electrodes. This is due to the relatively low resistance surface layer formed by the muscles girdling the thorax. As in our first paper, we use elementary models, and neglect the perturbing effects of other resistivity changes, e.g., the ribs, spine, sternum, liver, blood, blood vessels, pleural membranes, etc.

The complete heart-lead relationship in the einthoven triangle

The relationship between the source strength and the “manifest vector” in the Einthoven Triangle is derived for a line and a point dipole source and confirmed experimentally. The result permits the interpretation of the standard ECG leads in absolute terms and corrected for body size. The manifest vector is shown to be approximately\(\sqrt 3 \) times what it would be in an otherwise similar circular slab which circumscribes the triangle.

On the Independence of Magnetic and Electric Body Surface Recordings

Recent papers have shown that the ECG depends on the flow sources of the impressed field in the heart while the MCG is a function of its vortex source distribution. It has consequently been suggested that body surface electric and magnetic recordings yield completely independent information about the physiological generators. Such independence would be of enormous significance wherever present diagnostic procedures rely heavily on the ECG or EEG. This paper points out that the independence of the flow and vortex sources is only a mathematical possibility. It demonstrates that two important physical constraints are operating which require the flow and vortex sources to be, in effect, one-to-one with each other. Consequently, the electric and magnetic fields arising from excitable tissue in an assumed homogeneous volume conductor are fundamentally interdependent.

Computer Solution for Time‐Invariant Electric Fields

A general method suitable for computerized solution is described for finding the electric and/or current density fields due to time‐invariant sources. A charge distribution satisfying the boundary conditions is specified as the solution of an integral equation. The latter is approximated as a system of linear algebraic equations and solved by the computer. The field at any point is then obtained by Coulombs law and superposition from the boundary charge distribution and any impressed fields that exist. The method will, within the practical restrictions imposed by the computers capacity to solve simultaneous equations, solve any inhomogeneous, linear, isotropic, steady current flow and/or static field problem with any of the various boundary conditions which uniquely define a field. While the derivation is given here in terms of electric field quantities, it applies as well to the several analogous fields of engineering and physics. In addition to the theoretical formulation, data are given to guide in t...

Inhomogeneities as a Cause of Multiple Peaks of Heart Potential on the Body Surface: Theoretical Studies

This paper describes several studies that contribute to an understanding of the ways in which inhomogeneities and anisotropies may contribute to the formation of multiple peaks of potential on the body surface.

A Principle for Solving a Class of Anisotropic Current Flow Problems and Applications to Electrocardiography

In studies of electrocardiographic lead performance, theoretical analyses of the influence of the anisotropic heart and skeletal muscle are particularly difficult. In this paper, the basic differential equations of static fields and steady current flow are arranged to emphasize the field and conductivity dependent charge distributions which arise in anisotropic media. The equations are applied to two types of problems of immediate interest. Firstly, the equations are used to explain how anisotropic media may be included in current digital computer studies of the heart-lead relation and to conclude that the techniques which made the computer studies possible tend to lose their advantage when applied to arbitrary anisotropic configurations. Secondly, the equations are used to develop a principle which permits exact solutions for the fields of numerous simple anisotropic configurations. Three such configurations useful for heart-lead studies are analyzed with the following results: the anisotropic skeletal muscle can be treated in special cases such as a head-foot heart-vector lead approximately as isotropic with resistivity of 280 ohm cm; the closed dipolar layer in an anisotropic, inhomogeneous heart produces the same null electric field as it does in homogeneous isotropic media; bounds on the influence of the hearts anisotropy on a heart-vector lead field are estimated at plus or minus 12 percent of the average lead field intensity.