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Simulation of electromyograms showing interference patterns
Electroencephalography and Clinical Neurophysiology Elsevier Publishing Company, Amsterdam - Printed in The Netherlands
625
SIMULATION OF ELECTROMYOGRAMS SHOWING INTERFERENCE PATTERNS R. S. PERSON AND M. S. LIBKIND 1
Institute of Higher Nervous Activity and NeurophysiGlogy of the U.S.S.R. Academy of Sciences, and Moscow Phj~icoTechnical Institute, Moscow (U.S.S.R.) (Accepted for publication: October 1, 1969)
The interference pattern or global EMG is a summation of action potentials of a great number of motor units. The information about motor units latent in it is not easily revealed, because of the extremely complex pattern. For this reason the employment of surface electrodes in clinical EMG is limited. The purpose of this work was to study quantitative regularities of interference pattern formation by motor unit action potentials and to analyse the relationships between their parameters. The discharges of motor units may be asynchronous (independent) and synchronous. Asynchronous activity is observed during a weak contraction of the muscle (Bigland and Lippold 1954; Taylor 1962 and others). Synchronization was found during a strong contraction of the muscle as well as under fatigue or in diseases caused by lesions of motoneurones (Haas 1926; Adrian 1947; Buchthal and Madsen 1950; Zhukov and Zakharyantz 1959 and others). For a moderate range ofmuscle contraction in healthy people asynchronous activity was considered typical, but cross-correlation analysis of EMGs has given reason to believe that a tendency to synchronization exists even in this range of contraction and that synchronization increases with increase in the force of contraction (Person and Kudina 1968). Both asynchronous and synchronous activity of motor units has been investigated in this study. A preliminary report was published earlier (Person and Libkind 19~6). z It is deeply regretted that Dr. Libkind met with a fatal accident after this paper was submitted for publication.
METHOD
A stochastic model was used. Simulation was carried out graphically and mathematically, some of the parameters of the model being calculated with a digital computer. The characteristics of motor unit action potentials put into the model were partly borrowed from the literature on EMG and partly obtained in our own experiments on large human skeletal muscles with bipolar needle electrodes. A number of simplifying assumptions were made. Action potentials of an individual motor unit were assumed to have the shape shown in Fig. 1. In our model N was the number of active motor units, 0 was the mean frequency of motor unit discharge and the inter-spike intervals were independent random values having a normal distribution, with mean # and standard deviation a0. A single impulse, triangular in shape, diphasic and symmetrical with respect to the zero line, had the duration 2c and an amplitude from peak to peak 21,. Both for the graphical model and for computation it was assumed that 0=20 c/seo, p=40 mse¢, tT0ffi5 msec, 2cffil0 mse¢, on the accepted scale b= 3c. The mean duration of a diphasic wave of the interference pattern was 2d. The parameters of interference pattern were studied as depending on N, as an inorease in the number of active motor units was regarded as the main factor increasing a muscle contraction (Bigland and Lippold 1954; Hyde et al. 1954 and others). In graphical simulation the curve representing action potentials of an individual motor unit with all the parameters enumerated above was plotted £1ectroenceph. din. Newophysiol., 1970, 28:625--632
626
R. S. PERSON AND M. S. LIBKIND
on the scale of I mm to I msec (for the time of 500 msec). Next, trains of impulses randomly superimposed on one another were successively added up and the obtained synthetic EMGs photographed on the same time scale as the actttal EMG (Fig. 1). The graphical model was used to visualize the pattern of synthetic EMG and to calculate by direct measurement certain parameters of EMGs --frequency of fluctuations and integrated electrical activity. In the mathematical model the theory of random functions was employed. The activity of the i-th motor unit in the time interval from O to a certain time point T was regarded as a random process ~pl(t). The positive part of a single impulse has the following analytical expression: 0, 2b v,(t) ffi
when t < 0 and t > c; t,
when 0 ~
2b - 2b c
when c~<~t<~c.
(I)
The interference pattern, which is arrived at by summing up action potentials of active motor units, can be expressed as ,iV
fN(t)
=
(2)
The following characteristics of this simulated EMG were analytically studied: mean duration of fluctuations, autocorrelation function, frequency spectrum and integrated electrical activity. The mathematical procedure employed was earlier described by Person and Libkind (1967) and Libkind (1968, 1969). In the mathematical model synchronization was introduced in the following way: suppose N motor units are divided into p independent groups, each containing n motor units acting syn. chronously; i.e., N ffipn. Inside each group ira. pulses coincide either fully or partially, i.e., with a certain shift in time, the value of the shift not exceeding the duration of an impulse. By this we mean that there exists some difference in the time of conduction of impulses from different motoneurones to electrodes. Consequently, the re-
corded action potentials of synchronously discharging motor units are slightly shifted with respect to one another (out of phase synchronization). This seems to be near to what actually takes place in the physiological prototype. In the graphical model synchronization was introduced differently. Here N active motor units were divided into two groups, one of which contained mM motor units acting asynchronously and the other ms units acting synchronously. Thus,
N-ross+ms ms may be expressed as percentage content of synchronously acting motor units ft. Then
ms ffi 100 The content of synchronously acting motor units (as percentage) may be assumed either constant or increasing with N. For computation on a digital computer it was assumed that 1 time step = 1 msec; Tffi500 msec, N = 100. RESULTS
I. Asynchronoua activity of motor units 1. Visible pattern of graphical model. When a small number of motor units were added the impulses were frequently grouped with a "pileup" effect, which had been previously observed in similar conditions by Taylor 0962). With an increase in the number of active motor units the time intervals free of impulses gradually diminished. A "saturation" level was reached when 16-18 motor units acted. Fig. l shows that synthetic EMGs, especially the "saturated" ones, visually resemble the actual EMG.
2. Mean duration andfrequency of repetition of interference pattern fluctuations. For the determination of a mean duration of interference pattern fluctuations (2d) the mathematical model was used. The number of zero-line crossings in a unit of time was found. From this was found the mean duration of a monophasic wave d ffi
2:
~, - 0.91c
(3)
Thus, in the model there exists a simple, conElectroenceph. clin. Neurophysiol., 1970, 28:627~-632
627
SIMULATION OF EMGS
t205sec
c/~ tPO, 120'
2~
#/qU
3
101vlU
A
6O 00
I#0' B frO'
18FIu
IOO' 012
z/
6
a
io
15
lo~, kg
#OHU
Fig. 2 A: Relation of mean fluctuation frequency in the graphical model of EMG to the number of active motor units; B: relation of the mean fluctuation frequency of the actual EMG (m. biceps brachii) to the load (y axis - the frequency in ~ o" the initial).
RC~) t~ ~8
[---. l~O5sec' 751~V Fig. 1 Synthetic EMGs obtained by summation of increasing numbers of motor units (I-5) and the actual EMG of m. biceps brachii (6).
stant relationship between a mean duration of interference pattern fluctuations and duration of action potentials of motor units forming this pattern, the coeflioient of relationship being close to 1. These results agree very well with those obtained for d on a digital computer° The frequency of repetition for all fluctuations (the number of maxima) was determined from the graphical model. With an increase in N the number of fluctuations iacreased up to the level of "saturation" after which it became more stable, ranging between 130 and 140/sec. Thus, the frequencies of fluctuations in the model and in the actual EMG are values of the same order. The character of relationship between frequency of fluctuations and the number of active units in the model is like that between frequency and force of contraction in the actual EMG (Fig. 2). 3. Autocorrelation function and spectrum of
~6
~2 0
i"
B
. . . .
f
Fig. 3 A: Autocorrelation function of the actual EMG (m. biceps brachii); B: the same for the mathematical model of EMG. X axis - shift in msec; y axis - value of autocorrelation function.
interference pattern. The shape of an autocorrelation function, which was calculated from the mathematical model, was similar to that obtained earlier by Person and Mishin (1963) for an actual EMG (Fig. 3). The distance from the reference point to the first crossing of the function with x axis To may be regarded as one-fourth of a diphasic wave of autocorrelation function. It has been found that 2xo- 1.12cffi 1.24d
(4)
Thus, the parameter in question of the autocorrelation function is closely connected with the Electroenceph.din, NeurophysloL, 1970, 28:625-632
628
It. S. PERSON AND M. S. LIBKIND
the interferen~ pattern was investigated mathematically. It was found that, when motor units act independently of one another, the integrated electrical activity starting somewhere near N = 20 behaves on the average as
t
,. Xb where K I I
,
(5)
2
Assuming all the parameters, except N, constant we get
! |
!
o
T 'jAx 3¢
.Cot 3¢ Fig, 4
The spectrum of the mathematical model of EMG. X axis - frequency;y axis- spectral density.
These results, arrived at analytically, are illustrated by the graphical model (Fig. 5) where the area of EMG was directly measured. Similar results were obtained by Moore (1967a,b). Having computed a synthetic EMG as a sum of diphasic waves randomly superimposed on one another he obtained: r.m.s. --K[/Q
Nom~ ot /41/
sNkesl,~ee
Fig. 5 Relation of integrated EMG to the number of active motor units for the graphical model, Asynchronous activityof motor units. mean duration of main fluctuations of interference pattern, which agrees with observations on actual EMGs (Person and Mishin 1963). The spectrum of the model was calculated from the autocorrelation function (Fig. 4). It possessed one well pronounced peak whose maximum corresponded with the sinusoidal frequency 4ni3c, the other peaks being negligibly small. Thus, the dominating frequency is in a simple relationship with the duration of the elementary potential., For 2c=10 mse¢ the maximum of the spectrum corresponded with the frequency of 133 c/sec, which tallies with the results of spectral analysis for actual interference patterns led off by needle electrodes (Walton 1952). 4. Integrated electrical activity. The integral of
where r.m.s, is the root mean square of the resultant waves, Q is a number of summated waves and Kis a constant.
Thus, when motor units act asynchronously, the dependence of the integrated electrical activity on the number of active motor units is nonlinear, i.e., with an increase in the number of active motor units the integral increases at a gradually diminishing rate. The phenomenon may be understood from the following reasoning. When diphasic fluctuations are randomly superimposed on one another they may get "in phase" as well as "in antiphase". In the former ease the amplitude and the area of interference curve are increased, in the latter, diminished. This cancellation effect was noted by Adrian (1925). The relatioi~ between these two processes accounts for the character of dependence in question. The rate at whic~ the integral increases is considerably diminished when N-- 18-20. Turning to the graphical model we notice that at N m20 the synthetic EMG becomes "saturated", i,e. it no longer has any intervals free of impulses. A rapid growth of "unsaturated" E M G area is explained by the fact that some of the new impulses find themselves in vacant spaces and their areas are
Etectroenceph. din. Ne~ophy$iol., 1970, 28:625-632
SIMULATIONOF BMGS summated linearly. When no free intervals are left the probabilities of getting "in phase" or "in antiphase" for new fluctuations become almost equal. As has been said above, the chief factor increasing contraction is recruitment of new motor units. It is therefore interesting to compare changes in integrated electrical activity of the model caused by an increase in N with those in the actual EMG depending on the force of contraction. The comparison shows that in this respect the model behaves differently from a muscle. In the model the electrical activity reaches a certain level of saturation after which it increases only insignificantly, whereas in the muscle it is increased over the entire range of contraction force. Several authors have proved by needle as well as by surface electrodes that this relationship is almost linear or even sharper than linear (Lippold 1952: Gurfinkei et al. 1965; Meller 1966; Vredenbregt and Koster 1966). This disagreement deserves special consideration (see Discussion).
II. ,Synchronous activity of motor units I. Mean duration offluctuation, autocorrelation function and spectrum of interference pattern. Mathematical simulation has shown that in the case of synchronization without phase shift the mean duration of interference pattern fluctuation, as well as the positions of the frst zero of the autocorrelation function and of the spectral maximum, are virtually the same as in the case of asynchronous activity and the relationships de. scribed, above are relevant here. In the case of out of phase synchronization the picture is somewhat different. The mean duration of interference pattern fluctuation is increased I/i" times, becoming longer than the duration of a motor imit impulse: d' = 1.26c (7) The spectral maximum is shifted to the left as compared with its positioh in the case of asynchronous activity. Thus, simulation proves that increased duration of interference pattern fluctuation in the case of synchronization, which is known from physiological experiments (Fig. 6), and the dis-
629
~
/OOpV
25 ITISec I
'
Fig. 6 Actual EMG of m. biceps brachii; isometriccontraction, a load 5 kg. Surface electrodes. 1: the beginning of muscular activity; 2: the same after 5 rain of activity, under fatigue(synchronization).
/
/
/
/ / y 8 ~
//
o
"
1o
"
zo
I
zo
,
I
~o
!
~o
I
ao
Numberof Mt/ Fig. 7 Relation of integrated EMG to the number of active motor units for the graphical model. I: asynchronousactivity of motor units. 2: 20~ of motor units act synchro. nously; 3: proportion of synchronously acting motor units p increases with an increase in the number of active motor units N(/~ = 15 -t- 0.5 N); 4: synchronousactivity of motor urdts.
placement of the spectral maximum to the left (Kogi and Hakamada 1962 and others) are not caused by synchronization as such, but byshift in synchronization. 2. Integrated electrical activity. The study of integrated electrical activity of the mathematical model revealed that synchronization (both with and without phase shift) greatly affects its value. The relationship between integrated EMG and N becomes K,N~ (8) where Ks depends on the same parameters as Kx (b, c and 0), and the index ~ depends on the
£1ectroenceph.din. Neurophy~loL,19/0, 28:625-632
630
R. S. PERSON AND M. S. LIBKIND
degree of synchronization and varies from 0.5 to 1. If o~0.5, the relationship is close to that typical for asynchronous activity, whereas ifo,--, 1, it is close to being linear. A phase shift of impulses diminishes the rectifying influence of synchronization. Thus, simulation has shown that if synchronization is introduced into the model, the relationship is somewhat "rectified", i.e. the disagreement between physiological experimentation and the model is diminished or removed. Similar results were obtained by graphical simulation. If the percentage content of synchronously acting motor units increases with N, the relationship between integrated electrical activity of the interference pattern and the number of active motor units becomes closer to a straight line (Fig. 7). DISCUSSION
with the force of the motor unit contraction. This means that an increase in b cannot considerably affect the relationships between the force of contraction and EMG integral. One may assume that a considerable increase in the amplitude and area of the interference pattern during strong contraction is caused by activation of motor units which, being closest to the electrode, give action potentials of large amplitude (Rosenfalck 1969). However, the significance of this factor can hardly be evaluated at the present as it is impossible to apply these data to EMG led offwith surface electrodes without special studies. We suppose the integrated electrical activity is considerably increased over the entire range of muscular contraction because of the tendency to more and more increased synchronization of motor unit activity, which was not taken into consideration in the asynchronous model. Our synchronous model supports the supposition. Similar considerations were discussed by Moore (1967a,b). This hypothesis shows good agreement with the results of physiological experiments using cross-correlation analysis of EMGs, which showed increase in synchronization along with increased muscle contraction (Person and Kudina 1968). However, there is a possibility of multiple causes of the discussed phenomenon. It is generally believed that an EMG showing an interference pattern is a "bad" EMG, as one cannot discriminate in it separate action potentials of motor units and measure them. Simula~ tion has shown that when a great number of action potentials are summated, certain relation. ships are revealed between their parameters and tho~ of the interference pattern (in a statistical approach). Simulation enables us to realize the connection between the peculiarities of motor unit activity and the interference pattern. These data may be useful in revealing in the latter the information necessary in physiological and clinicai investigations. EMGs showing interference patterns may be obtained even when using surface electrodes, the advantage of this method being that the mean values of parameters for a large number of motor units are obtained at a single recording. The quantitative processing of interference patterns is easily carried out automati-
The discrepancy between the behaviour of the integrated EMG in the asynchronous model and in physiological experimentation might have been due to the fact that the force of contraction is increased not only by recruitment of new motor units but by increased frequency of motor unit discharges. However, as is seen from formula (5), an increase in the frequency of discharges 0 affects the integrated electrical activity according to the same law as an increase in N. In order to simplify the model, the potential amplitude of motor units b was assumed constant. In reality, however, action potentials of motor units recruited at a later stage of contraction have a larger amplitude (Norris and Gasteiger 1955 and others); i.e. b in formula (5) increases with an increase in N. Though this factor may play a certain part in the phenomenon dis. cussed, it alone cannot account for the disagree. ment between the model and the physiological prototype, because the amplitude of action potentials of high threshold motor units is only 3 or 4 times that of low threshold ones, whereas the amplitude of the interference pattern may increase 100-1000 times, ranging between several/~V and several mV. Besides, the ampli. tude of the motor unit action potential usually correlates with the size of the motor unit (the number of fibres in the unit) and, consequently, caily.
F.lectroenceph. clm. Heurophysiol., 19"/0,28:625--632
631
SIMULATION OF EMOS
Undoubtedly, any phenomenon is inevitably simplified in simulation, so that only the general regularities found in the model, rather than exact quantitative data, are applicable to the physiological prototype. We are convinced, nevertheless, that such an approach to the EMG showing an interference pattern, this Cinderella of clinical electromyography, will ensure it a more prominent place in clinical research, along with the classical form of motor unit EMG.
SUMMARY
The model of the EMG showing an interference pattern in which motor units are discharged independently (which is typical for relatively weak contractions) gives an adequate picture of this type of activity, agreeing with the actual EMG in appearance as well as in some important parameters (the number of maxima, frequency spectrum and autocorrelation function). Some of the interference pattern parameters (mean duration of fluctuation, the possition of the first zero of the autocorrelation function, frequency of the main spectral maximum) are in simple relation with the duration of motor unit action potentials composing this interference pattern. The study of dependence of integrated electrical activity of the asynchronous model on the number of active motor units N, revealed that after a certain level of activity is reached, the model disagrees with the results of physiological experimentation. The latter shows a more rapid increase in electrical activity with increased number of motor units (force of contraction). If synchronization increasing with increased N is put into the model, it becomes closer to the actual EMG in the range of activity of a great number of motor units. Simulation has shown that changes in the mean duration of interference pattern fluctuation and the frequency spectrum, observed during synchronization in physiological experiments, are not caused by synchronization but by the fact that this gtnchronization is not complete in the muscle. Small shifts in the impulses of motor units discharging synchronously account for an increase in mean duration of the interference
pattern fluctuation and also for the displacement of the main spectral maximum to the left. Rf~SUM~ SIMULATION D'I~LECTROMYOORAMMES MONTRANT DES PATTERNS D'INTERFi~-RENCE
Le mod61e de l'61ectromyogramme (EMG) montrant un pattern d'interf6rence dans lequel les unit6s motrices se d6chargent ind6pendamment (ce qui est typique de contractions relaUvement faibles) donne une repr6sentation ad6quate de ce type d'activit6, conforme ~tI'EMG r6el quant ~tson apparence en ce qui concerne certains param~tres importants (nombre des maxima, spectre de fr~quence et fonction d'autocorr61ation). Certains paxam6tres du pattern d'interf6rence (dur~e moyenne de fluctuation, position du premier z6ro de la fonction d'autocorr61ation, fr6quence du maximum principal du spectre) sent en relation simple avec la dur6e des potentiels d'action de l'unit6 motrice qui composent ce pattern d'interf6rence. L'6tude de la d6pendance entre l'activit6 61ectrique int6gr~e du mod~le asynchrone et le nombre JV des unit6s motrices actives, r6v~le qu'apr~s qu'un certain niveau d'activit6 soit atteint, le mod~le est en d6saccord avec les r~sultats de l'exp6rimentation physiologique. Cette derni~re montre une augmentation plus rapide de l'activit6 ~l~trique avec l'accroissement du nombre des unit~s motrices (force de contraction). Si une synchroni~tion augmentant aveu un nombre N accru est introduite clans le mod~le, il devient plus proche de I'EMG r~el clansla gamme d'activit6 d'un grand nombre d'unit6s motrices. La simulation montre que des modifications de la dur6e moyenne de la fluctuation du pattern d'interf~rence et du spectre de fr6quence, observ6es au cours de la synchronisation dans les exl~riences physiologiques, ne sent pas dues ~t la synchronisation, mais au fait que cette synchronisation n'est p~s complete dans le muscle. De petites variations dans les impulsions d'unit6s motrices se d~chargeant de fagon synchrone sent responsables d'un accroissement de la dur6e moyenne de la fluctuation du pattern d'interf6rence et ~galement du d6placement du maximum principal du sl~tre vers la gauche. Eiectroenceph. clin. Neurophysial., 1970, 28:625-632
632
R . S . PERSON AND M. S. LIBKIND
Moozuz, A. D. Synthesized EMG waves and their impliREFERENCES cations. Amer. J. phys. Med., 196To, 46: 1302.-1316. ADRIAN,E. D. Oliver-Sharpey lectures on the interpreta- Nozutis JR., F. H. and GAsr~I(;ER,E. L. Action potentials tion of ~l~celectromyogram. Lancet, 1923, 208: 1229of single motor units in normal muscle. Electroenceph. |233. din. NeurophysioL, 1955, 7:115-126. ADMAN, E. D. General principles of nervous activity. PERSON, R. S. and KUDINA, L. P. Cross-correlation of Brain, 1947, 70: 1-17. electromyograms showing interference patterns. EiecB~.~LAND,B. and Ln'POLD, O. Motor unit activity in the troenceph, din. Neurophysiol., 1968, 25: 58.68. voluntary contraction of human muscle. J'. Physiol. PEasoN, R. S. and LmrJND, M. S. [Modelling of interfer(Lond.) , 1954,12S: 322-335. ing bioelectrical activity]. In A. I. RorrBACK (Ed.), E~rCtrrtlAL, F. and MADSEN,A. Synchronous activity in [Electrophysioiogy of central nervous system. Papers of normal and atrophic muscle. Electroenceph. c/in. the Fifth USSR Conference.] (in Russian). Tbilisi, 1966: Neurophysiol., 1950, 2: 425-444. 237-238. GLqUlNKt~L,V. S., SinK, M. L. and KOTS,Y. M. [Postural PERSON,R. S. and LmKtND, M. S. [Modelling of interfercontrol.] (in Russian). Nauka, Moscow, 1965, 255 p. ing bioelectrical activity.] (in Russian). Bio)~zika, 1967, HAAS,E. Ober die Art der TfitiBkeitunserer Muskeln beim 12: 127-134. Halten verschiedener schwerer Gewichte. P/iiigers PEasoN, R. S. and MlSmN, L. N. [Auto-correlation and Arch. gas. PhysioL, 1926, 212: 651--656. cross-correlation analysis of the muscular electrnactivHYDE,J., Rt(~Ltz, C. M. and GELLHORN,C. The dependity.] (in Russian). Fizioi. Zh. (Leningr.), 1963, 49: ence of unit discharges on frequency and intensity 1050--1058. of stimulation of the motor cortex in the macaque..i. ROSENFALCK, P. Intra- and extracellular potential fields cell. comp. Physiol., 1954, 43: 293-310. of active nerve and muscle fibres. Actaphysiol. scand., KoGI, g. and HAKAMADA,T. Frequency analysis of the 1969, Suppl. 321,168 p. surface electromyogram in muscle fatigue. J. Sci. TAYt.OR, A. The significance of grouping of motor unit Labour (Tokyo), 1962, 38: 519-528. activity..I. Physiol. (Lond.), 1962,162: 259--269. LIBgnqZ), M. S. [Modelling of interfering bioelectrical acVREDFNntZOT, J. and Kos'r~R, W. G. Some aspects of tivity, 11.](in Russian). Bio:~:ika, 1968,13: 685-693. muscle mechanics in vivo. IPO ann. Progr. Rapt., LmKtND, M. S. [Modelling of interfering bioelectrical ac1966,1:94-100. tivity, Ill.] tin Russian). Bio.6:ika, 1969, 14: 369-372. WALTON,J. W. The electromy~3ram in myopathy: analLtppoz.t), O. C. J. The relation1 between integrated ~ction ysis with the audio frequency spectrometer. J. Neurol. potentials in a human muscle and its isometric tension. Neurosur8. Psychlat., 1952,1.5: 219-226. J. Physiol. (Land.), 1992,117: 492-499. ZHtnCOV, E. K. and ZAKHARYANTZ,Y. Z. [S,vllchroniz~ MOLI,ER, E. The chewing apparatus. Acre physiol, scand,, rhythm of action potentials during human muscular 1966, Suppl. 280, 292 p. activity.] (in Russian). Fi:lol. Zh. (Leainfr.), 1959, 45: MooRe, A. D. Synthetic EMG waves. In J. V. BASUAJIAN 1053-1059, (Ed.),Muscles alive. Second ed. ;Williams and Wilktns~ Baltimore, 1967a: 53-70.
Reference: P~SON, R. S. and LmgncD M. S. Simulation of electromyograrns showing interference patterns. Electroenceph, din. Neurophystol., 19"/0, 28: 62S--~2.
625
SIMULATION OF ELECTROMYOGRAMS SHOWING INTERFERENCE PATTERNS R. S. PERSON AND M. S. LIBKIND 1
Institute of Higher Nervous Activity and NeurophysiGlogy of the U.S.S.R. Academy of Sciences, and Moscow Phj~icoTechnical Institute, Moscow (U.S.S.R.) (Accepted for publication: October 1, 1969)
The interference pattern or global EMG is a summation of action potentials of a great number of motor units. The information about motor units latent in it is not easily revealed, because of the extremely complex pattern. For this reason the employment of surface electrodes in clinical EMG is limited. The purpose of this work was to study quantitative regularities of interference pattern formation by motor unit action potentials and to analyse the relationships between their parameters. The discharges of motor units may be asynchronous (independent) and synchronous. Asynchronous activity is observed during a weak contraction of the muscle (Bigland and Lippold 1954; Taylor 1962 and others). Synchronization was found during a strong contraction of the muscle as well as under fatigue or in diseases caused by lesions of motoneurones (Haas 1926; Adrian 1947; Buchthal and Madsen 1950; Zhukov and Zakharyantz 1959 and others). For a moderate range ofmuscle contraction in healthy people asynchronous activity was considered typical, but cross-correlation analysis of EMGs has given reason to believe that a tendency to synchronization exists even in this range of contraction and that synchronization increases with increase in the force of contraction (Person and Kudina 1968). Both asynchronous and synchronous activity of motor units has been investigated in this study. A preliminary report was published earlier (Person and Libkind 19~6). z It is deeply regretted that Dr. Libkind met with a fatal accident after this paper was submitted for publication.
METHOD
A stochastic model was used. Simulation was carried out graphically and mathematically, some of the parameters of the model being calculated with a digital computer. The characteristics of motor unit action potentials put into the model were partly borrowed from the literature on EMG and partly obtained in our own experiments on large human skeletal muscles with bipolar needle electrodes. A number of simplifying assumptions were made. Action potentials of an individual motor unit were assumed to have the shape shown in Fig. 1. In our model N was the number of active motor units, 0 was the mean frequency of motor unit discharge and the inter-spike intervals were independent random values having a normal distribution, with mean # and standard deviation a0. A single impulse, triangular in shape, diphasic and symmetrical with respect to the zero line, had the duration 2c and an amplitude from peak to peak 21,. Both for the graphical model and for computation it was assumed that 0=20 c/seo, p=40 mse¢, tT0ffi5 msec, 2cffil0 mse¢, on the accepted scale b= 3c. The mean duration of a diphasic wave of the interference pattern was 2d. The parameters of interference pattern were studied as depending on N, as an inorease in the number of active motor units was regarded as the main factor increasing a muscle contraction (Bigland and Lippold 1954; Hyde et al. 1954 and others). In graphical simulation the curve representing action potentials of an individual motor unit with all the parameters enumerated above was plotted £1ectroenceph. din. Newophysiol., 1970, 28:625--632
626
R. S. PERSON AND M. S. LIBKIND
on the scale of I mm to I msec (for the time of 500 msec). Next, trains of impulses randomly superimposed on one another were successively added up and the obtained synthetic EMGs photographed on the same time scale as the actttal EMG (Fig. 1). The graphical model was used to visualize the pattern of synthetic EMG and to calculate by direct measurement certain parameters of EMGs --frequency of fluctuations and integrated electrical activity. In the mathematical model the theory of random functions was employed. The activity of the i-th motor unit in the time interval from O to a certain time point T was regarded as a random process ~pl(t). The positive part of a single impulse has the following analytical expression: 0, 2b v,(t) ffi
when t < 0 and t > c; t,
when 0 ~
2b - 2b c
when c~<~t<~c.
(I)
The interference pattern, which is arrived at by summing up action potentials of active motor units, can be expressed as ,iV
fN(t)
=
(2)
The following characteristics of this simulated EMG were analytically studied: mean duration of fluctuations, autocorrelation function, frequency spectrum and integrated electrical activity. The mathematical procedure employed was earlier described by Person and Libkind (1967) and Libkind (1968, 1969). In the mathematical model synchronization was introduced in the following way: suppose N motor units are divided into p independent groups, each containing n motor units acting syn. chronously; i.e., N ffipn. Inside each group ira. pulses coincide either fully or partially, i.e., with a certain shift in time, the value of the shift not exceeding the duration of an impulse. By this we mean that there exists some difference in the time of conduction of impulses from different motoneurones to electrodes. Consequently, the re-
corded action potentials of synchronously discharging motor units are slightly shifted with respect to one another (out of phase synchronization). This seems to be near to what actually takes place in the physiological prototype. In the graphical model synchronization was introduced differently. Here N active motor units were divided into two groups, one of which contained mM motor units acting asynchronously and the other ms units acting synchronously. Thus,
N-ross+ms ms may be expressed as percentage content of synchronously acting motor units ft. Then
ms ffi 100 The content of synchronously acting motor units (as percentage) may be assumed either constant or increasing with N. For computation on a digital computer it was assumed that 1 time step = 1 msec; Tffi500 msec, N = 100. RESULTS
I. Asynchronoua activity of motor units 1. Visible pattern of graphical model. When a small number of motor units were added the impulses were frequently grouped with a "pileup" effect, which had been previously observed in similar conditions by Taylor 0962). With an increase in the number of active motor units the time intervals free of impulses gradually diminished. A "saturation" level was reached when 16-18 motor units acted. Fig. l shows that synthetic EMGs, especially the "saturated" ones, visually resemble the actual EMG.
2. Mean duration andfrequency of repetition of interference pattern fluctuations. For the determination of a mean duration of interference pattern fluctuations (2d) the mathematical model was used. The number of zero-line crossings in a unit of time was found. From this was found the mean duration of a monophasic wave d ffi
2:
~, - 0.91c
(3)
Thus, in the model there exists a simple, conElectroenceph. clin. Neurophysiol., 1970, 28:627~-632
627
SIMULATION OF EMGS
t205sec
c/~ tPO, 120'
2~
#/qU
3
101vlU
A
6O 00
I#0' B frO'
18FIu
IOO' 012
z/
6
a
io
15
lo~, kg
#OHU
Fig. 2 A: Relation of mean fluctuation frequency in the graphical model of EMG to the number of active motor units; B: relation of the mean fluctuation frequency of the actual EMG (m. biceps brachii) to the load (y axis - the frequency in ~ o" the initial).
RC~) t~ ~8
[---. l~O5sec' 751~V Fig. 1 Synthetic EMGs obtained by summation of increasing numbers of motor units (I-5) and the actual EMG of m. biceps brachii (6).
stant relationship between a mean duration of interference pattern fluctuations and duration of action potentials of motor units forming this pattern, the coeflioient of relationship being close to 1. These results agree very well with those obtained for d on a digital computer° The frequency of repetition for all fluctuations (the number of maxima) was determined from the graphical model. With an increase in N the number of fluctuations iacreased up to the level of "saturation" after which it became more stable, ranging between 130 and 140/sec. Thus, the frequencies of fluctuations in the model and in the actual EMG are values of the same order. The character of relationship between frequency of fluctuations and the number of active units in the model is like that between frequency and force of contraction in the actual EMG (Fig. 2). 3. Autocorrelation function and spectrum of
~6
~2 0
i"
B
. . . .
f
Fig. 3 A: Autocorrelation function of the actual EMG (m. biceps brachii); B: the same for the mathematical model of EMG. X axis - shift in msec; y axis - value of autocorrelation function.
interference pattern. The shape of an autocorrelation function, which was calculated from the mathematical model, was similar to that obtained earlier by Person and Mishin (1963) for an actual EMG (Fig. 3). The distance from the reference point to the first crossing of the function with x axis To may be regarded as one-fourth of a diphasic wave of autocorrelation function. It has been found that 2xo- 1.12cffi 1.24d
(4)
Thus, the parameter in question of the autocorrelation function is closely connected with the Electroenceph.din, NeurophysloL, 1970, 28:625-632
628
It. S. PERSON AND M. S. LIBKIND
the interferen~ pattern was investigated mathematically. It was found that, when motor units act independently of one another, the integrated electrical activity starting somewhere near N = 20 behaves on the average as
t
,. Xb where K I I
,
(5)
2
Assuming all the parameters, except N, constant we get
! |
!
o
T 'jAx 3¢
.Cot 3¢ Fig, 4
The spectrum of the mathematical model of EMG. X axis - frequency;y axis- spectral density.
These results, arrived at analytically, are illustrated by the graphical model (Fig. 5) where the area of EMG was directly measured. Similar results were obtained by Moore (1967a,b). Having computed a synthetic EMG as a sum of diphasic waves randomly superimposed on one another he obtained: r.m.s. --K[/Q
Nom~ ot /41/
sNkesl,~ee
Fig. 5 Relation of integrated EMG to the number of active motor units for the graphical model, Asynchronous activityof motor units. mean duration of main fluctuations of interference pattern, which agrees with observations on actual EMGs (Person and Mishin 1963). The spectrum of the model was calculated from the autocorrelation function (Fig. 4). It possessed one well pronounced peak whose maximum corresponded with the sinusoidal frequency 4ni3c, the other peaks being negligibly small. Thus, the dominating frequency is in a simple relationship with the duration of the elementary potential., For 2c=10 mse¢ the maximum of the spectrum corresponded with the frequency of 133 c/sec, which tallies with the results of spectral analysis for actual interference patterns led off by needle electrodes (Walton 1952). 4. Integrated electrical activity. The integral of
where r.m.s, is the root mean square of the resultant waves, Q is a number of summated waves and Kis a constant.
Thus, when motor units act asynchronously, the dependence of the integrated electrical activity on the number of active motor units is nonlinear, i.e., with an increase in the number of active motor units the integral increases at a gradually diminishing rate. The phenomenon may be understood from the following reasoning. When diphasic fluctuations are randomly superimposed on one another they may get "in phase" as well as "in antiphase". In the former ease the amplitude and the area of interference curve are increased, in the latter, diminished. This cancellation effect was noted by Adrian (1925). The relatioi~ between these two processes accounts for the character of dependence in question. The rate at whic~ the integral increases is considerably diminished when N-- 18-20. Turning to the graphical model we notice that at N m20 the synthetic EMG becomes "saturated", i,e. it no longer has any intervals free of impulses. A rapid growth of "unsaturated" E M G area is explained by the fact that some of the new impulses find themselves in vacant spaces and their areas are
Etectroenceph. din. Ne~ophy$iol., 1970, 28:625-632
SIMULATIONOF BMGS summated linearly. When no free intervals are left the probabilities of getting "in phase" or "in antiphase" for new fluctuations become almost equal. As has been said above, the chief factor increasing contraction is recruitment of new motor units. It is therefore interesting to compare changes in integrated electrical activity of the model caused by an increase in N with those in the actual EMG depending on the force of contraction. The comparison shows that in this respect the model behaves differently from a muscle. In the model the electrical activity reaches a certain level of saturation after which it increases only insignificantly, whereas in the muscle it is increased over the entire range of contraction force. Several authors have proved by needle as well as by surface electrodes that this relationship is almost linear or even sharper than linear (Lippold 1952: Gurfinkei et al. 1965; Meller 1966; Vredenbregt and Koster 1966). This disagreement deserves special consideration (see Discussion).
II. ,Synchronous activity of motor units I. Mean duration offluctuation, autocorrelation function and spectrum of interference pattern. Mathematical simulation has shown that in the case of synchronization without phase shift the mean duration of interference pattern fluctuation, as well as the positions of the frst zero of the autocorrelation function and of the spectral maximum, are virtually the same as in the case of asynchronous activity and the relationships de. scribed, above are relevant here. In the case of out of phase synchronization the picture is somewhat different. The mean duration of interference pattern fluctuation is increased I/i" times, becoming longer than the duration of a motor imit impulse: d' = 1.26c (7) The spectral maximum is shifted to the left as compared with its positioh in the case of asynchronous activity. Thus, simulation proves that increased duration of interference pattern fluctuation in the case of synchronization, which is known from physiological experiments (Fig. 6), and the dis-
629
~
/OOpV
25 ITISec I
'
Fig. 6 Actual EMG of m. biceps brachii; isometriccontraction, a load 5 kg. Surface electrodes. 1: the beginning of muscular activity; 2: the same after 5 rain of activity, under fatigue(synchronization).
/
/
/
/ / y 8 ~
//
o
"
1o
"
zo
I
zo
,
I
~o
!
~o
I
ao
Numberof Mt/ Fig. 7 Relation of integrated EMG to the number of active motor units for the graphical model. I: asynchronousactivity of motor units. 2: 20~ of motor units act synchro. nously; 3: proportion of synchronously acting motor units p increases with an increase in the number of active motor units N(/~ = 15 -t- 0.5 N); 4: synchronousactivity of motor urdts.
placement of the spectral maximum to the left (Kogi and Hakamada 1962 and others) are not caused by synchronization as such, but byshift in synchronization. 2. Integrated electrical activity. The study of integrated electrical activity of the mathematical model revealed that synchronization (both with and without phase shift) greatly affects its value. The relationship between integrated EMG and N becomes K,N~ (8) where Ks depends on the same parameters as Kx (b, c and 0), and the index ~ depends on the
£1ectroenceph.din. Neurophy~loL,19/0, 28:625-632
630
R. S. PERSON AND M. S. LIBKIND
degree of synchronization and varies from 0.5 to 1. If o~0.5, the relationship is close to that typical for asynchronous activity, whereas ifo,--, 1, it is close to being linear. A phase shift of impulses diminishes the rectifying influence of synchronization. Thus, simulation has shown that if synchronization is introduced into the model, the relationship is somewhat "rectified", i.e. the disagreement between physiological experimentation and the model is diminished or removed. Similar results were obtained by graphical simulation. If the percentage content of synchronously acting motor units increases with N, the relationship between integrated electrical activity of the interference pattern and the number of active motor units becomes closer to a straight line (Fig. 7). DISCUSSION
with the force of the motor unit contraction. This means that an increase in b cannot considerably affect the relationships between the force of contraction and EMG integral. One may assume that a considerable increase in the amplitude and area of the interference pattern during strong contraction is caused by activation of motor units which, being closest to the electrode, give action potentials of large amplitude (Rosenfalck 1969). However, the significance of this factor can hardly be evaluated at the present as it is impossible to apply these data to EMG led offwith surface electrodes without special studies. We suppose the integrated electrical activity is considerably increased over the entire range of muscular contraction because of the tendency to more and more increased synchronization of motor unit activity, which was not taken into consideration in the asynchronous model. Our synchronous model supports the supposition. Similar considerations were discussed by Moore (1967a,b). This hypothesis shows good agreement with the results of physiological experiments using cross-correlation analysis of EMGs, which showed increase in synchronization along with increased muscle contraction (Person and Kudina 1968). However, there is a possibility of multiple causes of the discussed phenomenon. It is generally believed that an EMG showing an interference pattern is a "bad" EMG, as one cannot discriminate in it separate action potentials of motor units and measure them. Simula~ tion has shown that when a great number of action potentials are summated, certain relation. ships are revealed between their parameters and tho~ of the interference pattern (in a statistical approach). Simulation enables us to realize the connection between the peculiarities of motor unit activity and the interference pattern. These data may be useful in revealing in the latter the information necessary in physiological and clinicai investigations. EMGs showing interference patterns may be obtained even when using surface electrodes, the advantage of this method being that the mean values of parameters for a large number of motor units are obtained at a single recording. The quantitative processing of interference patterns is easily carried out automati-
The discrepancy between the behaviour of the integrated EMG in the asynchronous model and in physiological experimentation might have been due to the fact that the force of contraction is increased not only by recruitment of new motor units but by increased frequency of motor unit discharges. However, as is seen from formula (5), an increase in the frequency of discharges 0 affects the integrated electrical activity according to the same law as an increase in N. In order to simplify the model, the potential amplitude of motor units b was assumed constant. In reality, however, action potentials of motor units recruited at a later stage of contraction have a larger amplitude (Norris and Gasteiger 1955 and others); i.e. b in formula (5) increases with an increase in N. Though this factor may play a certain part in the phenomenon dis. cussed, it alone cannot account for the disagree. ment between the model and the physiological prototype, because the amplitude of action potentials of high threshold motor units is only 3 or 4 times that of low threshold ones, whereas the amplitude of the interference pattern may increase 100-1000 times, ranging between several/~V and several mV. Besides, the ampli. tude of the motor unit action potential usually correlates with the size of the motor unit (the number of fibres in the unit) and, consequently, caily.
F.lectroenceph. clm. Heurophysiol., 19"/0,28:625--632
631
SIMULATION OF EMOS
Undoubtedly, any phenomenon is inevitably simplified in simulation, so that only the general regularities found in the model, rather than exact quantitative data, are applicable to the physiological prototype. We are convinced, nevertheless, that such an approach to the EMG showing an interference pattern, this Cinderella of clinical electromyography, will ensure it a more prominent place in clinical research, along with the classical form of motor unit EMG.
SUMMARY
The model of the EMG showing an interference pattern in which motor units are discharged independently (which is typical for relatively weak contractions) gives an adequate picture of this type of activity, agreeing with the actual EMG in appearance as well as in some important parameters (the number of maxima, frequency spectrum and autocorrelation function). Some of the interference pattern parameters (mean duration of fluctuation, the possition of the first zero of the autocorrelation function, frequency of the main spectral maximum) are in simple relation with the duration of motor unit action potentials composing this interference pattern. The study of dependence of integrated electrical activity of the asynchronous model on the number of active motor units N, revealed that after a certain level of activity is reached, the model disagrees with the results of physiological experimentation. The latter shows a more rapid increase in electrical activity with increased number of motor units (force of contraction). If synchronization increasing with increased N is put into the model, it becomes closer to the actual EMG in the range of activity of a great number of motor units. Simulation has shown that changes in the mean duration of interference pattern fluctuation and the frequency spectrum, observed during synchronization in physiological experiments, are not caused by synchronization but by the fact that this gtnchronization is not complete in the muscle. Small shifts in the impulses of motor units discharging synchronously account for an increase in mean duration of the interference
pattern fluctuation and also for the displacement of the main spectral maximum to the left. Rf~SUM~ SIMULATION D'I~LECTROMYOORAMMES MONTRANT DES PATTERNS D'INTERFi~-RENCE
Le mod61e de l'61ectromyogramme (EMG) montrant un pattern d'interf6rence dans lequel les unit6s motrices se d6chargent ind6pendamment (ce qui est typique de contractions relaUvement faibles) donne une repr6sentation ad6quate de ce type d'activit6, conforme ~tI'EMG r6el quant ~tson apparence en ce qui concerne certains param~tres importants (nombre des maxima, spectre de fr~quence et fonction d'autocorr61ation). Certains paxam6tres du pattern d'interf6rence (dur~e moyenne de fluctuation, position du premier z6ro de la fonction d'autocorr61ation, fr6quence du maximum principal du spectre) sent en relation simple avec la dur6e des potentiels d'action de l'unit6 motrice qui composent ce pattern d'interf6rence. L'6tude de la d6pendance entre l'activit6 61ectrique int6gr~e du mod~le asynchrone et le nombre JV des unit6s motrices actives, r6v~le qu'apr~s qu'un certain niveau d'activit6 soit atteint, le mod~le est en d6saccord avec les r~sultats de l'exp6rimentation physiologique. Cette derni~re montre une augmentation plus rapide de l'activit6 ~l~trique avec l'accroissement du nombre des unit~s motrices (force de contraction). Si une synchroni~tion augmentant aveu un nombre N accru est introduite clans le mod~le, il devient plus proche de I'EMG r~el clansla gamme d'activit6 d'un grand nombre d'unit6s motrices. La simulation montre que des modifications de la dur6e moyenne de la fluctuation du pattern d'interf~rence et du spectre de fr6quence, observ6es au cours de la synchronisation dans les exl~riences physiologiques, ne sent pas dues ~t la synchronisation, mais au fait que cette synchronisation n'est p~s complete dans le muscle. De petites variations dans les impulsions d'unit6s motrices se d~chargeant de fagon synchrone sent responsables d'un accroissement de la dur6e moyenne de la fluctuation du pattern d'interf6rence et ~galement du d6placement du maximum principal du sl~tre vers la gauche. Eiectroenceph. clin. Neurophysial., 1970, 28:625-632
632
R . S . PERSON AND M. S. LIBKIND
Moozuz, A. D. Synthesized EMG waves and their impliREFERENCES cations. Amer. J. phys. Med., 196To, 46: 1302.-1316. ADRIAN,E. D. Oliver-Sharpey lectures on the interpreta- Nozutis JR., F. H. and GAsr~I(;ER,E. L. Action potentials tion of ~l~celectromyogram. Lancet, 1923, 208: 1229of single motor units in normal muscle. Electroenceph. |233. din. NeurophysioL, 1955, 7:115-126. ADMAN, E. D. General principles of nervous activity. PERSON, R. S. and KUDINA, L. P. Cross-correlation of Brain, 1947, 70: 1-17. electromyograms showing interference patterns. EiecB~.~LAND,B. and Ln'POLD, O. Motor unit activity in the troenceph, din. Neurophysiol., 1968, 25: 58.68. voluntary contraction of human muscle. J'. Physiol. PEasoN, R. S. and LmrJND, M. S. [Modelling of interfer(Lond.) , 1954,12S: 322-335. ing bioelectrical activity]. In A. I. RorrBACK (Ed.), E~rCtrrtlAL, F. and MADSEN,A. Synchronous activity in [Electrophysioiogy of central nervous system. Papers of normal and atrophic muscle. Electroenceph. c/in. the Fifth USSR Conference.] (in Russian). Tbilisi, 1966: Neurophysiol., 1950, 2: 425-444. 237-238. GLqUlNKt~L,V. S., SinK, M. L. and KOTS,Y. M. [Postural PERSON,R. S. and LmKtND, M. S. [Modelling of interfercontrol.] (in Russian). Nauka, Moscow, 1965, 255 p. ing bioelectrical activity.] (in Russian). Bio)~zika, 1967, HAAS,E. Ober die Art der TfitiBkeitunserer Muskeln beim 12: 127-134. Halten verschiedener schwerer Gewichte. P/iiigers PEasoN, R. S. and MlSmN, L. N. [Auto-correlation and Arch. gas. PhysioL, 1926, 212: 651--656. cross-correlation analysis of the muscular electrnactivHYDE,J., Rt(~Ltz, C. M. and GELLHORN,C. The dependity.] (in Russian). Fizioi. Zh. (Leningr.), 1963, 49: ence of unit discharges on frequency and intensity 1050--1058. of stimulation of the motor cortex in the macaque..i. ROSENFALCK, P. Intra- and extracellular potential fields cell. comp. Physiol., 1954, 43: 293-310. of active nerve and muscle fibres. Actaphysiol. scand., KoGI, g. and HAKAMADA,T. Frequency analysis of the 1969, Suppl. 321,168 p. surface electromyogram in muscle fatigue. J. Sci. TAYt.OR, A. The significance of grouping of motor unit Labour (Tokyo), 1962, 38: 519-528. activity..I. Physiol. (Lond.), 1962,162: 259--269. LIBgnqZ), M. S. [Modelling of interfering bioelectrical acVREDFNntZOT, J. and Kos'r~R, W. G. Some aspects of tivity, 11.](in Russian). Bio:~:ika, 1968,13: 685-693. muscle mechanics in vivo. IPO ann. Progr. Rapt., LmKtND, M. S. [Modelling of interfering bioelectrical ac1966,1:94-100. tivity, Ill.] tin Russian). Bio.6:ika, 1969, 14: 369-372. WALTON,J. W. The electromy~3ram in myopathy: analLtppoz.t), O. C. J. The relation1 between integrated ~ction ysis with the audio frequency spectrometer. J. Neurol. potentials in a human muscle and its isometric tension. Neurosur8. Psychlat., 1952,1.5: 219-226. J. Physiol. (Land.), 1992,117: 492-499. ZHtnCOV, E. K. and ZAKHARYANTZ,Y. Z. [S,vllchroniz~ MOLI,ER, E. The chewing apparatus. Acre physiol, scand,, rhythm of action potentials during human muscular 1966, Suppl. 280, 292 p. activity.] (in Russian). Fi:lol. Zh. (Leainfr.), 1959, 45: MooRe, A. D. Synthetic EMG waves. In J. V. BASUAJIAN 1053-1059, (Ed.),Muscles alive. Second ed. ;Williams and Wilktns~ Baltimore, 1967a: 53-70.
Reference: P~SON, R. S. and LmgncD M. S. Simulation of electromyograrns showing interference patterns. Electroenceph, din. Neurophystol., 19"/0, 28: 62S--~2.