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A Model of the Magnetic Fields Created by Single Motor Unit Compound Action Potentials in Skeletal Muscle Kevin Kit Parker and John P

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A Model of the Magnetic Fields Created by Single Motor Unit Compound Action Potentials in Skeletal Muscle Kevin Kit Parker and John P 948 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 44, NO. 10, OCTOBER 1997 A Model of the Magnetic Fields Created by Single Motor Unit Compound Action Potentials in Skeletal Muscle Kevin Kit Parker and John P. Wikswo, Jr.,* Member, IEEE Abstract—We have developed a computationally simple model Of particular interest is the MMG, although much of what for calculating the magnetic-field strength at a point due to a we state about the MMG is applicable to the MNG in the single motor unit compound action potential (SMUCAP). The appropriate context. The EMG and the MMG have similarities, motor unit is defined only in terms of its anatomical features, and the SMUCAP is approximated using the tripole model. The as they are both measurements of the same phenomena of distributed current density J is calculated within the volume propagation of action signals along an excitable fiber, that defined by the motor unit. The law of Biot and Savart can then are documented in the literature [1], [2]. Recordings made be cast in a form necessitating that J be integrated only over in our laboratory on single motor units (SMU’s) of the extens the region containing current sources or conductivity boundaries. or digitorum longus in rats [1] and a subsequent analysis of The magnetic-field strength is defined as the summation of the contributions to the field made by every muscle fiber in the motor the magnetic recording technique [3] showed two important unit. Applying this model to SMUCAP measurements obtained features of the MMG: the AF was less sensitive to the position using a high-resolution SUper Conducting Quantum Interference of the magnetometer than the electric potential recording Device (SQUID) magnetometer may yield information regarding was to electrode placement, and the AF associated with AC the distribution of action currents (AC’s) and the anatomical properties of single motor units within a muscle bundle. in cylindrical fibers is less sensitive to the conductivities of the medium surrounding the bundle than is the surface Index Terms— Electromyogram, magnetomyogram, matemati- electric potential. Thus, magnetic recordings should provide cal model, SQUID. a clearer immediate picture of the bundle physiology than do the extracellular electric signals recorded on the skin. I. INTRODUCTION The MMG offers several additional advantages over the HE electroneurogram (ENG) and the electromyogram EMG. The noninvasive nature of the MMG is clinically attrac- T(EMG) are recordings of an electric potential difference tive over needle electrode recordings of the EMG and possibly between two points in the vicinity of an active nerve or more reliable than surface recorded electrodes [4]. Electrode muscle bundle. The measurements can be made using surface measurements are greatly affected by the surrounding tissue, electrodes on the skin over the active bundle, or with invasive whose conductivities can vary by orders of magnitude [5]. The electrodes in or around the bundle. Ionic currents propagating MMG does not show this sensitivity because the magnetic along the length of fibers in the bundle give rise to action permeability of all biological tissues is very close to that of potentials (AP’s) which are manifest as the electric potential free space. difference between the electrodes. These currents also produce Yet another advantage of the MMG over the EMG arises the magnetic fields that are recorded as the magnetoneurogram from the vector nature of the magnetic field. All three compo- (MNG) and the magnetomyogram (MMG). These signals may nents of the magnetic-field vector may be measured with the be defined as the measurement of one component of the same ease as can a single component. The surface-recorded magnetic field versus time, where the sources of the field EMG is restricted to the plane of the skin and yields no are the propagating action currents (AC’s). For bundles where such vector information. Such vector information could aid in multiple fibers are active at the same time and contribute to the the localization of the bioelectric source(s). It is important to measured signal, the summation of these signals is defined as note, however, that the magnetic field outside the body can be a compound signal, i.e., the action field (AF) measured from described by a scalar magnetic potential, so a complete map of a muscle bundle where multiple fibers are active is referred to a single field component would contain all of the information as the compound action field (CAF). present in the magnetic field. Previous models of the MMG have been computationally Manuscript received August 11, 1995; revised May 6, 1997. This work was exhaustive [6]. We have developed a simple model of a single, supported in part by the National Institutes of Health under Grants NS 24751 cylindrical fiber in an unbounded, homogeneous conductor to and NS 19794. Asterisk indicates corresponding author. K. K. Parker is with the Living State Physics Group, Department of Physics support inverse techniques capable of determining the position and Astronomy, Vanderbilt University, Nashville, TN 37235 USA. of the effective electrical axis of a motor unit and the number *J. P. Wikswo, Jr. is with the Living State Physics Group, Department of of fibers in the motor unit [7], [8]. Thus, we have reduced Physics and Astronomy, Vanderbilt University, Box 1807 Station B, Nashville, TN 37235 USA (e-mail: [email protected]). the description of the motor unit in the forward model to Publisher Item Identifier S 0018-9294(97)06900-0. anatomical features and other features based on single fiber 0018–9294/97$10.00 © 1997 IEEE Authorized licensed use limited to: IEEE Xplore. Downloaded on January 5, 2009 at 16:06 from IEEE Xplore. Restrictions apply. PARKER AND WIKSWO: MODEL OF MAGNETIC FIELDS CREATED BY SMUCAP’S IN SKELETAL MUSCLE 949 recordings in the laboratory environment. In this paper, we first develop a model of the extracellular potential based on our assumption regarding the nature of the transmembrane potential and then we develop a model of the magnetic fields produced by the intracellular currents. Simulations with this model indicate that the model is an accurate representation of the physiological phenomena we have observed. (a) II. A MODEL OF THE SINGLE MOTOR UNIT COMPOUND ACTION POTENTIAL (SMUCAP) A model of the AP propagating along an excitable fiber was introduced by Plonsey [9], who made assumptions about the transmembrane potential in order to derive the extracellular potential. These assumptions are summarized in Fig. 1, in which the transmembrane is approximated as a triangular waveform, Fig. 1(a), whose first and second spatial derivatives (b) are also shown in Fig. 1(b) and 1(c), respectively. We present the relevant results of Plonsey’s model. We can assume that the propagation of the AP along the length of an excitable fiber is quasi-static, so that the curl of the extracellular electric field is zero and, hence, it can be described as the negative gradient of a scalar potential, In the volume containing the excited fiber, the total current density can be separated into two components (c) (1) where is the impressed current density, which describes the current sources produced by the active fiber, and is the Ohmic-volume current density, i.e., the return current in the medium. We wish to determine the extracellular potential, , given a simplified description of from an active fiber. In the quasi-static limit, we can neglect displacement currents, so the divergence of the total current density is zero, and (1) becomes Fig. 1. The tripole model. Propagation is toward the left: (a) the trans- (2) membrane potential is approximated as a triangular waveform, (b) the first spatial derivative of the transmembrane potential is equal to the intracellular This is Poisson’s equation, which has the solution current, and (c) the second spatial derivative of the transmembrane potential is proportional to the membrane current. (Adapted from [8].) (3) and the membrane current is given by the continuity condition where is the field point at which is being measured and (5) is the source point. We immediately identify as the current source and the intracellular current, , are related by Ohm’s law density in the extracellular space, i.e., any impressed current density that disappears from the intracellular space must cross (6) the membrane as a current, , and then appear as a current where is the intracellular resistivity per unit length of the source in the extracellular space. fiber, defined by We now need to relate the current source density to the transmembrane potential, If we consider the extracellular (7) space to be extensive in comparison to the intracellular space, changes in will be much smaller than changes in the where is the intracellular conductivity and is the axon intracellular potential, and can, to first order, be ignored, radius. The three previous expressions can be combined to i.e., we limit ourselves to the core-conductor model [11]. In yield this case (8) (4) Authorized licensed use limited to: IEEE Xplore. Downloaded on January 5, 2009 at 16:06 from IEEE Xplore. Restrictions apply. 950 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 44, NO. 10, OCTOBER 1997 which shows that the current flowing outward from the mem- This integral can be divided into two parts, one for the brane element is directly proportional to the second spatial depolarization phase and the other for repolarization, and derivative of the transmembrane potential. the spatial extent of each phase determines the limits of Because we are using the triangular AP in Fig. 1(a), integration. Combining this with (6), we have is nonzero only at three points, as shown in Fig. 1(c), and the current source density is given by (9) where and are the strength and location, respectively, of the th current source and is the Dirac delta function (14) [10].
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