Estimation of Muscle Fiber Conduction Velocity ... - Semantic Scholar
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 9, SEPTEMBER 2007
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Estimation of Muscle Fiber Conduction Velocity With a Spectral Multidip Approach Dario Farina*, Member, IEEE, and Francesco Negro
Abstract—We propose a novel method for estimation of muscle fiber conduction velocity from surface electromyographic (EMG) signals. The method is based on the regression analysis between spatial and temporal frequencies of multiple dips introduced in the EMG power spectrum through the application of a set of spatial filters. This approach leads to a closed analytical expression of conduction velocity as a function of the auto- and cross-spectra of monopolar signals detected along the direction of muscle fibers. The performance of the algorithm was compared with respect to that of the classic single dip approach on simulated and experimental EMG signals. The standard deviation of conduction velocity estimates from simulated single motor unit action potentials was reduced from 1.51 m/s [10 dB signal-to-noise ratio (SNR)] and 1.06 m/s (20 dB SNR) with the single dip approach to 0.51 m/s (10 dB) and 0.23 m/s (20 dB) with the proposed method using 65 dips. When 200 active motor units were simulated in an interference EMG signal, standard deviation of conduction velocity decreased from 0.95 m/s (10 dB SNR) and 0.60 m/s (20 dB SNR) with a single dip to 0.21 m/s (10 dB) and 0.11 m/s (20 dB) with 65 dips. In experimental signals detected from the abductor pollicis brevis muscle, standard deviation of estimation decreased SD over 5 subjects) 1.25 0.62 m/s with one dip from (mean to 0.10 0.03 m/s with 100 dips. The proposed method does not imply limitation in resolution of the estimated conduction velocity and does not require any iterative procedure for the estimate since it is based on a closed analytical formulation. Index Terms—Conduction velocity, delay estimators, electromyographic (EMG), spectral analysis.
I. INTRODUCTION VERAGE muscle fiber conduction velocity can be estimated from surface electromyographic (EMG) signals [1], which comprise the contribution of the active superficial motor units. Several methods for the estimation of conduction velocity from the interference EMG have been proposed in the literature [2]. Usually they are based on the estimation of delay between two or more signals recorded along the direction of propagation of the action potentials. However, it can be shown that a single EMG signal recorded through the application of a spatial filter is theoretically sufficient for the estimation of propagation velocity [3]. The zeros (dips) in the spatial filter transfer function are introduced in the power spectrum of the recorded signal
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Manuscript received August 25, 2006; revised January 5, 2007. This work was supported by the Danish Technical Research Council, Centre for Neuroengineering (CEN) Project, under Contract 26-04-0100 (DF). Asterisk indicates corresponding author. *D. Farina is with the Center for Sensory-Motor Interaction (SMI), Department of Health Science and Technology, Aalborg University, Fredrik Bajers Vej 7D-3,Aalborg DK-9220, Denmark (e-mail: [email protected]). F. Negro is with the Center for Sensory-Motor Interaction (SMI), Department of Health Science and Technology, Aalborg University, Aalborg DK-9220, Denmark (e-mail: [email protected]). Digital Object Identifier 10.1109/TBME.2007.892928
since the spatial and temporal frequency domains are linearly related by conduction velocity (Fig. 1). Thus, conduction velocity can be estimated from the ratio between the temporal (detected from the power spectrum of the recorded signal) and spatial (imposed through the design of the spatial filter) frequency at which a dip occurs. The method has been applied with the single differential spatial filter [4] whose transfer function has a dip at the spatial frequency equal to the inverse of the interelectrode distance. There are two main problems associated with the spectral dip technique with single differential recording. First, the interelectrode distance should be large to introduce a dip in the bandwidth of the EMG signal. Second, the detection of the dip in the power spectrum of the recorded signal has large variance of estimation that is reflected in a large variability of conduction velocity estimates [4]. These issues could be overcome using spatial filters with more electrodes than the single differential and imposing more than one dip, so that results from different dips can be averaged and estimation variance reduced. For example, it is possible to locate a dip at any spatial frequency by the design of spatial filters from four detection points [2]. However, this approach has never been proposed and tested. Therefore, in this study we present a novel method for estimating conduction velocity from the surface EMG, based on the identification of a number of spectral dips introduced in the power spectrum of the detected signals through the application of a set of spatial filters. It is shown that this approach leads to a closed formula for the calculation of muscle fiber conduction velocity as a function of the auto- and cross-spectra of a set of detected monopolar signals. II. METHODS Surface EMG signals are spatially filtered by the linear combination of the monopolar signals detected from electrodes arranged in a specific geometrical pattern [5], [6]. The effect of the spatial filter on the recorded signal can be described by the transfer function of the filter in the spatial frequency domain. In the direction of propagation of the sources, the spatial frequency domain is related to the temporal frequency domain by the velocity of propagation (1) the temporal frewhere is the velocity of propagation, the spatial frequency in the direction of source quency, and propagation. The power spectrum in the temporal frequency domain is scaled with respect to the power spectrum in the spatial frequency domain. If the power spectrum is zero in one domain at
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 9, SEPTEMBER 2007
Fig. 1. Representative scheme of the introduction of dips in the surface EMG power spectrum. In the spatial domain z (bottom), three sinusoids [R (z)] with the same amplitude and spatial frequencies 25, 50, and 75 cycles/m move with a velocity v = 4 m=s and are recorded in four detection points, separated by the distance d . The dashed lines schematically show the sinusoids at a different time instant (translation). The output signals corresponding to the three sinusoids [P (z )] are the linear combinations of the signals at the four detection points. Specific selections of the coefficients of the linear combination null one of the spatial harmonics. This corresponds to locate a dip at a specific frequency f . Two examples of filter coefficient (a , i = 1, 2, 3, 4) selection are shown which correspond to null the first and third harmonics. The harmonics that are suppressed in the spatial domain have null amplitude also in the temporal domain (dips in the temporal frequency domain) (top). The frequency of the suppressed harmonic in the temporal domain is given by the frequency of the dip in the spatial domain scaled by conduction velocity. The y -axes of the power spectra (right) are in logarithmic normalized scale.
a specific frequency (dip), the dip in the other domain is at the same frequency scaled by conduction velocity (Fig. 1). Considering electrodes located along the direction of propagation of the action potentials, the power spectrum of the detected signal is the multiplication of the power spectrum of the monopolar signal and the transfer function of the spatial filter. Zeros in the power spectrum of the output signal can thus be introduced by the transfer function of the spatial filter. If the single differential filter is applied, one dip is introduced at dc and another (with its periodic repetitions) at a temporal frequency that depends on conduction velocity and interelectrode distance. Conduction velocity is thus estimated through (1) applied to the frequency of the dip in the spatial and temporal domains [3]. If more than one spatial filter is applied and each filter introduces dips at different frequencies within the signal bandwidth, conduction velocity can be estimated from (1) using all the dip frequencies, for example applying a least-squares regression analysis
(2)
is the th dip temporal frequency, the th dip spawhere tial frequency, and is the number of dips introduced. A. Spectral Multidip Estimator of Conduction Velocity In the following, the case of four monopolar recordings with constant interelectrode distance is considered and all the derivations are related to this case. Extensions to more than four detection points and variable interelectrode distance follow similar derivations. Moreover, the following derivations apply not only to monopolar recordings. The four detected signals may result from the application of any spatial filter at each detection point (preprocessing filter) under the condition that the filter is the same at the four detection points, so that the four detected signals have, in ideal conditions, the same shape and a relative delay between each other. However, for simplicity, in the following we will consider four monopolar recordings. of a spatial filter applied to four The transfer function detection points is
(3) where is the spatial frequency along the direction of acare the coefficients of the tion potential propagation,
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Fig. 2. Representation of the multidip approach for estimation of conduction velocity on simulated single motor unit action potentials. The motor unit action potential is generated by 200 fibers with a density of 20 bers=mm in the circular motor unit territory, 100-mm of fiber length, mean depth of 3 mm, and 5 mm, 5-mm interelectrode distance), conduction velocity v = 4 m=s. (a) Four monopolar signals are simulated as detected by rectangular electrodes (1 placed between the innervation zone and tendon region. (c) The monopolar signals are filtered with four spatial filters whose transfer function is chosen in order to introduce dips at spatial frequencies 100, 125, 150, and 175 cycles/m. The filter coefficients were computed from (7). (b) The dips introduced in the spatial frequency domain also appear in the power spectrum (temporal frequency domain) of the detected signals which was estimated with the periodogram approach. The locations of the dips in the two domains are linearly related [(1)]. (d) The slope of the regression line between dip frequencies in the two domains is an estimate of conduction velocity. Arbitrary units [au].
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linear combination (filter coefficients), and the constant distance between adjacent electrodes. Equation (3) can be rewritten as
. Under this condition, simple calculations (omitted) lead for to the following relations between the filter coefficients
(4) If the spatial filter introduces a zero at the spatial dc, as usually imposed to remove the spatial common mode [6], [7], the summation of the filter coefficients is zero (5)
(7) which were reported in [2] in an equivalent form. Equation (7) provides the filter coefficients to impose a zero at any frequency in the spatial domain (Fig. 2). The filtered signal in the temporal domain is (8)
and (4) can be simplified as
are the monopolar signals detected by where the four electrodes. From (8), the power spectrum of the signal is (6) All spatial filters considered in the following have transfer function equal to zero in dc. This also reduces the adverse effect of end-of-fiber components. The condition of a dip at frequency implies that the transfer function of the spatial filter is null
(9) where and
are the power spectra of the cross-spectra.
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 9, SEPTEMBER 2007
Imposing a dip at and thus using (7) for the filter coefficients, the expression of the power spectrum of the detected signal [(9)] can be simplified as
(10) given from (7) and dependent on the spatial frequency with . With the substitution of the dip
the dip in the spatial domain is a function of the location of the . It is thus possible to dip in the temporal frequency domain choose any location of the dip in the temporal frequency domain and derive the corresponding location in the spatial frequency domain without practically detect the dip, but with the use of a closed analytical formula. This can be done with an arbitrary number of dip locations. It is also noted that substituting (15) in (16) implies that the spatial filters are not applied to the signals. and is theoretically The obtained relation between linear with slope equal to conduction velocity [(1)]. Thus, using (2) (linear regression) and applying the analytical relation between spatial and temporal frequency [(16)], the following expression for conduction velocity is obtained:
(11) (17)
Equation (10) can be rewritten as (12) where , , and are linear combinations of the auto and cross-spectra of the monopolar signals [(11)]. Equation (12) indicates that the power spectrum of the spatially fil(and thus at the tered signal with dip at the spatial frequency ) is a second-order polynomial temporal frequency which is given by (7). function in the first filter coefficient Since and are real, the imaginary part in (12) is null and the following relation holds:
where , of rewritten as
,
, and n and
where is the th dip frequency. Equation (17) is an analytical expression that provides conduction velocity from the auto- and cross-spectra of four monopolar signals recorded along the direction of signal propagation. Once the auto- and cross-spectra are calculated, conduction velocity estimates can be computed from any frequency band. An alternative version of the proposed estimator consists in using only one dip frequency
(13)
(18)
are the imaginary parts , respectively. Thus, (12) can be
which is obtained from (17) with . Equation (18) expresses conduction velocity as a function of the dip frequency which can be arbitrarily chosen from the bandwidth of the signal.
(14) A spectral dip in the temporal domain corresponds to the minimum (in the ideal case equal to zero) of in . Solving for , the minimum of is obtained as (vertex of a parabola) (15) where is the temporal frequency at which the temporal dip occurs. Equation (15) thus provides the coefficient value as a function of the dip location in the temporal frequency domain. On the other hand, the location of the dip in the spatial frequency domain can be derived by inversion of the first equation in (7)
(16) Substituting (15) for in (16), a relation between and is obtained. If this equation is solved for , the location of
B. Simulations Simulated signals were generated to evaluate the performance of the proposed technique. The surface EMG model adopted was described in [8] and allows the generation of muscle fiber action potentials in a multilayer cylindrical volume conductor with muscle, subcutaneous and skin layers. Conductivities of the layers were the same as in previous studies [9]. In a first set of simulations, two single motor unit action potentials were generated from 200 fibers each with a density in the circular motor unit territory, 100-mm of 20 long (5-mm spread of the end-plates and tendon endings), and a mean depth of 3 or 6 mm. The motor units were located directly under the four recording electrodes (rectangular electrodes 5 mm, 5-mm interelectrode distance) and signals were 1 recorded as monopolar derivations. The center of the four electrode system was between the innervation zone and tendon. Simulated conduction velocity of all muscle fibers was 4 m/s.
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In a second set of simulations, the same EMG generation model was used to simulate the activity of a muscle (elliptical shape with axes 30 and 25.4 mm, cross-sectional area ; fibers 120-mm long with innervation zone located in the center) with 200 motor units (innervation ratio uniformly distributed in the range 50–1000) and Gaussian distribution of conduction velocity (mean 4 m/s, standard deviation 0.32 m/s [10]; smaller motor units had smaller conduction velocity [11]). All motor units were active with discharge rate uniformly distributed in the range 8–40 pulses/s (interpulse interval variability 20%), assigned so that the smaller motor units had higher discharge rate than the larger ones [12]. The duration of the simulated signal was 60 s. The signal was detected with four monopolar electrodes (5-mm interelectrode distance) located between the innervation zone and tendon, with the first at 5-mm distance from the innervation zone. In both sets of simulations, the signals were generated at 1024 Hz sampling rate. Independent white Gaussian noises were added to the simulated monopolar signals in order to obtain signal-to-noise ratios (SNRs) of 10 and 20 dB for 1000 signal realizations. C. Experimental Recordings Experimental surface EMG signals were detected from the abductor pollicis brevis muscle during isometric, constant force contractions. Five healthy subjects (three males and two females; age, mean SD, 29.6 1.6 yr), with no symptoms of neuromuscular disorders, participated in the experiment. The study was approved by the local Ethics Committee and written consent was obtained from all subjects prior to inclusion. Surface EMG signals were detected with a linear array of 16 electrodes (2.5 mm inter-electrode distance) in monopolar derivation. The EMG signals were amplified (multichannel EMG amplifier, LISIN-OT Bioelettronica, Torino, Italy), bandpass filtered (10–500 Hz), sampled at 2048 Hz, and converted in digital format on 12 b. Before applying the array, the skin was abraded with abrasive paste and then the location of the array was chosen by visual inspection of the quality of signals recorded in preliminary test contractions. The right hand of the subject was located on a support with fingers blocked by a strap and the thumb free to move. A weight of 1 kg was applied to the phalanx of the thumb and the subjects were instructed to maintain the thumb horizontal holding the weight. Signals were recorded in this condition for 60 s. D. Data Processing The proposed conduction velocity estimator [see (17)] was applied to the simulated and experimental signals. For both simulated and experimental signals, the auto- and cross-spectra of the four monopolar recordings were estimated using the periodogram approach [13] and conduction velocity was estimated applying (17) varying the number of frequencies used. The frequency range was 145–210 Hz (1 Hz increments for a total of maximum 65 dips) for the simulated signals and 100–200 Hz (1 Hz increments, maximum 100 dips) for the experimental signals, based on the bandwidth containing most of the signal power.
Fig. 3. Mean and standard deviation of conduction velocity estimates from simulated signals. Action potentials from one of the two simulated motor units (3-mm depth) were used for these results (see text for detailed simulation modalities) and noise was added with SNR of (a) 20 dB and (b) 10 dB. CV: conduction velocity.
In the first set of simulations, mean and standard deviation of conduction velocity estimates were computed from the 1000 conduction velocity values obtained from each simulated motor unit. In the second set of simulations, mean and standard deviation of the estimates were computed from the 60 values (1-s epochs) and the 1000 noise realizations. The experimental signals were down-sampled by two in both the spatial and temporal domain, so that sampling frequency and interelectrode distance matched the simulated conditions (1024 Hz and 5 mm). Moreover, only the central four monopolar recordings in the experimental case were used for further analysis. Each experimental recording was divided into 60 epochs of 1 s. Sixty conduction velocity estimates were thus obtained from each experimental recording. The 60 values were fit with a regression line, to compensate for changes in conduction velocity due to fatigue. The standard deviation of conduction velocity estimates was computed from the 60 values after subtraction of the regression line (trend over time). The initial value of conduction velocity was defined as the intercept of the regression line. III. RESULTS A. Simulations Fig. 3 shows the mean and standard deviation of conduction velocity estimates obtained applying the proposed algorithm to the simulated single motor unit action potential for depth 3 mm (first set of simulations). Results from one dip to 65 dips are shown. Standard deviation of estimation decreased with the
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Fig. 4. Representative experimental results on one subject. (a) Examples of experimental monopolar signals. (b) Values of the filter coefficient a for a range of temporal dip frequencies [see (15)]. (c) The spatial frequencies corresponding to the temporal frequencies in (b) [see (16)]. (d) Mean and standard deviation (over 60 epochs) of conduction velocity estimates for varying number of dip frequencies [see (17)]. In (a), (b), and (c) the data are obtained after averaging the power spectra over the 60 epochs of 1 s for clarity of representation. CV: conduction velocity.
number of dips used for both SNRs (from 1.51 m/s and 1.06 m/s with one dip to 0.51 m/s and 0.23 m/s with 65 dips in case of 10 and 20 dB SNR, respectively; Fig. 3). Similar results were obtained for the deeper motor unit (conduction velocity standard deviation 1.47 and 1.45 m/s with one dip, 0.75 m/s and 0.48 m/s with 65 dips). In the second set of simulations, the standard deviation of estimated conduction velocity for 10 dB SNR (20 dB) was 0.95 m/s (0.60 m/s) for one dip, 0.40 m/s (0.30 m/s) for 10 dips, and 0.21 m/s (0.11 m/s) for 65 dips. The mean value of the estimate was 4.23 m/s (4.21 m/s) for one dip, 3.88 m/s (4.06 m/s) for 10 dips, and 4.05 m/s (4.16 m/s) for 65 dips. B. Experiments Fig. 4 shows experimental results from one subject. The initial conduction velocity value over the five subjects was (mean SD) 3.65 0.59 m/s for one dip and 3.87 0.42 m/s for 100 dips. In agreement with the simulations, the standard deviation of conduction velocity estimation decreased with increasing number of dips [Fig. 4(d)]. Over the five subjects, standard deviation of estimation was 1.25 0.62 m/s with one dip and 0.10 0.03 m/s with 100 dips. IV. DISCUSSION In this study, a closed expression that relates conduction velocity to the auto- and cross-spectra of monopolar EMG signals recorded along the direction of the muscle fibers is proposed.
Thus, conduction velocity is expressed as a function of the temporal frequency. The derivation of the estimator is based on the concept of multiple spectral dips introduced in the signal power spectrum through the application of spatial filters. The estimator was tested on simulated and experimental signals and it was shown that the standard deviation of the estimates was substantially reduced using multiple dips with respect to the original single dip approach. The closed formula, (17), derived from the multidip approach provides an estimate of conduction velocity from the spectral analysis of monopolar signals and does not require iterative procedures to find local minima, as needed for maximum likelihood approaches [14]. This avoids problems of convergence to local rather than global minima and reduces the computational time. In addition, resolution in the estimation of conduction velocity is not limited by the sampling interval, as for cross-correlation methods that require resampling of the signal [15]. Moreover, most previous methods based on the estimation of delay from two or more signals depend on the spatial filters applied to detect the signals while the multidip approach is based on monopolar recordings, although other recording systems can be applied for preprocessing. With respect to the original single dip approach, the method does not require the detection of the dip in the power spectrum of the recorded signal since this is determined analytically [see (15)]. This avoids dependence of the estimate on the dip detection approach. Moreover, while the single dip approach is based on only one frequency and thus the information used from the power spectrum of the detected waveforms is limited to one point, the proposed method can use the entire frequency
FARINA AND NEGRO: ESTIMATION OF MUSCLE FIBER CONDUCTION VELOCITY WITH A SPECTRAL MULTIDIP APPROACH
bandwidth. This substantially reduces the standard deviation of estimation, as evidenced especially from the experimental results. As many other approaches, the method is based on the assumption that the velocity of propagation is constant over the length covered by the electrodes. If this does not occur [16], the theoretical relation between temporal and spatial frequency is disrupted and the method would provide an estimate on which the different velocity values are weighted. There are several potential extensions of the proposed approach. The number of electrodes can be increased to more than four in order to locate more than one dip (in addition to that in dc) for each spatial filter. The interelectrode distance can also vary among different pairs of adjacent electrode. The mathematical derivations would be similar in this case but the regression formula and the analytical derivation of the zeros in the power spectrum should be adapted. Another modification relates to the use of different interpolation methods [instead of that proposed in (2)]. Moreover, the spectral estimation approach (the simple periodogram in the results presented) may be optimized on the basis of the application. One important issue is the selection of the frequency bandwidth used for the location of the dips. The selected frequencies can also belong to discrete sets and not to a continuous band. In the results presented, the frequency band was fixed for both simulated and experimental signals and corresponded to the band of highest signal power. However, the selection of frequencies can be optimized on a signal basis. V. CONCLUSIONS A noninvasive method for estimation of conduction velocity from a multidip approach has been proposed. The method is the extension of the original proposal of single dip detection from bipolar recordings [3], [17]. The multidip approach led to a closed formula for conduction velocity estimate that depends on the set of frequencies chosen to locate the dips. The method allows reduction of the estimation variance with respect to the single dip approach, does not imply limitations in resolution of the estimated conduction velocity, and does not require any iterative procedure for the estimation. REFERENCES [1] L. Arendt-Nielsen and M. Zwarts, “Measurement of muscle fibre conduction velocity in humans: Techniques and applications,” J. Clin. Neurophysiol., vol. 6, pp. 173–190, 1989. [2] D. Farina and R. Merletti, “Methods for estimating muscle fibre conduction velocity from surface electromyographic signals,” Med. Biol. Eng. Comp., vol. 42, pp. 432–445, 2004. [3] L. Lindstrom, R. Magnusson, and I. Petersén, “The ‘dip phenomenon’ in power spectra of EMG signals,” Electroencephalogr. Clin. Neurophysiol., vol. 30, pp. 259–260, 1971. [4] G. N. McVicar and P. A. Parker, “Spectrum dip estimation of nerve conduction velocity,” IEEE Trans. Biomed. Eng., vol. BME-35, no. 12, pp. 1069–1076, Dec. 1988. [5] H. Broman, G. Bilotto, and C. J. De Luca, “A note on noninvasive estimation of muscle fiber conduction velocity,” IEEE Trans Biomed. Eng., vol. BME-32, no. 5, pp. 311–319, May 1985. [6] H. Reucher, G. Rau, and J. Silny, “Spatial filtering of noninvasive multielectrode EMG part I,” IEEE Trans Biomed. Eng., vol. 34, pp. 98–105, 1987. [7] H. Reucher, G. Rau, and J. Silny, “Spatial filtering of noninvasive multielectrode EMG part II,” IEEE Trans Biomed. Eng., vol. BME-34, no. 2, pp. 106–113, Feb. 1987.
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[8] D. Farina, L. Mesin, S. Martina, and R. Merletti, “A surface EMG generation model with multilayer cylindrical description of the volume conductor,” IEEE Trans. Biomed. Eng., vol. 51, no. 3, pp. 415–426, Mar. 2004. [9] D. Farina, M. Fosci, and R. Merletti, “Motor unit recruitment strategies investigated by surface EMG variables,” J. Appl. Physiol., vol. 92, pp. 235–247, 2002. [10] W. Troni, R. Cantello, and I. Rainero, “Conduction velocity along human muscle fibers in situ,” Neurology, vol. 33, pp. 1453–1459, 1983. [11] S. Andreassen and L. Arendt-Nielsen, “Muscle fibre conduction velocity in motor units of the human anterior tibial muscle: A new size principle parameter,” J. Physiol., vol. 391, pp. 561–571, 1987. [12] A. J. Fuglevand, D. A. Winter, and A. E. Patla, “Models of recruitment and rate coding organization in motor-unit pools,” J. Neurophysiol., vol. 70, pp. 2470–2488, 1993. [13] J. G. Proakis, C. M. Rader, F. Ling, and C. L. Nikias, Advanced Digital Signal Processing. New York: Macmillan, 1992. [14] K. C. McGill and L. J. Dorfman, “High-resolution alignment of sampled waveforms,” IEEE Trans. Biomed. Eng., vol. BME-31, no. 6, pp. 462–468, Jun. 1984. [15] M. Naeije and H. Zorn, “Estimation of the action potential conduction velocity in human skeletal muscle using the surface EMG cross-correlation technique,” Electromyogr. Clin. Neurophysiol., vol. 23, pp. 73–80, 1983. [16] N. A. Dimitrova, G. V. Dimitrov, and A. G. Dimitrov, “Calculation of spatially filtered signals produced by a motor unit comprising muscle fibres with non-uniform propagation,” Med. Biol. Eng. Comput., vol. 39, pp. 202–207, 2001. [17] L. Lindstrom and R. Magnusson, “Interpretation of myoelectric power spectra: A model and its applications,” Proc. IEEE, vol. 65, pp. 653–662, 1977. Dario Farina (M’01) graduated summa cum laude in electronics engineering (equivalent to the M.Sc. degree) and received the Ph.D. degree in electronics and communications engineering from the Politecnico di Torino, Torino, Italy, in 1998 and 2002, respectively, and the Ph.D. degree in automatic control and computer science from the Ecole Centrale de Nantes, Nantes, France, in 2001. During 1998, he was a Fellow of the Laboratory for Neuromuscular System Engineering, in Torino. From 1999 to 2004, he taught courses in electronics and mathematics at the Politecnico di Torino where, from 2002 to 2004, he was a Research Assistant Professor. Since 2004, he has been an Associate Professor in biomedical engineering at the Department of Health Science and Technology of Aalborg University, Aalborg, Denmark, where he teaches courses on biomedical signal processing, modeling, and neuromuscular physiology. He regularly acts as referee for approximately 20 scientific international journals. His main research interests are in the areas of signal processing applied to biomedical signals, modeling of biological systems, basic and applied physiology of the neuromuscular system, and brain-computer interfaces. Within these fields he has authored or coauthored more than 120 papers in peer-reviewed Journals. Dr. Farina is a Registered Professional Engineer in Italy. He is an Associate Editor of the IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING and is on the Editorial Boards of the Journal of Neuroscience Methods, the Journal of Electromyography and Kinesiology, and Medical & Biological Engineering & Computing. He is a member of the International Society of Electrophysiology and Kinesiology (ISEK) Council.
Francesco Negro received the M.Sc. degree in telecommunication engineering from the Politecnico di Torino, Torino, Italy, in November 2005. He is currently working toward the Ph.D. degree at the Center for Sensory-Motor Interaction (SMI), Department of Health Science and Technology, Aalborg University, Denmark. His research focuses on developing new methods for the evaluation of supraspinal factors in muscle control using joint high-density multichannel EMG and EEG recordings.
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Estimation of Muscle Fiber Conduction Velocity With a Spectral Multidip Approach Dario Farina*, Member, IEEE, and Francesco Negro
Abstract—We propose a novel method for estimation of muscle fiber conduction velocity from surface electromyographic (EMG) signals. The method is based on the regression analysis between spatial and temporal frequencies of multiple dips introduced in the EMG power spectrum through the application of a set of spatial filters. This approach leads to a closed analytical expression of conduction velocity as a function of the auto- and cross-spectra of monopolar signals detected along the direction of muscle fibers. The performance of the algorithm was compared with respect to that of the classic single dip approach on simulated and experimental EMG signals. The standard deviation of conduction velocity estimates from simulated single motor unit action potentials was reduced from 1.51 m/s [10 dB signal-to-noise ratio (SNR)] and 1.06 m/s (20 dB SNR) with the single dip approach to 0.51 m/s (10 dB) and 0.23 m/s (20 dB) with the proposed method using 65 dips. When 200 active motor units were simulated in an interference EMG signal, standard deviation of conduction velocity decreased from 0.95 m/s (10 dB SNR) and 0.60 m/s (20 dB SNR) with a single dip to 0.21 m/s (10 dB) and 0.11 m/s (20 dB) with 65 dips. In experimental signals detected from the abductor pollicis brevis muscle, standard deviation of estimation decreased SD over 5 subjects) 1.25 0.62 m/s with one dip from (mean to 0.10 0.03 m/s with 100 dips. The proposed method does not imply limitation in resolution of the estimated conduction velocity and does not require any iterative procedure for the estimate since it is based on a closed analytical formulation. Index Terms—Conduction velocity, delay estimators, electromyographic (EMG), spectral analysis.
I. INTRODUCTION VERAGE muscle fiber conduction velocity can be estimated from surface electromyographic (EMG) signals [1], which comprise the contribution of the active superficial motor units. Several methods for the estimation of conduction velocity from the interference EMG have been proposed in the literature [2]. Usually they are based on the estimation of delay between two or more signals recorded along the direction of propagation of the action potentials. However, it can be shown that a single EMG signal recorded through the application of a spatial filter is theoretically sufficient for the estimation of propagation velocity [3]. The zeros (dips) in the spatial filter transfer function are introduced in the power spectrum of the recorded signal
A
Manuscript received August 25, 2006; revised January 5, 2007. This work was supported by the Danish Technical Research Council, Centre for Neuroengineering (CEN) Project, under Contract 26-04-0100 (DF). Asterisk indicates corresponding author. *D. Farina is with the Center for Sensory-Motor Interaction (SMI), Department of Health Science and Technology, Aalborg University, Fredrik Bajers Vej 7D-3,Aalborg DK-9220, Denmark (e-mail: [email protected]). F. Negro is with the Center for Sensory-Motor Interaction (SMI), Department of Health Science and Technology, Aalborg University, Aalborg DK-9220, Denmark (e-mail: [email protected]). Digital Object Identifier 10.1109/TBME.2007.892928
since the spatial and temporal frequency domains are linearly related by conduction velocity (Fig. 1). Thus, conduction velocity can be estimated from the ratio between the temporal (detected from the power spectrum of the recorded signal) and spatial (imposed through the design of the spatial filter) frequency at which a dip occurs. The method has been applied with the single differential spatial filter [4] whose transfer function has a dip at the spatial frequency equal to the inverse of the interelectrode distance. There are two main problems associated with the spectral dip technique with single differential recording. First, the interelectrode distance should be large to introduce a dip in the bandwidth of the EMG signal. Second, the detection of the dip in the power spectrum of the recorded signal has large variance of estimation that is reflected in a large variability of conduction velocity estimates [4]. These issues could be overcome using spatial filters with more electrodes than the single differential and imposing more than one dip, so that results from different dips can be averaged and estimation variance reduced. For example, it is possible to locate a dip at any spatial frequency by the design of spatial filters from four detection points [2]. However, this approach has never been proposed and tested. Therefore, in this study we present a novel method for estimating conduction velocity from the surface EMG, based on the identification of a number of spectral dips introduced in the power spectrum of the detected signals through the application of a set of spatial filters. It is shown that this approach leads to a closed formula for the calculation of muscle fiber conduction velocity as a function of the auto- and cross-spectra of a set of detected monopolar signals. II. METHODS Surface EMG signals are spatially filtered by the linear combination of the monopolar signals detected from electrodes arranged in a specific geometrical pattern [5], [6]. The effect of the spatial filter on the recorded signal can be described by the transfer function of the filter in the spatial frequency domain. In the direction of propagation of the sources, the spatial frequency domain is related to the temporal frequency domain by the velocity of propagation (1) the temporal frewhere is the velocity of propagation, the spatial frequency in the direction of source quency, and propagation. The power spectrum in the temporal frequency domain is scaled with respect to the power spectrum in the spatial frequency domain. If the power spectrum is zero in one domain at
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Fig. 1. Representative scheme of the introduction of dips in the surface EMG power spectrum. In the spatial domain z (bottom), three sinusoids [R (z)] with the same amplitude and spatial frequencies 25, 50, and 75 cycles/m move with a velocity v = 4 m=s and are recorded in four detection points, separated by the distance d . The dashed lines schematically show the sinusoids at a different time instant (translation). The output signals corresponding to the three sinusoids [P (z )] are the linear combinations of the signals at the four detection points. Specific selections of the coefficients of the linear combination null one of the spatial harmonics. This corresponds to locate a dip at a specific frequency f . Two examples of filter coefficient (a , i = 1, 2, 3, 4) selection are shown which correspond to null the first and third harmonics. The harmonics that are suppressed in the spatial domain have null amplitude also in the temporal domain (dips in the temporal frequency domain) (top). The frequency of the suppressed harmonic in the temporal domain is given by the frequency of the dip in the spatial domain scaled by conduction velocity. The y -axes of the power spectra (right) are in logarithmic normalized scale.
a specific frequency (dip), the dip in the other domain is at the same frequency scaled by conduction velocity (Fig. 1). Considering electrodes located along the direction of propagation of the action potentials, the power spectrum of the detected signal is the multiplication of the power spectrum of the monopolar signal and the transfer function of the spatial filter. Zeros in the power spectrum of the output signal can thus be introduced by the transfer function of the spatial filter. If the single differential filter is applied, one dip is introduced at dc and another (with its periodic repetitions) at a temporal frequency that depends on conduction velocity and interelectrode distance. Conduction velocity is thus estimated through (1) applied to the frequency of the dip in the spatial and temporal domains [3]. If more than one spatial filter is applied and each filter introduces dips at different frequencies within the signal bandwidth, conduction velocity can be estimated from (1) using all the dip frequencies, for example applying a least-squares regression analysis
(2)
is the th dip temporal frequency, the th dip spawhere tial frequency, and is the number of dips introduced. A. Spectral Multidip Estimator of Conduction Velocity In the following, the case of four monopolar recordings with constant interelectrode distance is considered and all the derivations are related to this case. Extensions to more than four detection points and variable interelectrode distance follow similar derivations. Moreover, the following derivations apply not only to monopolar recordings. The four detected signals may result from the application of any spatial filter at each detection point (preprocessing filter) under the condition that the filter is the same at the four detection points, so that the four detected signals have, in ideal conditions, the same shape and a relative delay between each other. However, for simplicity, in the following we will consider four monopolar recordings. of a spatial filter applied to four The transfer function detection points is
(3) where is the spatial frequency along the direction of acare the coefficients of the tion potential propagation,
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Fig. 2. Representation of the multidip approach for estimation of conduction velocity on simulated single motor unit action potentials. The motor unit action potential is generated by 200 fibers with a density of 20 bers=mm in the circular motor unit territory, 100-mm of fiber length, mean depth of 3 mm, and 5 mm, 5-mm interelectrode distance), conduction velocity v = 4 m=s. (a) Four monopolar signals are simulated as detected by rectangular electrodes (1 placed between the innervation zone and tendon region. (c) The monopolar signals are filtered with four spatial filters whose transfer function is chosen in order to introduce dips at spatial frequencies 100, 125, 150, and 175 cycles/m. The filter coefficients were computed from (7). (b) The dips introduced in the spatial frequency domain also appear in the power spectrum (temporal frequency domain) of the detected signals which was estimated with the periodogram approach. The locations of the dips in the two domains are linearly related [(1)]. (d) The slope of the regression line between dip frequencies in the two domains is an estimate of conduction velocity. Arbitrary units [au].
2
linear combination (filter coefficients), and the constant distance between adjacent electrodes. Equation (3) can be rewritten as
. Under this condition, simple calculations (omitted) lead for to the following relations between the filter coefficients
(4) If the spatial filter introduces a zero at the spatial dc, as usually imposed to remove the spatial common mode [6], [7], the summation of the filter coefficients is zero (5)
(7) which were reported in [2] in an equivalent form. Equation (7) provides the filter coefficients to impose a zero at any frequency in the spatial domain (Fig. 2). The filtered signal in the temporal domain is (8)
and (4) can be simplified as
are the monopolar signals detected by where the four electrodes. From (8), the power spectrum of the signal is (6) All spatial filters considered in the following have transfer function equal to zero in dc. This also reduces the adverse effect of end-of-fiber components. The condition of a dip at frequency implies that the transfer function of the spatial filter is null
(9) where and
are the power spectra of the cross-spectra.
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Imposing a dip at and thus using (7) for the filter coefficients, the expression of the power spectrum of the detected signal [(9)] can be simplified as
(10) given from (7) and dependent on the spatial frequency with . With the substitution of the dip
the dip in the spatial domain is a function of the location of the . It is thus possible to dip in the temporal frequency domain choose any location of the dip in the temporal frequency domain and derive the corresponding location in the spatial frequency domain without practically detect the dip, but with the use of a closed analytical formula. This can be done with an arbitrary number of dip locations. It is also noted that substituting (15) in (16) implies that the spatial filters are not applied to the signals. and is theoretically The obtained relation between linear with slope equal to conduction velocity [(1)]. Thus, using (2) (linear regression) and applying the analytical relation between spatial and temporal frequency [(16)], the following expression for conduction velocity is obtained:
(11) (17)
Equation (10) can be rewritten as (12) where , , and are linear combinations of the auto and cross-spectra of the monopolar signals [(11)]. Equation (12) indicates that the power spectrum of the spatially fil(and thus at the tered signal with dip at the spatial frequency ) is a second-order polynomial temporal frequency which is given by (7). function in the first filter coefficient Since and are real, the imaginary part in (12) is null and the following relation holds:
where , of rewritten as
,
, and n and
where is the th dip frequency. Equation (17) is an analytical expression that provides conduction velocity from the auto- and cross-spectra of four monopolar signals recorded along the direction of signal propagation. Once the auto- and cross-spectra are calculated, conduction velocity estimates can be computed from any frequency band. An alternative version of the proposed estimator consists in using only one dip frequency
(13)
(18)
are the imaginary parts , respectively. Thus, (12) can be
which is obtained from (17) with . Equation (18) expresses conduction velocity as a function of the dip frequency which can be arbitrarily chosen from the bandwidth of the signal.
(14) A spectral dip in the temporal domain corresponds to the minimum (in the ideal case equal to zero) of in . Solving for , the minimum of is obtained as (vertex of a parabola) (15) where is the temporal frequency at which the temporal dip occurs. Equation (15) thus provides the coefficient value as a function of the dip location in the temporal frequency domain. On the other hand, the location of the dip in the spatial frequency domain can be derived by inversion of the first equation in (7)
(16) Substituting (15) for in (16), a relation between and is obtained. If this equation is solved for , the location of
B. Simulations Simulated signals were generated to evaluate the performance of the proposed technique. The surface EMG model adopted was described in [8] and allows the generation of muscle fiber action potentials in a multilayer cylindrical volume conductor with muscle, subcutaneous and skin layers. Conductivities of the layers were the same as in previous studies [9]. In a first set of simulations, two single motor unit action potentials were generated from 200 fibers each with a density in the circular motor unit territory, 100-mm of 20 long (5-mm spread of the end-plates and tendon endings), and a mean depth of 3 or 6 mm. The motor units were located directly under the four recording electrodes (rectangular electrodes 5 mm, 5-mm interelectrode distance) and signals were 1 recorded as monopolar derivations. The center of the four electrode system was between the innervation zone and tendon. Simulated conduction velocity of all muscle fibers was 4 m/s.
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In a second set of simulations, the same EMG generation model was used to simulate the activity of a muscle (elliptical shape with axes 30 and 25.4 mm, cross-sectional area ; fibers 120-mm long with innervation zone located in the center) with 200 motor units (innervation ratio uniformly distributed in the range 50–1000) and Gaussian distribution of conduction velocity (mean 4 m/s, standard deviation 0.32 m/s [10]; smaller motor units had smaller conduction velocity [11]). All motor units were active with discharge rate uniformly distributed in the range 8–40 pulses/s (interpulse interval variability 20%), assigned so that the smaller motor units had higher discharge rate than the larger ones [12]. The duration of the simulated signal was 60 s. The signal was detected with four monopolar electrodes (5-mm interelectrode distance) located between the innervation zone and tendon, with the first at 5-mm distance from the innervation zone. In both sets of simulations, the signals were generated at 1024 Hz sampling rate. Independent white Gaussian noises were added to the simulated monopolar signals in order to obtain signal-to-noise ratios (SNRs) of 10 and 20 dB for 1000 signal realizations. C. Experimental Recordings Experimental surface EMG signals were detected from the abductor pollicis brevis muscle during isometric, constant force contractions. Five healthy subjects (three males and two females; age, mean SD, 29.6 1.6 yr), with no symptoms of neuromuscular disorders, participated in the experiment. The study was approved by the local Ethics Committee and written consent was obtained from all subjects prior to inclusion. Surface EMG signals were detected with a linear array of 16 electrodes (2.5 mm inter-electrode distance) in monopolar derivation. The EMG signals were amplified (multichannel EMG amplifier, LISIN-OT Bioelettronica, Torino, Italy), bandpass filtered (10–500 Hz), sampled at 2048 Hz, and converted in digital format on 12 b. Before applying the array, the skin was abraded with abrasive paste and then the location of the array was chosen by visual inspection of the quality of signals recorded in preliminary test contractions. The right hand of the subject was located on a support with fingers blocked by a strap and the thumb free to move. A weight of 1 kg was applied to the phalanx of the thumb and the subjects were instructed to maintain the thumb horizontal holding the weight. Signals were recorded in this condition for 60 s. D. Data Processing The proposed conduction velocity estimator [see (17)] was applied to the simulated and experimental signals. For both simulated and experimental signals, the auto- and cross-spectra of the four monopolar recordings were estimated using the periodogram approach [13] and conduction velocity was estimated applying (17) varying the number of frequencies used. The frequency range was 145–210 Hz (1 Hz increments for a total of maximum 65 dips) for the simulated signals and 100–200 Hz (1 Hz increments, maximum 100 dips) for the experimental signals, based on the bandwidth containing most of the signal power.
Fig. 3. Mean and standard deviation of conduction velocity estimates from simulated signals. Action potentials from one of the two simulated motor units (3-mm depth) were used for these results (see text for detailed simulation modalities) and noise was added with SNR of (a) 20 dB and (b) 10 dB. CV: conduction velocity.
In the first set of simulations, mean and standard deviation of conduction velocity estimates were computed from the 1000 conduction velocity values obtained from each simulated motor unit. In the second set of simulations, mean and standard deviation of the estimates were computed from the 60 values (1-s epochs) and the 1000 noise realizations. The experimental signals were down-sampled by two in both the spatial and temporal domain, so that sampling frequency and interelectrode distance matched the simulated conditions (1024 Hz and 5 mm). Moreover, only the central four monopolar recordings in the experimental case were used for further analysis. Each experimental recording was divided into 60 epochs of 1 s. Sixty conduction velocity estimates were thus obtained from each experimental recording. The 60 values were fit with a regression line, to compensate for changes in conduction velocity due to fatigue. The standard deviation of conduction velocity estimates was computed from the 60 values after subtraction of the regression line (trend over time). The initial value of conduction velocity was defined as the intercept of the regression line. III. RESULTS A. Simulations Fig. 3 shows the mean and standard deviation of conduction velocity estimates obtained applying the proposed algorithm to the simulated single motor unit action potential for depth 3 mm (first set of simulations). Results from one dip to 65 dips are shown. Standard deviation of estimation decreased with the
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Fig. 4. Representative experimental results on one subject. (a) Examples of experimental monopolar signals. (b) Values of the filter coefficient a for a range of temporal dip frequencies [see (15)]. (c) The spatial frequencies corresponding to the temporal frequencies in (b) [see (16)]. (d) Mean and standard deviation (over 60 epochs) of conduction velocity estimates for varying number of dip frequencies [see (17)]. In (a), (b), and (c) the data are obtained after averaging the power spectra over the 60 epochs of 1 s for clarity of representation. CV: conduction velocity.
number of dips used for both SNRs (from 1.51 m/s and 1.06 m/s with one dip to 0.51 m/s and 0.23 m/s with 65 dips in case of 10 and 20 dB SNR, respectively; Fig. 3). Similar results were obtained for the deeper motor unit (conduction velocity standard deviation 1.47 and 1.45 m/s with one dip, 0.75 m/s and 0.48 m/s with 65 dips). In the second set of simulations, the standard deviation of estimated conduction velocity for 10 dB SNR (20 dB) was 0.95 m/s (0.60 m/s) for one dip, 0.40 m/s (0.30 m/s) for 10 dips, and 0.21 m/s (0.11 m/s) for 65 dips. The mean value of the estimate was 4.23 m/s (4.21 m/s) for one dip, 3.88 m/s (4.06 m/s) for 10 dips, and 4.05 m/s (4.16 m/s) for 65 dips. B. Experiments Fig. 4 shows experimental results from one subject. The initial conduction velocity value over the five subjects was (mean SD) 3.65 0.59 m/s for one dip and 3.87 0.42 m/s for 100 dips. In agreement with the simulations, the standard deviation of conduction velocity estimation decreased with increasing number of dips [Fig. 4(d)]. Over the five subjects, standard deviation of estimation was 1.25 0.62 m/s with one dip and 0.10 0.03 m/s with 100 dips. IV. DISCUSSION In this study, a closed expression that relates conduction velocity to the auto- and cross-spectra of monopolar EMG signals recorded along the direction of the muscle fibers is proposed.
Thus, conduction velocity is expressed as a function of the temporal frequency. The derivation of the estimator is based on the concept of multiple spectral dips introduced in the signal power spectrum through the application of spatial filters. The estimator was tested on simulated and experimental signals and it was shown that the standard deviation of the estimates was substantially reduced using multiple dips with respect to the original single dip approach. The closed formula, (17), derived from the multidip approach provides an estimate of conduction velocity from the spectral analysis of monopolar signals and does not require iterative procedures to find local minima, as needed for maximum likelihood approaches [14]. This avoids problems of convergence to local rather than global minima and reduces the computational time. In addition, resolution in the estimation of conduction velocity is not limited by the sampling interval, as for cross-correlation methods that require resampling of the signal [15]. Moreover, most previous methods based on the estimation of delay from two or more signals depend on the spatial filters applied to detect the signals while the multidip approach is based on monopolar recordings, although other recording systems can be applied for preprocessing. With respect to the original single dip approach, the method does not require the detection of the dip in the power spectrum of the recorded signal since this is determined analytically [see (15)]. This avoids dependence of the estimate on the dip detection approach. Moreover, while the single dip approach is based on only one frequency and thus the information used from the power spectrum of the detected waveforms is limited to one point, the proposed method can use the entire frequency
FARINA AND NEGRO: ESTIMATION OF MUSCLE FIBER CONDUCTION VELOCITY WITH A SPECTRAL MULTIDIP APPROACH
bandwidth. This substantially reduces the standard deviation of estimation, as evidenced especially from the experimental results. As many other approaches, the method is based on the assumption that the velocity of propagation is constant over the length covered by the electrodes. If this does not occur [16], the theoretical relation between temporal and spatial frequency is disrupted and the method would provide an estimate on which the different velocity values are weighted. There are several potential extensions of the proposed approach. The number of electrodes can be increased to more than four in order to locate more than one dip (in addition to that in dc) for each spatial filter. The interelectrode distance can also vary among different pairs of adjacent electrode. The mathematical derivations would be similar in this case but the regression formula and the analytical derivation of the zeros in the power spectrum should be adapted. Another modification relates to the use of different interpolation methods [instead of that proposed in (2)]. Moreover, the spectral estimation approach (the simple periodogram in the results presented) may be optimized on the basis of the application. One important issue is the selection of the frequency bandwidth used for the location of the dips. The selected frequencies can also belong to discrete sets and not to a continuous band. In the results presented, the frequency band was fixed for both simulated and experimental signals and corresponded to the band of highest signal power. However, the selection of frequencies can be optimized on a signal basis. V. CONCLUSIONS A noninvasive method for estimation of conduction velocity from a multidip approach has been proposed. The method is the extension of the original proposal of single dip detection from bipolar recordings [3], [17]. The multidip approach led to a closed formula for conduction velocity estimate that depends on the set of frequencies chosen to locate the dips. The method allows reduction of the estimation variance with respect to the single dip approach, does not imply limitations in resolution of the estimated conduction velocity, and does not require any iterative procedure for the estimation. REFERENCES [1] L. Arendt-Nielsen and M. Zwarts, “Measurement of muscle fibre conduction velocity in humans: Techniques and applications,” J. Clin. Neurophysiol., vol. 6, pp. 173–190, 1989. [2] D. Farina and R. Merletti, “Methods for estimating muscle fibre conduction velocity from surface electromyographic signals,” Med. Biol. Eng. Comp., vol. 42, pp. 432–445, 2004. [3] L. Lindstrom, R. Magnusson, and I. Petersén, “The ‘dip phenomenon’ in power spectra of EMG signals,” Electroencephalogr. Clin. Neurophysiol., vol. 30, pp. 259–260, 1971. [4] G. N. McVicar and P. A. Parker, “Spectrum dip estimation of nerve conduction velocity,” IEEE Trans. Biomed. Eng., vol. BME-35, no. 12, pp. 1069–1076, Dec. 1988. [5] H. Broman, G. Bilotto, and C. J. De Luca, “A note on noninvasive estimation of muscle fiber conduction velocity,” IEEE Trans Biomed. Eng., vol. BME-32, no. 5, pp. 311–319, May 1985. [6] H. Reucher, G. Rau, and J. Silny, “Spatial filtering of noninvasive multielectrode EMG part I,” IEEE Trans Biomed. Eng., vol. 34, pp. 98–105, 1987. [7] H. Reucher, G. Rau, and J. Silny, “Spatial filtering of noninvasive multielectrode EMG part II,” IEEE Trans Biomed. Eng., vol. BME-34, no. 2, pp. 106–113, Feb. 1987.
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[8] D. Farina, L. Mesin, S. Martina, and R. Merletti, “A surface EMG generation model with multilayer cylindrical description of the volume conductor,” IEEE Trans. Biomed. Eng., vol. 51, no. 3, pp. 415–426, Mar. 2004. [9] D. Farina, M. Fosci, and R. Merletti, “Motor unit recruitment strategies investigated by surface EMG variables,” J. Appl. Physiol., vol. 92, pp. 235–247, 2002. [10] W. Troni, R. Cantello, and I. Rainero, “Conduction velocity along human muscle fibers in situ,” Neurology, vol. 33, pp. 1453–1459, 1983. [11] S. Andreassen and L. Arendt-Nielsen, “Muscle fibre conduction velocity in motor units of the human anterior tibial muscle: A new size principle parameter,” J. Physiol., vol. 391, pp. 561–571, 1987. [12] A. J. Fuglevand, D. A. Winter, and A. E. Patla, “Models of recruitment and rate coding organization in motor-unit pools,” J. Neurophysiol., vol. 70, pp. 2470–2488, 1993. [13] J. G. Proakis, C. M. Rader, F. Ling, and C. L. Nikias, Advanced Digital Signal Processing. New York: Macmillan, 1992. [14] K. C. McGill and L. J. Dorfman, “High-resolution alignment of sampled waveforms,” IEEE Trans. Biomed. Eng., vol. BME-31, no. 6, pp. 462–468, Jun. 1984. [15] M. Naeije and H. Zorn, “Estimation of the action potential conduction velocity in human skeletal muscle using the surface EMG cross-correlation technique,” Electromyogr. Clin. Neurophysiol., vol. 23, pp. 73–80, 1983. [16] N. A. Dimitrova, G. V. Dimitrov, and A. G. Dimitrov, “Calculation of spatially filtered signals produced by a motor unit comprising muscle fibres with non-uniform propagation,” Med. Biol. Eng. Comput., vol. 39, pp. 202–207, 2001. [17] L. Lindstrom and R. Magnusson, “Interpretation of myoelectric power spectra: A model and its applications,” Proc. IEEE, vol. 65, pp. 653–662, 1977. Dario Farina (M’01) graduated summa cum laude in electronics engineering (equivalent to the M.Sc. degree) and received the Ph.D. degree in electronics and communications engineering from the Politecnico di Torino, Torino, Italy, in 1998 and 2002, respectively, and the Ph.D. degree in automatic control and computer science from the Ecole Centrale de Nantes, Nantes, France, in 2001. During 1998, he was a Fellow of the Laboratory for Neuromuscular System Engineering, in Torino. From 1999 to 2004, he taught courses in electronics and mathematics at the Politecnico di Torino where, from 2002 to 2004, he was a Research Assistant Professor. Since 2004, he has been an Associate Professor in biomedical engineering at the Department of Health Science and Technology of Aalborg University, Aalborg, Denmark, where he teaches courses on biomedical signal processing, modeling, and neuromuscular physiology. He regularly acts as referee for approximately 20 scientific international journals. His main research interests are in the areas of signal processing applied to biomedical signals, modeling of biological systems, basic and applied physiology of the neuromuscular system, and brain-computer interfaces. Within these fields he has authored or coauthored more than 120 papers in peer-reviewed Journals. Dr. Farina is a Registered Professional Engineer in Italy. He is an Associate Editor of the IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING and is on the Editorial Boards of the Journal of Neuroscience Methods, the Journal of Electromyography and Kinesiology, and Medical & Biological Engineering & Computing. He is a member of the International Society of Electrophysiology and Kinesiology (ISEK) Council.
Francesco Negro received the M.Sc. degree in telecommunication engineering from the Politecnico di Torino, Torino, Italy, in November 2005. He is currently working toward the Ph.D. degree at the Center for Sensory-Motor Interaction (SMI), Department of Health Science and Technology, Aalborg University, Denmark. His research focuses on developing new methods for the evaluation of supraspinal factors in muscle control using joint high-density multichannel EMG and EEG recordings.
