Analysis of Motor Unit Action Potentials during Constant Force Isometric Contractions, Study Guides, Projects, Research of Statistics

Download Analysis of Motor Unit Action Potentials during Constant Force Isometric Contractions and more Study Guides, Projects, Research Statistics in PDF only on Docsity! Kvbernet ik 12. 16()-168(19731 'C;' b) Spnngcr·\·crlag 197:\ Some Properties of Motor Unit Action Potential Trains Recorded during Constant Force Isometric Contractions in Man Carlo J. De Luca and William J. Forrest Anatomy Department. Queen's University. Kingston. Ontario. Canada Received: December 3. 1972 Abstract A specially designed needle electrode was used to record motor unit action potentials for the complete time duration of constant force isometric contractions varying in discrete steps from minimum to maximum force levels. A total of 70 motor unit action potential trains were recorded and analyzed Several properties of the motor unit action potentials were observed The inter-pulse intervals between adjacent motor unit action potentials of a particular motor unit action potential train were measured and subsequently analyzed as a real continuous random variable. The distribution of the values of the inter-pulse intervals was described by the Weibull probability distribution function with time and force dependent parameters. Furthermore' the Survivor function and the Hazard function of the Weibull probability distribution function described certain characteristics of the motor unit firing intervals. Most important of all. it became possible to derive an equation that would generate a real continu­ ous random variable whose properties would be identical to those of the inter-pulse intervals. Introduction A muscle contraction is the result of concurrent contractions of several motor units. A motor unit consists of a group of muscle fibers and their inner­ vating terminal branches of one nerve fiber whose cell body is located in the anterior horn of the spinal gray matter. When a motor unit is stimulated, an extra-cellularly placed electrode will record the current distribution in the territory of the motor unit. The recorded pulse is called the motor unit action potential. A sequence of motor unit action potentials is known as a motor unit action potential train (MUAPT); the time interval between adjacent pulses will be referred to as the inter-pulse intercal (lPI). MUAPT's from human skeletal muscles have been analyzed under various conditions by numerous investigators (Bigland and Lippold. 1954; Buchthal et al.. 1954: Clamann, 1967; Gilson and Mills, 1941; Gurfinkel et al.; 1970: Kaiser and Petersen, 1965; Larsson et al.. 1965; Masland et al.. 1969: Person and Kudina, 19'7~) This paper will deal with the properties of MU APT's recorded at various levels of constant force isometric contraction for the complete time duration of a human skeletal muscle contraction. Materials and Methods The following equipment arrangement was used to record the MUAPT's. A sturdy wooden chair with a high back was modified as follows. Two adjustable Velcro straps were fastened to both sides of the back of the chair. A force gauge capable of measuring 50 kg of force with a displacement of 0.1 mm was secured to the chair. A cuff consisting of an adjustable band of cotton webbing 5 ern in width and two pieces of Velcro was connected to the force gauge by a flexible steel cable. The output of the force gauge was attached to one channel of a dual-beam oscilloscope (oscilloscope I). The differential preamplifier was cascaded with a single-ended amplifier. The output of the amplifier was connected to a separate oscilloscope. The outputs of the amplifier and the force gauge were fed to an FM tape recorder. An audio amplifier and a speaker were cascaded to the differential preamplifier. Acoustic representation of motor unit action potentials assisted significantly in detecting the presence of different MUAPT's. A block diagram for the equipment arrangement is presented in Fig 1. Tape Recorde rForce Gauge i ~ Fig. I. Block diagram of the equipment arrangement of the experiment , 161 • • C. J. De Luca and W. J. Forrest: Properties of Motor Unit Action Potential Trains Four right-handed male subjects volunteered for the experiment. Their ages varied from 22-32 years. with an average of 25.3 years. \ II -ubjccts denied past injury to the right shoulder region. A ,,,hlect was seated in the chair and the two Velcro straps were t.i-tcncd over his shoulders. The straps kept the torso of the subject in a fixed position with respect to the force gauge and prevented shrugging of the shoulder by restricting the elevation of the scapula. but did not impede the rotation of the scapula. The cuff was secured snugly about the distal part of the right arm just proximal to the elbow joint The maximum force output of the isometric abduction of each subject was measured. A specially designed quadri-filar electrode (De Luca and Forrest. 1972) was inserted into the central area of the middle fibers of the deltoid muscle. The r electrode was capable of recording distinct MUAPTs from a muscle contracting at any force level (including maximal force). The tip of the electrode penetrated to the middle of the depth of the deltoid muscle. Clamann (1967) pointed out that this region of a muscle contains motor units having a gradation of thresholds from low to high. The electrode was connected to the input of the differential preamplifier. The :1 dB points of the bandwidth of the preamplifier were set at 100 and IAOO Hz. The gain of the preamplifier and amplifier was regulated \(\ give the largest possible signal output that could be stored on Ill.lgnetic tape. thereby optimizing the signal-to-noise ratio. The trace of the isolated second channel of oscilloscope I w"s placed at the equivalent voltage representation of 5 kg above the trace of the other channel which displayed the output of the force gauge. Each subject was instructed to abduct the upper limb in the coronal plane with the arm medialIy rotated and the forearm pronated. Then he was asked to superimpose the two traces on oscilloscope I as quickly as possible with a minimal amount of overshoot. When the desired level of isometric abduction was achieved. the subject was requested to maintain the force out­ put constant until he was no longer capable of doing so. At the end of the contraction. the electrode was removed from the muscle. Each subject had a minimal rest period of two hours between successive contractions. Prior to each contraction, the electrode was reinserted into the deltoid muscle. Hence. different MUAPTs were recorded for each contraction. MUAPTs were obtained for contractions with monitored force outputs of 5, 10, 15, 20 and 25 kg. The recorded MUA PTs were photographed on a 35 mm film moving at a speed of 250 mm/sec. A 50 Hz square-wave calibration Signal was photographed to check the true speed of the camera. A l(\t,,1 of 70 MUAPT's were recorded from the four subjects. Two persons independently interpreted the records, thus reducing the probability of allocating a motor unit action potential to the \\rong MUAPT. The IPl's of all the MUAPTs were measured. The accuracy of the measurement was ± 0.1 msec. Analysis of Data A recent study (De Luca, 1972) showed that the relative force contribution of the anterior. middle and posterior fibers of the deltoid muscle and that of the supraspinatus muscle remains constant during isometric abduction. The force contribution of the middle fibers of the deltoid muscle during isometric abduction was calculated from the values of the measured force of abduction by employing a special technique described by De Luca (1972). The force output of the middle fibers of the deltoid muscle was found I,) he linearly proportional to the measured force of abduction. The IPI's of a MUAPT were analvzed as a random variable. The following terminology will he used to describe the various tests: X = a real continuous random variable representing the IPI. x = the range of all possible values (outcomes) which can be assumed by X. x, = a specific outcome of X: the value of a specific I PI. Xi array = a complete series of outcomes of X: hence. it represents all the IPI's of one M UAPT. Stationarit v The mean value and the standard deviation of the l Pl's for every 5 sec interval of a MUAPT were calculated. These values were calculated for all 70 MUAPTs. The mean values were plotted against the corresponding time. In addition. for each MUAPT. the mean values were plotted against the corresponding standard deviations. and a polynomial least-square regression was perfor­ med on all the values for each MUAPT. All the MUAPT's were fitted with a Znd, 3rd or 4th degree polynomial. The degree of the polynomial which provided the best fit for the values of a parti ­ cular MUAPT was determined by calculating the residual sum-of­ squares between two successive degrees of the polynomial. This procedure has been described by Ostle (1954). The accepted degree was obtained when the values of the residuals was less than 5 x 10- 6 . The histograms of the l Pl's for each of the 70 MUAPTs were plotted by a computer program. The mean, standard devia­ tion, skewness, minimum value, maximum value and total number of IPI's were calculated for each MUAPT. Histograms were also plotted for sections of the MUAPT's. Each MUAPT was divided into 10 equal time-sections; 700 histograms were formed. Probability Distribution Function The following three probability distribution functions (PDF's) were fitted to the histograms of the [PI's of each MUAPT: Lognormal fx(x) = 1 , exp { [In ( x; a )f} (x - a) (2rrK)· - .. 2K Gamma fx(x) = f! )U<S [_x_;_a. r- I exp [__Ix_;_a_l] ~-- -~--- ..)~~ ~l""\ 'l:: Weibull fx(x) = 4[_x_;_ar- ., [- (X;~)l The PDF's are described by three parameters K. {3 and x; where K = shape parameter. {3 = scale parameter and x = location param­ eter. The parameter K has no units, {3 and a have the units of the IPI's (msec). The parameter a was evaluated by finding the minimum value of Xi' The «best" estimates of K and fJ were obtained by the Maximum Likelihood method which maximized the function In L(K. {3) = I In ftx, IK. f3) i = 1 The goodness-of-fit of the three PDF's with the «best" estimates of the parameters was measured by the Kolmogorov-Smirnov test. The three PDF's were also fitted to the IPI's of sections of a MUAPT. Each MUAPT was divided into consecutive time­ sections each containing at least 1501Prs; this is the minimum number of I PI's that should be used to obtain a meaningful test of the goodness-of-fit of a PDF (Pearson and Hartley. 19541. A total of 225 sections was formed. 80 10 1M C J De Luca and Vi J Forrest: Properties of Motor Unit Action Potential Trains rectus femoris muscle. Person and Kudina (19721 maintained that for motor unit firing rates greater than 10-13 pulses per sec. the IPI histograms were symmetrical. Clamann (1967). recording from the human biceps brachii muscle. found that the IPI histograms had a Gaussian distribution. No evidence of symmetry was detected in the IPI histograms obtained in this study. It is interesting to note that IPI"s of neural motor activity in the central nervous 70 MEAN 138·9 MS STO OEV 115·3 MS 60 SKEW 1·87 VMIN 5·3 MS _ 50 VMAX 866·9 MS a. NO OF VALUES 756 ::, 40 C( W <II 30 ::; ~ z 20 75 150 225 3X 3?S 450 525 fIX 675 750 825 T MEl N MS FIg. 5. Histogram of the inter-pulse intervals of a motor unit action potential train which was recorded during an isometric contraction. The contraction was sustained until the pre-set constant force could no longer be maintained. The force from the monitored muscle was 26 kg 2525 25 ,11-11946 p-1l6.94 2020 2050- 8639 5. D.- 89.99 15 10 ),-15290 5.0-10172 100 300 6 ,LL-19l57 5.0-163.44 5 ),-13292 5 :)'112 91 100 300 500 7JO 9 system of mammals have histograms with large positive skewness and a shape similar to the histo­ grams of Figs. 5 and 6. Martin and Branch (1958) obtained histograms of spontaneous activity from single Betz cells in the motor cortex of anesthetized cats with midbrain lesions. Evarts (1964) recorded action potentials from pyramidal tract neurons in the precentral gyrus of intact. unanesthetized mon­ keys at rest and during movements. The time dependence of the mean. standard deviation and shape of the histograms of the IPI's strongly indicate that the l Pl's of a MUAPT are nonstationary. This result agrees with that of Masland et al. (1969). Probability Distribution Function The results of the goodness-of-fit of the three PDF's to the IPl's of the complete MUAPT arc listed in Table 2. A large Kolrnogorov-Srnirnov probability level indicates a good fit. The Gamma PDF provides by far the worst fit for the IPl"s of the complete MUAPT's: 83 0 of the 0 MUAPT's had a p ~ 0.05. The results for the Log­ normal and Weibull PDF's are quite similar with the Weibull PDF providing a slightly better fit than the Lognormal. Even in the best case. 39'\) of the MUAPT's have a p~0.05 which indicates that none of the three PDF's provides an acceptable fit for the IPI's of the complete MUAPT's. This is not a sur­ p-12571 p-137 07 5.0-10028 50'112.77 100 30e 500 IDe 300 500 3 25 ,lL-122 00 "p-14446 50'107 78~~L5'0'Cl01'76 10 5 o 10C: 300 5C:: 10C 3:10 500 25 7 B 20 '5 10 .1,1" "'~lJwIJLLI'L.-~LL...l--U'---.L IJJ 30J ::00 10 Fig. 6. Histograms of len cqu.i] and consecutive time-sections of a motor unit action potential train that was recorded during an isometric contract Inn. The contraction was sustained until the pre-set constant force could no longer be maintained. The abscissa is scaled in milli­ seconds 165 III C .I. De luca and \\ .I Forrest: Properties of Motor Unit AClinll Potcnt iul Tr.uns prisint! result because the lPls of ~l i"fLJAPT are nl111statil1nary: therefore, one would evpcct that the '., Iests made I1n the sectioned MlJAPT would be more indicat ive. The results or such a test ITahle 3) show a clear delineation between the goodness-of-fit of the Wei bull PDF and of the other two PDF's. A -/ test with nine degrees of freedom was performed to 11- measure the uniformity of the MUAPT sections at various probability levels. The Gamma and the d Lognormal PDF's still provide very bad fits with 1'<0.00001. The results for the Weibull PDF have a significance level of l' = 0.41. This is very strong evidence that the Wei bull PDF provides a good fit Iable 2. Pcrccntuue ,'I ",'mplcte motor unit action potential trains at various levels of I""h"hilily determined from the Kolmogorov­ xmirnov statistics " Probability level Gamma fin 110) Lognormal (in "cd Weibull (in ~,,) II I' ~ 0.0) r~O.IO r ~ 0.90 f' ~ O'J) li3 X9 0 0 44 54 IA 0 39 53 0 0 Total number of motor unit action potential trains = 70. Table :1. Number and percentage of motor unit action potential train sections at ten levels of probability determined from the II· Kolmogorov-Srnirnov statistics Probability Gamma Lognormal level 00 -0.1 98 44 91 41 "I'" 0 uO! 0.2 37 17 30 13u " 0'(l.20.3 26 12 17 7.6°"'0 0.3-0A 23 10 ~n tt 4.9"" OAO.) 8 3.6 1 \ , 12 5A''. 0.) -0.6 6 2.7 ~o to 4.5°" 0.6 -0.7 12 SAoo 12 5.4 o~". (l7 OS 4 l.R°0 to 4.)", OS 09 6 2.7 °0 13 ).8°, 0<,) 10 4 I.R°0 18 8.0°, 1 ota] number of sections = 224. Weibull '7025 tt 19 8.5% 21 9A% 30 13 % 26 t2 '70 0/ ,'028 13 t8 X.O?o 23 10 '70 19 8.5 ~il J5 6.7% for the IPI's of the MUAPTs recorded at all levels of constant force isometric contraction. The above results do not prove that the Weibull PDF provides the best possible fit. It is conceivable that some other PDF that provides a better fit exists, However. the fit provided by the Weibull PDF is highly significant. Furthermore. the Weibull PDF is a relatively simple mathematical expression which c~n be manipulated to investigate some properties of the MUAPT, which will be discussed in subsequent sections, Trends ill the Parameters "I tn« Weibllll Probability Distribut io/: Funct ion A multiple linear least-square regression was performed on the values of % and the natural logarithm of II calculated for the sectioned MUAPT's with respect to time and constant force, MUAPT's recorded from all four subjects were used, The time duration of the MUAPT's varied considerably for dilTerent subjects and force levels of the sustained contraction (see Table 1). Hence, it was necessary to normalize the time and force, Time was expressed as a fraction of the total time duration of the MUAPT. and the constant force as a fraction of the maximum force output of the middle fibers of the deltoid muscle, Table 4 lists the results of the multiple linear least­ square regression, The parameter Yo is unitless: fJ has units of msec. The mean and standard error of the time and constant-force coefficients were calculated, A r-test was performed to establish the significance level of the time and constant-force coefficients; in all cases, the r-test indicated highly significant results, To elucidate the validity of the linear relationship, the residuals of Yo and In fJ were separately plotted against their corresponding (a) predicted value, (b) normalized time, and (c) normalized constant-force, In all six plots, the plotted values were randomly dispersed indicating that a linear relationship is as good as any other relationship. Table 4. Time and force dependence of the Weibull probability distribution function parameters Parameter Constant Regression coefficients -~ - ~-_ ...~-- .._-------~------ ------­--=--:------­ Time coefficient Constant-force coefficient -­ -------"-­ - -_.­ -~-------- -------------­ Standard p value Standard p value Mean error of I-test Mean error of r-test J!I.­ :\1:· /' In I{i) 1.16 460 -0.\9 0.67 0.03 0,12 0000001 0.000001 0.18 - 1.16 0.05 o.t7 0,0001 nOOOOOI lot> C .J De Luca and W.J Forrcst : Properties of Motor L'nu Action Potential Trains The average time and force dependence of %and Ii can be expressed by the following equations: KlI.cPl= 1.16-0.19T+OI8cP O<T<I for (l(I. cP) = expl-+.60 - 0,67 T - 1.16~IJI msec O<cP<1 where r = normalized time duration of the MUAPT, cP = normalized constant force. The above equations are general expressions valid for all ML'APTs, rand cP were found to be independent with no significant interaction term. The average value of the parameter. 1.. calculated for the complete \1LAPT's \\as found to be 389± 2,82msec. Stociiastic Properties of' Motor l'llir Acrioll Potential Trains Some properties of the Weibull PDF yield useful inforrna t ion about the M L'APT. The following proper­ ties are valid for all MLAPT's recorded during a voluntary constant force isometric contraction from t he deltoid muscle The mean value of the time and force dependent Wcibull PDF is given by: ! 1)p(I.cP)=IJ(I.cPlr!1 +------ +1. • K(r.cP) where F = the Gamma function {J(I. cP) time and force dependent parameters of the %II.cPl= V,'eibul1 PDF 'J. = minimum value of the I PI's. The qeneralized t"'illg rate may be expressed as the inverse of the mean 1000 q(T. cP) = . pulses per sec. . (3IT. cP) r( 1 + -KI+ 'J.1 -h , (I. '1') The qcneraiizrd firing rare represents the expected dependence of the firing rate of a typical motor unit with respect to time during a constant force isometric contraction. The family of curves for the qeneralized [iritu; ratt' is plotted in Fig. 7. Near the end of a very weak contraction T~ 1 and </>::::.0. then %(I.</»::::.I and Il(T.cP)::::. 195 msec. At these parameter values the Weibull PDF approaches the Exponential PDF and the scale parameter {J(I.</>l becomes the mean value of the IPI. Hence the lowest firing rate of a typical motor unit is approximately 5 pulses per sec. This result corresponds with ob­ servations made by Bigland and Lippold (1954) and Person and Kudina (19721. 5 o o 2 D.• 0.6 0.8 '. 0 Normalized contraction -t.rne Fig. 7. Generalized firing rate of motor unit action potenuals as a function of normalized contraction-time at various normalized constant-force levels. The force was normalized with re-peci to the maximum isometric contraction The Survivor function of the time and force de­ pendent Weibull PDF is 30 r I I r ~ I This function gives the probability that a motor unit has not fired up to time 17 measured from the time of the previous firing. The equation indicates that the probability of a motor unit not having fired after a previous firing decreases exponentially with respect to the amount of time that elapses. The negative derivative with respect io n of the logarithm of the Survivor function describes another useful function known as the Hazard function that can be expressed as K(r, </>1 ( '7 - 'J. ')"lr.4>'" J 0H(II, T, </» = --~. --_. {J(T, </>l {J(I. </>l This function gives the probability per unit time of an immediate firing when no firing has occurred for 17 time. The quantity 0H(Il, T, </»:1'1 is the probability that a motor unit will fire during the small time interval :1'7, given that the motor unit has not fired for '7 lime.