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Fractal analysis of the electromyographic interference pattern
Journal of Neuroscience Methods 58 (1995) 103-108
ractal analysis of the electromyographic interference J. Andrew Gitter a,b,M. Joseph Czerniecki aa b Physical
Medicine
a Rehabilitation Medicine, Uniuersity of Washington, and Rehabilitation, Seattle Veterans Administration
Seattle, WA 98108, USA Medical Center, Seattle,
WA 981@
USA
Received 25 January 1994;revised 29 August 1994;accepted6 September 1994
Abstract Evaluation of motor unit recruitment is an important component of the clinical EMG exam. Typically this is assessed qualitatively using auditory features and estimates of the visual complexity of the EMG waveform. Recent advances in nonlinear dynamics have led to the development of the concept of fractals which can be used to quantify complexity and space filling features of various structures. This study was undertaken to determine if the normal EMG interference pattern (IF) has fractal characteristics that might be helpful in quantitative analysis. EMG activity was recorded from the 9 normal biceps muscles as force was varied from 10 to 90% of maximal. Using a box count algorithm, the fractal dimension was calculated. The EMG IP displays fractal characteristics with a dimension that is highly correlated with force and ranges from 1.1 to 1.4 as force increases from 10 to 90% MVC. The fractal dimension (FD)--force relationship is similar to that observed with other methods of IP analysis and suggests that the fractal dimensioncan be usedto quantify and capture the essenceof the ‘complexity’ of motor unit recruitment patterns. Keywords:
Electromyography; Muscle contraction; Chaos;Fractal; Recruitment; Models, biological
1. Introduction Evaluation of the recruitment or interference pattern (IP) 3s an important component of the clinical electromyographic (EMG) study of a muscle. The EMG IP is generated by the summation of the motor units within the recording volume of the EMG electrode as the number of units and their firing rates vary with force generation. Most commonly, recruitment is subjectively assessedby monitoring its auditory qualities and its visual appearance. Because of the importance of the IP evaluation in the diagnosis of neuromuscular disease, objective and quantitative methods for assessing recruitment have been developed (Nandedkar et al., 1986a,b; Stalberg et al., 1983). This study explores the possible role of a new technique of evaluating the PP in clinical electrophysiologic studies. Recent advances in nonlinear dynamics has led to the concepts of fractals and fractal dimension. The term fractal was first used by Mandelbrot (1980) to describe shapes or structures which exhibit the property of self similarity, that is, complexity and detail which is retained with successive levels of scaling or magnification. The classic example of a natural fractal 01650270/95/$09.50 SSDIOl65-0270(94)00164-2
0 1995ElsevierScienceB.V. All rightsreserved
is a coastline (Peitgen et al., 1992a). If the length of a coastline is measured from a low resolution photograph, a unique value will be obtained. ith a magnified or higher resolution image, new details, i.e. inlets, bays and peninsulas not previously seen, are exposed. Accurate measurement of the length now requires a smaller scale or ruler and the length measurement becomes greater. This process can be carried out indefinitely, at each Ievel of magnification, new detail emerges as smaller and smaller scales are used to measure the coastline length. Fractal structures can be quantified by determining the so-called fractal dimension (FD). The FD is a single value which can characterize the complexity and space filling propensity of a structure. An intuitive understanding of the significance of an object’s fractal dimension can be gained by relating the FD to the traditional notion of dimension, the Euclidean dimension. Using Euclidean definitions, lines, planes, and volumes have dimension of 1, 2 and 3 respectively. A fractal planar waveform will have a dimension between 1 and 2, with higher values occurring as it becomes increasing convoluted and fills more of the plane that contains it. Similarly, structures which
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ran branch in all directions, will have a fractal dimension between 2 and 3 depending on their complexity and volume filling potential. The fractal concept can be extended from its use in characterizing the complexity, convolutedness and space filling nature of structures and waveforms to characterizing similar properties in the temporal changes of a system’s behavior (Glenny et al., 1991), (Goldberger et al., 19901, (Katz, 1988). Self similarity that can exist over a range of scale sizes (length) in a structure can also exist as fluctuations in the output of a dynamic process over multiple scales of time (Goldberger and West, 1987). Based on our empirical observation that the EMG IP has a similar visual appearance at several different gain and sweep settings (analogous to viewing the waveform at different scales), limited pilot data (Gitter et al., 19911, and motivated by evidence of fractal structures and behaviors occurring in the nervous system (Arle and Simon, 1990; Bullimore et al., 1992; Goldberger et al., 1985; Kobayashi and Musha, 1982; Mpitsos et al., 1988; Schierwagen, 1986; Smith et al., 1989; Teich, 1989) this study was undertaken to explore whether normal motor unit recruitment (as manifested by the EMG interference pattern) has fractal characteristics. In addition, we hypothesized if the IP exhibits fractal characteristics, the dimension of the IP would be positively correlated with muscle force. This relationship would reflect a change from a relatively simple waveform consisting of one or two motor units discharging at minimal force to a much more complex and space filling interference pattern at high force levels.
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relative to the target force was given to the subjects to aid in the maintenance of a constant force contraction. The middle one second of EMG data was analyzed. The order of force production was randomized and a 1 min rest period between each force was used, For each muscle studied, the above protocol was repeated three times with a 5 min rest between trials to minimize fatigue effects. 2.3. EMG recording procedures
Standard disposable monopolar needle electrodes were used to record the EMG IP. Subjects were seated with their arm supported by their side and the elbow flexed to 90 degrees with the forearm supinated. The needle electrode was positioned to maximize the motor unit amplitudes during a low force contraction and secured in place. The EMG signal was recorded using a Nicolet Viking EMG machine which was interfaced to a personal computer using an analog to digital converter (ADC). Data was collected and stored for off line analysis. Total EMG signal gain from electrode to input of the ADC was 1400. The EMG signal was bandpass filtered with frequency cutoffs of 20 Hz and 2.5 Hz and digitized with a sampling rate of 5k Hz using a la-bit ADC (National Instruments NB-MIO-16). Followiag signal acquisition, the EMG data was reduced in resolution to eliminate any potential influence of ADC nonlinearity and quantization error on the fractal calculations. This resulted in a data resolution of 27.9 microvolts (PV) per ADC bit. 2.4. Force measurements
The EMG interference pattern was recorded from 9 biceps brachii muscles from 7 subjects. The subjects had no prior history of neuromuscular disorders and ranged in age from 22 to 38 years. Informed consent was obtained from each subject and the studies performed were approved by the institutional human subjects review board.
Isometric force was measured using a strain gauge force dynamometer. The dynamometer was secured to the floor and a leather strap attached to the dynamometer was placed across the wrist of the test arm. The strap was adjusted to prevent elbow joint motion during muscle contraction. The force signal was digitized at 500 Hz, simultaneously with the EMG signal. The middle one second of force data was averaged to determine the force level produced during the contraction.
2.2. Experimental protocol
2.5. Data analysis
The EMG interference pattern (EMG IP) was collected during isometric contractions of the biceps muscle. Using a force dynamometer, the maximal voluntary contraction (MVC) force was determined during a 1 s maximal effort contraction. EMG data was then recorded during a three second contraction at forces corresponding to lo%, 30%, 50%, 70%, and 90% of the MVC. Visual feedback of the force generated
The fractal dimension of the EMG EP was e~~~rnated using a box counting method (Arle and Simon, 1990; Peitgen et al., 1992a; Pickover, 1986a). This procedure is illustrated in Fig. 1. Determining the ‘box~~o~nti~g’ fractal dimension can be envisioned as the process of superimposing the EMG IP wave form onto a regular grid consisting of square elements (or ‘boxes’). The number of boxes which the waveform passes through is
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sions (Saupe, 1988). Fractal Bro~~ia~ motion data of dimension 1.0, 1.1, 1.2, 1.3, 1.4, and 1.5 were used covering the range of fractal estimates for the E data. When analyzed by the algorithms used in this study the calculated fractal dimensions of t sets were 0.98, 1.08, 1.19, 1.28, 1.37, and 1.4 tively. The selection of box sizes used in determining the fractal dimension should reflect the range of scale sizes over which self similarity exists in the waveform. Since a priori knowledge of the appropriate box sizes to use was lacking, log N vs. log l/S plots for an wide range of box sizes (2-500) were made from 7 randomly chosen trials, one from each subject. From this analysis a range of box sizes was chosen to incorporate the region over which a linear relationship existed between log N and log l/S. The box sizes used were chosen to be multiples of the ADC bit resultion and sa~l~~~~g rate to speed computations. As such a unit box had a physical dimension of 5580 ,uV/ps.
3 Continue
successive
grid
scaling
Fig. 1. Diagrammatic representation of the algorithm used to determine the fractal dimension of the EMG IP waveform. The total number of boxes entered (shaded region) by the waveform are counted as the size of the overlying grid is successively reduced in size.
counted. This procedure is repeated as the size of the individual boxes vary from large to small sizes. In this way the relationship between the size of the individual grid elements or boxes and the total number of boxes entered can be determined. Because the box count can vary somewhat, depending on the alignment of the grid relative to the EMG IP waveform, the box counts were made with three grid alignments relative to the waveform baseline and the results averaged. Fractals display an inverse power law relationship between the number of boxes entered (N) and the box size (A’). In this type of relationship, the number of boxes entered will increase exponentially as the box size decreases. By plotting the log of the number of boxes entered (log N) versus the log of the inverse of the box size (log l/S), the exponential relationship will become linear. The slope of the resulting line is the fractal dimension. For natural fractal structures, this linear relationship usually exists only over a limited range of scale sizes (Peitgen et al., 1992a). Custom software was written to automate this task. The average of the fractal dimensions obtained for the three trials at each force for each muscle was calculated and was used for qualitative and statistical analyses. The accuracy of the computer algorithm used to estimate actal dimension was verified by creating a series wnian motion data sets of known fractal dimen-
3. Results
Subjects were able to achieve constant force contractions which varied less than 10% from the target force, except for the 10% MVC trials actual force varied from 9.5% to 16% 90% MVC force, clipping of portions waveform occurred during a few trials. Since this effect appeared minimal these trials were included in the analysis. However, excessive clipping has the potential to alter waveform complexity and thus the fractal dimension calculated for the interference p The relationship between the total nu entered and box size for a wide range of box sizes is shown for the random trials in Fig. 2. For small box sizes, sizes, ranging from 2 to 8, the dimension decreases rapidly and approachs the value of 1. This was most apparent for the higher force trials where a change in slope was more readily apparent. A slope of a log-log plot which exactly equals 1 occurs for structures which are nonfractal, e.g. straight lines, circles and other regular geometric shapes. Tllese waveforms do not show complexity over multiple scales and have a fractal dimension of one which is equal to their Euclidean dimension. The reduction in the slope of the log-log plot for the EMG waveform at small box sizes implies that fractal characteristics do not exist at those scale sizes or at least that the self similiarity and complexity is reduced. The ability to fully explore the scaling relationship at small sizes was limited by the resolution of the data available for analysis. Similarly at box sizes greater than ap~ro~mat~ly 125, the slope of the log N vs. log l/S plots decreased and ultimately approached the value of 1. These effects were seen in
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100 Box
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I
I
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1
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Fig. 2. Relationship between the mean total number of boxes entered and the size of the box for 7 randomly chosen trials. (Note logarithmic graph axes). Region B of the graph spans box sizes ranging from 8 to 125 units which were used to estimate the fractal dimension in this study. At larger box sizes (region A) or smaller box sizes (region B) the slope of the relationship between total box count and box size was reduced and approached the value of 1. This indicates a reduction or loss of fractal characteristic at those scales of magnitude. (Note: a unit box has dimensions of 5580 pV/ps).
the other randomly chosen trials. As a result of this analysis, a range of box sizes from 8 to 125 units was used for further fractal dimension calculations (unit box has dimensions of 5580 pV/ps). The relationship between the fractal dimension and force for each subject is shown in Fig. 3. All subjects demonstrated a similar basic relationship. The fractal dimension increases in a nearly linear fashion over the force range of lo-50% of the MVC. Above 50% MVC, the rate of increase of the FD with force is reduced
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l-L-?-0
20
40
% MVC
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Force
Fig. 4. The average fractal dimension (mean rt: 1 SD) as a function of force. Between 10 and 70% MVC the fractal dimension is highly correlated to force (r = 0.94). Above 70% tbe fractal dimension plateaus and is poorly correlated to force (I = 0.351.
and in some subjects a plateau or small decline occurs above 70% (3 subjects). The average fractal dimension increased from a value of 1.1 at 10% MVC to 1.40 at 90% MVC (Fig. 4). The fractal dimension was highly correlated to force was most (Y = 0.92, P < 0.0001). This correlation marked for the force range of lo-70% MVC 4~ = 0.94, P < 0.0001) and decreased substantially at forces above 70% (r = 0.35, P = 0.1533). The overall relationship between FD and force appeared nonlinear and could be modelled by a second order polynomial relationship. The fractal dimension values were grouped by %MVC, and a one-way ANOVA was calculated with an F value of 119.71 (df = 4,40) which was significant at P < 0.001. To further explore whether the FD at each force level was different, a Studenr-Newman-Keuls post hoc test was performed. At the P = 0.05 level, the FD at all forces except for comparisons of 7’0% MVC and 90% MVC were statistically different.
1.4-
4. Discussion g ‘R s f a x z LF
l.3-
l.Z-’
l.T-
14 0
I 20
I 40 % MVC
I 60
I 80
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Force
Fig. 3. Practal dimension for each individual muscle studied (n = 9) as a function of force.
During the clinical EMG exam, rec~itme~t is analyzed as muscle force is varied from low to relatively high levels. As force generation exceeds minimal contraction levels, the visual complexity of the recruitment pattern increases rapidly and the ability to visually identify and track individual MUAP discharges is lost. Indeed the complexity of MUAP summation and overlap is reflected by the use of the term ‘interference’ pattern to describe its visual appearance. Although the interpretation of recruitment relies on various features including onset and recruitment rates of WJAP, esti-
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mates of motor unit number, size and morphology and auditory features: the subjective assessment of the visual complexity and use of clinical rules of thumb such as ‘the IP should normally fill the screen and obscure the baseline’ remain an important component of the clinical assessment. Capturing the essence of this visual estimate of complexity may improve both qualitative and quantitative clinical assessment of recruitment. Subjectively the fractal dimension appears related to the degree of convolution and space filling attributes of various waveforms (Katz, 1988; Miyashita et al., 1991). Indeed, in studies of visual perception, the fractal dimension of planar curves and textured surfaces correlate well to visual estimates of complexity and roughness (Cutting and Garvin, 1987; Marchak, 1990; Pentland, 1984). The results of this study indicates that the EMG recruitment pattern has fractal characteristics over the range of scale sizes used for analysis. In addition, the fractal dimension is positively correlated to force and increases from a dimension of 1.1 to 1.4 as the force varies from 10% to 90% of maximal. Thus as an estimate of complexity and space filling, the relationship of FD to force may be a useful addition to assessments based on quantitative turns and amplitude (Stalberg et al., 1983) or EQUIP (expert quantitative analysis of the interference pattern) analysis (Nandedkar et al., 1986a,b). Although this study did not attempt to correlate conventional quantitative IP analysis parameters to the fractal dimension, it is interesting to note that several previously described quantitative parameters share a similar relationship with force as does the FD. Several studies (Bril and Fuglsang-Frederiksen, 1984; Christensen et al., 1984; Fuglsang-Frederiksen, 1981) have reported a nonlinear relationship between the turns and baseline crossings per second and force. As force exceeds approximately 50% of maximal, the number of turns or baseline crossings plateau or declines. A similar nonlinear relationship is apparent in the data presented by Nandedkar (1986a,b) for the number of small segments (NSS) and to a lesser degree the activity parameter for their EQUIP analysis. Likewise, the various box sizes used in this analysis can not be directly related to conventional measures used in quantitative IP analysis, but some general comparisons are useful. The smallest box sizes used for analysis was 8 by 8 units. A box this size would span a vertical amplitude of approximately 220 /JV and a horizontal time period of 1.6 ms. This amplitude is close to the amplitude associated with a ‘turn’ in turns and amplitude analysis which is defined to be a change in the waveform direction with an amplitude of at least 100 PV relative to the previous and susequent turn (Nandedkar et al., 1986a). The time scale of 1.6 ms compares to the 1.5 to 5 ms used in defining the
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waveform segments that constitiuted the ‘activity’ measurement for EQUIP analysis The largest box size used was 125 units in size which corresponds to an amplitude of 3.4 millivolts (mV>. This is close to the maximal EMG envelope amplitudes of 2-4 mV for the normal EMG pattern as reported by Kamieniecka (1982). Thus it seems reasonable that parameters used in other quantitative analysis methods may represent features and charcteristics of the signal that contibute to its apparent fractal structure. Fractal analysis has the potential advantage of being able to succinctly summarize with a single value the self similarity, complexity and space filling nature of the IP. Additional study of the relationships between the fractal dimension and various parameters of HP analysis are needed to better understand wllich features of the EMG IP contribute to the waveform complexity and fullness. There are several limitations regarding the use of fractal analysis. Biological processes and signals can never be truly fractal in the mathematical sense since the self similarity does not exist over an infinite range of scale sizes (Glenny et al., 1991). However, it is often possible to describe and analyze the fractal characteristics of these processes over a more limited range of scales. This is the approach taken in this study. In addition, most natural structures and biological processes do not posess strict self similarity at different scales but rather ‘statistical self similarity’ (Peitgen et al., 1992b; Pickover and Kborasani, 1986b). Under differing degrees of magnification, the structure of a waveform does not appear as an exact replica of a less magnified version but rather it continues to manifest a similar visual appearance and have similar statistical properties. Although the linear relationship of the log-log plot of box count and box size is highly characteristic of a fractal relationship, it does not eliminate the possibility that the variability and complexity of a process can be explained by other nonfractal but exponentially declining models. Additional supportive evidence for a fractal relationship of the EMG recruitment pattern comes from Fourier analysis studies. The EMG interference pattern has a frequency power spectrum that is broad band and shows an exponential decline in power with frequency (Christensen and F~~s~~g-Frederi~se~, 1986; Fuglsang-Frederiksen, 1990; Mills, 199 processes which have self similar variation over a range of time scales will also generate a broad ban quency spectrum (Goldberger and West, 198 1991). Signal energy is distributed over a broad frequencies with a inverse power-law distribution; that is; the power exponentially declines as the frequency increases. A final practical constraint in the accuracy of the FD estimates is the choice of scale sizes used in the
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box counting algorithm. The range of scale sizes (box sizes) used in this study was chosen empirically by observing the range of sizes over which a linear relationship existed between the log N and log l/S. For scale sizes (box sizes) larger than those used in this analysis, the calculated FD approached the value of 1. This represents the lack of scaling that occurs at very iarge measurements. At these scale sizes a single box could often span the peak to peak amplitude of the waveforms. The choice of box size limits for use at the small end of the scale size was based on the systematic deviation of box count from a linear regression line. -Again this may reflect an intrinsic property of the 9MG interference waveform, but the influence of the waveform quantification during the ADC process must also be considered. If insufficient resolution of the waveform occurs during analog to digital conversion, information will be lost that may limit the range of scale sizes over which self similarity exists. These encouraging early results suggest that the fractal dimension and other related measures of nonEnear dynamical system behavior may be useful tools in characterizing the patterns of motor unit recruitment seen in health and disease.
ferences Arle, J.E. and Simon, R.H. (1990) An application of fractal dimension to the detection of transients in the electroencephalogram, Electroencephalogr. Clin. Neurophysiol., 75: 296-305. Bril, V. and Fuglsang-Frederiksen, A. (1984) Number of potential reversals (turns) and amplitude of the pattern of electrical activity of theg abductor pollicis brevis muscle in patients with neurogenie diseases, Acta Neurol. Stand., 70: 169-175. Buchthal, F. and Kamieniecka, Z. (1982) The diagnostic yield of quantified electromyography and quantified muscle biopsy in neuromuscular disorders, Muscle Nerve, 5: 265-280. Bullimore, E., Brammer, M., Binnie, C. and Alarcon, G. (1992) Fractal eiectroencephalography (Letter), Lancet, 339: 618-619 Christensen, H., Monaco, M.L., Dahl, K. and Fuglsang-Frederiksen, A. (1984) Processing of eiectrical activity in human muscle during a gradual increase in force, Electroencephalogr. Clin. Neurophysiol., 58: 230-239. Christensen, H. and Fuglsang-Frederiksen, A. (1986) Power spectIilrn and turns analysis of EMG at different voluntary efforts in normai subjects, Electroencephalogr. Clin. Neurophysiol., 64: 528-535. rtitting, J.E. and Garvin, J.J. (1987) Fractal curves and complexity, Perception Psychophysics, 42: 365-370. Fuglsang-Frederiksen, A. (1990) Power spectrum of the Needle EMG in normal and diseased muscles, Methods Clin. Neurophysiol., 2: 1-8. Fuglsang-Frederiksen, A, (1981) Electrical activity and force during voluntary contraction of normal and diseased muscle, Acta Neurol. Stand., 63 (Suppl. 83): l-60. Gitter, A.J., Czerniecki, J.C. and DeGroot, D. (1991) Muscle force
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and EMG activity relationship using fractal dimension zna!ysis, Muscle Nerve, 14: 884. Glenny, R.W., Robertson, H.T., Yamashiro, S. and Bassingthwaighte, J.B. (1991) Applications of fractal analysis to physiology, J. Appl. Physiol., 70: 2351-2367. Goldberger, A.L., Bhargava, V., West, B.J. and Mandell, A.J. (1985) On the mechanism of cardiac electrical stability, Biophys. f., 48: 525-528. Goldberger, A.L. and West, B.J. (1987) Fractals in physiology and medicine, Yale J. Biol. Med., 60: 42.1-435. Goldberger, A.L., Rigney, D.R. and West, B.J. (1990) Chaos and fractals in human physiology, Scient. Am., 262: 42-49. King, C.C. (1991) Fractal and chaotic dynamics in nervous systems, Prog. Neurobiol., 36: 279-308. Katz, M.J. (1988) Fractals and the analysis of waveforms, Comput. Biol. Med., 18: 145-156. Kobayashi, M. and Musha, T. (1982) l/f fluctuation of heartbear period, IEEE Trans. Biomed. Eng., 29: 456-458. Mandelbrot, B. (1980) Fractal aspects of the iteration of z to Az(l -z) for complex A and z, Ann. NY Acad. Sci., 357: 249-259. Marchak, F.M. (1990) Fractal models in the visual percepticm of textures and surfaces in nature, Dissert. Abstr. Int., 51: 1020-B. Miyashita, Y., Higuchi, S., Sakai, K. and Masui, N. (1991) Generation of fractal patterns for probing the visual memory, Neurosci. Res., 12: 307-311. Mpitsos, G.J., Burton, R.M., Creech, H.C. and Soinila, S.Q. (1988) Evidence for chaos in spike trains of neurons that generate rhythmic motor patterns, Brain Res. Bull., 21: 529-538. Nandedkar, S., Sanders, D.B. and Stalberg, E.V. (1986&l Automatic analysis of the electromyographic interference pattern. Part I: Development of quantitative features, Muscle Nerve, 9: 431-439. Nandedkar, S., Sanders, D.B. and Stalberg, E.V. (1986b) Simulation and analysis of the electromyographic interference pattern in normal muscle. Part II: Activity, upper Gentile amplitude and number of small segments, Muscle Nerve, 9: 486-490. Peitgen, H., Jurgens, H. and Saupe, D. (1992a) Chaos and Fractals, New Frontiers of Science, Springer-Verlag, New York, pp. 183228. Peitgen, H., Jurgens, H. and Saupe, D. (1992b) Chaos and Fractals, New Frontiers of Science, Springer-Verlag, New York, pp. 457505. Pentland, A.P. (1984) Fractal-based description of natural scenes, IEEE Trans Pattern Analysis and Machine Intelligence, PAMI-6: 661-674. Pickover, C.A. (1986a) A Monte Carlo approach for e placement in fractal-dimension calculations for waveform graphs, Comput. Graphics Forum, 5: 203-210. Pickover, C.A. and Khorasani, A. (1986b) Fractal characterization of speech waveform graphs, Comput. Graphic, 10: 51-61. Saupe, D. (1988) Algorithms for random fractals. In H.O. Peitgenand and D.H. Suape (Eds.), The Science of Fractal Images, SpringerVerlag, New York, pp. 74-90. Schierwagen, A.K. (1986) Dendritic branching patterns. In H. Degn, A.V. Holden and L.F. Olsen (Eds.), Chaos in Biologicai Systems, Plenum Press, New York, pp. 191-195. Smith, T.G., Marks, W.B., Lange, G.D., Sherriff, W.H. and Neale, E.A. (1989) A fractal analysis of ceil images, J. Neurosci. Methods, 27: 173-180. Stalberg, E., Chu, J., Nandekar, S., Stalberg, S. and Ericsson, h?r~ (1983) Automatic analysis of the EMG interference pattern, Electroencephalogr. Clin. Neurophysiol., 56: 672-681. Teich, M.C. (1989) Fractal character of the auditory neural spike train, IEEE Trans. Biomed. Eng., 36: 150-160.
ractal analysis of the electromyographic interference J. Andrew Gitter a,b,M. Joseph Czerniecki aa b Physical
Medicine
a Rehabilitation Medicine, Uniuersity of Washington, and Rehabilitation, Seattle Veterans Administration
Seattle, WA 98108, USA Medical Center, Seattle,
WA 981@
USA
Received 25 January 1994;revised 29 August 1994;accepted6 September 1994
Abstract Evaluation of motor unit recruitment is an important component of the clinical EMG exam. Typically this is assessed qualitatively using auditory features and estimates of the visual complexity of the EMG waveform. Recent advances in nonlinear dynamics have led to the development of the concept of fractals which can be used to quantify complexity and space filling features of various structures. This study was undertaken to determine if the normal EMG interference pattern (IF) has fractal characteristics that might be helpful in quantitative analysis. EMG activity was recorded from the 9 normal biceps muscles as force was varied from 10 to 90% of maximal. Using a box count algorithm, the fractal dimension was calculated. The EMG IP displays fractal characteristics with a dimension that is highly correlated with force and ranges from 1.1 to 1.4 as force increases from 10 to 90% MVC. The fractal dimension (FD)--force relationship is similar to that observed with other methods of IP analysis and suggests that the fractal dimensioncan be usedto quantify and capture the essenceof the ‘complexity’ of motor unit recruitment patterns. Keywords:
Electromyography; Muscle contraction; Chaos;Fractal; Recruitment; Models, biological
1. Introduction Evaluation of the recruitment or interference pattern (IP) 3s an important component of the clinical electromyographic (EMG) study of a muscle. The EMG IP is generated by the summation of the motor units within the recording volume of the EMG electrode as the number of units and their firing rates vary with force generation. Most commonly, recruitment is subjectively assessedby monitoring its auditory qualities and its visual appearance. Because of the importance of the IP evaluation in the diagnosis of neuromuscular disease, objective and quantitative methods for assessing recruitment have been developed (Nandedkar et al., 1986a,b; Stalberg et al., 1983). This study explores the possible role of a new technique of evaluating the PP in clinical electrophysiologic studies. Recent advances in nonlinear dynamics has led to the concepts of fractals and fractal dimension. The term fractal was first used by Mandelbrot (1980) to describe shapes or structures which exhibit the property of self similarity, that is, complexity and detail which is retained with successive levels of scaling or magnification. The classic example of a natural fractal 01650270/95/$09.50 SSDIOl65-0270(94)00164-2
0 1995ElsevierScienceB.V. All rightsreserved
is a coastline (Peitgen et al., 1992a). If the length of a coastline is measured from a low resolution photograph, a unique value will be obtained. ith a magnified or higher resolution image, new details, i.e. inlets, bays and peninsulas not previously seen, are exposed. Accurate measurement of the length now requires a smaller scale or ruler and the length measurement becomes greater. This process can be carried out indefinitely, at each Ievel of magnification, new detail emerges as smaller and smaller scales are used to measure the coastline length. Fractal structures can be quantified by determining the so-called fractal dimension (FD). The FD is a single value which can characterize the complexity and space filling propensity of a structure. An intuitive understanding of the significance of an object’s fractal dimension can be gained by relating the FD to the traditional notion of dimension, the Euclidean dimension. Using Euclidean definitions, lines, planes, and volumes have dimension of 1, 2 and 3 respectively. A fractal planar waveform will have a dimension between 1 and 2, with higher values occurring as it becomes increasing convoluted and fills more of the plane that contains it. Similarly, structures which
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/Journal
ran branch in all directions, will have a fractal dimension between 2 and 3 depending on their complexity and volume filling potential. The fractal concept can be extended from its use in characterizing the complexity, convolutedness and space filling nature of structures and waveforms to characterizing similar properties in the temporal changes of a system’s behavior (Glenny et al., 1991), (Goldberger et al., 19901, (Katz, 1988). Self similarity that can exist over a range of scale sizes (length) in a structure can also exist as fluctuations in the output of a dynamic process over multiple scales of time (Goldberger and West, 1987). Based on our empirical observation that the EMG IP has a similar visual appearance at several different gain and sweep settings (analogous to viewing the waveform at different scales), limited pilot data (Gitter et al., 19911, and motivated by evidence of fractal structures and behaviors occurring in the nervous system (Arle and Simon, 1990; Bullimore et al., 1992; Goldberger et al., 1985; Kobayashi and Musha, 1982; Mpitsos et al., 1988; Schierwagen, 1986; Smith et al., 1989; Teich, 1989) this study was undertaken to explore whether normal motor unit recruitment (as manifested by the EMG interference pattern) has fractal characteristics. In addition, we hypothesized if the IP exhibits fractal characteristics, the dimension of the IP would be positively correlated with muscle force. This relationship would reflect a change from a relatively simple waveform consisting of one or two motor units discharging at minimal force to a much more complex and space filling interference pattern at high force levels.
of Neuroscience
Methods
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relative to the target force was given to the subjects to aid in the maintenance of a constant force contraction. The middle one second of EMG data was analyzed. The order of force production was randomized and a 1 min rest period between each force was used, For each muscle studied, the above protocol was repeated three times with a 5 min rest between trials to minimize fatigue effects. 2.3. EMG recording procedures
Standard disposable monopolar needle electrodes were used to record the EMG IP. Subjects were seated with their arm supported by their side and the elbow flexed to 90 degrees with the forearm supinated. The needle electrode was positioned to maximize the motor unit amplitudes during a low force contraction and secured in place. The EMG signal was recorded using a Nicolet Viking EMG machine which was interfaced to a personal computer using an analog to digital converter (ADC). Data was collected and stored for off line analysis. Total EMG signal gain from electrode to input of the ADC was 1400. The EMG signal was bandpass filtered with frequency cutoffs of 20 Hz and 2.5 Hz and digitized with a sampling rate of 5k Hz using a la-bit ADC (National Instruments NB-MIO-16). Followiag signal acquisition, the EMG data was reduced in resolution to eliminate any potential influence of ADC nonlinearity and quantization error on the fractal calculations. This resulted in a data resolution of 27.9 microvolts (PV) per ADC bit. 2.4. Force measurements
The EMG interference pattern was recorded from 9 biceps brachii muscles from 7 subjects. The subjects had no prior history of neuromuscular disorders and ranged in age from 22 to 38 years. Informed consent was obtained from each subject and the studies performed were approved by the institutional human subjects review board.
Isometric force was measured using a strain gauge force dynamometer. The dynamometer was secured to the floor and a leather strap attached to the dynamometer was placed across the wrist of the test arm. The strap was adjusted to prevent elbow joint motion during muscle contraction. The force signal was digitized at 500 Hz, simultaneously with the EMG signal. The middle one second of force data was averaged to determine the force level produced during the contraction.
2.2. Experimental protocol
2.5. Data analysis
The EMG interference pattern (EMG IP) was collected during isometric contractions of the biceps muscle. Using a force dynamometer, the maximal voluntary contraction (MVC) force was determined during a 1 s maximal effort contraction. EMG data was then recorded during a three second contraction at forces corresponding to lo%, 30%, 50%, 70%, and 90% of the MVC. Visual feedback of the force generated
The fractal dimension of the EMG EP was e~~~rnated using a box counting method (Arle and Simon, 1990; Peitgen et al., 1992a; Pickover, 1986a). This procedure is illustrated in Fig. 1. Determining the ‘box~~o~nti~g’ fractal dimension can be envisioned as the process of superimposing the EMG IP wave form onto a regular grid consisting of square elements (or ‘boxes’). The number of boxes which the waveform passes through is
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sions (Saupe, 1988). Fractal Bro~~ia~ motion data of dimension 1.0, 1.1, 1.2, 1.3, 1.4, and 1.5 were used covering the range of fractal estimates for the E data. When analyzed by the algorithms used in this study the calculated fractal dimensions of t sets were 0.98, 1.08, 1.19, 1.28, 1.37, and 1.4 tively. The selection of box sizes used in determining the fractal dimension should reflect the range of scale sizes over which self similarity exists in the waveform. Since a priori knowledge of the appropriate box sizes to use was lacking, log N vs. log l/S plots for an wide range of box sizes (2-500) were made from 7 randomly chosen trials, one from each subject. From this analysis a range of box sizes was chosen to incorporate the region over which a linear relationship existed between log N and log l/S. The box sizes used were chosen to be multiples of the ADC bit resultion and sa~l~~~~g rate to speed computations. As such a unit box had a physical dimension of 5580 ,uV/ps.
3 Continue
successive
grid
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Fig. 1. Diagrammatic representation of the algorithm used to determine the fractal dimension of the EMG IP waveform. The total number of boxes entered (shaded region) by the waveform are counted as the size of the overlying grid is successively reduced in size.
counted. This procedure is repeated as the size of the individual boxes vary from large to small sizes. In this way the relationship between the size of the individual grid elements or boxes and the total number of boxes entered can be determined. Because the box count can vary somewhat, depending on the alignment of the grid relative to the EMG IP waveform, the box counts were made with three grid alignments relative to the waveform baseline and the results averaged. Fractals display an inverse power law relationship between the number of boxes entered (N) and the box size (A’). In this type of relationship, the number of boxes entered will increase exponentially as the box size decreases. By plotting the log of the number of boxes entered (log N) versus the log of the inverse of the box size (log l/S), the exponential relationship will become linear. The slope of the resulting line is the fractal dimension. For natural fractal structures, this linear relationship usually exists only over a limited range of scale sizes (Peitgen et al., 1992a). Custom software was written to automate this task. The average of the fractal dimensions obtained for the three trials at each force for each muscle was calculated and was used for qualitative and statistical analyses. The accuracy of the computer algorithm used to estimate actal dimension was verified by creating a series wnian motion data sets of known fractal dimen-
3. Results
Subjects were able to achieve constant force contractions which varied less than 10% from the target force, except for the 10% MVC trials actual force varied from 9.5% to 16% 90% MVC force, clipping of portions waveform occurred during a few trials. Since this effect appeared minimal these trials were included in the analysis. However, excessive clipping has the potential to alter waveform complexity and thus the fractal dimension calculated for the interference p The relationship between the total nu entered and box size for a wide range of box sizes is shown for the random trials in Fig. 2. For small box sizes, sizes, ranging from 2 to 8, the dimension decreases rapidly and approachs the value of 1. This was most apparent for the higher force trials where a change in slope was more readily apparent. A slope of a log-log plot which exactly equals 1 occurs for structures which are nonfractal, e.g. straight lines, circles and other regular geometric shapes. Tllese waveforms do not show complexity over multiple scales and have a fractal dimension of one which is equal to their Euclidean dimension. The reduction in the slope of the log-log plot for the EMG waveform at small box sizes implies that fractal characteristics do not exist at those scale sizes or at least that the self similiarity and complexity is reduced. The ability to fully explore the scaling relationship at small sizes was limited by the resolution of the data available for analysis. Similarly at box sizes greater than ap~ro~mat~ly 125, the slope of the log N vs. log l/S plots decreased and ultimately approached the value of 1. These effects were seen in
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Fig. 2. Relationship between the mean total number of boxes entered and the size of the box for 7 randomly chosen trials. (Note logarithmic graph axes). Region B of the graph spans box sizes ranging from 8 to 125 units which were used to estimate the fractal dimension in this study. At larger box sizes (region A) or smaller box sizes (region B) the slope of the relationship between total box count and box size was reduced and approached the value of 1. This indicates a reduction or loss of fractal characteristic at those scales of magnitude. (Note: a unit box has dimensions of 5580 pV/ps).
the other randomly chosen trials. As a result of this analysis, a range of box sizes from 8 to 125 units was used for further fractal dimension calculations (unit box has dimensions of 5580 pV/ps). The relationship between the fractal dimension and force for each subject is shown in Fig. 3. All subjects demonstrated a similar basic relationship. The fractal dimension increases in a nearly linear fashion over the force range of lo-50% of the MVC. Above 50% MVC, the rate of increase of the FD with force is reduced
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Fig. 4. The average fractal dimension (mean rt: 1 SD) as a function of force. Between 10 and 70% MVC the fractal dimension is highly correlated to force (r = 0.94). Above 70% tbe fractal dimension plateaus and is poorly correlated to force (I = 0.351.
and in some subjects a plateau or small decline occurs above 70% (3 subjects). The average fractal dimension increased from a value of 1.1 at 10% MVC to 1.40 at 90% MVC (Fig. 4). The fractal dimension was highly correlated to force was most (Y = 0.92, P < 0.0001). This correlation marked for the force range of lo-70% MVC 4~ = 0.94, P < 0.0001) and decreased substantially at forces above 70% (r = 0.35, P = 0.1533). The overall relationship between FD and force appeared nonlinear and could be modelled by a second order polynomial relationship. The fractal dimension values were grouped by %MVC, and a one-way ANOVA was calculated with an F value of 119.71 (df = 4,40) which was significant at P < 0.001. To further explore whether the FD at each force level was different, a Studenr-Newman-Keuls post hoc test was performed. At the P = 0.05 level, the FD at all forces except for comparisons of 7’0% MVC and 90% MVC were statistically different.
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l.Z-’
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Fig. 3. Practal dimension for each individual muscle studied (n = 9) as a function of force.
During the clinical EMG exam, rec~itme~t is analyzed as muscle force is varied from low to relatively high levels. As force generation exceeds minimal contraction levels, the visual complexity of the recruitment pattern increases rapidly and the ability to visually identify and track individual MUAP discharges is lost. Indeed the complexity of MUAP summation and overlap is reflected by the use of the term ‘interference’ pattern to describe its visual appearance. Although the interpretation of recruitment relies on various features including onset and recruitment rates of WJAP, esti-
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mates of motor unit number, size and morphology and auditory features: the subjective assessment of the visual complexity and use of clinical rules of thumb such as ‘the IP should normally fill the screen and obscure the baseline’ remain an important component of the clinical assessment. Capturing the essence of this visual estimate of complexity may improve both qualitative and quantitative clinical assessment of recruitment. Subjectively the fractal dimension appears related to the degree of convolution and space filling attributes of various waveforms (Katz, 1988; Miyashita et al., 1991). Indeed, in studies of visual perception, the fractal dimension of planar curves and textured surfaces correlate well to visual estimates of complexity and roughness (Cutting and Garvin, 1987; Marchak, 1990; Pentland, 1984). The results of this study indicates that the EMG recruitment pattern has fractal characteristics over the range of scale sizes used for analysis. In addition, the fractal dimension is positively correlated to force and increases from a dimension of 1.1 to 1.4 as the force varies from 10% to 90% of maximal. Thus as an estimate of complexity and space filling, the relationship of FD to force may be a useful addition to assessments based on quantitative turns and amplitude (Stalberg et al., 1983) or EQUIP (expert quantitative analysis of the interference pattern) analysis (Nandedkar et al., 1986a,b). Although this study did not attempt to correlate conventional quantitative IP analysis parameters to the fractal dimension, it is interesting to note that several previously described quantitative parameters share a similar relationship with force as does the FD. Several studies (Bril and Fuglsang-Frederiksen, 1984; Christensen et al., 1984; Fuglsang-Frederiksen, 1981) have reported a nonlinear relationship between the turns and baseline crossings per second and force. As force exceeds approximately 50% of maximal, the number of turns or baseline crossings plateau or declines. A similar nonlinear relationship is apparent in the data presented by Nandedkar (1986a,b) for the number of small segments (NSS) and to a lesser degree the activity parameter for their EQUIP analysis. Likewise, the various box sizes used in this analysis can not be directly related to conventional measures used in quantitative IP analysis, but some general comparisons are useful. The smallest box sizes used for analysis was 8 by 8 units. A box this size would span a vertical amplitude of approximately 220 /JV and a horizontal time period of 1.6 ms. This amplitude is close to the amplitude associated with a ‘turn’ in turns and amplitude analysis which is defined to be a change in the waveform direction with an amplitude of at least 100 PV relative to the previous and susequent turn (Nandedkar et al., 1986a). The time scale of 1.6 ms compares to the 1.5 to 5 ms used in defining the
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waveform segments that constitiuted the ‘activity’ measurement for EQUIP analysis The largest box size used was 125 units in size which corresponds to an amplitude of 3.4 millivolts (mV>. This is close to the maximal EMG envelope amplitudes of 2-4 mV for the normal EMG pattern as reported by Kamieniecka (1982). Thus it seems reasonable that parameters used in other quantitative analysis methods may represent features and charcteristics of the signal that contibute to its apparent fractal structure. Fractal analysis has the potential advantage of being able to succinctly summarize with a single value the self similarity, complexity and space filling nature of the IP. Additional study of the relationships between the fractal dimension and various parameters of HP analysis are needed to better understand wllich features of the EMG IP contribute to the waveform complexity and fullness. There are several limitations regarding the use of fractal analysis. Biological processes and signals can never be truly fractal in the mathematical sense since the self similarity does not exist over an infinite range of scale sizes (Glenny et al., 1991). However, it is often possible to describe and analyze the fractal characteristics of these processes over a more limited range of scales. This is the approach taken in this study. In addition, most natural structures and biological processes do not posess strict self similarity at different scales but rather ‘statistical self similarity’ (Peitgen et al., 1992b; Pickover and Kborasani, 1986b). Under differing degrees of magnification, the structure of a waveform does not appear as an exact replica of a less magnified version but rather it continues to manifest a similar visual appearance and have similar statistical properties. Although the linear relationship of the log-log plot of box count and box size is highly characteristic of a fractal relationship, it does not eliminate the possibility that the variability and complexity of a process can be explained by other nonfractal but exponentially declining models. Additional supportive evidence for a fractal relationship of the EMG recruitment pattern comes from Fourier analysis studies. The EMG interference pattern has a frequency power spectrum that is broad band and shows an exponential decline in power with frequency (Christensen and F~~s~~g-Frederi~se~, 1986; Fuglsang-Frederiksen, 1990; Mills, 199 processes which have self similar variation over a range of time scales will also generate a broad ban quency spectrum (Goldberger and West, 198 1991). Signal energy is distributed over a broad frequencies with a inverse power-law distribution; that is; the power exponentially declines as the frequency increases. A final practical constraint in the accuracy of the FD estimates is the choice of scale sizes used in the
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box counting algorithm. The range of scale sizes (box sizes) used in this study was chosen empirically by observing the range of sizes over which a linear relationship existed between the log N and log l/S. For scale sizes (box sizes) larger than those used in this analysis, the calculated FD approached the value of 1. This represents the lack of scaling that occurs at very iarge measurements. At these scale sizes a single box could often span the peak to peak amplitude of the waveforms. The choice of box size limits for use at the small end of the scale size was based on the systematic deviation of box count from a linear regression line. -Again this may reflect an intrinsic property of the 9MG interference waveform, but the influence of the waveform quantification during the ADC process must also be considered. If insufficient resolution of the waveform occurs during analog to digital conversion, information will be lost that may limit the range of scale sizes over which self similarity exists. These encouraging early results suggest that the fractal dimension and other related measures of nonEnear dynamical system behavior may be useful tools in characterizing the patterns of motor unit recruitment seen in health and disease.
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