Journal of Electromyography and Kinesiology 20 (2010) 1249–1258

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Journal of Electromyography and Kinesiology

journal homepage: www.elsevier .com/locate / je lek in

Analysis of the relationship between the rise-time and the amplitudeof single-fibre potentials in human muscles

Javier Rodríguez a,*, Javier Navallas a, Luis Gila b, Ignacio Rodríguez a, Armando Malanda a

a Public University of Navarra, Department of Electrical and Electronical Engineering, 31006 Pamplona, Spainb Virgen del Camino Hospital, Department of Clinical Neurophysiology, 31008 Pamplona, Spain

a r t i c l e i n f o

Article history:Received 1 February 2010Received in revised form 19 April 2010Accepted 2 July 2010

Keywords:SFAP convolutional modelSFAP rise-timeIAP spatial profileCore-conductor theoryNeedle movement

1050-6411/$ - see front matter � 2010 Published bydoi:10.1016/j.jelekin.2010.07.005

* Corresponding author. Address: Universidad PCampus de Arrosadía s/n. 31006 Pamplona, Spain. Te948 169720.

E-mail addresses: javier.rodriguez.falces@gmail.eRodríguez).

a b s t r a c t

Using the core-conductor theory, a single fibre action potential (SFAP) can be expressed as the convolu-tion of a biolectrical source and a weight function. In the Dimitrov–Dimitrova (D–D) SFAP convolutionalmodel, the first temporal derivative of the intracellular action potential (IAP) is used as the source. Thepresent work evaluates the relationship between the SFAP peak-to-peak amplitude (Vpp) and peak-to-peak interval (rise-time, RT) at different fibre-to-electrode distances using simulated signals obtainedby the D–D model as well as real recordings. With a single fibre electrode, we recorded 63 sets of consec-utive SFAPs from the m. tibialis anterior of four normal subjects. The needle was intentionally movedwhilst recording each SFAP set. We used the observed changes in RT and Vpp within each SFAP set as apoint of reference with which to evaluate how closely the relationship between RT and Vpp providedby the D–D model reflects real data. We found that half of the recorded SFAP sets had rise-times higherthan those generated by the D–D model. We also showed the influence of the IAP spatial length on thesensitivity of RT and Vpp with radial distance. The study reveals some inaccuracies in simulated SFAPswhose origin might be related to the assumptions made in the core-conductor theory.

� 2010 Published by Elsevier Ltd.

1. Introduction suitable approach for dealing with the excitation onset and extinc-

Traditionally, electromyography (EMG) studies have drawnattention to the calculation of single fibre action potentials (SFAPs)produced by excitable fibres and especially by fibres of finitelength. On the basis of the volume conductor theory, Plonsey(1974, 1977) and Andreassen and Rosenfalck (1981) showed howa SFAP can be expressed as a convolution of an excitation sourceand a weight function. A simplification of this model based onthe core-conductor theory, introduced by Andreassen andRosenfalck (1981), assumed that the transmembrane current wasdistributed and concentrated along the axis of the fibre (line sourcemodel). A further simplification, introduced by Nandedkar andStalberg (N–S) in 1983, consisted in replacing the complex weightfunction (that involved Bessel functions) with a simplerexpression: a potential generated by a current source in a linesource model. The current formulation of the N–S SFAP convolu-tional model is the result of these approximations in the excitationand in the weight functions (Nandedkar and Stalberg, 1983).

Although the N–S model was found computationally more effi-cient and more accurate than previous SFAP models, it still lacked a

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s, malanda@unavarra.es (J.

tion. To overcome this limitation, it was necessary to represent theexcitation source as two stacks of double-layer disks distributedequidistantly along the fibre axis, each stack generating a dipolefield (Wilson and MacLeod, 1933). Following this idea, Dimitrovand Dimitrova (D–D) replaced the stacks of distributed current di-poles by dipoles lumped along the axis of the fibre, therefore,assuming the line source model (Dimitrov and Dimitrova, 1989).The weight function was computed as the potential produced bytwo lumped current dipoles propagating in opposite directionsfrom the endplate toward the fibre ends.

Since the introduction of the D–D and N–S convolutional models,few studies have addressed the accuracy of the underlying SFAPapproximations (Albers et al., 1989; Van Veen et al., 1993; Rodriguezet al., 2006). The accuracy of SFAP models is predominantly relatedto their ability to generate potentials as similar as possible to exper-imental recordings. The present work evaluates the resemblance be-tween measured and computed SFAPs by analysing the relationshipbetween two SFAP parameters: the peak-to-peak amplitude (Vpp)and peak-to-peak interval (rise-time). Parameters Vpp and rise-timeare very important for clinical diagnosis and EMG quantitative anal-ysis. In fact, in single fibre, concentric needle and macro EMG studiesmost acceptance criteria of experimental potentials are based on therise-time and/or Vpp (Stålberg and Trontelj, 1979).

Empirical determination of how the rise-time (RT) of a SFAP isrelated to its corresponding Vpp at different radial distances r in

1250 J. Rodríguez et al. / Journal of Electromyography and Kinesiology 20 (2010) 1249–1258

non-diseased muscles is very valuable since it can be used as a ref-erence to establish a comparison with Vpp–RT relationships ob-tained from SFAP models or from muscles suffering somedisorder. In close proximity to the fibre, for example, SFAPs withthin spikes (low rise-times) and high amplitudes are expected. Inthis scenario SFAPs with unduly wide spikes would reflect somekind of neurogenic disease (Stålberg and Trontelj, 1979), whereasSFAPs with overly narrow spikes would reflect a myopathic processNandedkar and Sanders (1988).

In order to obtain experimental data related to the effect of ra-dial distance on Vpp and rise-time, we recorded numerous sets ofSFAPs with needle movement. According to theory, when the nee-dle is moved during the recording process, the only variable thatcan change between consecutive discharges is the radial distance:the excitation source, the anisotropy of the volume conductor, andthe propagation velocity are not affected by needle movement.Thus, the relationships between Vpp and rise-time in recordingswith needle movement can be compared with the effects of radialdistance on Vpp and rise-time in simulated SFAPs in order to eval-uate how closely simulated data reflects experimental data.

When studying the Vpp–RT relationship, one should take intoaccount the effect of the spatial extension of the intracellular ac-tion potential (IAP) on the SFAP. In fact, within the 300 lm-radiuswhere SFAPs are usually recorded, changes in the shape of the IAPcould affect the SFAP Vpp and rise-time more than changes in fibrediameter and/or propagation velocity (Dimitrov and Dimitrova,1989). We studied the sensitivity of Vpp and RT to variations in ra-dial distance for different IAP spatial lengths.

The goals of the present work were: (1) to calculate the rela-tionship between Vpp and rise-time in human muscle fibres fromvarious subjects, and (2) to analyse the accuracy of the D–D modelat simulating SFAPs in the proximity of muscle fibres by comparingthe Vpp–RT relationships of SFAPs generated by this model withthose observed in consecutive SFAPs recorded whilst the needlewas under movement, and (3) to show the importance of consider-ing the IAP spatial length when studying the variation of Vpp and RTwith radial distance.

2. Material

2.1. SF-EMG recordings under needle movement

SF-EMG signals were recorded from the right tibialis anteriormuscle of four subjects (three men and one woman) aged between

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Fig. 1. (a) A set of consecutive SFAPs recorded with needle movement. The gradual changtotal of 99 potentials. (b) Definition of SFAP rise-time (RT) and Vpp.

28 and 41 years (mean 34 years). No subject had known symptomsof neuromuscular disorders. The study was conducted in accor-dance with the declaration of Helsinki and approved by the ClinicalInvestigation Ethics Committee of Navarra. Written informed con-sent was obtained from all participants before inclusion.

Experimental signals were recorded with an electromyograph(Counterpoint, Dantec Co., Skovlunde, Denmark) using a SF elec-trode (core diameter of 25 lm, needle diameter of 0.46 mm, needlelength of 37 mm; Viasys Neurocare). The bandwidth of the EMGrecording system was 2 Hz–10 kHz. Recordings were stored digi-tally after sampling at a rate of 50 kHz and digitization at 16 bitsper sample.

With the subject sat and the leg stretched and positioned on aflat surface, the muscle was slightly contracted. No attempts weremade to make the contraction strictly isometrical and the forcewas not measured. The slightest possible effort was used to facili-tate maintenance of the contraction for a long period and to lessenthe effects of fatigue. During the contractions subjects had a feed-back on EMG, i.e. they were able to see in the electromyograph’sscreen the potentials that were being recorded. The SF electrodewas inserted in the muscle about half way between the tendonand the endplate region. SFAPs were recorded from different inser-tion points in each subject. For each insertion point, the position ofthe SF electrode was changed several times in order to detect se-quences of potentials. Specifically, when a SFAP was found, we ex-tracted up to 99 consecutive discharges of it. We will refer to eachcollection of such discharges as a SFAP set.

The criteria used to accept a recorded SFAP set for further studywere: (1) the SFAPs have a biphasic or triphasic morphology, (2)continuous monitoring of the discharges shows a consistent SFAPshape, (3) there is no other potentials overlapping the SFAP, and(4) the amount of noise is not so high as to make the measure-ments of SFAP rise-time unreliable.

In accordance with the purposes of the study, the electromyog-raphist intentionally moved the needle whilst recording each SFAPset so that each discharge was registered at a different radial dis-tance. Thus, each of our SFAP sets comprises a variable numberof consecutive discharges with different values of Vpp, as shownin Fig. 1(a). For each subject, we recorded about 35 different SFAPsets. After applying the abovementioned criteria, we had 63 sets ofSFAPs for subsequent analysis. The durations of such sets werefound to range from 2.7 to 6.9 s with mean and SD of 4.9 ± 1.2 s.

In line with other authors (Wallinga et al., 1985; Bronzino,1996), the rise-time was calculated as the time interval between10% and 90% of the transition between the positive and negative

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Definition of SFAP rise-time (RT)t

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es in SFAP Vpp reflect the changes in the position of the needle. This set comprises a

J. Rodríguez et al. / Journal of Electromyography and Kinesiology 20 (2010) 1249–1258 1251

peaks of a SFAP (see Fig. 1(b)). This definition of rise-time is sup-posed to be more robust against noise and sampling frequency er-rors than the classical inter-peak time interval. Nevertheless, aninevitable intrinsic error affects the calculation of the 10–90%rise-time: the rise-time can only be as accurate as the length ofone time step and so depends on the exact position of the startingsample corresponding to 0.1 Vpp. At a sampling frequency of50 kHz, one time step corresponds to 20 ls.

3. Methods

3.1. Dimitrov–Dimitrova (D–D) SFAP convolutional model

We performed simulations of SFAPs using the model proposedby Dimitrov and Dimitrova (1998). In this model the fibre is consid-ered as a time-shift invariant system and SFAPs can be expressed asa convolution of the input signal and impulse response of the corre-sponding system:

SFAPðtÞ ¼ Can �@IAPðtÞ@t

� IRDD ð1Þ

As can be seen, the coefficient of proportionality is Can = d2Kan -ri/16ran, where d is the fibre diameter (in lm), Kan the anisotropyratio, ri the intracellular conductivity, and ran the tissue conduc-tivity. The input signal is the first temporal derivative of the IAP,(IAP(t))/t. The impulse response (IRDD) is computed as the potentialproduced at a detection point by two current dipoles propagatingalong the fibre in opposite directions from the endplate towardthe fibre ends. In these conditions, the analytical expression ofthe D–D impulse response (IRDD) can be expressed as

IRDDðr;vtÞ ¼ v � ðz0 � vtÞ½ðz0 � vtÞ2 þ kan � r2�

32

ð2Þ

where z0 is the longitudinal distance of the electrode in respect tothe endplate (in mm), v the propagation velocity (in m/s), and rthe radial distance (in mm). We shall refer to z0, Kan, d, v and r asIR parameters and, except when we vary one of them in the courseof our study, their values will be set to z0 = 20 mm, Kan = 5(Rosenfalck, 1969; Griep et al., 1978; Andreassen and Rosenfalck,1981), d = 55 lm (Nandedkar and Stålberg, 1983; Nandedkar andSanders, 1988), and r = 100 lm. Note that in Eq. (2) only the sourcepropagating to the right is considered. Calculation of thepropagation velocity of a fibre with a certain diameter d wasperformed according to the equation (Nandedkar and Stålberg,1983; Nandedkar and Sanders, 1988)

vðm=sÞ ¼ 3:5þ 0:05 � ðd� 55Þ ð3Þ

We considered a 90 mm long fibre with a right semilength of40 mm. We used a reference velocity of 3.5 m/s and a time step(sampling interval) of 1 ls (1000 kHz). This means that the spacestep used for the simulations is 3.5 lm, which is suitable for the ra-dial distances of interest (20–300 lm).

Dimitrov and Dimitrova (1998) proposed an IAP approximationthat provides independent changes of different phases. The timecourse of a typical IAP with a negative afterpotential is divided intofour portions: the rising phase, the rapidly falling phase, the tran-sition phase and the slowly falling phase. The depolarization por-tion (rising phase) is characterized by three parameters, namelyA1, A2 and A3 Eq. (4).

RPðsÞ ¼ A1 � sA2 � e�A3s 0 6 s 6 Tdep ð4Þ

The duration of this phase Tdep can be calculated as the quotientA2/A3. The second, third and fourth IAP phases model the repolari-zation process occurring at the membrane of the cell after thedepolarization. The duration of this repolarization portion is con-

trolled by parameter Tspl (Dimitrov and Dimitrova, 1998;Rodriguez et al., 2006). We shall refer to A1, A2, A3 and Tspl as exci-tation parameters. The coefficient values used for normal musclefibres were A1 = 125�21242, A2 = 3.75, A3 = 15 and Tspl = 2. Theyprovide an IAP with Tdep = 250 ls, a value close to that measuredby Ludin (1973) in humans or by Wallinga (1985) in rats.

The main drawback of this description is that the IAP first deriv-ative is not sufficiently smooth. An alternative IAP approximationthat overcomes this problem has been proposed recently(Arabadzhiev et al., 2008). Nevertheless, since the target of thepresent study is to analyse the effect of each of the IAP phases onthe SFAP rise-time, the Dimitrov and Dimitrova IAP approximationis preferable here.

3.2. Analysis of the factors affecting SFAP rise-time

In Fig. 2 we show the dependence of SFAP rise-time (RT) onexcitation parameters (first row) and on IR parameters (secondrow).

The effect of A2 and A3 on rise-time is significant, being morepronounced in the case of A2 (Fig. 2(b) and (c), respectively). Incontrast, rise-time is insensitive to parameters A1 and Tspl

(Fig. 2(a) and (d), respectively). This means that rise-time is verysensitive to changes in the duration of the IAP depolarizationphase, whereas it is largely unaffected by the IAP repolarizationphase. This result is not surprising: at very short radial distancesSFAP main spike is essentially determined by the IAP depolariza-tion portion (Dimitrova and Dimitrov, 2006).

Parameter z0 does not affect the SFAP rise-time except when theelectrode is close to the neuromuscular or fibre–tendon junctions(Fig. 2(e)). In these cases, SFAPs tend to be biphasic, which makesrise-time change abruptly. Parameters Kan, and v have a slight ef-fect on rise-time (Fig. 2(f) and (g), respectively). However, the sen-sitivity of rise-time to changes in radial distance is considerable(Fig. 2(h)), especially when r is greater than 100 lm, which is ingood accordance to previous studies (Nandedkar and Stålberg,1983; Dimitrov and Dimitrova, 1989, 2006).

From the above analysis, we see that not all the parameters ofthe D–D model affect the rise-time in the same manner. In orderto simplify the study of rise-time, we can make some reasonableassumptions regarding the muscle fibres and recording conditions.First, since A1 and Tspl have no effect on rise-time we can removethem from our study. Second, if we assume that our SFAPs havebeen recorded far from the neuromuscular or fibre/tendon junc-tions and that Kan does not vary significantly within the same mus-cle, then the effect of z0, and Kan could also be neglected. Inaddition, changes in propagation velocity can be considered to beexclusively due to changes in fibre diameter, as described in Eq.(3). Finally, parameters A2, A3 will be substituted by their quotient,Tdep; in this way, another parameter is removed from the studyand, more importantly, the duration of the IAP rising phase canbe restricted to the range of 200–400 ls, as reported by severalauthors (Ludin, 1973; Wallinga et al., 1985). Under these assump-tions, SFAP rise-time can be regarded as depending on only threeparameters: Tdep, v and r.

3.3. Analysis of the factors affecting SFAP Vpp

In Fig. 3 we show the dependence of SFAP Vpp on excitationparameters (first row) and on IR parameters (second row).

Parameters that determine the IAP repolarization phase (Tspl)have no influence on Vpp, as shown in Fig. 3(d). In contrast, Vpp issensitive to changes in parameters that control the IAP depolariza-tion portion, i.e. A2 and A3, as can be seen in Fig. 3(b) and (c).Parameter A1 acts as a scale factor in Eq. (4) and, therefore, has alinear effect on Vpp (Fig. 3(a)).

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Vpp

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pp(m

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Fig. 3. Effect of excitation (first row) and IR (second row) parameters on SFAP Vpp. Default values have been used for excitation and IR parameters with the exception of theparameter under study in each case. Changes in the propagation velocity were introduced by varying the fibre diameter from 25 to 85 lm according to Eq. (3). The ranges ofvariation of excitation and IR parameters have been chosen to be the same as or similar to those used in previous works (Nandedkar and Stålberg, 1983; Rodriguez et al.,2007; Arabadzhiev et al., 2008).

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RT

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Fig. 2. Rise-time dependence on excitation (first row) and IR (second row) parameters. Default values have been used for excitation and IR parameters with the exception ofthe parameter under study in each case. Changes in the propagation velocity were introduced by varying the fibre diameter from 25 to 85 lm according to Eq. (3). The rangesof variation of excitation and IR parameters have been chosen to be the same as or similar to those used in previous works (Nandedkar and Stålberg, 1983; Rodriguez et al.,2007; Arabadzhiev et al., 2008).

1252 J. Rodríguez et al. / Journal of Electromyography and Kinesiology 20 (2010) 1249–1258

As in the case of rise-time, z0 does not affect Vpp except whenthe electrode is close to the neuromuscular or fibre-tendon junc-tions (Fig. 3(e)). Parameters Kan and v have some influence on Vpp

(Fig. 3(f) and (g), respectively). Finally, the radial decline of Vpp,shown in Fig. 3(h), demonstrates that the amplitude of SFAPs is

strongly dependent on the distance from the electrode, as foundexperimentally by Ekstedt and Stalberg (1973).

From the latter analysis we conclude that the different D–Dmodel parameters affect Vpp in different ways. However, by makingthe same assumptions as we made for the rise-time, analysis of the

J. Rodríguez et al. / Journal of Electromyography and Kinesiology 20 (2010) 1249–1258 1253

effects of the excitation and impulse response functions on Vpp

could be simplified as to consider only the influence of parametersTdep, v and r.

3.4. Simulation and experimental studies

3.4.1. Simulation studiesBy changing the values of Tdep and v, we investigated how the

length of the IAP rising phase affects the relationships RT-r(Fig. 4, first row), Vpp-r (Fig. 4, second row), and RT–Vpp (Fig. 4, thirdrow). In these simulations, Tdep was allowed to vary between 200and 400 ls and v ranged from 2 to 5 m/s.

3.4.2. Experimental studiesFor each consecutive discharge in a SFAP set, we measured rise-

time and Vpp. Then, we depicted the rise-time (RT) and Vpp valuesof these discharges in a RT–Vpp space. The resulting RT–Vpp dia-grams allowed us to study the variations of rise-time that followed

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v=2.5 m/s (d=35 m) v=3.5 m/s

r (µr (µm)

r (µr (µm)

Fig. 4. Diagrams showing simulation rise-time dependence on radial distance (first row)Vpp (third row) for different propagation velocities (v) and IAP depolarization durations (4.5 m/s, respectively. In each diagram solid lines, dashed lines, dotted lines, line with cro400 ls, respectively. Simulation parameters (except v and Tdep) were set to their defaul

the changes in Vpp within each set of SFAPs. Because RT–Vpp dia-grams are formed by a set of points, the comparison of variousSFAP sets represented in the same RT–Vpp space is not easy. Toovercome this problem we used polynomial regression to fit aregression line to each set of points within a RT–Vpp diagram.

We assessed how well relationships between rise-time and Vpp

found in SFAPs generated using the D–D model reflects these rela-tionships in recorded SFAPs. To do this, we used the simulated-fit-ting curves generated with different values of Tdep and v to fit themto the experimental RT–Vpp diagrams. The least square fittingmethod was used to determine the combination of Tdep and v thatprovided the best match. The coefficient of determination (R2) wascalculated for the Tdep-v curve of best fit in each RT–Vpp diagram;R2 expressed the proportion of the variation observed in a RT–Vpp

diagram that could be explained by the D–D model. We then con-ducted a statistical analysis of the R2 values obtained and, moreimportantly, identified those SFAP sets for which the fit wasinaccurate.

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— 200 us--- 250 us··· 300 us

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o o o

(d=55 m) v=4.5 m/s (d=75 m)

r (µm)m)

r (µm)m)

, the radial decline of Vpp (second row), and the relationship between rise-time andTdep). Diagrams in the first, second and third columns correspond to v of 2.5, 3.5 andsses, and line with opened circles correspond to Tdep values 200, 250, 300, 350 and

t values. In each diagram, we indicate in brackets the fibre diameter d.

1254 J. Rodríguez et al. / Journal of Electromyography and Kinesiology 20 (2010) 1249–1258

4. Results

4.1. Simulation results

4.1.1. Rise-time dependence on radial distance in the D–D modelIn each of the diagrams in the first row of Fig. 4, rise-time is

plotted as a function of radial distance for a fixed propagationvelocity and different durations of the IAP depolarization phase.Curves in each of these diagrams have some features in common.First, they all increase monotonically with radial distance, asFig. 2(h) suggested they should. Second, curves are shifted towardshigher values in the rise-time axis as Tdep increases, an observationthat can be predicted from Fig. 2(b) and (c). Third, no crossings areobserved between the curves for any value of Tdep or v.

As long as the propagation velocity is kept constant, the depen-dence of rise-time on radial distance is largely unaffected by thevalue of Tdep. This is evidenced by the fact that the slopes of thecurves in Fig. 4(a–c) are very similar. This means that the durationof the IAP depolarization phase has little effect on the sensitivity ofrise-time to changes in radial distance. The propagation velocity,however, has a considerable influence on this sensitivity. In fact,in Fig. 4(a–c) we can see that the slopes of the curves decrease withv.

4.1.2. Radial decline of Vpp in the D–D modelIn each of the diagrams in the second row of Fig. 4 the radial de-

cline of Vpp is plotted for a fixed v and different Tdep. Curves in eachof these diagrams have some features in common. First, they are allshifted towards lower values in the Vpp axis as Tdep increases. Thisis because an increase in Tdep is followed by an increase in the spa-tial length of the IAP depolarization phase, producing potentialswith lower amplitudes. Second, in each of the diagrams, curve dif-ferences attributable to Tdep are more pronounced at short radialdistances (less than approximately 100 lm). In fact, for greater ra-dial distances curves become very similar. This is a result of thedistance-dependent effect of the IAP profile length on Vpp: closeto the fibre, the spatial length of the IAP depolarization portion(proportional to Tdep) has a considerable effect on the SFAP ampli-tude, but, as the electrode is moved further away from the fibre,this effect becomes less important (Dimitrova and Dimitrov, 2006).

In contrast to the case with the rise-time, changes in Tdep have asignificant influence on the sensitivity of Vpp with radial distance.Specifically, the shorter Tdep, the steeper the radial decline of Vpp.The propagation velocity also has a considerable influence on thedecline of Vpp with radial distance. Although in Fig. 4(d–f) the com-parison between the curves with the same Tdep and different v isdifficult, it can be shown that the smaller the propagation velocity,

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R 2 = 43% R 2 = 85%

Accurate fit to a

Fig. 5. (a) and (b) Illustrative examples of two RT–Vpp diagrams corresponding to two setregression curves corresponding to each RT–Vpp diagram. Solid lines represent the simulavalues of the determination coefficients (R2) are shown inside the diagrams. (c) Histapproximates to a normal distribution, with a mean of 50% and a standard deviation of

the more abrupt is the radial decline of Vpp. From the foregoing, weconclude that the shorter the spatial length of the IAP depolariza-tion phase (determined by the product of Tdep and v), the steeperthe Vpp radial decline.

4.1.3. Relationship between rise-time and Vpp in the D–D modelDiagrams in the third row of Fig. 4 show how the rise-time and

the Vpp of SFAPs generated by the D–D model are related. The sep-aration between the different curves depends on the value of Tdep

and is roughly the same for all propagation velocities. Note thatthe scale of the Vpp axis is not the same in Fig. 4(g–i). The curvesof the latter figures were fitted to experimental RT–Vpp diagramsto assess the accuracy of the D–D model.

4.2. Analysis of the accuracy of the D–D SFAP model

For each of our 63 sets of SFAPs we calculated its correspondingRT–Vpp diagram. Two examples of such diagrams are shown inFig. 5(a) and (b). In these figures, the dashed line represents thepolynomial regression curve that describes the set of points withina RT–Vpp diagram, and the solid line corresponds to the simulated-fitting curve of highest R2 calculated by the D–D model. Two exam-ples of fittings with R2 of 43% and 85% are shown in the RT–Vpp dia-grams of Fig. 5(a) and (b), respectively.

From the analysis of all diagrams we noticed that when R2 wasgreater than about 50%, then the difference between the regressionand the simulated curves was not large and therefore the fittingmight be considered acceptable. For our 63 SFAP sets, we foundout that in only 46% of the cases was there a simulation curve offit with R2 greater than 50%.To obtain more insight into the degreeof accuracy with which SFAPs simulated by the D–D model fitexperimental RT–Vpp diagrams we calculated the histogram of R2,shown in Fig. 5(c). We will comment on two aspects of thishistogram:

– First, there were 10 sets of SFAPs (15% of the total) for which thebest R2 value was below 10%. This suggests that the approxima-tion offered to RT–Vpp diagrams by the D–D model is far fromaccurate for a small proportion of cases.

– Second, excluding the aforementioned 10 SFAP sets, the R2 val-ues have, approximately, a normal distribution with a mean of50% and a standard deviation of 15%. Thus, the majority of SFAPsets are associated with determination coefficients within therange 30–70%. This means that the fit was not precise for a largenumber of SFAP sets (see Fig. 5(a) and (b) for an example withR2 = 43%). In fact, only in 8% of all RT–Vpp diagrams was the bestR2 higher than 70% (see Fig. 5(c) for an example with R2 = 85%).

1 1.5 2 0 10 20 30 40 50 60 70 80 900

5

10

15

(b) (c)

Freq

uenc

y

R 2 ( % )p (mV)

Histogram of R2

10 20 30 40 50 60 70 80 900

RT- Vpp diagram

s of SFAPs recorded using needle movement. Dashed lines represent the polynomialted-fitting curves generated by the D–D model that best fit the RT–Vpp diagrams. Theogram of R2 values for 63 sets of SFAPs. Note that the profile of the histogram15%.

J. Rodríguez et al. / Journal of Electromyography and Kinesiology 20 (2010) 1249–1258 1255

The distributions of Tdep and v used to obtain the best fits areshown in Fig. 6(a) and (b), respectively. By observing the histogramcorresponding to Tdep, one realizes that it approximately follows anormal distribution with a mean and an SD of 297.2 ± 50.2 m/s.The histogram of propagation velocity can also be assumed to fol-low a normal distribution but skewed to the left (the mean value is3.65 m/s and the most frequent value is 3.9 m/s).

In order to identify the characteristics of those sets of SFAPs forwhich the D–D model could not provide a curve with an accuratefit, we plotted, in the same RT–Vpp space, the polynomial regres-sion lines corresponding to the SFAP sets (dashed lines), togetherwith the simulated-fitting curves obtained using different combi-nations of Tdep and v (solid lines). These graphs, one for SFAP setswith R2 less than 50% and another for sets with R2 greater than50%, are shown in Fig. 7.

As can be seen, regression lines for SFAP sets with R2 less than50% are totally or partially beyond the area limited by the simu-lated-fitting curves (hereafter referred to as acceptance area). Notethat the vast majority of these lines fall out the acceptance area be-cause of their high values of rise-time (only three regression lineslie below this area). In fact, the ten SFAP sets with R2 less than 10%had regression lines entirely above the acceptance area: none ofthese lines crossed into the acceptance area at all. In contrast,Fig. 7(b) shows that the polynomial regression lines for SFAP sets

200 225 250 275 300 325 350 375 4000

2

4

6

8

10

12

14

16

18

Nº

of r

epet

ition

s

(a)

Tdep (msec)

Histogram of Tdep

Fig. 6. Histogram of Tdep (a) and v (b) corresponding to the simu

50

70

90

110

130

150

170

190

210

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250

0 1 2 3 4 5

RT

(µs

)

(a)

Vpp (mV)

SFAP sets with R 2 < 50%

Fig. 7. (a) Comparison between the polynomial regression lines corresponding to the SFAusing the D–D model (solid lines). Note that most of these regression lines are beyond tlines representing the polynomial regression lines of the SFAP sets with R2 greater than 50curves.

with R2 greater than 50% (dashed lines) fall well within the accep-tance area.

In summary, the relationships between rise-time and Vpp ofSFAPs generated by the D–D model accurately reflect the relation-ships observed in about half of our sets of experimental recordings.

5. Discussion

5.1. Strengths and weaknesses of the D–D SFAP convolutional model

The Dimitrov and Dimitrova SFAP convolutional model iswidely used and well-known for its simplicity and computationalefficiency. Another big advantage of this model is that it affordsthe possibility of making independent changes in excitation andimpulse response functions, as shown in Eq. (1). This allows theseparation of the constants and parameters related to the volumeconductor (included in the IR) from those related to the source ofexcitation (i.e. the IAP). This flexibility enables study of the effectof the model’s parameters on the waveform of the potential, pro-viding insight into the relationships between the anatomical and/or physiological properties of the fibre and the shape of the poten-tial. In the present study we have analysed the effects of the D–Dmodel parameters on the SFAP rise-time and Vpp (see Sections3.2 and 3.3 for details).

2 3 4 50

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4

6

8

10

12

14

16

18

v (m/sec)

(b)

Histogram of v

lated curves of the D–D model that produced the best fits.

50

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190

210

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0 1 2 3 4 5

Vpp (mV)

(b)

RT

(µs

)

SFAP sets with R 2 > 50%

P sets with R2 less than 50% (dashed lines) and the simulated-fitting curves obtainedhe area limited by the simulated curves. (b) The same as in (a) but with the dashed%. In this case, the regression lines fall well within the area limited by the simulated

1256 J. Rodríguez et al. / Journal of Electromyography and Kinesiology 20 (2010) 1249–1258

A strong point of the D–D model is its ability to reflect the effectof the IAP spatial length on the Vpp radial decline. An example ofthis can be appreciated in Section 4.1, where the Vpp radial declinebecomes steeper as spatial length of the IAP depolarization phasegets shorter. This is essential to model the distance-dependent ef-fects of the IAP profile, IAP duration, propagation velocity, andafterpotentials on EMG signals, as shown in Dimitrova and Dimit-rov (2002, 2006).

The present work has evaluated the accuracy of the D–D modelat simulating SFAPs recorded close to the fibre by comparing sim-ulated signals with potentials recorded using a SF electrode. Specif-ically, our approach has been to compare how the relationshipbetween Vpp and rise-time changes with radial distance in simu-lated and experimental SFAPs. A similar strategy was followed byVan Veen et al. in 1993 to analyse the resemblance between mea-sured and computed SFAPs. Despite the fact that Van Veen’s groupused Nandedkdar and Stalberg SFAP convolutional model to simu-late SFAPs, their main conclusion coincide with ours: generatedSFAPs have slightly lower rise-time that experimental ones.

Before considering the possible reasons that may explain thisfinding, we should remember the two main sources of error arisingfrom the simplifications adopted in SFAP convolutional models.First, in these models the excitation source is assumed to be dis-tributed along the axis of the fibre. From the studies of Plonsey(1974) and Andreassen and Rosenfalck (1981), we know that errorgenerated from this assumption increases as the electrode gets clo-ser to the fibre and that it is less than 14%. At radial distancesgreater than approximately 150 lm, the error is known to be lessthan 5% (Wilson et al., 1933).

Second, SFAP convolutional approaches are based on the linearcore-conductor model, under which the influence of the surround-ing medium is neglected. Theoretically, however, the excitations inD–D or N–S models depend on both the intracellular and extracel-lular potential fields and, therefore, the effect of the extracellularpotential distribution, although small, should be considered whenmodeling excitation functions (Henriquez and Plonsey, 1988; VanVeen et al., 1993). As a consequence of this failure to accommodateextracellular potential fields, the calculation of sources, as the firsttemporal derivative of the IAP in the D–D model and as the secondin the N–S model, will be accompanied by a certain amount oferror.

5.2. Possible reasons to explain the narrow-spike SFAPs

The RT–Vpp diagram of Fig. 7(a) clearly demonstrates that sim-ulation curves generated by the D–D model do not completely cov-er the area occupied by our experimental SFAPs. Specifically, theacceptance area delimited by the D–D model curves only containsa fraction of all recorded SFAPs: those with lower rise-time values.This suggests that a certain proportion of individual muscle fibresgenerate potentials that are wider than predicted by SFAP convolu-tional models. Could this limitation in the D�D SFAP approach beattributed to an incorrect modeling of the excitation source or toinadequate values for the impulse response parameters? Whatother factors could account for these simulated SFAPs with undulynarrow spikes?

The fact that several different IAP approximations have beenused in recent studies (Nandedkar and Stålberg, 1983; Dimitrovand Dimitrova, 1989; Arabadzhiev et al., 2008) raises the questionof whether the low rise-times might not be due to an inappropriateduration of the IAP depolarization portion. However, in the presentwork the duration of the IAP rising phase varied within the range200–400 ls, which should be sufficient to model IAPs from bothnormal muscle fibres, with a Tdep of approximately 230 ls (Ludin,1973), and pathological muscle fibres, with a Tdep of about 350 ls

(Ludin, 1973). Thus, we conclude that the IAP description usedEq. (1) cannot be responsible for overly narrow SFAPs.

With regard to the impulse response, we have no means tocheck whether the Vpp radial declines shown in Fig. 4 actuallyresemble those obtained experimentally. However, the large rangeof fibre diameters considered in our simulations (from 25–85 lm,which is in good agreement with the diameter variability reportedby other authors (Stålberg, 1986; Blijham et al., 2006) enables gen-eration of a large variety of Vpp radial declines. Note that, in orderto enlarge the acceptance area to include higher rise-times, itwould be necessary to increase fibre diameter beyond 85 lm,which would give rise to very unrealistic radial declines of Vpp.Hence, the impulse response is not likely to be the explanationfor narrow-spiked SFAPs either.

As we mentioned in Section 5.1, the error derived from theassumption that the source of excitation is distributed along the fi-bre axis only becomes significant if the electrode is in the immedi-ate vicinity of the fibre. Therefore, inaccuracies introduced by suchan error should primarily affect SFAPs with very high Vpp. However,Fig. 7 shows that most large SFAPs (with Vpp greater than 3 mV) fallwell within the acceptance area. Consequently, there is no reasonto believe that concentration of the source in the fibre axis leadsto unduly narrow SFAPs.

Without having further evidence, the assumption that the exci-tation source can be modelled as the first temporal derivative ofthe IAP, as in Eq. (1), could be responsible for the low values ofrise-time. The inclusion of the effect of the extracellular potentialdistribution when calculating the excitation source could modifyslightly its waveform and duration, which in turn would increaseslightly the duration of the SFAP main spike. Although this inter-pretation is highly speculative and not based in the data of thepresent work, the experimental studies of Van Veen et al. (1993)point in the same direction. In fact, Van Veen’s group examinedand compared the degree of accuracy with which four differentexcitation sources (two of them calculated directly from the IAPand two calculated from the transmembrane current) approximatea RT–Vpp diagram obtained from recorded SFAPs. His resultsshowed that the determination coefficients corresponding to thetwo sources based on the transmembrane current, both of whichincluded the effect of the extracellular potential distribution, werenoticeably higher than those corresponding to the simulatedsources.

Other considerations that may explain to some extent whysome recorded SFAPs present higher rise-times than simulatedones are the electrode settings, the needle shaft (Theeuwen et al.,1993) and/or the angulation or rotation of the cannula. The useof a band-pass filter with a low frequency of 3 Hz allows low-fre-quency components from distant fibres to be picked up and re-corded. Such components would add to the SFAP detected fromthe closest fibre, generating a potential with a slightly smootherand possibly wider spike. The cannula of the electrode would alsorecord activity from distant fibres, producing the same smoothingeffect described above. However, as the muscle was contracted at avery low level and therefore a small number of motor units werelikely to be activated, the effect of the filter and the cannula onthe SFAP rise-time should not be relevant. Finally, the angulationor rotation of the cannula, which mainly affects the SFAP Vpp, mightchange the characteristics of some RT–Vpp diagrams and shouldalso be considered as a possible explanation for the observed mis-match in rise-times seen in our recorded and simulated data.

5.3. Considerations for future SFAP and MUP modeling

Of the proposed explanations for the low rise-time values ofsimulated SFAPs, we believe incorrect modelling of the bioelectri-cal source to be the most plausible. Further study is required to

J. Rodríguez et al. / Journal of Electromyography and Kinesiology 20 (2010) 1249–1258 1257

establish a correct model for the excitation function. Simultaneousrecording, from the same fibre, of the intracellular action potentialand the transmembrane current is essential to establish how accu-rately the latter can be calculated as the second derivative of theformer.

The fact that some muscle fibres generate wider SFAPs than ex-pected could explain why some motor unit potentials (MUPs) re-corded within the motor unit territory, although having a crispsound and a sharp appearance, present rise-time values higherthan 500 ls, as reported by Barkhaus and Nandedkar (1996),Dumitru et al. (1997), and Masakado et al. (2000). In this respect,further study of rise-times of both SFAPs and MUPs would beuseful.

As reported by some authors (Stålberg, 1986; Blijham et al.,2006), the diameters and propagation velocities of the different fi-bres within a muscle can differ considerably. Recent studies(Blijham et al., 2006) have found that diameters range from 30 to70 lm in the biceps brachii and from 50–95 lm in the quadriceps.Consequently, most recent MUP models in literature (Dimitrovaand Dimitrov, 2002; Arabadzhiev et al., 2008) include certain var-iability in the fibre diameter. Similarly, Van Veen et al. (1993)found that IAPs belonging to different fibres from the same muscleare not identical (Van Veen et al., 1993), but to date most MUPmodels describe the excitation source with an analytical functionof constant profile. The results of the present work suggest thatwe should admit some variability in the duration of the IAP depo-larization portion in order to be able to generate the range of rise-times observed experimentally. This proposal is in agreement withseveral other studies that have demonstrated that to explain thechanges of certain SFAP parameters it is necessary to considerthe excitation source as a variable function (Rodriguez et al.,2006, 2007, 2009, 2010).

5.4. Factors affecting the propagation velocity within each SFAP set

Several authors (Stålberg, 1966; Nishizono et al., 1989) haveshown that changes in the motor unit discharge rate have an effecton propagation velocity. Similarly, changes in the level of voluntarycontraction were found to alter propagation velocity (Nielsen et al.,1984). It would be interesting to determine whether the motor unitdischarge rate and/ or the contraction level may change during therecording of each SFAP set.

The durations of our SFAP sets were found to be 4.9 ± 1.2 s. Thepossibility that the motor unit discharge rate change significantlythroughout such short-duration sets is most unlikely. In addition,the variations in the level of contraction are expected to be smallwithin each of our SFAP sets, provided that subjects performedthe slightest possible effort to facilitate maintenance of the con-traction for a long period. Besides, Nishizono et al. (1989) andNielsen et al. (1984) demonstrated that a large variation in motorunit discharge rate and in contraction level was necessary to pro-duce a noticeably change in conduction velocity. Therefore, thereis no reason to believe that propagation velocity can change sig-nificantly within each SFAP set due to the abovementioned ef-fects; rather conduction velocity can be assumed to be roughlyconstant.

It must be mentioned that in our experiments, as muscles wereslightly contracted, only small motor units are expected to be re-cruited, and not a broad range of motor unit types. Therefore, themeasures of rise-times and Vpp reported here are likely to corre-spond mainly to muscle fibres of small diameters. It may be plau-sible that an even larger distribution of the experimental datashown in Fig. 7(a) could be expected if a broader range of musclefibre types was investigated, which would only strengthen ourmain conclusion.

6. Conclusions

1. The relationships between SFAP rise-time and Vpp as generatedby the D–D SFAP convolutional approach accurately reflect therelationships observed in about half of our sets of experimentalrecordings. Approximately half of our experimental SFAPs arewider than predicted by this model.

2. The assumptions adopted in the core-conductor model (includ-ing those of the line source model) may introduce inaccuraciesin simulated SFAPs regarding the relationship between rise-time and Vpp.

3. The D–D SFAP convolutional model reflects the effect of the IAPspatial length on the radial decline of SFAP amplitude.

Conflict of interest statement

We the undersigned declare that this manuscript is original, hasnot been published before and is not currently being considered forpublication elsewhere.

We wish to confirm that there are no known conflicts of interestassociated with this publication and there has been no significantfinancial support for this work that could have influenced itsoutcome.

We confirm that the manuscript has been read and approved byall named authors and that there are no other persons who satis-fied the criteria for authorship but are not listed. We further con-firm that the order of authors listed in the manuscript has beenapproved by all of us.

We confirm that we have given due consideration to the protec-tion of intellectual property associated with this work and thatthere are no impediments to publication, including the timing ofpublication, with respect to intellectual property. In so doing weconfirm that we have followed the regulations of our institutionsconcerning intellectual property.

Acknowledgement

This work was supported by the Spanish Ministry of Educationand Science under the project SAF2007-65383.

References

Albers BA, Put JHM, Wallinga W, Wirtz P. Quantitative analysis of single musclefibre action potentials recorded at known distances. Electroencephalogr ClinNeurophysiol 1989;73:245–53.

Andreassen S, Rosenfalck A. Relationship of intracellular and extracellular actionpotentials of skeletal muscle fibers. CRC Crit Rev Bioeng 1981;7:267–306.

Arabadzhiev T, Dimitrov GV, Chakarov V, Dimitrov A, Dimitrova NA. Effects ofchanges in intracellular action potential on potentials recorded by single-fiber,macro, and belly-tendon electrodes. Muscle Nerve 2008;37(6):700–12.

Barkhaus PE, Nandedkar SD. On the selection of concentric needle electromyogrammotor unit action potentials: is the rise time criterion too restrictive? MuscleNerve 1996;19(12):1554–60.

Blijham PJ, ter Laak HJ, Schelhaas HJ, van Engelen BG, Stegeman DF, Zwarts MJ.Relation between muscle fiber conduction velocity and fiber size inneuromuscular disorders. J Appl Physiol 2006;100(6):1837–41.

Bronzino JD. The Biomedical Engineering Handbook (Book Reviews). In IEEEengineering in medicine and biology magazine, vol. 15; 1996. p. 8 [chapter 14].

Dimitrov GV, Dimitrova NA. Extracellular potentials produced by a transitionbetween an inactive and active regions of an excitable fibre. Electromyogr ClinNeurophysiol 1989;29:265–71.

Dimitrov GV, Dimitrova NA. Precise and fast calculation of the motor unit potentialsdetected by a point and rectangular plate electrode. Med Eng Phys1998;20:374–81.

Dimitrova NA, Dimitrov GV. Amplitude-related characteristics of motor unit and M-wave potentials during fatigue. A simulation study using literature data onintracellular potential changes found in vitro. J Electromyogr Kinesiol2002;12:339–49.

Dimitrova NA, Dimitrov GV. Electromyography (EMG) modeling. In: Metin A, editor.Wiley encyclopedia of biomedical engineering. Hoboken, NJ: John Wiley andSons; 2006.

1258 J. Rodríguez et al. / Journal of Electromyograph

Dumitru D, King JC, Nandedkar SD. Comparison of single-fiber and macro electroderecordings: Relationship to motor unit action potential duration. Muscle Nerve1997;20:1381–8.

Ekstedt J, Stålberg E. How the size of the needle electrode leading-off surface influencesthe shape of the single muscle fibre action potential in electromyography. CompProg Biomed 1973;3:204–12.

Griep PAM, Boon KL, Stegeman DF. A study of the motor unit action potential bymeans of computer simulation. Biol Cybern 1978;30:221–30.

Henriquez CS, Plonsey R. The effect of the extracellular potential on propagation inexcitable tissue. Comments Theor Biol 1988;1:47–64.

Ludin H. Action potentials of normal and dystrophic human muscle fibers. In:Desmedt JE, editor. New development in electromyography and clinicalneurophysiology. Basel: Karger; 1973. p. 400–6.

Masakado Y, Akaboshi K, Kimura A, Chino N. Rise time or rise rate: which should bea criterion for analysing motor unit action potentials? Electromyogr ClinNeurophysiol 2000;40(6):381–3.

Nandedkar S, Sanders DB. Simulation of concentric needle EMG motor unit actionpotentials. Muscle Nerve 1988;11:151–9.

Nandedkar S, Stalberg E. Simulation of Macro EMG motor unit action potentials.EEG Clin Neurophysiol 1983;56:52–62.

Nandedkar S, Stålberg E. Simulation of single muscle fibre action potentials. MedBiol Eng Comput 1983;21:158–65.

Nielsen LA, Foster A, Mills KR. The relationship between muscle-fibre conductionvelocity and force in the human vastus lateralis. J Physiol 1984;353:6P.

Nishizono H, Kurata H, Miyashita M. Muscle fibre conduction velocity related tostimulation rate. Electroencephalogr Clin Neurophysiol 1989;72:529–34.

Plonsey R. The active fiber in a volume conductor. IEEE Trans Biomed Eng1974;21:371–81.

Plonsey R. Action potential sources and their volume conductor fields. Proc IEEE1977;65:601–11.

Rodriguez J, Malanda A, Gila A, Rodríguez I, Navallas J. Analysis of the peak-to-peakratio of extracellular potentials in the proximity of excitable fibres. JElectromyogr Kinesiol. Accepted 2009 August 24 [Epub ahead of print].

Rodriguez J, Malanda A, Gila A, Rodríguez I, Navallas J. Modelling fibrillationpotentials–a new analytical description for the muscle intracellular actionpotential. IEEE Trans Biomed Eng 2006;53(4):581–92.

Rodriguez J, Malanda A, Gila L, Rodríguez I, Navallas J. Modelling fibrillationpotentials–Analysis of time parameters in the muscle intracellular actionpotential. IEEE Trans Biomed Eng 2007;54(8):1361–70.

Rodriguez J, Malanda A, Gila A, Rodríguez I, Navallas J. Identification procedure in amodel of single fibre action potential – part I: estimation of fibre diameter andradial distance. J Electromyogr Kinesiol 2010;20:264–73.

Rosenfalck P. Intra- and extracellular fields of active nerve and muscle fibers. Aphysico-mathematical analysis of different models. Acta Physiol Scand1969(Suppl. 321):1–168.

Stålberg E. Propagation velocity in human muscle fibres in situ. Acta Physiol Scand1966;287:1–112.

Stålberg E. Single fibre EMG, Macro EMG, Scanning EMG. New ways of looking at themotor unit. Crc Crit Rev Clin Neurobiol 1986;2(2):125–67.

Stålberg E, Trontelj J. Single fibre electromyography. Old woking. UK: MirvallePress; 1979.

Theeuwen MM, Gootzen TH, Stegeman DF. Muscle Electric Activity I: a model studyof the effect of needle electrodes on single fibre action potentials. Ann BiomedEng 1993;21:377–89.

Van Veen BK, Wolters H, Wallinga W. The bioelectrical source in computing singlemuscle fiber action potentials. Biophys J 1993;64:1492–8.

Wallinga W, Gielen FLH, Wirtz P, de Jong P, Broenink J. The different intracellularaction potentials of fast and slow muscle fibres. Electromyogr Clin Neurophysiol1985;60:539–47.

Wilson F, MacLeod A, Barker P. The distribution of the action currents produced byheart muscle and other excitable tissues immersed in extensive conductingmedia. J Gen Physical 1933;16:423–56.

Javier Rodríguez Falces was born in Pamplona in 1979.He graduated in 2003, and obtained the PhD in 2007 inTelecommunication Engineering from the Public Uni-versity of Navarra, Pamplona, Spain. He worked as aConsultant Engineer (2004-2005) and as a SystemEngineer (2005-2006) in the private sector. He has alsoworked for the Higher Scientific Investigation Council ofSpain during one year (2006). In 2007 he becameAssistant Professor in the Electrical and ElectronicsEngineering Department of the Public University ofNavarra. During this period he has been teaching severalsubjects related to digital signal processing, imageprocessing and biomedical engineering. His research

focuses on signal processing applied to biomedical signals, modeling of biologicalsystems, electromyography and sensory-motor interaction studies.

Javier Navallas Irujo was born in Pamplona in 1976. Hegraduated in 2002, and he obtained the PhD in 2008 in

Telecommunication Engineering from the Public Uni-versity of Navarra, Pamplona, Spain. He has also workedas a software engineer. He is presently Assistant Pro-fessor of the Electrical and Electronics EngineeringDepartment of this University. His research interests aremodeling of biological systems and neurosciences.

y and Kinesiology 20 (2010) 1249–1258

Luis Gila Useros received his MD degree from theComplutense University, Madrid, Spain in 1983. In 1988he completed his specialization in Neurology at the‘‘Ramón y Cajal” Hospital, Madrid. Between 1989 and1998 he worked as a neurologist at the ‘‘San Millán”Hospital, Logroño, Spain. From 1998 to 2001 he carriedout his specialization training in Clinical Neurophysiologyat the ‘‘Virgen del Camino” Hospital, Pamplona, Spain,where at the present time he is a staff member at theDepartment of Clinical Neurophysiology. His researchinterests include quantitative electromyography and theautomatic analysis of electromyographic signals.

Ignacio Rodriguez Carreño was born in Madrid in1976. In 2000 he obtained his degree in Telecommuni-cation Engineering from the Public University of Nava-rre, Pamplona, Spain. In 2001 he worked as an engineerin the development of software and hardware forcommunications systems. In 2002 he joined the PublicUniversity of Navarre as Assistant Professor of theDepartment of Electrical and Electronics Engineeringand he started his doctoral courses. Since 2003 he hasbeen granted a research fellowship by the NavarraGovernment to complete his Ph.D. His area of researchis the development of software and algorithmic meth-ods for the processing of EMG signals.

Armando Malanda Trigueros was born in Madrid,Spain, in 1967. In 1992 he graduated in Telecommuni-cation Engineering at the Madrid Polytechnic Univer-sity. In 1999 he received his Ph.D. degree from theCarlos III University, Madrid. In 1992 he joined theSchool of Telecommunication and Industrial Engineer-ing of the Public University of Navarra. In 2003 hebecame Associate Professor in the Electrical and Elec-tronics Engineering Department of this University.During all this period he has been teaching severalsubjects related to digital signal processing, imageprocessing and biomedical engineering. His areas ofinterest comprise the analysis, modeling and simulation

of bioelectrical signals, particularly EEG and EMG.