BioelectromagneticsSupplement 1 :191-204 (1992)
Factors Affecting Neural Stimulation With Magnetic Fields -
Maria A. Stuchly and Karu P. Esselle Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia, Canada (M.A.S.), and School of Mathematics, Physics, Computing and Electronics, MacQuarie University, Sydney, New South Wales, Austrtalia, (K.P.E) High amplitude magnetic field pulses produced by coils external to the body have been used for medical diagnosis since the mid- 1980s to stimulate motor neurons in the brain cortex and peripheral nerves. While successful applications have since blossomed, it has only been during the last three years that quantitative dosimetric data have become available. The factors affecting neural stimulation can be divided into three categories broadly related to the characteristics of (i) the stimulus, (ii) the neuron, and (iii) the induced electric field as related to the configuration of the stimulating coil. The stimulus, in the case of magnetic field stimulation, has the form of an exponentially decaying pulse with a small overshoot of the opposite polarity. Physical and electrical properties of a neuron affect its electrical stimulation. Dosimetric considerations are limited to the linear model describing the threshold phenomena, where passive electrical properties and the cable model provide a reasonable approximation of neuron behavior. The electromagnetic variable responsible for stimulation is the spatial derivative of the induced electric field along the neuron axis. This paper examines the factors involved in eliciting threshold excitation of motor neurons by magnetic fields. The description of various factors is largely based on published data except for the analysis of the electromagnetic stimuli induced by various Coils. 0 1992 Wiley-Liss. Inc. Key words: magnetic stimulation, neural excitation, cable model, induced electric field, stimulating coil
INTRODUCTION
Electrical excitation of motor neurons in the brain cortex, of peripheral neurons, and observation of evoked responses have been extensively used in research and medical practice. Until recently excitation has been obtained with current pulses produced either by implanted or external electrodes applied near the neuron. Magnetic field stimulation offers the advantages that it is a non-invasive, non-contact method which produces minimal discomfort to the patient as only low density current flows through skin pain receptors during the procedure [Amassian et a]., 1989; Barker et al., 1985, 1986; Freeston eta]., 19841. Magnetic stimulation of brain, spinal cord, and peripheral nerves has been used to diagnose various medical conditions associated with the abnormal conduction of Address reprint requests to Maria A. Stuchly, Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia, V8W 3P6 Canada.
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motor pathways. Detailed information and references can be found in recent publications, e.g., Chokroverty [ 19901; J. Clin. Neurophys. [1991]. Magnetic stimulation has recently expanded to other medical applications such as stimulation of inspiratory motor nerves [Mouchawar et al., 19901, and the heart [Bourland et al., 19901. The developments in magnetic stimulation of neurons for diagnostic purposes provide an interesting illustration of dosimetric problems encountered in quantification of the interaction. In the context of searching for dosimetric measures of interactions of extremely low frequency magnetic fields with biological systems it is interesting to examine the parameters involved in eliciting a relatively well understood effect of neural stimulation. It is also interesting to note that while diagnostic applications of magnetic stimulation have blossomed since the mid- 1980s, it is only during the last two years that quantitative evaluations of the induced electric fields responsible for the excitation have been made. Only recently has the awareness of some of the critical parameters of magnetic stimulation emerged, even though many clinical studies still remain highly qualitative in nature (Amassian et al., 1989; CIaus et al., 1990; Day et al., 1989). In this paper we examine the published data to assess the factors involved in eliciting a threshold excitation of motor neurons by time-varying magnetic fields. We also present the results of our analysis of the induced electric fields from various coil configurations above a stratified tissue half-space. This analysis allows us to gain physical insight into several parameters defining magnetic stimulation. MAGNETIC STIMULUS
A typical magnetic neural stimulator consists of a capacitor C which is charged to a voltage V and then discharged through the stimulating coil of inductance L, and a resistance R [Barker et al., 1987; Geddes, 1987, 19881. The coil current is equal to
ve-(
RrI2L )
I(t)=
sino,,t
a ,L where
The spatial derivative of the magnetically induced electric field is ultimately responsible for neural stimulation. The electric field strength is proportional to the time derivative of the magnetic field and therefore the time derivative of the coil current, that itself may have an oscillatory or non-oscillatory (overdamped) character. For a typical RCL stimulator, even an overdamped current pulse results in a biphasic stimulus. The amplitude of the undesired pulse phase can be made small by a proper selection of a damping factor. For an overdamped pulse, the time derivative of the current is [Basser and Roth, 19911:
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where w l = R/2L. and the stimulus duration is
Waveforms other than overdamped or underdamped pulses described by Equation 1 have not been utilized so far in magnetic stimulators. However, sinusoidal pulses or even pulses closely resembling rectangular pulses are feasible, although technically more difficult to obtain. NEURAL STIMULATION
Neurons are divided to two categories: myelinated, i.e., covered with a fatty sheath called myelin, and unmyelinated. Most human motor neurons in the cortex and large peripheral neurons are myelinated. The two categories of neurons differ in size (diameter) and electrical characteristics. Myelinated neurons are 2 to 20 p m in diameter, while unmyelinated neurons are typically 0.3 to 1.3 pm. Thresholds for depolarization (stimulation) depend on the neuron dimensions and passive electrical properties, and are described in terms of the space constant (also called the length constant and the characteristic length). The space constant for myelinated fibers is approximately equal to the internodal distance (for an accurate expression for the length constant see Eq. 12). The intermodal distance is approximately 200 times the inner diameter of the axon. The stimulation threshold is inversely proportional to the neuron diameter. The temporal response of neurons depends on their time constant, which depends on the membrane and axoplasm passive electric properties. Passive electrical behavior of neurons is described by the cable equation that has been used to compute the response characteristics of various neurons to internally or externally applied stimuli, by the current applied through extracellular electrodes positioned close to the neuron [McNeal, 1976; Plonsey and Altman, 1988; Rattay 1986, 1988; Reilly et al., 1985; Reilly 1988; Rubinstein and Spelman, 1988; Tranchina and Nicholson, 19861, and by the electric field induced by an externally applied magnetic field [Basser and Roth, 1991; Roth and Basser, 1990; Roth et al., 1990, 1991a, b]. Considering only linear passive properties of the membrane, the total current at the node n, as illustrated in Figurel, is given in terms of the Vi,n, the intracelVeJl 1
1
veJl+l OUTSIDE
9
l
v% Vi,"+l
+%fl
w
INSIDE (AXOPIASM)
-Ax-
Fig. 1. A cable model of the passive electrical properties of a myelinated nerve fibre [adapted from McNeal, 19761.
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lular potentials at nodes n, and Vn, the transmembrane potential at node n, [McNeal, 19761 (5) where the nodal capacitance is Cm= c m n d 6, , cmis the membrane capacitance per unit area, d, is the axon diameter at the node, 6 is the width of the nodal gap. The axial internodal resistance is rI = 4 p I A x / n d d 1p2,,is the resistivity of the axoplasm, Ax is the internodal distance. The current at node n is I,.,,= Vl/rm,the nodal membrane resistance is r,,, = ( g , n d l a)-',and gInis the membrane conductance per unit area. By substituting into Equation 5 , I,," = Vn/rmand V,,"= Vn + Ve,,, the following set of linear, first-order differential equations describing responses of the nerve fibre is obtained
for n = ... -2, -1, 0, 1, 2, ... For subthreshold stimuli, the membrane resistance, stant. Equation 6 can be rearranged as follows:
z,
can be assumed con-
indicating that the stimulation function can be considered as the second difference quotient of the node voltages Ve,n. For unmyelinated fibers this function is expressed as a second differential quotient [Rattay, 19871. Stimulation starts at the node at which Vc is maximum. Equation 7 indicates that the spatial derivative of the electric field along the axon axis is the main source of change in the transmembrane potential. The McNeal model was further extended by Reilly [1988, 19891 to include nonlinear behavior to several adjacent nodes. Reilly evaluated the strength-duration curves for several waveforms of the stimulus, including biphasic pulses and multiple pulses. The most interesting results for magnetic stimulation are summarized in Figure 2. Stimulation (membrane depolarization) thresholds are higher for an exponentially decaying stimulus with the time constant equal to the duration of a rectangular pulse. Similarly, the thresholds for a sinusoidal pulse are also higher. Multiple monophasic pulses result in lowering the thresholds with the amount of decrease dependent on the interpulse interval, pulse duration and number of pulses. In an axially uniform electric field, stimulation occurs at the axon's end. The minimum electric field required for stimulation decreases proportionally to the axon diameter. This can also be seen from Equation 7, since Ax, the internodal separation is proportional to the fiber diameter (typically Ax = 100 do, where do is fiber outer diameter, and d, = 0.6 to 0.7 d,,). The main limitation of the models mentioned above is that the myelin sheath i s assumed to be a perfect insulator, and therefore not contribute to the current across the membrane. Taking into account myelin conductivity, the transmembrane potential,
Magnetic Field Neural Stimulation
1
10
lo2
lo5
195
10'
Stimulus Duration or Time Constant us)
Fig. 2. Stimulation thresholds with a rectangular pulse (solid line), exponential pulse (dot-dashed line) and sinusoidal pulse (dotted line) for a 20 pn myelinated neuron as a function of pulse duration (rectangular, period (sinusoidal) or time constant (exponential) [adapted from Reilly, 19881.
V, along the axon can be described by a cable equation [Basser and Roth, 19911 as follows:
The space constant of the internodal region, Ao, is equal to:
and the time constant, zo, is
r,, = & , & ' P o
(10)
where p , is the resistivity of myelin, and E' is the dielectric constant of myelin. Basser and Roth [ 199 11 performed an analysis of neuron stimulation using a model comprising the myelin contribution (Eq. 8) and a nonlinear current source at the node of the highest external potential. Figure 3 illustrates the dependence of the threshold electric field gradient (taken with respect to the direction of the neuron axis) on the exponential pulse decay time for neurons of various diameters.
1
-
I
I
I
10"
100
10'
Q
?oJ
lo4
rc (ms)
Fig. 3. Stimulation thresholds vs. decay time constant for exponential pulses in neurons of three diameters, 5, 12.5, and 20 pm [adapted from Basser and Roth, 19911.
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If only subthreshold responses are of interest, and the distance over which the electric field varies is large compared with the neuron internodal separation, the set of equations describing an axon (Eq. 7) can be simplified to an equation of an equivalent uniform axon [Basser and Roth, 199 I]. When the membrane impedances are averaged over the nodal and internodal regions, and Rushton’s [ 195 I ] scaling laws for axon of different diameters (Add,, = 100, dl/dl,= 0.6) are used, a cable equation for subthreshold electromagnetic stimulation is obtained [Basser and Roth, 19911:
where the equivalent length and time constants are
z = ( c,,,+ 6 5 2 ~ , ~ ’ / S g,,, ) / (+ 6 5 2 / ~6),
where p , is the resistivity of axoplasm and gm is t h e membrane conductance per unit area. INDUCED ELECTRIC FIELDS Calculations of the electric field induced by a magnetic field and its spatial derivatives in the actual biological tissues comprising neurons are very complex and can only be accomplished by numerical methods requiring considerable computer resources. Even analyses of such simplified models as homogeneous cylinders and layered concentric spheres require about two hours of computational time on large computers [Roth et al., 1991a, b]. To be able to gain physical insight how the stimulating coil configuration and dimensions affect the spatial characteristics of the stimulus and to be able to optimize the coil geometry, we selected a simplified tissue model for analysis. The configuration of the model is shown in Figure 4. Since stimulating pulses contain frequencies usually below 10 kHz, the following quasi-static approximations are made: (i) the displacement current in tissue is negligibly small compared with the conduction current, (ii) the penetration (skin) depth is much greater than the distances of interest, and (iii) time-retardation of
Stimulating Coil Air
4=
Y Tissue-Air
Planar
Interface
////////////////////////////////////////////////// Homogeneous Tissue Half-Space
Fig. 4. A stimulating coil of an arbitrary shape and N turns above a tissue half-space.
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electromagnetic fields is negligible [Roth et al., 1991al. The electric field E induced in tissue consists of two components: the primary field E'' directly induced by the coil current, and the secondary field E" due to surface charges at the tissue-air interface. The primary electric field is
where po is the permeability of free space, A is the magnetic vector potential, d i is an infinitesimal element of the coil, r is the distance between the coil-element and the point where the electric field is calculated, and the line-integral is along the length of the coil. The secondary electric field is derived from an electric scalar potential, according to
w,
The function
w
-
E" =-Vv is a solution of Laplace's equation
(16)
v21y=o
(17) subjected to the boundary condition that the normal electric field is zero at the tissueair interface, i.e.,
where 2 is the unit vector normal to the interface. For the configuration shown in Figure 4 the terms in Equation 18 can be expanded using Bessel integrals and the corresponding coefficients can be equated, resulting in the following analytical expression for the electric field d 2 produced by the coil-element dt [Esselle and Stuchly, 19911:
where
dlx,dly,dlz are the x, y, z components of dt ,respectively, and i and j are the unit vectors in x- and y- directions. The total electric field produced by the coil is obtained by numerically integrating Equation 19 along the length of the coil.
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The spatial derivatives of the electric field in x- and y-directions, produced by the coil-element d , can be obtained from Equation 19 as
The spatial derivatives d Ex/ d x and dEy/ d y are obtained by numerically integrating above expressions along the length of the coil. Because of the boundary condition (Eq. 18) the electric field component normal to the interface (EL)is zero everywhere inside tissue. Therefore, the analysis and Equations 19, 22, and 23 also apply to heterogeneous multi-layered tissue, if all the interfaces between the layers are parallel to the tissue-air interface, and also as a limiting case, for heterogeneous biological systems where the conductivity varies continuously but only in the direction perpendicular to the tissue-air interface. Various coil configurations, including square, circular, double-square and double-circular coils, oriented parallel, at an angle, and perpendicular to the interface have been considered [Esselle and Stuchly, 19911. Figure 5 illustrates results for the electric field (E ) and its spatial derivative, dEy/dy, for a square coil presented as contour and $-dimensional surface plots. Since EYis symmetrical about the x-axis, only the y20 region is shown in the plots. A contour or 3D plot of Ex can be obtained by rotating the corresponding Ey-plot,clockwise, by 90". The spatial derivatives satisfy the relationship d Ex/ d x = - d Ey/ d y anywhere in the tissue halfspace, so d Ex/d x can also be obtained from Figure 5c, d. The contour interval on the graphs is 10% of the peak value, hence the smallest contours (closest to the coils) show 90% of the peak values. The electric field EYand its derivative for a double square coil are illustrated in Figure 6. The other component of the electric field, Ex,has four smaller maxima of 203.9 V/m at x = k3.04 cm, y = k2.56 cm. However, as indicated earlier, d E I / d x is always equal to - d E / d y everywhere in the tissue half space, despite the different spatial profiles of , E! and Ey. Figure 7a shows how the peak (maximum) value of ( d E x / d x ) or ( d EJd y) varies with depth, for circular, square and double-square coils. In order to -make the graphs universal, both axes have been normalized. The locations of the peak value are shown in Figure 7b and the widths of the 90% contour in the x- and ydirection are given in Figure 7c. It can be seen that double coils produce asymmetrical contours of 90% values of d Ex/ d x and d Ey/ d y.
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3
Fig. 5 . Induced electric field and its derivative for a square coil of a dimension a = 5 cm, centered at x = 0, y = 0 (Fig. 4), parallel to the interface; the coil has 10 turns, and carries current rising at 100 A/ ps. A: Contours of Eyon a plane in tissue 1 cm below the coil; max E = 236 V/m at x = +2.5 cm, = 0 . B: 3-dimensional plot of Ev. C, D: a( d Eyld y) in tissue in a plane 1 cm below the coil, max 374 V/m at x = y'= k2.5 cm.
I ?I
DISCUSSION
The spatial derivative of the electric field in the axial direction along an axon, as evident from Equation 8, is the critical parameter for magnetic field induced threshold stimulation. The analysis of fields induced in tissue by various coils leads to several observations that are important from the dosimetnc point of view. For all coil configurations considered in our analysis, d Ex/dx is always equal to - d Ey/ d y. The maximum of the derivative for some coil configurations is located where the field is equal to zero (e.g., for a single loop perpendicular to the air-tissue interface). When considering the placement of the coil with respect to the neuron to be stimulated, the 90% contour widths (Figure 7c) can be considered as somewhat arbitrarily selected metrics for the region of subthreshold stimulation. For most coil configurations the widths in the x and y directions are approximately equal; however a square coil perpendicular to the interface produces an elongated contour. If a minimum number of neurons is to be stimulated, the coil should be oriented parallel to the neuron. The main disadvantage of this configuration is that the magnitude of the field derivative is much lower than for the same coil parallel to the interface.
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Fig. 6 . Induced electric field and its derivative for a double square coil with each side dimension a = of turns 20, current 100 Alps. A, B: Ey, rnax IEy = 470 Vlm. C, D: a( J Eyl d y), = 748 Vlm at x = 0, y = k2.5 cm.
I
Single coils parallel to the interface produce a total of four “hot spots” (Fig. 5). It is apparent, that the same neuron is at one point depolarized (where the derivative is negative) and hyperpolarized at a distant point depending on the coil dimensions. In practice, this means the direction of current in the coil must be properly selected for a given neuron so that the action potential can propagate. This problem can be avoided for double coils in one direction, and for quadruple coils in both directions, as in the latter case only one hot spot is produced. Subthreshold electromagnetic stimulation, as described by Equation 1 1, can be compared to other types of stimulation, namely the stimulation by extracellular electrodes or by a microelectrode. The factor on the right side of the equation, i.e., A2( d Ex/ d x), is analogous to - A2( d *V! d xz) for extracellular electrodes separated from the membrane, where Ve is the extracellular potential produced by the electrodes [Rattay 1986, 19881, and to - A2raifor intracellular microelectrodes, where ra is the resistance per unit length of the axoplasm and i is the inward applied current per unit length [Plonsey, 19691.
Magnetic Field Neural Stimulation
0.0
0.2
0.4
0.6
0.8
201
1.O
h/a
Fig. 7. The normalized maximum values of the electric field derivatives (a), their locations (x,,,. y,,,) (b), and the size of the region within which the value is 90% of the maximum (Ax, By) (c); a is the dimension of one coil, h is the separation between the coil and the plane at which the value of the normalized d E l a x is given, po= 4 n x lo7H/m, N is the total number of turns, dI/dt is the rate of change of coil current.
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Three simplifying assumptions have been made in all our considerations. The neuron is assumed to be placed i n a homogeneous, isotropic conductive medium of high conductivity. This in many cases is not a correct representation, and complexities resulting from the fact that the neuron is in a bundle with other neurons must be considered, as has been evaluated for stimulation with extracellular electrodes [Altman and Plonsey, 19881. The electric field is assumed to be unperturbed by the neuron. This assumption is correct in a simplified case analyzed here of a semi-infinite tissue half-space with no changes in electrical properties other than those perpendicular to the air-tissue interface and the neuron axis. The induced transmembrane potential is assumed not to vary across the axon cross-section. The last assumption is reasonable, provided the electric field is not significantly altered by the presence of the neuron. CONCLUSIONS
In numerous publications on diagnostic applications of magnetic stimulation it is stipulated that the induced electric field is the key dosimetric parameter responsible for stimulation. On the other hand, neural behavior modeling and neurophysiological data clearly indicate that t h e spatial derivative of the electric field in the direction along the axon is the parameter quantitatively describing subthreshold stimulation. The success of clinical applications, despite the limitations of the physical models employed, can at least partly be explained by the use of stimuli well above threshold. Furthermore, neurons are not straight cables, which may aid stimulation in a uniform field. Our analysis of a simplified model leads to several general conclusions which are likely to hold even for more complex models and provide at least qualitative understanding of neural stimulation with magnetic fields. For instance, the electric field component perpendicular to the tissue boundary is also zero for the spherical tissue model and very small for the cylindrical tissue model [Roth et al., 1991bl. The decrease of the field derivative with the distance from interface and the number of hot spots are general enough to be applicable to other configurations. The main limitation of the simplified analysis is the neglection of tissue heterogeneity in the direction parallel to the interface. Such variations in tissue conductivity has previously been shown to affect the magnitude and direction of induced electric fields [Polk and Song, 19901. ACKNOWLEDGMENTS
This work was performed while the authors were with the Bureau of Radiation and Medical Devices, Health and Welfare. The authors are grateful to Dr. Bradley Roth from the U.S. National Institute of Health for supplying prepublication copies of his manuscripts. We also thank Dr. Stan s. Stuchly of the University of Victoria for stimulating discussions and advice, and Pave1 Dvorak and Jack McLean of Health and Welfare Canada for their review of the manuscript. Constructive comments provided by unnamed reviewers are appreciated.
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List of Symbols membrane capacitance per unit area nodal capacitance axon diameter nerve diameter, external-axon plus myelin electric field strength along the axis of a nerve membrane conductance per unit area inwards current from a microelectrode per unit length membrane current at n-th node resistance per unit length of axoplasm stimulating coil radius axial internodal resistance nodal membrane resistance transmembrane potential external nodal potential, n-th node internal nodal potential, n-th node membrane resting potential threshold membrane potential direction along the nerve axis internodal distance width of the node the permittivity of vacuum myelin dielectric constant space constant of an (equivalent) axon myelin space constant resistivity of axoplasm resistivity of myelin time constant of an (equivalent) axon myelin time constant stimulus duration
REFERENCES Altman KW, Plonsey R (1988): Development of a model for point source electrical fibre bundle stimulation. Med Biol Eng Comput 26:466-475. Ainassian VE, Cracco RQ, Maccabee PJ (1989): Focal stimulation of human cerebral cortex with the magnetic coil: A comparison with electrical stimulation. Electroencephalogr Clin Neurophysiol 74:401-416. Barker AT, Jalinous R, Freeston 1L (1985): Non-invasive stimulation of human motor cortex. Lancet i:1106. Barker AT, Freeston IL, Jalinous R, Jarratt JA (1986): Clinical evaluation of conduction time measurements in central motor pathways using magnetic stimulation of the human brain. Lancet i: 1106. Barker AT, Freeston IL, Jalinous R, Jarratt JA (1987): Magnetic stimulation of the human brain and peripheral nervous system: An introduction and the results of an initial clinical evaluation. J Neurosurg 20: 100-1 09. Basser PJ, Roth BJ (199 1) Stimulation of a myelinated nerve axon by electromagnetic induction. Med Biol Eng Comput 29:261-268. Bourland JD, Mouchawar GA, Nyenhuis JA, Geddes LA, Foster KS, Jones JT, Graber JP (1990): Transchest magnetic (eddy-current) stimulation of the dog heart. Med Biol Eng Comput 28: 196198. Chokroverty S (1990): "Magnetic Stimulation in Clinical Neurophysiology." Boston: Butterworks Publishing. CIaus D, Harding AE, Hess CW, Mills KR, Murray NMF, Thomas PK (1988): Central motor conduction i n degenerative ataxic disorders: A magnetic stimulation study. J Neurol Neurosurg Psychiatry 5 I :790-795.
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Day BL, Dressler D, De Noordhout AM, Marsden CD, Nakashima K, Rothwell JC, Thompson PD (1989): Electric and magnetic stimulation of human motor cortex: Surface emg and single motor unit responses. J Physiol 41 2:449-473. Esselle, KP, Stuchly MA (1992): Neural stimulation with magnetic fields: Analysis of induced electric fields. IEEE Trans Biomed Eng 39: 693-700. Freeston IL, Barker AT, Jalinous R, Polson MJR (1984): Nerve stimulation using magnetic fields. Proc 6th Annual Conf of IEEE Eng in Med Biol (Semmlow JL, Wellowitz W, eds, pp 557-561. Geddes LA ( I 987): Optimal stimulus duration for extracranial cortical stimulation. Neurosurgery 20:9499. Geddes LA (1988): Stimulation of excitable tissue with time varying magnetic fields. Proc. 10th Annual Intern Conf IEEE Eng Med Biol SOC918-921. Journal of Clinical Neurophysiology (199 I): Special issue on magnetic stimulation of the nervous system. 8:no. 1. McNeal DR (1976) Analysis of a model for excitation of myelinated nerve. lEEE Trans Biomed Eng BME-231329-337. Mouchawar G, Bourland JD, Voorhees WD, Geddes LA (1990): Stimulation of inspiratory motor nerves with a pulsed magnetic field. Med Biol Eng Comput 28:613. Plonsey R, (1969): "Bioelectric Phenomena." New York: McGraw-Hill. Plonsey R, Altman KW (1988): Electrical stimulation of excitable cells - A model approach. Proc IEEE 76: 1122-1 129. Polk C, Song JH ( 1 990): Electric fields induced by low frequency magnetic fields in inhomogeneous biological structures that are surrounded by an electric insulator. Bioelectromagnetics I 1 :235249. Rattay F ( 1 986): Analysis of models for external stimulation of axons. IEEE Trans Biomed Eng BME33:974-971. Rattay F (1987): Ways to approximate current-distance relations for electrically stimulated fibers. J Theor Biol 125:339-349. Rattay F (1988): Modelling the excitation of fibers under surface electrodes. IEEE Trans Biomed Eng 35: 199-202. Reilly JP, Freeman VT, Larken WD (1985) Sensory effects of transient electrical stimulation -Evaluation with a neuroelectric model. IEEE Trans Biomed Eng BME-32:1001-1011. Reilly JP (1988) Electrical models for neural excitation studies. Johns Hopkins APL Technical Digest 9:44-59. Reilly JP ( 1989) Peripheral nerve stimulation by induced electric currents: Exposure to time-varying magnetic fields. Med Biol Eng Comput 27:lOl-1 10. Roth BJ, Basser PJ (1990): A model of the stimulation of a nerve fiber by electromagnetic induction. IEEE Trans Biomed Eng 37588-597. Roth BJ, Cohen LG, Hallett M, Friauf W, Basser PJ (1990): A theoretical calculation of the electric field induced by magnetic stimulation of a peripheral nerve. Muscle Nerve 13:734-741. Roth BJ, Cohen LG, Hallett M (1991a): The electric field induced during magnetic stimulation. Electroencephalogr Clin Neurophysiol (accepted for publication). Roth BJ, Saypol JM, Hallett M, Cohen LG, (I991 b): A theoretical calculation of the electric field induced in the cortex during magnetic stimulation. Electroencephalogr Clin Neurophysiol 8 I :47-56. Rubinstein JT, Spelman FA (1988): Analytical theory for extracellular electrical stimulation of nerve with focal electrodes. 1. Passive unmyelinated axon. Biophys J 54:975-981. Rushton WAH (1951): A theory of the effects of fibre size in myelinated nerve. J Physiol (London) I15:101-122. Tranchina D, Nicholson C (1986) A model for the polarization of neurons by extrinsically applied electric fields. Biophys J SO: 1139-1 156.
Factors Affecting Neural Stimulation With Magnetic Fields -
Maria A. Stuchly and Karu P. Esselle Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia, Canada (M.A.S.), and School of Mathematics, Physics, Computing and Electronics, MacQuarie University, Sydney, New South Wales, Austrtalia, (K.P.E) High amplitude magnetic field pulses produced by coils external to the body have been used for medical diagnosis since the mid- 1980s to stimulate motor neurons in the brain cortex and peripheral nerves. While successful applications have since blossomed, it has only been during the last three years that quantitative dosimetric data have become available. The factors affecting neural stimulation can be divided into three categories broadly related to the characteristics of (i) the stimulus, (ii) the neuron, and (iii) the induced electric field as related to the configuration of the stimulating coil. The stimulus, in the case of magnetic field stimulation, has the form of an exponentially decaying pulse with a small overshoot of the opposite polarity. Physical and electrical properties of a neuron affect its electrical stimulation. Dosimetric considerations are limited to the linear model describing the threshold phenomena, where passive electrical properties and the cable model provide a reasonable approximation of neuron behavior. The electromagnetic variable responsible for stimulation is the spatial derivative of the induced electric field along the neuron axis. This paper examines the factors involved in eliciting threshold excitation of motor neurons by magnetic fields. The description of various factors is largely based on published data except for the analysis of the electromagnetic stimuli induced by various Coils. 0 1992 Wiley-Liss. Inc. Key words: magnetic stimulation, neural excitation, cable model, induced electric field, stimulating coil
INTRODUCTION
Electrical excitation of motor neurons in the brain cortex, of peripheral neurons, and observation of evoked responses have been extensively used in research and medical practice. Until recently excitation has been obtained with current pulses produced either by implanted or external electrodes applied near the neuron. Magnetic field stimulation offers the advantages that it is a non-invasive, non-contact method which produces minimal discomfort to the patient as only low density current flows through skin pain receptors during the procedure [Amassian et a]., 1989; Barker et al., 1985, 1986; Freeston eta]., 19841. Magnetic stimulation of brain, spinal cord, and peripheral nerves has been used to diagnose various medical conditions associated with the abnormal conduction of Address reprint requests to Maria A. Stuchly, Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia, V8W 3P6 Canada.
0 1992 Wiley-Liss, Inc.
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motor pathways. Detailed information and references can be found in recent publications, e.g., Chokroverty [ 19901; J. Clin. Neurophys. [1991]. Magnetic stimulation has recently expanded to other medical applications such as stimulation of inspiratory motor nerves [Mouchawar et al., 19901, and the heart [Bourland et al., 19901. The developments in magnetic stimulation of neurons for diagnostic purposes provide an interesting illustration of dosimetric problems encountered in quantification of the interaction. In the context of searching for dosimetric measures of interactions of extremely low frequency magnetic fields with biological systems it is interesting to examine the parameters involved in eliciting a relatively well understood effect of neural stimulation. It is also interesting to note that while diagnostic applications of magnetic stimulation have blossomed since the mid- 1980s, it is only during the last two years that quantitative evaluations of the induced electric fields responsible for the excitation have been made. Only recently has the awareness of some of the critical parameters of magnetic stimulation emerged, even though many clinical studies still remain highly qualitative in nature (Amassian et al., 1989; CIaus et al., 1990; Day et al., 1989). In this paper we examine the published data to assess the factors involved in eliciting a threshold excitation of motor neurons by time-varying magnetic fields. We also present the results of our analysis of the induced electric fields from various coil configurations above a stratified tissue half-space. This analysis allows us to gain physical insight into several parameters defining magnetic stimulation. MAGNETIC STIMULUS
A typical magnetic neural stimulator consists of a capacitor C which is charged to a voltage V and then discharged through the stimulating coil of inductance L, and a resistance R [Barker et al., 1987; Geddes, 1987, 19881. The coil current is equal to
ve-(
RrI2L )
I(t)=
sino,,t
a ,L where
The spatial derivative of the magnetically induced electric field is ultimately responsible for neural stimulation. The electric field strength is proportional to the time derivative of the magnetic field and therefore the time derivative of the coil current, that itself may have an oscillatory or non-oscillatory (overdamped) character. For a typical RCL stimulator, even an overdamped current pulse results in a biphasic stimulus. The amplitude of the undesired pulse phase can be made small by a proper selection of a damping factor. For an overdamped pulse, the time derivative of the current is [Basser and Roth, 19911:
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where w l = R/2L. and the stimulus duration is
Waveforms other than overdamped or underdamped pulses described by Equation 1 have not been utilized so far in magnetic stimulators. However, sinusoidal pulses or even pulses closely resembling rectangular pulses are feasible, although technically more difficult to obtain. NEURAL STIMULATION
Neurons are divided to two categories: myelinated, i.e., covered with a fatty sheath called myelin, and unmyelinated. Most human motor neurons in the cortex and large peripheral neurons are myelinated. The two categories of neurons differ in size (diameter) and electrical characteristics. Myelinated neurons are 2 to 20 p m in diameter, while unmyelinated neurons are typically 0.3 to 1.3 pm. Thresholds for depolarization (stimulation) depend on the neuron dimensions and passive electrical properties, and are described in terms of the space constant (also called the length constant and the characteristic length). The space constant for myelinated fibers is approximately equal to the internodal distance (for an accurate expression for the length constant see Eq. 12). The intermodal distance is approximately 200 times the inner diameter of the axon. The stimulation threshold is inversely proportional to the neuron diameter. The temporal response of neurons depends on their time constant, which depends on the membrane and axoplasm passive electric properties. Passive electrical behavior of neurons is described by the cable equation that has been used to compute the response characteristics of various neurons to internally or externally applied stimuli, by the current applied through extracellular electrodes positioned close to the neuron [McNeal, 1976; Plonsey and Altman, 1988; Rattay 1986, 1988; Reilly et al., 1985; Reilly 1988; Rubinstein and Spelman, 1988; Tranchina and Nicholson, 19861, and by the electric field induced by an externally applied magnetic field [Basser and Roth, 1991; Roth and Basser, 1990; Roth et al., 1990, 1991a, b]. Considering only linear passive properties of the membrane, the total current at the node n, as illustrated in Figurel, is given in terms of the Vi,n, the intracelVeJl 1
1
veJl+l OUTSIDE
9
l
v% Vi,"+l
+%fl
w
INSIDE (AXOPIASM)
-Ax-
Fig. 1. A cable model of the passive electrical properties of a myelinated nerve fibre [adapted from McNeal, 19761.
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lular potentials at nodes n, and Vn, the transmembrane potential at node n, [McNeal, 19761 (5) where the nodal capacitance is Cm= c m n d 6, , cmis the membrane capacitance per unit area, d, is the axon diameter at the node, 6 is the width of the nodal gap. The axial internodal resistance is rI = 4 p I A x / n d d 1p2,,is the resistivity of the axoplasm, Ax is the internodal distance. The current at node n is I,.,,= Vl/rm,the nodal membrane resistance is r,,, = ( g , n d l a)-',and gInis the membrane conductance per unit area. By substituting into Equation 5 , I,," = Vn/rmand V,,"= Vn + Ve,,, the following set of linear, first-order differential equations describing responses of the nerve fibre is obtained
for n = ... -2, -1, 0, 1, 2, ... For subthreshold stimuli, the membrane resistance, stant. Equation 6 can be rearranged as follows:
z,
can be assumed con-
indicating that the stimulation function can be considered as the second difference quotient of the node voltages Ve,n. For unmyelinated fibers this function is expressed as a second differential quotient [Rattay, 19871. Stimulation starts at the node at which Vc is maximum. Equation 7 indicates that the spatial derivative of the electric field along the axon axis is the main source of change in the transmembrane potential. The McNeal model was further extended by Reilly [1988, 19891 to include nonlinear behavior to several adjacent nodes. Reilly evaluated the strength-duration curves for several waveforms of the stimulus, including biphasic pulses and multiple pulses. The most interesting results for magnetic stimulation are summarized in Figure 2. Stimulation (membrane depolarization) thresholds are higher for an exponentially decaying stimulus with the time constant equal to the duration of a rectangular pulse. Similarly, the thresholds for a sinusoidal pulse are also higher. Multiple monophasic pulses result in lowering the thresholds with the amount of decrease dependent on the interpulse interval, pulse duration and number of pulses. In an axially uniform electric field, stimulation occurs at the axon's end. The minimum electric field required for stimulation decreases proportionally to the axon diameter. This can also be seen from Equation 7, since Ax, the internodal separation is proportional to the fiber diameter (typically Ax = 100 do, where do is fiber outer diameter, and d, = 0.6 to 0.7 d,,). The main limitation of the models mentioned above is that the myelin sheath i s assumed to be a perfect insulator, and therefore not contribute to the current across the membrane. Taking into account myelin conductivity, the transmembrane potential,
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1
10
lo2
lo5
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10'
Stimulus Duration or Time Constant us)
Fig. 2. Stimulation thresholds with a rectangular pulse (solid line), exponential pulse (dot-dashed line) and sinusoidal pulse (dotted line) for a 20 pn myelinated neuron as a function of pulse duration (rectangular, period (sinusoidal) or time constant (exponential) [adapted from Reilly, 19881.
V, along the axon can be described by a cable equation [Basser and Roth, 19911 as follows:
The space constant of the internodal region, Ao, is equal to:
and the time constant, zo, is
r,, = & , & ' P o
(10)
where p , is the resistivity of myelin, and E' is the dielectric constant of myelin. Basser and Roth [ 199 11 performed an analysis of neuron stimulation using a model comprising the myelin contribution (Eq. 8) and a nonlinear current source at the node of the highest external potential. Figure 3 illustrates the dependence of the threshold electric field gradient (taken with respect to the direction of the neuron axis) on the exponential pulse decay time for neurons of various diameters.
1
-
I
I
I
10"
100
10'
Q
?oJ
lo4
rc (ms)
Fig. 3. Stimulation thresholds vs. decay time constant for exponential pulses in neurons of three diameters, 5, 12.5, and 20 pm [adapted from Basser and Roth, 19911.
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If only subthreshold responses are of interest, and the distance over which the electric field varies is large compared with the neuron internodal separation, the set of equations describing an axon (Eq. 7) can be simplified to an equation of an equivalent uniform axon [Basser and Roth, 199 I]. When the membrane impedances are averaged over the nodal and internodal regions, and Rushton’s [ 195 I ] scaling laws for axon of different diameters (Add,, = 100, dl/dl,= 0.6) are used, a cable equation for subthreshold electromagnetic stimulation is obtained [Basser and Roth, 19911:
where the equivalent length and time constants are
z = ( c,,,+ 6 5 2 ~ , ~ ’ / S g,,, ) / (+ 6 5 2 / ~6),
where p , is the resistivity of axoplasm and gm is t h e membrane conductance per unit area. INDUCED ELECTRIC FIELDS Calculations of the electric field induced by a magnetic field and its spatial derivatives in the actual biological tissues comprising neurons are very complex and can only be accomplished by numerical methods requiring considerable computer resources. Even analyses of such simplified models as homogeneous cylinders and layered concentric spheres require about two hours of computational time on large computers [Roth et al., 1991a, b]. To be able to gain physical insight how the stimulating coil configuration and dimensions affect the spatial characteristics of the stimulus and to be able to optimize the coil geometry, we selected a simplified tissue model for analysis. The configuration of the model is shown in Figure 4. Since stimulating pulses contain frequencies usually below 10 kHz, the following quasi-static approximations are made: (i) the displacement current in tissue is negligibly small compared with the conduction current, (ii) the penetration (skin) depth is much greater than the distances of interest, and (iii) time-retardation of
Stimulating Coil Air
4=
Y Tissue-Air
Planar
Interface
////////////////////////////////////////////////// Homogeneous Tissue Half-Space
Fig. 4. A stimulating coil of an arbitrary shape and N turns above a tissue half-space.
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electromagnetic fields is negligible [Roth et al., 1991al. The electric field E induced in tissue consists of two components: the primary field E'' directly induced by the coil current, and the secondary field E" due to surface charges at the tissue-air interface. The primary electric field is
where po is the permeability of free space, A is the magnetic vector potential, d i is an infinitesimal element of the coil, r is the distance between the coil-element and the point where the electric field is calculated, and the line-integral is along the length of the coil. The secondary electric field is derived from an electric scalar potential, according to
w,
The function
w
-
E" =-Vv is a solution of Laplace's equation
(16)
v21y=o
(17) subjected to the boundary condition that the normal electric field is zero at the tissueair interface, i.e.,
where 2 is the unit vector normal to the interface. For the configuration shown in Figure 4 the terms in Equation 18 can be expanded using Bessel integrals and the corresponding coefficients can be equated, resulting in the following analytical expression for the electric field d 2 produced by the coil-element dt [Esselle and Stuchly, 19911:
where
dlx,dly,dlz are the x, y, z components of dt ,respectively, and i and j are the unit vectors in x- and y- directions. The total electric field produced by the coil is obtained by numerically integrating Equation 19 along the length of the coil.
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The spatial derivatives of the electric field in x- and y-directions, produced by the coil-element d , can be obtained from Equation 19 as
The spatial derivatives d Ex/ d x and dEy/ d y are obtained by numerically integrating above expressions along the length of the coil. Because of the boundary condition (Eq. 18) the electric field component normal to the interface (EL)is zero everywhere inside tissue. Therefore, the analysis and Equations 19, 22, and 23 also apply to heterogeneous multi-layered tissue, if all the interfaces between the layers are parallel to the tissue-air interface, and also as a limiting case, for heterogeneous biological systems where the conductivity varies continuously but only in the direction perpendicular to the tissue-air interface. Various coil configurations, including square, circular, double-square and double-circular coils, oriented parallel, at an angle, and perpendicular to the interface have been considered [Esselle and Stuchly, 19911. Figure 5 illustrates results for the electric field (E ) and its spatial derivative, dEy/dy, for a square coil presented as contour and $-dimensional surface plots. Since EYis symmetrical about the x-axis, only the y20 region is shown in the plots. A contour or 3D plot of Ex can be obtained by rotating the corresponding Ey-plot,clockwise, by 90". The spatial derivatives satisfy the relationship d Ex/ d x = - d Ey/ d y anywhere in the tissue halfspace, so d Ex/d x can also be obtained from Figure 5c, d. The contour interval on the graphs is 10% of the peak value, hence the smallest contours (closest to the coils) show 90% of the peak values. The electric field EYand its derivative for a double square coil are illustrated in Figure 6. The other component of the electric field, Ex,has four smaller maxima of 203.9 V/m at x = k3.04 cm, y = k2.56 cm. However, as indicated earlier, d E I / d x is always equal to - d E / d y everywhere in the tissue half space, despite the different spatial profiles of , E! and Ey. Figure 7a shows how the peak (maximum) value of ( d E x / d x ) or ( d EJd y) varies with depth, for circular, square and double-square coils. In order to -make the graphs universal, both axes have been normalized. The locations of the peak value are shown in Figure 7b and the widths of the 90% contour in the x- and ydirection are given in Figure 7c. It can be seen that double coils produce asymmetrical contours of 90% values of d Ex/ d x and d Ey/ d y.
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3
Fig. 5 . Induced electric field and its derivative for a square coil of a dimension a = 5 cm, centered at x = 0, y = 0 (Fig. 4), parallel to the interface; the coil has 10 turns, and carries current rising at 100 A/ ps. A: Contours of Eyon a plane in tissue 1 cm below the coil; max E = 236 V/m at x = +2.5 cm, = 0 . B: 3-dimensional plot of Ev. C, D: a( d Eyld y) in tissue in a plane 1 cm below the coil, max 374 V/m at x = y'= k2.5 cm.
I ?I
DISCUSSION
The spatial derivative of the electric field in the axial direction along an axon, as evident from Equation 8, is the critical parameter for magnetic field induced threshold stimulation. The analysis of fields induced in tissue by various coils leads to several observations that are important from the dosimetnc point of view. For all coil configurations considered in our analysis, d Ex/dx is always equal to - d Ey/ d y. The maximum of the derivative for some coil configurations is located where the field is equal to zero (e.g., for a single loop perpendicular to the air-tissue interface). When considering the placement of the coil with respect to the neuron to be stimulated, the 90% contour widths (Figure 7c) can be considered as somewhat arbitrarily selected metrics for the region of subthreshold stimulation. For most coil configurations the widths in the x and y directions are approximately equal; however a square coil perpendicular to the interface produces an elongated contour. If a minimum number of neurons is to be stimulated, the coil should be oriented parallel to the neuron. The main disadvantage of this configuration is that the magnitude of the field derivative is much lower than for the same coil parallel to the interface.
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Fig. 6 . Induced electric field and its derivative for a double square coil with each side dimension a = of turns 20, current 100 Alps. A, B: Ey, rnax IEy = 470 Vlm. C, D: a( J Eyl d y), = 748 Vlm at x = 0, y = k2.5 cm.
I
Single coils parallel to the interface produce a total of four “hot spots” (Fig. 5). It is apparent, that the same neuron is at one point depolarized (where the derivative is negative) and hyperpolarized at a distant point depending on the coil dimensions. In practice, this means the direction of current in the coil must be properly selected for a given neuron so that the action potential can propagate. This problem can be avoided for double coils in one direction, and for quadruple coils in both directions, as in the latter case only one hot spot is produced. Subthreshold electromagnetic stimulation, as described by Equation 1 1, can be compared to other types of stimulation, namely the stimulation by extracellular electrodes or by a microelectrode. The factor on the right side of the equation, i.e., A2( d Ex/ d x), is analogous to - A2( d *V! d xz) for extracellular electrodes separated from the membrane, where Ve is the extracellular potential produced by the electrodes [Rattay 1986, 19881, and to - A2raifor intracellular microelectrodes, where ra is the resistance per unit length of the axoplasm and i is the inward applied current per unit length [Plonsey, 19691.
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0.2
0.4
0.6
0.8
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1.O
h/a
Fig. 7. The normalized maximum values of the electric field derivatives (a), their locations (x,,,. y,,,) (b), and the size of the region within which the value is 90% of the maximum (Ax, By) (c); a is the dimension of one coil, h is the separation between the coil and the plane at which the value of the normalized d E l a x is given, po= 4 n x lo7H/m, N is the total number of turns, dI/dt is the rate of change of coil current.
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Three simplifying assumptions have been made in all our considerations. The neuron is assumed to be placed i n a homogeneous, isotropic conductive medium of high conductivity. This in many cases is not a correct representation, and complexities resulting from the fact that the neuron is in a bundle with other neurons must be considered, as has been evaluated for stimulation with extracellular electrodes [Altman and Plonsey, 19881. The electric field is assumed to be unperturbed by the neuron. This assumption is correct in a simplified case analyzed here of a semi-infinite tissue half-space with no changes in electrical properties other than those perpendicular to the air-tissue interface and the neuron axis. The induced transmembrane potential is assumed not to vary across the axon cross-section. The last assumption is reasonable, provided the electric field is not significantly altered by the presence of the neuron. CONCLUSIONS
In numerous publications on diagnostic applications of magnetic stimulation it is stipulated that the induced electric field is the key dosimetric parameter responsible for stimulation. On the other hand, neural behavior modeling and neurophysiological data clearly indicate that t h e spatial derivative of the electric field in the direction along the axon is the parameter quantitatively describing subthreshold stimulation. The success of clinical applications, despite the limitations of the physical models employed, can at least partly be explained by the use of stimuli well above threshold. Furthermore, neurons are not straight cables, which may aid stimulation in a uniform field. Our analysis of a simplified model leads to several general conclusions which are likely to hold even for more complex models and provide at least qualitative understanding of neural stimulation with magnetic fields. For instance, the electric field component perpendicular to the tissue boundary is also zero for the spherical tissue model and very small for the cylindrical tissue model [Roth et al., 1991bl. The decrease of the field derivative with the distance from interface and the number of hot spots are general enough to be applicable to other configurations. The main limitation of the simplified analysis is the neglection of tissue heterogeneity in the direction parallel to the interface. Such variations in tissue conductivity has previously been shown to affect the magnitude and direction of induced electric fields [Polk and Song, 19901. ACKNOWLEDGMENTS
This work was performed while the authors were with the Bureau of Radiation and Medical Devices, Health and Welfare. The authors are grateful to Dr. Bradley Roth from the U.S. National Institute of Health for supplying prepublication copies of his manuscripts. We also thank Dr. Stan s. Stuchly of the University of Victoria for stimulating discussions and advice, and Pave1 Dvorak and Jack McLean of Health and Welfare Canada for their review of the manuscript. Constructive comments provided by unnamed reviewers are appreciated.
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List of Symbols membrane capacitance per unit area nodal capacitance axon diameter nerve diameter, external-axon plus myelin electric field strength along the axis of a nerve membrane conductance per unit area inwards current from a microelectrode per unit length membrane current at n-th node resistance per unit length of axoplasm stimulating coil radius axial internodal resistance nodal membrane resistance transmembrane potential external nodal potential, n-th node internal nodal potential, n-th node membrane resting potential threshold membrane potential direction along the nerve axis internodal distance width of the node the permittivity of vacuum myelin dielectric constant space constant of an (equivalent) axon myelin space constant resistivity of axoplasm resistivity of myelin time constant of an (equivalent) axon myelin time constant stimulus duration
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Day BL, Dressler D, De Noordhout AM, Marsden CD, Nakashima K, Rothwell JC, Thompson PD (1989): Electric and magnetic stimulation of human motor cortex: Surface emg and single motor unit responses. J Physiol 41 2:449-473. Esselle, KP, Stuchly MA (1992): Neural stimulation with magnetic fields: Analysis of induced electric fields. IEEE Trans Biomed Eng 39: 693-700. Freeston IL, Barker AT, Jalinous R, Polson MJR (1984): Nerve stimulation using magnetic fields. Proc 6th Annual Conf of IEEE Eng in Med Biol (Semmlow JL, Wellowitz W, eds, pp 557-561. Geddes LA ( I 987): Optimal stimulus duration for extracranial cortical stimulation. Neurosurgery 20:9499. Geddes LA (1988): Stimulation of excitable tissue with time varying magnetic fields. Proc. 10th Annual Intern Conf IEEE Eng Med Biol SOC918-921. Journal of Clinical Neurophysiology (199 I): Special issue on magnetic stimulation of the nervous system. 8:no. 1. McNeal DR (1976) Analysis of a model for excitation of myelinated nerve. lEEE Trans Biomed Eng BME-231329-337. Mouchawar G, Bourland JD, Voorhees WD, Geddes LA (1990): Stimulation of inspiratory motor nerves with a pulsed magnetic field. Med Biol Eng Comput 28:613. Plonsey R, (1969): "Bioelectric Phenomena." New York: McGraw-Hill. Plonsey R, Altman KW (1988): Electrical stimulation of excitable cells - A model approach. Proc IEEE 76: 1122-1 129. Polk C, Song JH ( 1 990): Electric fields induced by low frequency magnetic fields in inhomogeneous biological structures that are surrounded by an electric insulator. Bioelectromagnetics I 1 :235249. Rattay F ( 1 986): Analysis of models for external stimulation of axons. IEEE Trans Biomed Eng BME33:974-971. Rattay F (1987): Ways to approximate current-distance relations for electrically stimulated fibers. J Theor Biol 125:339-349. Rattay F (1988): Modelling the excitation of fibers under surface electrodes. IEEE Trans Biomed Eng 35: 199-202. Reilly JP, Freeman VT, Larken WD (1985) Sensory effects of transient electrical stimulation -Evaluation with a neuroelectric model. IEEE Trans Biomed Eng BME-32:1001-1011. Reilly JP (1988) Electrical models for neural excitation studies. Johns Hopkins APL Technical Digest 9:44-59. Reilly JP ( 1989) Peripheral nerve stimulation by induced electric currents: Exposure to time-varying magnetic fields. Med Biol Eng Comput 27:lOl-1 10. Roth BJ, Basser PJ (1990): A model of the stimulation of a nerve fiber by electromagnetic induction. IEEE Trans Biomed Eng 37588-597. Roth BJ, Cohen LG, Hallett M, Friauf W, Basser PJ (1990): A theoretical calculation of the electric field induced by magnetic stimulation of a peripheral nerve. Muscle Nerve 13:734-741. Roth BJ, Cohen LG, Hallett M (1991a): The electric field induced during magnetic stimulation. Electroencephalogr Clin Neurophysiol (accepted for publication). Roth BJ, Saypol JM, Hallett M, Cohen LG, (I991 b): A theoretical calculation of the electric field induced in the cortex during magnetic stimulation. Electroencephalogr Clin Neurophysiol 8 I :47-56. Rubinstein JT, Spelman FA (1988): Analytical theory for extracellular electrical stimulation of nerve with focal electrodes. 1. Passive unmyelinated axon. Biophys J 54:975-981. Rushton WAH (1951): A theory of the effects of fibre size in myelinated nerve. J Physiol (London) I15:101-122. Tranchina D, Nicholson C (1986) A model for the polarization of neurons by extrinsically applied electric fields. Biophys J SO: 1139-1 156.
