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Relevance of stimulus duration for activation of motor and sensory fibers: implications for the study of H-reflexes and magnetic stimulation
Electroencephalography and clinical Neurophysiology, 85 (1992) 22-29 © 1992 Elsevier Scientific Publishers Ireland, Ltd. 0924-980X/92/$05.00
22
ELMOCO 91035
Relevance of stimulus duration for activation of motor and sensory fibers: implications for the study of H-reflexes and magnetic stimulation * Marcela Panizza, Jan Nilsson, Bradley J. Roth, Peter J. Basser and Mark Hallett Human Motor control Section (M.H.), National Institute of Neurological Disorders and Stroke, National Institutes of Health, Bethesda, MD (U.S.A.); EMG Laboratory (M.P. and ZN.), Fondazione Clinica del Lavoro, Campoli MT (BN) (Italy); and Biomedical Engineering and Instrumentation Program (B.J.R. and P.J.B.), National Institutes of Health, Bethesda, MD (U.S.A.) (Accepted for publication: 15 July 1991)
Summary
Electric stimuli with durations of 0.5-1.0 msec are optimal for studies of H-reflexes. It is more difficult to obtain H-reflexes with shorter duration stimuli or with magnetic stimulation. In order to understand this behavior, we studied the excitation thresholds for motor and sensory fibers in the ulnar, median and tibial nerves using both electric and magnetic stimulation. For short duration electrical stimuli (0.1 msec) the threshold for motor fibers is lower than for sensory fibers. For longer duration electric stimuli (1.0 msec) the threshold for sensory fibers is lower. For magnetic stimulation the threshold for motor fibers is much lower than for sensory fibers. Thus, stimulus duration is a critical parameter for sensory fiber excitation, and current magnetic stimulators are not optimal.
Key words: Threshold; Stimulus duration; Motor fibers; Sensory fibers; Magnetic stimulation; H-reflexes
When stimulating peripheral nerves electrically, the relative recruitment 'of sensory and motor fibers depends on the stimulus duration. Erlanger and Blair (1938) first showed that threshold for sensory fibers is lower than for motor fibers for long stimulus durations and that this relation reverses for short stimulus durations. Their results were based on a study of the strength duration curve of the sciatic nerve in a bull frog. Veale et al. (1973) showed the same phenomenon in motor and sensory fibers in the ulnar nerve in humans. Paillard (1955) described the preferential activation of the H-reflex over motor fibers in the tibial nerve in humans when using a stimulus duration of 1 msec, and the inverse when using a stimulus duration of 0.1 msec. Recently, Panizza et al. (1989) demonstrated that stimulus duiations shorter than 0.2 msec were not optimal to elicit H-reflexes either in the tibial or median nerve because the motor fibers were often activated before or simultaneously with the H-reflex. Furthermore, the recruitment of sensory and motor fibers depends on the shape of the stimulation pulse, since for short duration stimuli changing the stimulus from a monophasic to a biphasic pulse changes the
Correspondence to: Mark Hallett, NINDS, NIH, Building 10, Room 5N226, Bethesda, MD 20892 (U.S.A.). * Presented in part at the joint Symposium AAEE-AEEGS, San Diego, CA, October 1988.
excitability pattern for different fiber diameters (Reilly et al. 1985). Recently, Olney et al. (1990) found that electrical stimulation (duration unspecified) at 25% of maximal activated sensory fibers preferentially to motor f~ers, but magnetic stimulation at 25% of maximal did not share this effect. Studies of threshold magnetic stimulation have not previously been done. The purpose of this study is (1) to highlight by examples that the thresholds of the sensory and motor fibers vary as a function of duration of the electrical stimulation, (2) to study the relative recruitment of sensory and motor fibers using magnetic stimulation, and (3) to use our experimental results and a mathematical model to compare electric and magnetic stimulation.
Material and methods Six healthy volunteers (aged 22-51 years, mean 34 years) were studied by stimulating the ulnar nerve when recording motor and sensory fiber responses. The median nerve was studied in 8 healthy volunteers (aged 28-36 years, mean 31 years) by recording motor, sensory and H-reflex responses, and the tibial nerve was studied in 8 healthy volunteers (aged 26-39 years, mean 33 years) recording motor fiber responses and H-reflexes. All volunteers gave their informed consent for participation in this study.
MOTOR AND SENSORY FIBERS
Stimulation Electric. Constant current stimuli (Dantec 15E07), with durations from 0.05 to 1 msec (square wave), were delivered from surface electrodes mounted in a plastic mold (2.5 cm between anode and cathode) and placed at the elbow for ulnar nerve stimulation, at the cubital fossa for the median nerve, and at the popliteal fossa for the tibial nerve stimulation. Magnetic. A time-varying magnetic field, produced by either a Dantec magnetic stimulator (Dantec Electronics, Tonsbakken 16-18, DK-2740 Skovlunde, Denmark) or a Cadwell MES-10 Magnetic Stimulator (Cadwell Laboratories, Inc., 909 N. Kellogg Street, Kennewick, W A 99336, U.S.A.), was used to induce current near the ulnar, median or tibial nerve. For the Dantec stimulator 3 prototype coils of different inductances (5.8, 50, and 160/xH) were used; the durations of the induced currents, measured from the start of the pulse to the end of its first phase, were 105, 294, and 574 ~sec, respectively. The coils were all tightly wound flat spirals with inner diameters of 7 cm and outer diameters ranging from 9.7 to 12.3 cm. For the Cadwell stimulator a coil with an inductance of 20 /zH was used, producing a stimulus duration of 80 ~sec measured to the end of the first phase (920 ~sec measured to the end of the underdamped oscillating wave form), This coil was tightly wound and circular, with an inner diameter of 4.6 cm and outer diameter of 9.2 cm. The outer edge of each coil was placed over the optimal position for electrical stimulation, with the plane of the coil tangent to the axis of the arm or leg, and with the handle perpendicular to the axis of the limb.
23
case, a ground electrode was placed just distal to the stimulating electrode.
Threshold criterion The threshold for sensory fibers was defined as the minimum stimulus current required to evoke a recognizable sensory potential with an amplitude of at least 0.5 /xV, after averaging from 10 to 50 responses. In determining the threshold of motor fibers an amplitude criterion of 25 tzV was used.
Results
Ulnar nerve Electric stimulation.
For a stimulus duration of 0.1 msec the threshold for sensory fibers was 6 + 26% (mean + S.D.) higher than for motor fibers. For a stimulus duration of 0.3 msec the threshold was 11 + 18% lower for sensory fibers than for motor fibers, a behavior already reversed with respect to 0.1 msec. For a 1 msec duration stimulus the difference was even larger, the threshold for sensory fibers being 27 + 15% lower than for motor fibers (Fig. 1). Magnetic stimulation. Responses from sensory and motor fibers were recorded in all subjects. When using the Cadwell stimulator the threshold for sensory fibers was 107 + 91% higher than for motor fibers, whereas when using the Dantec stimulator with the coil producing the shortest duration pulse, the threshold for sensory fibers was 52 + 38% higher than for motor fibers.
Median nerve Electric stimulation. Recordings Sensory fibers.
Two ring electrodes were placed 1 cm apart near the tip of digit V when recording from the ulnar nerve, and on digit III when recording from the median nerve, Motor fibers. For recording from muscles innervated by the ulnar nerve, a hooked-wire electrode was inserted in the abductor digiti quinti (ADQ) referenced to a surface electrode on digit II, or a surface electrode was placed over the belly of A D Q referenced to a surface electrode on digit V. For recording from muscles innervated by the median nerve, a hooked-wire electrode was inserted in the abductor pollicis brevis (APB) referenced to a surface electrode on digit V, and a hooked-wire electrode ~,as inserted in flexor carpi radialis (FCR) referenced to a surface electrode placed 4 cm away, or surface electrodes were placed 4 cm apart over FCR and over the belly of APB referenced to the 2nd phalange of the thumb. For recordings from muscles innervated by the tibial nerve, surface electrodes were placed 4 cm apart over the soleus muscle with the active electrode proximal. In each
For stimulus durations equal to or less than 0.3 msec the threshold for sensory fibers was higher than for motor fibers, being 56 + 49% higher at 0.1 msec, and 24 + 47% higher at 0.3 msec. For stimulus durations equal to or greater than 0.5 msec the threshold for sensory fibers was lower than for motor fibers, being 2 + 35% lower for 0.5 msec, and 21 + 15% lower for 1 msec (Fig. 2). For a stimulus duration of 0.1 msec the threshold for H-reflexes was 49 + 56% higher than for motor fibers, whereas at a stimulus duration of 1 msec the threshold for H-reflexes was 14 + 23% lower than for motor fibers (Fig. 2). In 3 of the 8 subjects a stimulus duration of 0.05 msec was used, and the thresholds for sensory fibers and H-reflexes were 63 + 13% and 85 + 23% higher than for motor fibers. Magnetic stimulation. Studies were done only with the Dantec magnetic stimulator, and it was possible to obtain sensory and motor responses in all subjects. The threshold for sensory fibers was 103 + 71% higher than for motor fibers, the range being the same no matter which coil was used. Also H-reflexes were recorded in all the subjects, and the threshold for H-reflexes was
24
M. PANIZZA ET AL.
110 + 100% higher than for motor fibers, independent of the pulse duration,
Tibial nerve Electric stimulation. For a stimulus duration of 0.1 msec the threshold for H-reflexes was 12 + 15% higher than for motor fibers; for a duration of 1 msec it was 16 ± 23% lower, Magnetic stimulation. H-reflexes were evoked by the Cadwell or the Dantec magnetic stimulator, and recorded in all the subjects. When using the Cadwell stimulator the H-reflex threshold was 133 ± 24% higher than for motor fibers, whereas for the Dantec magnetic stimulator the H-reflex threshold was 140 ± 54% higher than for motor fibers, independent of the pulse duration. Sensory fiber response compared to H-reflex response for stimulus intensities just below motor threshold It is often desirable to produce H-reflexes without an M-response. The ability to produce such an H-reflex should depend on the ability to stimulate Ia sensory afferents without stimulating motor fibers. Thus, the maximum amplitude of an H-reflex produced at a particular electrical stimulus duration should be predictable on the basis of the amplitude of the sensory nerve action potential produced just below motor threshold. Fig. 3A shows the amplitude of the mean Ulnar
sensory fiber response in the median nerve (digit III) for stimuli just below motor threshold (flexor carpi radialis) as a function of stimulus duration. Fig. 3B shows the recruitment curve of the H-reflex amplitude (flexor carpi radialis) just below motor threshold for different durations of the electrical stimuli to the median nerve. Sensory fibers and H-reflex show the same tendencies when changing stimulus durations.
Is there a difference in recruitment when eliciting responses by different durations and wave shapes of magnetic stimulation? No differences were found in the relative thresholds of motor and sensory fibers when using the Dantec magnetic stimulator with each of its 3 coils, and therefore the relative recruitment of motor and sensory fibers was independent of stimulus duration (Fig. 4A). Threshold for motor fibers occurred at a setting of 15-20% on the magnetic stimulator, resulting in a magnetic field changing at a peak rate of 4.1, 2.1 and 1.2 k T e s l a / s e c respectively for the coils with durations of 105, 294 and 574 /zsec (the stimulus intensity is proportional to the rate of change of the magnetic field). Apparently the increasing duration of the stimulus was compensated by the decreasing intensity. In 4 subjects, results obtained using the Cadwell stimulator (oscillating wave s h a p e ) w e r e compared to those found using the Dantec stimulator (triangular nerve
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Fig. 1. A: responses from one healthy volunteer evoked by electric or magnetic stimulation of the ulnar nerve at the elbow while recording from abductor digiti V and digit V. At 0.1 msec stimulus duration the motor threshold is lower than the sensory threshold, similar to responses elicited by magnetic stimulation. At 1 msec stimulus duration the sensory threshold is lower than motor threshold. B: the mean normalized threshold intensity as a function of the electric stimulus duration for the sensory and motor responses in the ulnar nerve.
MOTOR AND SENSORY FIBERS
25
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Duration (ms) Fig. 2. A: responses from one healthy volunteer evoked by electric or magnetic stimulation of the median nerve at the cubital fossa while recording from APB, FCR and digit Ill. At 0.I msec stimulus duration the motor threshold is lower than the sensory threshold, and a motor response can be seen in APB. At I msec stimulus duration an H-reflex can be seen in FCR, and a sensory response in digit Ill without motor response. For magnetic stimulation the motor threshold appears below the sensory and H-reflex thresholds, and the motor responses can he seen in both APB and FCR. B: the mean normalized threshold intensity as a function of the electric stimulus duration for sensory, H-reflex and motor responses. Sensory and H-reflex thresholds behave in the same way all along the curve.
w a v e s h a p e ) w i t h t h e coil p r o d u c i n g t h e s h o r t e s t d u r a t i o n p u l s e (Fig. 4B). T h e s e r e c o r d i n g s w e r e p e r f o r m e d in 4 o f t h e s u b j e c t s . S e t t i n g s o f 4 5 - 6 0 % o n t h e m a g n e t i c s t i m u l a t o r s r e s u l t e d in c a l i b r a t e d s t i m u l u s i n t e n sities o f a b o u t 17.0 k T e s l a / s e c f o r t h e C a d w e l l s t i m u l a -
tor, a n d tor. Ulnar Cadwell w a s 150
a b o u t 12 k T e s l a / s e c
nerve. W h e n r e s p o n s e s w e r e e v o k e d by t h e stimulator, the threshold for sensory fibers _+ 7 0 % h i g h e r t h a n f o r m o t o r fibers. W h e n
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Duration (ms) Fig. 3. Median nerve: histogram (A) of the mean sensory amplitude (digit l i d below motor threshold (Tin) as a function of the electrical stimulus duration, and histogram (B) of the mean H-reflex amplitude below motor threshold (flexor carpi radialis) for the same stimulus durations. The two histograms show similar behavior for all stimulus durations and have maximal values between 0.5 and I msec duration of the stimuli.
26
M. PANIZZA ET AL.
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F~. 4. A: responses from one normal volunteer evoked by magnetic stimulation of the median nerve at the cubital fossa using different durations of the stimulus (DantCc m ~ t ~ stimulator), while recording motor from F C R and sensory from digit III. Under the recordings of the motor and sensory responses; traces resulting from the induced voltage for the different durations (obtained at a setting on the magnetic stimulator of 30%) can be seen. B: responses from the same control recorded when using the two different magnetic stimulators. On the left the Cadwell magnetic stimulator (biphasic wave shape), and on the right the Dantec stimulator (monophasic wave shape). The recordings show the presence of sensory, motor and H-reflex responses, and under these traces can be seen recordings of the induced voltages for the two different stimulators, measured at a setting on the magnetic stimulator of 30%. In the bottom part of the figure are 2 histograms showing the' mean normalized threshold intensity for the motor and sensory thresholds for each magnetic stimulator.
MOTOR AND SENSORY FIBERS responses were evoked by the Dantec magnetic stimulator, the threshold for sensory fibers was only 52 + 38% higher than for motor fibers. For both stimulators the thresholds for motor fibers were similar, at a setting of 2 0 - 2 5 % on the magnetic stimulators, Tibial nerve, H-reflex. The threshold for H-reflexes was 133 + 24% higher than for motor fibers when responses were elicited by the Cadwell stimulator, and 112 + 85% higher when responses were elicited by the Dantec stimulator. For both stimulators the threshold for motor fibers was the same, at a setting of 20-25% on the magnetic stimulators,
Discussion
When using electric stimulation, preferential activation of the sensory fibers over the motor fibers was observed for long stimulus duration and the inverse behavior for short stimulus duration in the median, ulnar, and tibial nerves. The point of reversal in the behavior was slightly different between individuals and between different nerves in the same individual, ranging from a stimulus duration of 0.2 msec for the ulnar nerve to 0.5 msec for the median nerve, The strength-duration relationship for a given fiber can be characterized by the rheobase stimulus strength and the chronaxie duration. The difference in stimulus threshold at long durations (the difference in rheobase) may be due to sensory fibers having a larger diameter than motor fibers. McNeal's model of electric stimulation indicates that threshold varies inversely with fiber diameter for a given pulse duration (McNeal 1976). Basser and Roth (1991) predicted that threshold is inversely proportional to the square of the fiber diameter during magnetic stimulation. Indirect evidence of differences in diameter between motor and sensory fibers is that the sensory conduction velocity in the periphery is significantly faster than the motor conduction velocity (Kimura 1983, 1984). The dependence of the relative recruitment of sensory and motor fibers on stimulus duration cannot be explained by a difference in fiber diameter, or rheobase, To explain this behavior, the strength-duration curves for sensory and motor fibers must have different chronaxies. One explanation of the different chronaxies is that the two classes of fibers have different membrane time constants (see Appendix). Barker et al. (1990) have reported preliminary data that the time constant for ulnar nerve motor fibers was about 150 tzsec and concluded that the optimal stimulus duration for motor fibers should be short. There is evidence that the membrane properties of sensory and motor fibers differ (Neumcke et al. 1980; Palti et al. 1980), although we know of no experimental measurements comparing time constants in the two fiber classes. Thus the lower
27 threshold of the m o t o r fibers with short stimulus durations remains unexplained. The behavior of the H-reflex and the sensory response was similar over a wide range of stimulus durations (Fig. 2B). Furthermore, when analyzing the amplitudes of the sensory potential and H-reflex below motor threshold in the median nerve both amplitudes increased with stimulus duration (Fig. 3) (Panizza et al. 1989). The similar dependence of the sensory fibers and the H-reflexes on stimulus duration can be interpreted as the activation of the H-reflex depends on the activation of the Ia fibers and the sensory threshold depends on the activation of the largest cutaneous fibers the diameter of which is probably similar to Ia afferents. The hypothesis that fibers of similar diameter react the same way to different stimulus durations can explain the fact that the behavior of sensory threshold in relation to stimulus duration can be used to predict the behavior of H-reflex and vice versa. When using magnetic stimulation the motor threshold was lower than the sensory and H-reflex thresholds for all the nerves studied and for all the coils used. These results may look contradictory since a magnetic stimulus duration of about 0.6 msec behaves in the same way as an electric stimulus duration of 0.1 msec. In order to understand such a behavior it is necessary to analyze the wave form of the magnetic stimulus. As can be seen in Fig. 4A the wave form of the induced current produced by the Dantec magnetic stimulator is nearly triangular. Both the duration and pulse shape influence the strength-duration curve. In the Appendix it is shown that a triangular pulse must have a duration over 3 times as long as a square pulse to produce a similar strength-duration curve. Because of this difference between the electric and magnetic strength-duration curves, the "measured duration" of a triangular wave shape must be divided by a constant of 3.625 in order to obtain the "effective duration" comparable to a square wave pulse (see Appendix). This conversion reduces the duration of the magnetic stimulus from 105, 294 and 574/xsec to 29, 81 and 158/.tsec, respective!y. Therefore considering these new "effective durations" of the magnetic stimulus, it is reasonable that all 3 Dantec coils behave in a way similar to an electric stimulus shorter than 0.2 msec. The difference of intensity necessary to stimulate sensory fibers and to obtain H-reflexes was larger with the magnetic stimulator using a oscillating wave form than with the stimulator using a nearly triangular wave form, even when sensory potentials or H-reflexes could be elicited at higher intensities with both of them; This observation suggests once more an influence of the magnetic wave shape for the threshold of different fibers. Early studies with electrical stimulation showed that biphasic stimuli alter the threshold of fibers. The alteration depends on the diameter of the fiber, the
28
M. P A N I Z Z A E T AL.
duration of the stimulus, and the delay between each phase of the biphasic stimulus. The alterations are maximal with a shorter stimulus duration, the shorter the delay between the stimuli, and for larger diameter fibers (Gorman et al. 1980; Reilly et al. 1985). Thus biphasic stimulation requires a higher stimulus level to be effective for the larger diameter sensory fibers, but requires a comparable stimulus level for motor fibers. In conclusion, magnetic stimulation can activate motor fibers easily, but is at a disadvantage in stimulating sensory fibers, and this is particularly true for magnetic stimulation with an oscillatory wave form.
Appendix The magnetic stimulation strength-duration curve In this appendix we use a mathematical model to show that the strength-duration curves produced by electric and magnetic stimulation are different. Specifically, electric and magnetic stimulation result in strength-duration curves with identical rheobases but different chronaxies. We base our calculations on the model of magnetic stimulation derived by Roth and Basser (1990). We assume that the wave form of the induced current is triangular; a more general treatment can be found in Basser and Roth (1991). First, let us review how the strength-duration curve can be derived for electric stimulation (Plonsey 1969). The tronsmembrane potential, V, along a spaceclamped ~ axon as a function of time, t, is governed by the differential equation dV ~---~-+V=S(t) (1) where S(t) is the stimulus strength as a function of time and z is the membrane time constant. Consider a square pulse of duration d, so that S(t) is S(t)=(~0
0
0
(3)
1 This analysis also holds if the axon is not space clamped but the spatial variations in the stimulus are small over distances on the o r d e r of a length constant. This is the case in most applications of magnetic stimulation. It also holds for myelinated fibers if the spatial extent of the induced current is large compared to the distance between nodes (Basser and Roth 1991). In that case, the m e m b r a n e properties must be taken as the average properties along the fiber (Andrietti and Bernardini 1984).
The membrane potential reaches a maximum at the end of the stimulus, t = d. In Eq. (3) let us set V equal to the threshold potential, Vthr~shoL~,and t equal to the pulse duration. The resulting equation between S O and d is called the "strength-duration curve"
Vthreshold SO
. (4) 1 - e -d/* For durations d that are long compared to z, the strength So is a constant (So = Vthresho~d), called the rheobase stimulus. For durations that are short compared to ~-, the strength-duration curve reduces to S O= Vthr,shold •'/d, so that the product of the stimulus strength and duration is constant. Thus, for short duration stimuli it is the area under the stimulus curve, not its amplitude or duration individually, that determines whether threshold is reached. The chronaxie, dchronaxie, is a measure of the duration at which the strengthduration curve changes from its long duration to short duration behavior. Chronaxie is defined as the stimulus duration required to activate the nerve with a stimulus strength equal to twice rheobase. For the strengthduration curve in Eq. (4), dchronaxi~ can be calculated analytically dchrona~i ~ = --Z In(0.5) = 0.693 ~-. (5) Having reviewed the well-known properties of the electric strength-duration curve, we now turn our attention to the analogous relationship in magnetic stimulation. We assume that the main stimulus pulse can be represented as a triangular function; the induced current has a very fast rise and then fails linearly to zero 2. We can express the triangular form of the stimulus wave form, S(t), as S(t) =
{~(d) 0 1-
otherwise / 0< t
(6)
Eq. (1), governing the t r a n s m e m b r a n e p o t e n t i a l a s a function of time, can be solved analytically using the stimulus in Eq. (6) and initial condition V(0) = 0, yielding [ ~._t ( d ) ] V(t) = S o 1 + T - 1+ e -t/r . (7) To obtain the strength-duration curve, in Eq. (7) we set V equal to Vthr~ho~d and t equal to the time that V(t) reaches its maximum, tm~. Unlike electrical stimulation, V(t) does not reach a maximum at the end of
2 Actually the fall is not quite linear and there is always a second phase of the induced current of opposite polarity that follows the first phase. If the second phase has a low amplitude and a duration that is longer than ~-, then it does not stimulate the axon.
MOTOR AND SENSORY FIBERS
29
the pulse. To find tmax, we set d V / d t equal to zero, finding
If we substitute this result into Eq. (7) and rearrange terms, we get the strength-duration curve appropriate for magnetic stimulation Vthreshold
So=
1 - -~ In --~. + 1
)
S 0 = 2 Wthreshold q'//d. T h e f a c t o r o f 2 a r i s e s b e c a u s e a
magnetic stimulus pulse of amplitude SO and duration d has half the area under the curve as does the electric stimulus pulse of the same strength and duration. For d >> z we obtain the same rheobase condition, S O= Wthreshold, as f o r electric stimulation. Finally, to deter-
mine the chronaxie for magnetic stimulation w e set the strength equal to twice rheobase in Eq. (9) and solve for the duration corresponding to chronaxie, dchronaxi e. We find that dchronaxi e satisfies the equation ( dchronaxie ) + 1 1" '
In 21"
(10)
which can be solved numerically, yielding dchronaxi e = 2.513 t".
(11) The chronaxie for electric stimulation was 0.693 ~-, so that the ratio for the magnetic to electric chronaxie for the same membrane time constant is 3.625. The magnetic strength duration curve is much wider than 100
~ g n e t i c 10 So
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1 I 0.1
We thank Dr. Roger Gilliatt and Dr. I. Tasaki for helpful discussions. The prototype coils and contributions from Mr. Stig Wanding Andersen and Mr. Maurizio Bonneau from Dantec Electronics, Inc. and Dr. John Cadwell from Cadwell Laboratories, Inc. are acknowledged.
References
For d << ~" the strength-duration curve reduces to
dchronaxie
the electric curve, as shown in Fig. 5. In essence, an electric stimulus of duration d acts like a magnetic stimulus with over 3.5 times the duration.
t 1 "
I 10
I 100
d/x Fig. 5. Strength-duration curve for an electric stimulus (rectangular wave shape), and a magnetic stimulus (assumed triangular wave shape). S o is the stimulus strength divided by the rheobase stimulus strength, and d/~" is the pulse duration divided by the time constant of the axon membrane.
Andrietti, F. and Bernardini, G. Segmented and "equivalent" representation of the cable equation. Biophys. J., 1984, 46: 615-623. Barker, A.T., Freeston, I.L. and Garnham, C.W. Measurement of cortical and peripheral neural membrane time constants in man using magnetic nerve stimulation. J. Physiol. (Lond.), 1990, 423:
66P. Basser, P.J. and Roth, B.J. Stimulationof a myelinatednerve axon by electromagnetic induction. Med. Biol. Eng. Comput., 1991, 29: 261-268. Erlanger, J. and Blair, E.A. Comparative observations on motor and sensory fibers with special reference to repetitiousness. Am. J.
Physiol., 1938, 121: 431-453. Gorman, P.H., Mortimer, J.T. and Van den Honert, C. Enhancing excitability differences between nerve fibers. Proc. 33rd ACEMB, Washington, DC, 30 September-3 October, 1980: 63. Kimura, J. Electrodiagnosis in Diseases of Nerve and Muscle: Principies and Practice. Davis, Philadelphia, PA, 1983. Kimura, J. Nerve conduction studies and electromyography. In: P.J. Dyck, P.K. Thomas, E.D. Lambert and R. Bunge (Eds.), Peripheral Neuropathy, Vol. I. Saunders, Philadelphia, PA, 1984: 919966. McNeal, D.R. Analysis of a model for excitation of myelinated nerve.
IEEE Trans. Biomed.Eng., 1976, 23: 329-337.
Neumcke, B., Schwarz, W. and Stampfli, R. Differences between K channels in motor and sensory nerve fibers of the frog as revealed by fluctuation analysis. Pfliigers Arch., 1980, 387: 9-16. Olney, R.K., So, Y.T., Goodin, D.S. and Aminoff, M.J. A comparison of magnetic and electrical stimulation of peripheral nerves. Muscle Nerve, 1990, 13: 957-963. Paillard, J. R6flexes et R6gulations d'Origine Proprioceptive chez l'Homme. Arnette, Paris, 1955. Palti, Y., Moran, N. and Stampfli, R. Potassium currents and conductance: comparison between motor and sensory myelinated fibers. Biophys. J., 1980, 32: 955-966. Panizza, M., Nilsson, J. and HaUett, M. Optimal stimulus duration for H-reflexes. Muscle Nerve, 1989, 12: 576-579. Plonsey, R. Bioelectric Phenomena. McGraw-Hill, New York, 1969: 18-21. Reilly, J.P., Freeman, V.T. and Larkin, W.D. Sensory effects of transient electrical stimulation - evaluation with a neuroelectric model. IEEE Trans. Biomed. Eng., 1985, 32: 1001-1011. Roth, B.J. and Basser, P.J. A model of the stimulation of a nerve fiber by electromagnetic induction. IEEE Trans. Biomed. Eng., 1990, 37: 588-597. Veale, J.L., Mark, R.F. and Rees, S. Differential sensitivity of motor and sensory fibres in human ulnar nerve. J. Neurol. Neurosurg. Psychiat., 1973, 36: 75-86.
22
ELMOCO 91035
Relevance of stimulus duration for activation of motor and sensory fibers: implications for the study of H-reflexes and magnetic stimulation * Marcela Panizza, Jan Nilsson, Bradley J. Roth, Peter J. Basser and Mark Hallett Human Motor control Section (M.H.), National Institute of Neurological Disorders and Stroke, National Institutes of Health, Bethesda, MD (U.S.A.); EMG Laboratory (M.P. and ZN.), Fondazione Clinica del Lavoro, Campoli MT (BN) (Italy); and Biomedical Engineering and Instrumentation Program (B.J.R. and P.J.B.), National Institutes of Health, Bethesda, MD (U.S.A.) (Accepted for publication: 15 July 1991)
Summary
Electric stimuli with durations of 0.5-1.0 msec are optimal for studies of H-reflexes. It is more difficult to obtain H-reflexes with shorter duration stimuli or with magnetic stimulation. In order to understand this behavior, we studied the excitation thresholds for motor and sensory fibers in the ulnar, median and tibial nerves using both electric and magnetic stimulation. For short duration electrical stimuli (0.1 msec) the threshold for motor fibers is lower than for sensory fibers. For longer duration electric stimuli (1.0 msec) the threshold for sensory fibers is lower. For magnetic stimulation the threshold for motor fibers is much lower than for sensory fibers. Thus, stimulus duration is a critical parameter for sensory fiber excitation, and current magnetic stimulators are not optimal.
Key words: Threshold; Stimulus duration; Motor fibers; Sensory fibers; Magnetic stimulation; H-reflexes
When stimulating peripheral nerves electrically, the relative recruitment 'of sensory and motor fibers depends on the stimulus duration. Erlanger and Blair (1938) first showed that threshold for sensory fibers is lower than for motor fibers for long stimulus durations and that this relation reverses for short stimulus durations. Their results were based on a study of the strength duration curve of the sciatic nerve in a bull frog. Veale et al. (1973) showed the same phenomenon in motor and sensory fibers in the ulnar nerve in humans. Paillard (1955) described the preferential activation of the H-reflex over motor fibers in the tibial nerve in humans when using a stimulus duration of 1 msec, and the inverse when using a stimulus duration of 0.1 msec. Recently, Panizza et al. (1989) demonstrated that stimulus duiations shorter than 0.2 msec were not optimal to elicit H-reflexes either in the tibial or median nerve because the motor fibers were often activated before or simultaneously with the H-reflex. Furthermore, the recruitment of sensory and motor fibers depends on the shape of the stimulation pulse, since for short duration stimuli changing the stimulus from a monophasic to a biphasic pulse changes the
Correspondence to: Mark Hallett, NINDS, NIH, Building 10, Room 5N226, Bethesda, MD 20892 (U.S.A.). * Presented in part at the joint Symposium AAEE-AEEGS, San Diego, CA, October 1988.
excitability pattern for different fiber diameters (Reilly et al. 1985). Recently, Olney et al. (1990) found that electrical stimulation (duration unspecified) at 25% of maximal activated sensory fibers preferentially to motor f~ers, but magnetic stimulation at 25% of maximal did not share this effect. Studies of threshold magnetic stimulation have not previously been done. The purpose of this study is (1) to highlight by examples that the thresholds of the sensory and motor fibers vary as a function of duration of the electrical stimulation, (2) to study the relative recruitment of sensory and motor fibers using magnetic stimulation, and (3) to use our experimental results and a mathematical model to compare electric and magnetic stimulation.
Material and methods Six healthy volunteers (aged 22-51 years, mean 34 years) were studied by stimulating the ulnar nerve when recording motor and sensory fiber responses. The median nerve was studied in 8 healthy volunteers (aged 28-36 years, mean 31 years) by recording motor, sensory and H-reflex responses, and the tibial nerve was studied in 8 healthy volunteers (aged 26-39 years, mean 33 years) recording motor fiber responses and H-reflexes. All volunteers gave their informed consent for participation in this study.
MOTOR AND SENSORY FIBERS
Stimulation Electric. Constant current stimuli (Dantec 15E07), with durations from 0.05 to 1 msec (square wave), were delivered from surface electrodes mounted in a plastic mold (2.5 cm between anode and cathode) and placed at the elbow for ulnar nerve stimulation, at the cubital fossa for the median nerve, and at the popliteal fossa for the tibial nerve stimulation. Magnetic. A time-varying magnetic field, produced by either a Dantec magnetic stimulator (Dantec Electronics, Tonsbakken 16-18, DK-2740 Skovlunde, Denmark) or a Cadwell MES-10 Magnetic Stimulator (Cadwell Laboratories, Inc., 909 N. Kellogg Street, Kennewick, W A 99336, U.S.A.), was used to induce current near the ulnar, median or tibial nerve. For the Dantec stimulator 3 prototype coils of different inductances (5.8, 50, and 160/xH) were used; the durations of the induced currents, measured from the start of the pulse to the end of its first phase, were 105, 294, and 574 ~sec, respectively. The coils were all tightly wound flat spirals with inner diameters of 7 cm and outer diameters ranging from 9.7 to 12.3 cm. For the Cadwell stimulator a coil with an inductance of 20 /zH was used, producing a stimulus duration of 80 ~sec measured to the end of the first phase (920 ~sec measured to the end of the underdamped oscillating wave form), This coil was tightly wound and circular, with an inner diameter of 4.6 cm and outer diameter of 9.2 cm. The outer edge of each coil was placed over the optimal position for electrical stimulation, with the plane of the coil tangent to the axis of the arm or leg, and with the handle perpendicular to the axis of the limb.
23
case, a ground electrode was placed just distal to the stimulating electrode.
Threshold criterion The threshold for sensory fibers was defined as the minimum stimulus current required to evoke a recognizable sensory potential with an amplitude of at least 0.5 /xV, after averaging from 10 to 50 responses. In determining the threshold of motor fibers an amplitude criterion of 25 tzV was used.
Results
Ulnar nerve Electric stimulation.
For a stimulus duration of 0.1 msec the threshold for sensory fibers was 6 + 26% (mean + S.D.) higher than for motor fibers. For a stimulus duration of 0.3 msec the threshold was 11 + 18% lower for sensory fibers than for motor fibers, a behavior already reversed with respect to 0.1 msec. For a 1 msec duration stimulus the difference was even larger, the threshold for sensory fibers being 27 + 15% lower than for motor fibers (Fig. 1). Magnetic stimulation. Responses from sensory and motor fibers were recorded in all subjects. When using the Cadwell stimulator the threshold for sensory fibers was 107 + 91% higher than for motor fibers, whereas when using the Dantec stimulator with the coil producing the shortest duration pulse, the threshold for sensory fibers was 52 + 38% higher than for motor fibers.
Median nerve Electric stimulation. Recordings Sensory fibers.
Two ring electrodes were placed 1 cm apart near the tip of digit V when recording from the ulnar nerve, and on digit III when recording from the median nerve, Motor fibers. For recording from muscles innervated by the ulnar nerve, a hooked-wire electrode was inserted in the abductor digiti quinti (ADQ) referenced to a surface electrode on digit II, or a surface electrode was placed over the belly of A D Q referenced to a surface electrode on digit V. For recording from muscles innervated by the median nerve, a hooked-wire electrode was inserted in the abductor pollicis brevis (APB) referenced to a surface electrode on digit V, and a hooked-wire electrode ~,as inserted in flexor carpi radialis (FCR) referenced to a surface electrode placed 4 cm away, or surface electrodes were placed 4 cm apart over FCR and over the belly of APB referenced to the 2nd phalange of the thumb. For recordings from muscles innervated by the tibial nerve, surface electrodes were placed 4 cm apart over the soleus muscle with the active electrode proximal. In each
For stimulus durations equal to or less than 0.3 msec the threshold for sensory fibers was higher than for motor fibers, being 56 + 49% higher at 0.1 msec, and 24 + 47% higher at 0.3 msec. For stimulus durations equal to or greater than 0.5 msec the threshold for sensory fibers was lower than for motor fibers, being 2 + 35% lower for 0.5 msec, and 21 + 15% lower for 1 msec (Fig. 2). For a stimulus duration of 0.1 msec the threshold for H-reflexes was 49 + 56% higher than for motor fibers, whereas at a stimulus duration of 1 msec the threshold for H-reflexes was 14 + 23% lower than for motor fibers (Fig. 2). In 3 of the 8 subjects a stimulus duration of 0.05 msec was used, and the thresholds for sensory fibers and H-reflexes were 63 + 13% and 85 + 23% higher than for motor fibers. Magnetic stimulation. Studies were done only with the Dantec magnetic stimulator, and it was possible to obtain sensory and motor responses in all subjects. The threshold for sensory fibers was 103 + 71% higher than for motor fibers, the range being the same no matter which coil was used. Also H-reflexes were recorded in all the subjects, and the threshold for H-reflexes was
24
M. PANIZZA ET AL.
110 + 100% higher than for motor fibers, independent of the pulse duration,
Tibial nerve Electric stimulation. For a stimulus duration of 0.1 msec the threshold for H-reflexes was 12 + 15% higher than for motor fibers; for a duration of 1 msec it was 16 ± 23% lower, Magnetic stimulation. H-reflexes were evoked by the Cadwell or the Dantec magnetic stimulator, and recorded in all the subjects. When using the Cadwell stimulator the H-reflex threshold was 133 ± 24% higher than for motor fibers, whereas for the Dantec magnetic stimulator the H-reflex threshold was 140 ± 54% higher than for motor fibers, independent of the pulse duration. Sensory fiber response compared to H-reflex response for stimulus intensities just below motor threshold It is often desirable to produce H-reflexes without an M-response. The ability to produce such an H-reflex should depend on the ability to stimulate Ia sensory afferents without stimulating motor fibers. Thus, the maximum amplitude of an H-reflex produced at a particular electrical stimulus duration should be predictable on the basis of the amplitude of the sensory nerve action potential produced just below motor threshold. Fig. 3A shows the amplitude of the mean Ulnar
sensory fiber response in the median nerve (digit III) for stimuli just below motor threshold (flexor carpi radialis) as a function of stimulus duration. Fig. 3B shows the recruitment curve of the H-reflex amplitude (flexor carpi radialis) just below motor threshold for different durations of the electrical stimuli to the median nerve. Sensory fibers and H-reflex show the same tendencies when changing stimulus durations.
Is there a difference in recruitment when eliciting responses by different durations and wave shapes of magnetic stimulation? No differences were found in the relative thresholds of motor and sensory fibers when using the Dantec magnetic stimulator with each of its 3 coils, and therefore the relative recruitment of motor and sensory fibers was independent of stimulus duration (Fig. 4A). Threshold for motor fibers occurred at a setting of 15-20% on the magnetic stimulator, resulting in a magnetic field changing at a peak rate of 4.1, 2.1 and 1.2 k T e s l a / s e c respectively for the coils with durations of 105, 294 and 574 /zsec (the stimulus intensity is proportional to the rate of change of the magnetic field). Apparently the increasing duration of the stimulus was compensated by the decreasing intensity. In 4 subjects, results obtained using the Cadwell stimulator (oscillating wave s h a p e ) w e r e compared to those found using the Dantec stimulator (triangular nerve
Electric stimulation s
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Fig. 1. A: responses from one healthy volunteer evoked by electric or magnetic stimulation of the ulnar nerve at the elbow while recording from abductor digiti V and digit V. At 0.1 msec stimulus duration the motor threshold is lower than the sensory threshold, similar to responses elicited by magnetic stimulation. At 1 msec stimulus duration the sensory threshold is lower than motor threshold. B: the mean normalized threshold intensity as a function of the electric stimulus duration for the sensory and motor responses in the ulnar nerve.
MOTOR AND SENSORY FIBERS
25
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nerve
Electric stimulation 0.I ms --
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Duration (ms) Fig. 2. A: responses from one healthy volunteer evoked by electric or magnetic stimulation of the median nerve at the cubital fossa while recording from APB, FCR and digit Ill. At 0.I msec stimulus duration the motor threshold is lower than the sensory threshold, and a motor response can be seen in APB. At I msec stimulus duration an H-reflex can be seen in FCR, and a sensory response in digit Ill without motor response. For magnetic stimulation the motor threshold appears below the sensory and H-reflex thresholds, and the motor responses can he seen in both APB and FCR. B: the mean normalized threshold intensity as a function of the electric stimulus duration for sensory, H-reflex and motor responses. Sensory and H-reflex thresholds behave in the same way all along the curve.
w a v e s h a p e ) w i t h t h e coil p r o d u c i n g t h e s h o r t e s t d u r a t i o n p u l s e (Fig. 4B). T h e s e r e c o r d i n g s w e r e p e r f o r m e d in 4 o f t h e s u b j e c t s . S e t t i n g s o f 4 5 - 6 0 % o n t h e m a g n e t i c s t i m u l a t o r s r e s u l t e d in c a l i b r a t e d s t i m u l u s i n t e n sities o f a b o u t 17.0 k T e s l a / s e c f o r t h e C a d w e l l s t i m u l a -
tor, a n d tor. Ulnar Cadwell w a s 150
a b o u t 12 k T e s l a / s e c
nerve. W h e n r e s p o n s e s w e r e e v o k e d by t h e stimulator, the threshold for sensory fibers _+ 7 0 % h i g h e r t h a n f o r m o t o r fibers. W h e n
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Duration (ms) Fig. 3. Median nerve: histogram (A) of the mean sensory amplitude (digit l i d below motor threshold (Tin) as a function of the electrical stimulus duration, and histogram (B) of the mean H-reflex amplitude below motor threshold (flexor carpi radialis) for the same stimulus durations. The two histograms show similar behavior for all stimulus durations and have maximal values between 0.5 and I msec duration of the stimuli.
26
M. PANIZZA ET AL.
A
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F~. 4. A: responses from one normal volunteer evoked by magnetic stimulation of the median nerve at the cubital fossa using different durations of the stimulus (DantCc m ~ t ~ stimulator), while recording motor from F C R and sensory from digit III. Under the recordings of the motor and sensory responses; traces resulting from the induced voltage for the different durations (obtained at a setting on the magnetic stimulator of 30%) can be seen. B: responses from the same control recorded when using the two different magnetic stimulators. On the left the Cadwell magnetic stimulator (biphasic wave shape), and on the right the Dantec stimulator (monophasic wave shape). The recordings show the presence of sensory, motor and H-reflex responses, and under these traces can be seen recordings of the induced voltages for the two different stimulators, measured at a setting on the magnetic stimulator of 30%. In the bottom part of the figure are 2 histograms showing the' mean normalized threshold intensity for the motor and sensory thresholds for each magnetic stimulator.
MOTOR AND SENSORY FIBERS responses were evoked by the Dantec magnetic stimulator, the threshold for sensory fibers was only 52 + 38% higher than for motor fibers. For both stimulators the thresholds for motor fibers were similar, at a setting of 2 0 - 2 5 % on the magnetic stimulators, Tibial nerve, H-reflex. The threshold for H-reflexes was 133 + 24% higher than for motor fibers when responses were elicited by the Cadwell stimulator, and 112 + 85% higher when responses were elicited by the Dantec stimulator. For both stimulators the threshold for motor fibers was the same, at a setting of 20-25% on the magnetic stimulators,
Discussion
When using electric stimulation, preferential activation of the sensory fibers over the motor fibers was observed for long stimulus duration and the inverse behavior for short stimulus duration in the median, ulnar, and tibial nerves. The point of reversal in the behavior was slightly different between individuals and between different nerves in the same individual, ranging from a stimulus duration of 0.2 msec for the ulnar nerve to 0.5 msec for the median nerve, The strength-duration relationship for a given fiber can be characterized by the rheobase stimulus strength and the chronaxie duration. The difference in stimulus threshold at long durations (the difference in rheobase) may be due to sensory fibers having a larger diameter than motor fibers. McNeal's model of electric stimulation indicates that threshold varies inversely with fiber diameter for a given pulse duration (McNeal 1976). Basser and Roth (1991) predicted that threshold is inversely proportional to the square of the fiber diameter during magnetic stimulation. Indirect evidence of differences in diameter between motor and sensory fibers is that the sensory conduction velocity in the periphery is significantly faster than the motor conduction velocity (Kimura 1983, 1984). The dependence of the relative recruitment of sensory and motor fibers on stimulus duration cannot be explained by a difference in fiber diameter, or rheobase, To explain this behavior, the strength-duration curves for sensory and motor fibers must have different chronaxies. One explanation of the different chronaxies is that the two classes of fibers have different membrane time constants (see Appendix). Barker et al. (1990) have reported preliminary data that the time constant for ulnar nerve motor fibers was about 150 tzsec and concluded that the optimal stimulus duration for motor fibers should be short. There is evidence that the membrane properties of sensory and motor fibers differ (Neumcke et al. 1980; Palti et al. 1980), although we know of no experimental measurements comparing time constants in the two fiber classes. Thus the lower
27 threshold of the m o t o r fibers with short stimulus durations remains unexplained. The behavior of the H-reflex and the sensory response was similar over a wide range of stimulus durations (Fig. 2B). Furthermore, when analyzing the amplitudes of the sensory potential and H-reflex below motor threshold in the median nerve both amplitudes increased with stimulus duration (Fig. 3) (Panizza et al. 1989). The similar dependence of the sensory fibers and the H-reflexes on stimulus duration can be interpreted as the activation of the H-reflex depends on the activation of the Ia fibers and the sensory threshold depends on the activation of the largest cutaneous fibers the diameter of which is probably similar to Ia afferents. The hypothesis that fibers of similar diameter react the same way to different stimulus durations can explain the fact that the behavior of sensory threshold in relation to stimulus duration can be used to predict the behavior of H-reflex and vice versa. When using magnetic stimulation the motor threshold was lower than the sensory and H-reflex thresholds for all the nerves studied and for all the coils used. These results may look contradictory since a magnetic stimulus duration of about 0.6 msec behaves in the same way as an electric stimulus duration of 0.1 msec. In order to understand such a behavior it is necessary to analyze the wave form of the magnetic stimulus. As can be seen in Fig. 4A the wave form of the induced current produced by the Dantec magnetic stimulator is nearly triangular. Both the duration and pulse shape influence the strength-duration curve. In the Appendix it is shown that a triangular pulse must have a duration over 3 times as long as a square pulse to produce a similar strength-duration curve. Because of this difference between the electric and magnetic strength-duration curves, the "measured duration" of a triangular wave shape must be divided by a constant of 3.625 in order to obtain the "effective duration" comparable to a square wave pulse (see Appendix). This conversion reduces the duration of the magnetic stimulus from 105, 294 and 574/xsec to 29, 81 and 158/.tsec, respective!y. Therefore considering these new "effective durations" of the magnetic stimulus, it is reasonable that all 3 Dantec coils behave in a way similar to an electric stimulus shorter than 0.2 msec. The difference of intensity necessary to stimulate sensory fibers and to obtain H-reflexes was larger with the magnetic stimulator using a oscillating wave form than with the stimulator using a nearly triangular wave form, even when sensory potentials or H-reflexes could be elicited at higher intensities with both of them; This observation suggests once more an influence of the magnetic wave shape for the threshold of different fibers. Early studies with electrical stimulation showed that biphasic stimuli alter the threshold of fibers. The alteration depends on the diameter of the fiber, the
28
M. P A N I Z Z A E T AL.
duration of the stimulus, and the delay between each phase of the biphasic stimulus. The alterations are maximal with a shorter stimulus duration, the shorter the delay between the stimuli, and for larger diameter fibers (Gorman et al. 1980; Reilly et al. 1985). Thus biphasic stimulation requires a higher stimulus level to be effective for the larger diameter sensory fibers, but requires a comparable stimulus level for motor fibers. In conclusion, magnetic stimulation can activate motor fibers easily, but is at a disadvantage in stimulating sensory fibers, and this is particularly true for magnetic stimulation with an oscillatory wave form.
Appendix The magnetic stimulation strength-duration curve In this appendix we use a mathematical model to show that the strength-duration curves produced by electric and magnetic stimulation are different. Specifically, electric and magnetic stimulation result in strength-duration curves with identical rheobases but different chronaxies. We base our calculations on the model of magnetic stimulation derived by Roth and Basser (1990). We assume that the wave form of the induced current is triangular; a more general treatment can be found in Basser and Roth (1991). First, let us review how the strength-duration curve can be derived for electric stimulation (Plonsey 1969). The tronsmembrane potential, V, along a spaceclamped ~ axon as a function of time, t, is governed by the differential equation dV ~---~-+V=S(t) (1) where S(t) is the stimulus strength as a function of time and z is the membrane time constant. Consider a square pulse of duration d, so that S(t) is S(t)=(~0
0
0
(3)
1 This analysis also holds if the axon is not space clamped but the spatial variations in the stimulus are small over distances on the o r d e r of a length constant. This is the case in most applications of magnetic stimulation. It also holds for myelinated fibers if the spatial extent of the induced current is large compared to the distance between nodes (Basser and Roth 1991). In that case, the m e m b r a n e properties must be taken as the average properties along the fiber (Andrietti and Bernardini 1984).
The membrane potential reaches a maximum at the end of the stimulus, t = d. In Eq. (3) let us set V equal to the threshold potential, Vthr~shoL~,and t equal to the pulse duration. The resulting equation between S O and d is called the "strength-duration curve"
Vthreshold SO
. (4) 1 - e -d/* For durations d that are long compared to z, the strength So is a constant (So = Vthresho~d), called the rheobase stimulus. For durations that are short compared to ~-, the strength-duration curve reduces to S O= Vthr,shold •'/d, so that the product of the stimulus strength and duration is constant. Thus, for short duration stimuli it is the area under the stimulus curve, not its amplitude or duration individually, that determines whether threshold is reached. The chronaxie, dchronaxie, is a measure of the duration at which the strengthduration curve changes from its long duration to short duration behavior. Chronaxie is defined as the stimulus duration required to activate the nerve with a stimulus strength equal to twice rheobase. For the strengthduration curve in Eq. (4), dchronaxi~ can be calculated analytically dchrona~i ~ = --Z In(0.5) = 0.693 ~-. (5) Having reviewed the well-known properties of the electric strength-duration curve, we now turn our attention to the analogous relationship in magnetic stimulation. We assume that the main stimulus pulse can be represented as a triangular function; the induced current has a very fast rise and then fails linearly to zero 2. We can express the triangular form of the stimulus wave form, S(t), as S(t) =
{~(d) 0 1-
otherwise / 0< t
(6)
Eq. (1), governing the t r a n s m e m b r a n e p o t e n t i a l a s a function of time, can be solved analytically using the stimulus in Eq. (6) and initial condition V(0) = 0, yielding [ ~._t ( d ) ] V(t) = S o 1 + T - 1+ e -t/r . (7) To obtain the strength-duration curve, in Eq. (7) we set V equal to Vthr~ho~d and t equal to the time that V(t) reaches its maximum, tm~. Unlike electrical stimulation, V(t) does not reach a maximum at the end of
2 Actually the fall is not quite linear and there is always a second phase of the induced current of opposite polarity that follows the first phase. If the second phase has a low amplitude and a duration that is longer than ~-, then it does not stimulate the axon.
MOTOR AND SENSORY FIBERS
29
the pulse. To find tmax, we set d V / d t equal to zero, finding
If we substitute this result into Eq. (7) and rearrange terms, we get the strength-duration curve appropriate for magnetic stimulation Vthreshold
So=
1 - -~ In --~. + 1
)
S 0 = 2 Wthreshold q'//d. T h e f a c t o r o f 2 a r i s e s b e c a u s e a
magnetic stimulus pulse of amplitude SO and duration d has half the area under the curve as does the electric stimulus pulse of the same strength and duration. For d >> z we obtain the same rheobase condition, S O= Wthreshold, as f o r electric stimulation. Finally, to deter-
mine the chronaxie for magnetic stimulation w e set the strength equal to twice rheobase in Eq. (9) and solve for the duration corresponding to chronaxie, dchronaxi e. We find that dchronaxi e satisfies the equation ( dchronaxie ) + 1 1" '
In 21"
(10)
which can be solved numerically, yielding dchronaxi e = 2.513 t".
(11) The chronaxie for electric stimulation was 0.693 ~-, so that the ratio for the magnetic to electric chronaxie for the same membrane time constant is 3.625. The magnetic strength duration curve is much wider than 100
~ g n e t i c 10 So
electric ~
~
1 I 0.1
We thank Dr. Roger Gilliatt and Dr. I. Tasaki for helpful discussions. The prototype coils and contributions from Mr. Stig Wanding Andersen and Mr. Maurizio Bonneau from Dantec Electronics, Inc. and Dr. John Cadwell from Cadwell Laboratories, Inc. are acknowledged.
References
For d << ~" the strength-duration curve reduces to
dchronaxie
the electric curve, as shown in Fig. 5. In essence, an electric stimulus of duration d acts like a magnetic stimulus with over 3.5 times the duration.
t 1 "
I 10
I 100
d/x Fig. 5. Strength-duration curve for an electric stimulus (rectangular wave shape), and a magnetic stimulus (assumed triangular wave shape). S o is the stimulus strength divided by the rheobase stimulus strength, and d/~" is the pulse duration divided by the time constant of the axon membrane.
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