Journal of Electromyography and Kinesiology 22 (2012) 88–97

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Journal of Electromyography and Kinesiology

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Effects of changes in the shape of the intracellular action potential on thepeak-to-peak ratio of single muscle fibre potentials

Javier Rodríguez a,⇑, Javier Navallas a, Luis Gila b, Ibán Latasa a, Armando Malanda a

a Public University of Navarra, Department of Electrical and Electronic 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 22 February 2011Received in revised form 22 June 2011Accepted 28 June 2011

Keywords:Intracellular action potentialPeak-to-peak ratioIAP spike durationIAP de-repolarization timeIAP asymmetry

1050-6411/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.jelekin.2011.06.005

Abbreviations: A1, point of steepest rise along thesteepest decay along the IAP repolarization phase; Cality; Cas, coefficient of asymmetry defined as Irep/Id

Dimitrov and Dimitrova IAP model; C2, second coeDimitrova IAP model; C3, third coefficient of the DimitCM1, centre of mass of the positive phase of the IAP fimass of the negative phase of the IAP first derivativetemporal derivative of the IAP; D–D, Dimitrov and Dimto-repolarization time; IAP, intracellular action potentas Tdep–TM1; Irep, Time interval defined as TM2–Tdep

anisotropy ratio; M1, point along the IAP rising phaspoint along the IAP rising phase corresponding to CMelectrode distance); RP, rising phase; ran, the tissueconductivity; SFAP, single fibre action potential; TA1, tthe point of steepest rise along the IAP rising phase; TB

to the point of steepest decay along the IAP repolarizthe depolarization (rising) phase of the intracellulcoordinate or time instant corresponding to CM1; TM2

corresponding to CM2; Tspl, parameter of the Dimcontrolling the character of the repolarization phase;peak-to-peak voltage; V1, voltage of the SFAP first posSFAP negative phase; V3, voltage of the SFAP second pdistance of the electrode.⇑ Corresponding author. Address: Universidad Púb

pus de Arrosadía s/n, 31006 Pamplona, Spain. Tel.: +3169720.

E-mail addresses: javier.rodriguez.falces@gmail.cora.es, malanda@unavarra.es (J. Rodríguez).

a b s t r a c t

In situ recording of the intracellular action potential (IAP) of human muscle fibres is not yet feasible, andconsequently, knowledge about certain IAP characteristics of these IAPs is still limited. The ratio betweenthe amplitudes of the second and first phases (the so-called peak-to-peak ratio, PPR) of a single fibreaction potential (SFAP) is known to be closely related to the IAP profile. The PPR of experimentallyrecorded SFAPs has been found to be largely independent of changes in the fibre-to-electrode (radial) dis-tance. The main goal of this paper is to analyze the effect of changes in different aspects of the IAP spikeon the relationship between PPR and radial distance. Based on this analysis, we hypothesize about thecharacteristics of IAPs obtained experimentally. It was found that the sensitivity of the SFAP PPR tochanges in radial distance is essentially governed by the duration of the IAP spike. Assuming that, formammals, the duration of the IAP rising phase lies within the range 0.2–0.4 ms, we tentatively suggestthat the duration of the IAP spike should be over approximately 0.75 ms, with the shape of the spikestrongly asymmetric. These IAP characteristics broadly coincide with those observed in mammal IAPs.

� 2011 Elsevier Ltd. All rights reserved.

ll rights reserved.

IAP rising phase; B1, point ofan, coefficient of proportion-

ep; C1, first coefficient of thefficient of the Dimitrov androv and Dimitrova IAP model;rst derivative; CM2, centre of

; d, fibre diameter; IAP/t, firstitrova; DRT, depolarization-

ial; Idep, Time interval defined; IR, impulse response; Kan,e corresponding to CM1; M2,2; r, radial distance (or fibre–conductivity; ri, intracellularime instant corresponding to1, time instant correspondingation phase; Tdep, duration ofar action potential; TM1, x-, x-coordinate or time instantitrov and Dimitrova modelv, propagation velocity; Vpp,

itive phase; V2, voltage of theositive phase; z0, longitudinal

lica de Navarra D.I.E.E., Cam-4 948 169094; fax: +34 948

m, javier.rodriguez@unavar-

1. Introduction

Characteristics of single fibre action potentials (SFAPs) areknown to depend essentially on the shape of the intracellular ac-tion potential (IAP) (Plonsey, 1974, 1977; Andreassen and Rosen-falck, 1981) as well as on the fibre-to-electrode (radial) distanceand volume conductor parameters (Lorente de Nó, 1947; Plonsey,1974; Miller-Larson, 1985). Mathematical formulations of SFAPsas a function of the IAP, as they are presented in convolutionalmodels, require a precise knowledge of the time course of theIAP. In fact, for intramuscular recordings, the IAP waveform andduration can potentially affect SFAP features more than the elec-trode position and volume conductor properties (Dimitrov andDimitrova, 1979, 1987, 1989; Dimitrova and Dimitrov, 2003).Trayanova and Dimitrov (1982), for example, showed that asym-metry of the IAP waveform affects considerably the relationshipbetween the amplitudes of the first positive phase (V1) and secondnegative phase (V2) of SFAPs [see Fig. 1(a)]. Later, Rodríguez-Falceset al. (2006) demonstrated that changes in the profile of the IAPdepolarization phase are sufficient to modify the peak-to-peak ra-tio (PPR, defined as the ratio between the amplitudes of the secondand first phases) of an SFAP [see Fig. 1(a)].

According to the volume conductor theory, the IAP profiledetermines how the characteristics of SFAPs change with electrodeposition. In a recent work, Rodríguez et al. (2011a) studied the

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(a)Definition of SFAP parameters

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Fig. 1. (a) Definition of SFAP parameters V1, V2, V3, Vpp, and PPR. (b) Set of consecutive SFAPs recorded with needle movement. The gradual changes in SFAP Vpp reflect thechanges in the position of the needle. (c) Representation of the PPR versus the Vpp for the SFAP set shown in (b). The solid line is the linear regression line. Note that the PPR iskept relatively unchanged.

J. Rodríguez et al. / Journal of Electromyography and Kinesiology 22 (2012) 88–97 89

changes in the PPRs of consecutive potentials recorded at varyingradial distances. As a main result, the PPR was found to be largelyindependent of positional changes of the electrode within the 0.3-mm radius within which SFAPs are usually recorded [see Fig. 1(b)and (c) for an example]. This result is in good accordance withother single-fibre studies incorporating intentional needle move-ment (Håkansson, 1957; Rosenfalck 1969; Stålberg and Trontelj,1979). The finding is also supported by simulation studies (Dimit-rov and Dimitrova, 1979,; Dimitrova and Dimitrova, 2006), whichestablish that, within the 0.3-mm radius, the field generated bythe IAP spatial profile can be assumed to have a dipole character.However, do all the IAP descriptions proposed in the literature giverise to PPRs that follow this experimentally observed behaviour?

Subsequent to the IAP analytical function introduced by Rosen-falck in 1969, a number of IAP mathematical expressions have beenproposed to improve the resemblance between measured andcomputed SFAPs (Nandedkar and Stålberg, 1983; Fleisher, 1984;McGill and Lateva, 2001; Arabadzhiev et al., 2008, 2009). In gen-eral, authors use the same IAP profile to model extracellular poten-tials irrespective of the fibre type, pathology or functional state ofthe muscle. Only recent studies (Dimitrova and Dimitrov, 2002;Arabadzhiev et al., 2008, 2009) have included different IAP profilesto account for changes in the membrane properties produced bymyopathy and dystrophy (Arabadzhiev et al., 2008) or to modeldifferent states of muscle fatigue (Dimitrova and Dimitrov, 2002;Arabadzhiev et al., 2009). In these latter studies, although the vari-

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M1

InfluenceofIAP phasesonSFAP phases

Spatiallength(mm)

SFAP

(a)

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+ −

+−−

− +−+−

+−+−

+−

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ts

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Fig. 2. (a) Superimposed representation of the spatial profiles of the IAP and its correspsequence of lumped dipoles. (b) Superimposed representation of an IAP and its first tempcentres of mass of the positive and negative phases, respectively, of the IAP/t, and M1 anderivative calculated as in (b). Parameters Tdep, Idep, and Irep are indicated.

ations introduced in IAP profiles are based on physiological data,the choice of the specific shape with which to model the sourceis arbitrary to a large extent. Moreover, since in situ recording ofthe IAP from human muscle fibres has not yet been achieved,knowledge about certain parameters of the IAP spike (such asduration and asymmetry) is still insufficient.

The ultimate objective of the present study is to assess the ef-fects of changes in the characteristics of the IAP spike on the sen-sitivity of the SFAP PPR to changes in radial distance. The paperbegins with the identification of parameters of the IAP spike thatcan potentially affect the value of the SFAP PPR. Afterwards, usingthe Dimitrov and Dimitrova (D–D) SFAP convolutional, we investi-gate systematically how changes in each of these parameters affectthe relationship between the PPR and radial distance. On the basisof this analysis, and considering the PPR behaviour observedexperimentally, we tentatively suggest some characteristics thatshould be found in IAPs recorded in mammals.

2. Methods

2.1. Determination of the IAP characteristics that could affect the valueof the SFAP PPR

In Fig. 2(a), we superimpose the spatial profiles of an SFAP andits corresponding IAP. As can be seen in this figure, the IAP spatial

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s)

M2

IAP

∂IAP/ ∂t

(b)

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∂IAP/ ∂t

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IrepIdep

TM1 TM2Tdep

onding SFAP (normalized for the sake of clarity). The IAP profile is represented as aoral derivative (normalized for the sake of clarity). Points CM1 and CM2 represent thed M2 are the corresponding points on the IAP profile. (c) IAP and its first temporal

90 J. Rodríguez et al. / Journal of Electromyography and Kinesiology 22 (2012) 88–97

profile is represented as a sequence of lumped dipoles distributedequidistantly along its profile. Details of such an IAP presentationcan be found in Rodríguez et al. (2011a). At short radial distances,the first two phases of the SFAP are essentially generated by the di-poles lying on the IAP depolarization phase (Dumitru, 1994). How-ever, Fig. 2(a) evidences that the SFAP negative phase could also beaffected by the negative fields produced by the dipoles lying on theIAP repolarization phase.

From the above it follows that, at any radial distance, the valueof V2 (and therefore that of PPR) will depend on the separation (dis-tance) between the dipoles lying on the de- and re-polarizationphases. A means to estimate this separation in the time domainis by defining a duration for the spike of the IAP that takes into ac-count its fundamental properties. Based on multipole analysis, wecalculated the centres of mass of the positive (CM1) and negative(CM2) phases of the first derivative of the IAP, as shown inFig. 2(b). As established in Rodríguez et al. (2011b), the temporalseparation between the x-coordinates of CM1 and CM2 (or, equiva-lently, the interval between the corresponding points along the IAPprofile, M1 and M2) will be taken as the estimate of the IAP spikeduration. This duration was defined by Rodríguez et al. (2011b)and is known as the depolarization-to-repolarization time (DRT).

In addition to the DRT, other parameters of the IAP spike couldaffect the value of PPR. In the proximity of the fibre, the spatial pro-file of the depolarization phase of the IAP essentially determinesthe first two phases of the SFAP and, therefore, the duration of thisIAP portion [Tdep, Fig. 2(c)] may also play a role on how the PPRchanges with the electrode position. The time intervals Idep and Irep

defined as Tdep�TM1 and TM2�Tdep, respectively [Fig. 2(c)] are alsorelevant characteristics of the IAP spike and can be used to assessits coefficient of asymmetry (Cas = Irep/Idep).

2.2. Calculation of SFAPs using the convolutional model of Dimitrovand Dimitrova

Simulations of SFAPs were performed using the model proposedby Dimitrov and Dimitrova (1988). In this model, the transfer func-tion between the electrical activity in the fibre and the extracellu-lar potential can be assumed to be a time-shift invariant linearsystem, and SFAPs are expressed as a convolution of the input sig-nal and impulse response of the corresponding system:

SFAPðtÞ ¼ Can �@IAPðtÞ@t

� IRðtÞ ð1Þ

(a)

Fig. 3. (a) Time course of a D-D IAP divided into four portions and its first temporal deri(B1) of the IAP spike are indicated. The de–repolarization time (DRT) and the intervals Id

were C1 = 16.1 � 107, C2 = 4.9, C3 = 14, and Tspl = 0.9 ms. (b) D–D IAP and its first derivativeshown together with their corresponding points on the IAP (M1 and M2, respectively). Nrespectively, of the IAP/t.

As can be seen, the coefficient of proportionality is Can = d2-

Kan�ri/16ran, where d is the fibre diameter (in lm), Kan the anisot-ropy ratio, ri the intracellular conductivity, and ran the tissueconductivity. The input signal (i.e., the excitation source) is the firsttemporal derivative of the IAP, IAP(t)/t. The impulse response (IR)represents the potential produced at a detection point by two cur-rent dipoles propagating along the fibre in opposite directions fromthe endplate toward the fibre ends. In these conditions, the analyt-ical expression of the D–D impulse response (IR) was calculated as:

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

32

ð2Þ

where z0 is the longitudinal distance of the electrode (in mm), v thepropagation velocity (in m/s), and r the radial distance (in mm). Weshall refer to z0, Kan, v and r as impulse response parameters and, ex-cept when we varied one of them in the course of our study, theirvalues were set to z0 = 20 mm, Kan = 5 (as suggested by Rosenfalck,1969; Griep et al., 1978; Andreassen and Rosenfalck, 1981),v = 3.5 m/s (as suggested by Nandedkar and Stålberg, 1983; Nan-dedkar and Sanders, 1988), and r = 0.1 mm.

2.2.1. Calculation of parameters DRT, Tdep, Idep, and Irep for the IAPmodel proposed by Dimitrov and Dimitrova

In our simulations we used the IAP approximation proposed byDimitrov and Dimitrova in 1998, which divides the IAP time courseinto four portions [Fig. 3(a)]. Such an approach was convenient forour study because it allows the de- and re-polarization portions ofthe IAP to change independently. The first phase, the rising phase(RP), models the depolarization process triggered by the openingof the sodium channels and is characterized by three coefficients:C1, C2 and C3 (3).

RPðtÞ ¼ C1 � tC2t � e�C3t 0 6 t 6 Tdep ð3Þ

The duration of the depolarization phase can be calculated asTdep = C2/C3. The point of steepest rise (A1) along this phase occursat TA1, and is approximately Tdep/2 [Fig. 3(a)].

The second, third and fourth portions of the IAP, taken together,model the repolarization process occurring at the membrane of thecell after the depolarization. In the D–D IAP, the characteristics ofthe re-polarization phase are essentially controlled by the param-eter Tspl, which approximately equals the duration of the IAP with-out the after-potential [Fig. 3(a)]. The point of steepest decay (B1)

(b)

vative (normalized for the sake of clarity). The points of steepest rise (A1) and decayep and Irep of the spike are also shown. Coefficient values used to synthesize the IAPcalculated as in (a). The centres of mass of the IAP first derivative (CM1 and CM2) areote that the points CM1 and CM2 line approximately at the first and second peaks,

J. Rodríguez et al. / Journal of Electromyography and Kinesiology 22 (2012) 88–97 91

of the repolarization portion occurs at TB1 [Fig. 3(a)], and can becalculated as

TB1 ¼ Tdep þðTspl � TdepÞ

2ð4Þ

From the above definition of TB1 (4), it follows that the durationsof the second and third phases are the same. Note that this is a par-ticular characteristic of the D–D IAP that is not necessarily found inevery real IAP.

As established in Rodríguez et al. (2011b), because of the specialarchitecture of the D–D IAP, the centres of mass of its first temporalderivative, IAP(t)/t (CM1 and CM2) and the peaks of this function areclosely related [Fig. 3(b)]. Specifically, CM1 calculated on the IAP(t)/t lines approximately at the first peak of this function and thereforethe locations of points A1 and M1 are rather close [Fig. 3(b)]. Simi-larly, the x-coordinate of CM2 nearly coincides with the secondpeak of this function and, as a result, M2 � B1. Thus, for the partic-ular case of the D–D IAP, the DRT of the spike can be approximatelycalculated as the time separation between points of steepest riseand decay of the spike. From the definitions of TA1 from TB1 (4),one realizes that the time interval between A1 and B1 is Tspl/2. Thismeans that, for the D–D IAP description, the DRT of the spike canbe easily modified by changing the value of Tspl. Additionally, theintervals Idep and Irep can be obtained as Tdep�TA1 and TB1�Tdep,respectively. The coefficient of asymmetry for the spike of the D–D IAP is calculated as the ratio Irep over Idep.

2.3. Simulation studies

Three sets of simulations were carried out in order to analysehow changes in the above-defined parameters of the IAP spike af-fect the sensitivity of PPR to changes in radial distance. Usually, a

Table 1Coefficients for modelling the IAPs used in the first set of simulations (the IAPs are repres

First series of IAPs

Line with circles Line with crosses Dotted line Dashed line Solid line

C1 7.5 � 107 10.0 � 107 12.6 � 107 16.1 � 107 19.3 � 107

C2 2.8 3.5 4.2 4.9 5.6C3 14 14 14 14 14Tspl 0.97 1.00 1.02 1.04 1.06

Table 2Coefficients for modelling the IAPs used in the second set of simulations (these IAPs are p

First series of IAPs

Line with circles Line with crosses Dotted line Dashed line Solid line

C1 12.6 � 107 12.6 � 107 12.6 � 107 12.6 � 107 12.6 � 107

C2 4.2 4.2 4.2 4.2 4.2C3 14 14 14 14 14Tspl 1.0 1.05 1.09 1.13 1.17

Table 3Coefficients for modelling the IAPs used in the third set of simulations (these IAPs are rep

First series of IAPs

Line with circles Line with crosses Dotted line Dashed line Solid line

C1 7.5 � 107 7.5 � 107 7.5 � 107 7.5 � 107 7.5 � 107

C2 2.8 2.8 2.8 2.8 2.8C3 14 14 14 14 14Tspl 1.0 1.3 1.6 1.9 2.2

change in one of the IAP parameters investigated implies a changein the others, and so it is difficult to assess their effect on the PPR-rrelationship in isolation of each other. This complicates the inter-pretation of the changes in the PPR sensitivity.

The first set of simulations comprised two tests. In the first wevaried the DRT parameter by introducing controlled changes in Irep

(while keeping the depolarization phase constant) and analysedthe variation observed in the PPR sensitivity. In the second test,the DRT parameter was varied through changes in Idep (while keep-ing the repolarization phase unchanged). The coefficients of theIAPs used in these two tests are given in Table 1.

In the second set of simulations, we investigated whether a col-lection of IAPs with the same spike width, but different risingphase and asymmetry give rise to SFAP PPRs with different sensi-tivities to changes in radial distance. To do this, we simulatedtwo series of IAPs, each of them comprising various IAPs with thesame values of DRT but different values of Idep (Tdep) and Irep, andcompared the variations of the PPR sensitivity. Specifically, eachseries consisted of 5 IAPs with Tdep ranging from 0.20 to 0.40 msin steps of 0.05 ms. DRT values were 0.5 and 1.0 ms for the firstand second series, respectively (Fig. 5). The coefficients for theseIAPs can be found in Table 2.

The third group of simulations studied the effect of a progres-sive widening of the IAP spike with a view to determining whetherit is possible to generate SFAP PPRs that can be considered largelyindependent of positional changes of the electrode. The DRT valuewas increased from 0.45 to 1.05 ms in steps of 0.15 ms (Fig. 6). Thecorresponding coefficients of the IAPs are presented in Table 3.

All simulations included variation of radial distance between0.3 and 0.3 mm (the range within which SFAPs are usually re-corded in reality). All other impulse response parameters (z0, Kan,and v) were set to the default values stated in Section 2.1.

ented in Fig. 4).

Second series of IAPs

Line with circles Line with crosses Dotted line Dashed line Solid line

7.5 � 107 10.0 � 107 12.6 � 107 16.1 � 107 19.3 � 107

2.8 3.5 4.2 4.9 5.614 14 14 14 141.97 2.00 2.02 2.04 2.06

lotted in Fig. 5).

Second series of IAPs

Line with circles Line with crosses Dotted line Dashed line Solid line

7.5 � 107 10.0 � 107 12.6 � 107 16.1 � 107 19.3 � 107

2.8 3.5 4.2 4.9 5.614 14 14 14 141.0 1.06 1.10 1.16 1.20

resented in Fig. 6).

Second series of IAPs

Line with circles Line with crosses Dotted line Dashed line Solid line

19.3 � 107 19.3 � 107 19.3 � 107 19.3 � 107 19.3 � 107

5.6 5.6 5.6 5.6 5.614 14 14 14 141.0 1.3 1.6 1.9 2.2

Table 4Values of the IAP spike parameters used in the simulations of Fig. 4.

Changes in Irep Changes in Idep

Tdep Idep Irep DRT Cas Tdep Idep Irep DRT Cas

sss 0.30 0.14 0.36 0.50 2.57 0.20 0.12 0.40 0.52 3.33��� 0.30 0.14 0.38 0.52 2.71 0.25 0.13 0.40 0.53 3.07ddd 0.30 0.14 0.40 0.54 2.85 0.30 0.14 0.40 0.54 2.85--- 0.30 0.14 0.42 0.56 3.00 0.35 0.15 0.40 0.55 2.66–– 0.30 0.14 0.44 0.58 3.14 0.40 0.16 0.40 0.56 2.50

92 J. Rodríguez et al. / Journal of Electromyography and Kinesiology 22 (2012) 88–97

3. Results

3.1. Effect of the IAP parameters on the PPR-r relationship

In Fig. 4(a) we show a set of five IAPs with the same rising phase(note that Tdep and Idep have identical values in Table 4) but differ-ent profiles for the repolarization phase. As indicated by the data inthis table, when fixing the rising phase, an increase in Irep bringsabout an increase in the asymmetry of the IAP spike (Cas goesup) and a slowing of the falling portion of the IAP. The IAP profilesof Fig. 4(a) give rise to the PPR-r curves shown in Fig. 4(b). As canbe seen, the sensitivity of PPR to changes in radial distance de-creases as DRT (Irep) increases (see Table 4).

In the five IAP profiles of Fig. 4(c), the repolarization phase isapproximately the same (the time interval Irep is fixed), while thetime parameters of the depolarization portion (Idep and Tdep)change. With a largely conserved profile for the IAP falling phase,an increase in Tdep results in a decrease of asymmetry of the IAPspike (Cas drops) and comparatively steeper (and stronger) repolar-ization phases. The PPR-r curves of Fig. 4(d) demonstrate that,again, the sensitivity of PPR to changes in radial distance decreasesas DRT (Idep) increases. Note that this sensitivity decreases eventhough the spikes become less asymmetric (Cas drops).

In order to assess the effect of changes in the IAP Tdep and asym-metry on the PPR-r relationship in isolation from the influence ofthe IAP spike duration, we simulated two series of IAPs, each ofthem consisting of five IAP spikes with the same IAP DRT, as thoseshown in Fig. 5(a and c). To keep the DRT unchanged within eachseries, an increase in Tdep (specifically in Idep) is accompanied bya shortening of the IAP repolarization phase (Irep decreases asshown in Table 5). Such an opposite variation in Idep and Irep bringsabout a reduction in the asymmetry of the IAP spike (see Table 5).

The PPR-r curves contained in the diagrams of Fig. 5(b and d)are largely parallel, showing that, if the DRT of an IAP is kept con-stant, changes in the duration of the IAP depolarization phase (or inthe asymmetry of the IAP spike) do not necessarily imply notice-able alterations in the sensitivity of PPR to changes in radial dis-tance. Note also that PPR-r curves are shifted towards highervalues in the PPR axis as Tdep increases and Irep decreases, i.e., as

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Time (ms)

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ge (

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ge (

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Fig. 4. Simulation of the effects of changes in the IAP repolarization phase, Irep (a), andcorresponding values of parameters Tdep, Idep, Irep, DRT, and Cas, are given in Table 4.

the repolarization phase becomes shorter as compared to thedepolarization phase.

3.2. Properties of the IAP necessary to obtain PPRs largely unaffectedby radial distance

The series of IAPs in Fig. 6(a and d) show a progressive length-ening (in time) of IAPs with a Tdep of 0.2 and 0.4 ms, respectively. Inboth series, the widening of the spike results in a rise of the DRTfrom 0.45 to 1.05 ms. Also, in both series, the lengthening occursat the expense of an slowing of the repolarization phase: the risingphase remains unchanged (parameters Tdep and Idep have identicalvalues). As can be seen in Table 6, if Tdep is kept untouched, an in-crease in Irep is invariably followed by an increase in the asymme-try of the spike.

The IAP series of Fig. 6(a and c) give rise to the PPR-r diagramsshown in Fig. 6(b and d), respectively. In both diagrams we can seethat the sensitivity of PPR to changes in radial distance decreasesas DRT increases. The PPR-r curves of these diagrams present othersimilarities: their slopes decrease abruptly when the DRT is in-creased from 0.45 to 0.75 ms, but a further increase in DRT (from0.75 to 1.05 ms) does not result in a comparable big change ofslope. In fact, for DRT values higher than 0.75 ms, IAPs generateSFAPs whose PPRs are relatively little-influenced by changes in ra-dial distance, this being more true for IAPs with longer depolariza-tion phases. Thus, irrespective of the duration of the IAP risingphase, the width of the spike has a considerable effect on howmuch the PPRs of resulting SFAPs vary with electrode position.

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Table 5Values of the IAP spike parameters used in the simulations of Fig. 5.

Changes in Tdep and Cas with DRT = 0.5 ms Changes in Tdep and Cas with DRT = 1.0 ms

Tdep Idep Irep DRT Cas Tdep Idep Irep DRT Cas

sss 0.20 0.12 0.38 0.50 3.16 0.20 0.12 0.88 1.00 7.33��� 0.25 0.13 0.37 0.50 2.84 0.25 0.13 0.87 1.00 6.69ddd 0.30 0.14 0.36 0.50 2.57 0.30 0.14 0.86 1.00 6.14--- 0.35 0.15 0.35 0.50 2.33 0.35 0.15 0.85 1.00 5.66–– 0.40 0.16 0.34 0.50 2.12 0.40 0.16 0.84 1.00 5.25

0 0.1 0.2 0.30.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.50

20

40

60

80

100

0 0.5 1 1.50

20

40

60

80

100

0 0.1 0.2 0.30.8

1

1.2

1.4

1.6

1.8

2

Time (ms)

Volta

ge (

mV)

r (mm)

PPR

PPR

Volta

ge (

mV)

(a) (b)

(c) (d)

Fig. 5. Simulation of two collections of IAPs (a and c), each of them comprising five IAPs with the same spike duration (DRT) but different values of Idep and Irep, and theircorresponding PPR-r relationships (b and d). For each series, the values of parameters Tdep, Idep, Irep, DRT, and Cas, are given in Table 5.

0 0.5 1 1.5 20

20

40

60

80

100

0 0.5 1 1.5 20

20

40

60

80

100

0 0.1 0.2 0.3

1

1.5

2

0 0.1 0.2 0.3

1

1.5

2

Time (ms)

(a)

Volta

ge (m

V)

r (mm)

PPR

(b)

PPR

(c) (d)

Volta

ge (m

V)

Fig. 6. Simulation of two collections of IAPs (a and c), each of them comprising five IAPs with the same Tdep but different values of DRT, and their corresponding PPR-rrelationships (b and d). The values of IAP parameters Tdep, Idep, Irep, DRT, and Cas, are given in Table 6. For the two collections of IAPs, if the DRT is made longer than 0.75 ms, thePPRs of the corresponding SFAPs are little affected by changes in radial distance.

J. Rodríguez et al. / Journal of Electromyography and Kinesiology 22 (2012) 88–97 93

Table 6Values of the IAP spike parameters used in the simulations of Fig. 6.

Changes in DRT with Tdep = 0.20 ms Changes in DRT with Tdep = 0.40 msTdep Idep Irep DRT Cas Tdep Idep Irep DRT Cas

sss 0.20 0.10 0.35 0.45 3.16 0.40 0.16 0.29 0.45 1.81��� 0.20 0.10 0.50 0.60 2.84 0.40 0.16 0.44 0.60 2.75ddd 0.20 0.10 0.65 0.75 2.57 0.40 0.16 0.59 0.75 3.68--- 0.20 0.10 0.80 0.90 2.33 0.40 0.16 0.74 0.90 4.65–– 0.20 0.10 0.95 1.05 2.12 0.40 0.16 0.89 1.05 5.56

94 J. Rodríguez et al. / Journal of Electromyography and Kinesiology 22 (2012) 88–97

4. Discussion

4.1. Factors affecting the sensitivity of the SFAP PPR to changes inradial distance

Although it is well established that the PPR of SFAPs is largelyindependent of positional changes of the needle within the 0.3-mm radius within which SFAPs are usually detected (Håkansson,1957; Rosenfalck 1969; Stålberg and Trontelj, 1979; Rodríguezet al., 2011a), the present work is the first attempt to search forthe IAP characteristics requisite to generate SFAPs consistent withthis observed independence. In intramuscular recordings, theamplitude of the phases of an SFAP and therefore its PPR are pri-marily determined by the properties of the spike of its correspond-ing IAP (Dimitrova and Dimitrov, 2002). Accordingly, we haveproposed a number of parameters (DRT, Tdep, Idep, Irep, and Cas) tocharacterize different aspects of the IAP spike and evaluated theirimpact on the sensitivity of PPR to changes in radial distance.

In Section 3.1 we investigated, the effects on the PPR-r relation-ship of changes in Idep and Irep, in isolation of each other. We foundthat an increase in any of them (with the concomitant increase inthe DRT) brings about a decrease in the sensitivity of PPR tochanges in radial distance [Fig. 4(b and d)]. More interesting isthe analysis of simultaneous changes in the asymmetry of theIAP spike and in its de- repolarization time, also carried out in Sec-tion 3.1. We observed that if the increase in Cas is accompanied by awidening of the spike [the DRT gets higher, as in Fig. 4(a)], thenthere is a reduction in the PPR sensitivity [Fig. 4(b)]. However, ifthe increase in Cas is accompanied by shortening of the spike [theDRT gets lower, as in Fig. 4(c)], then the PPR sensitivity is seen toincrease [Fig. 4(d)]. Thus, variations in the PPR sensitivity areapparently governed by changes in the width of the IAP spike. Thisresult is supported by the fact that alterations in the asymmetry ofthe spike that leave its DRT unchanged [as in Fig. 5(a and c)] do notaffect substantially the PPR dependence on radial distance [Fig. 5(band d)].

The impact of simultaneous changes in parameters Tdep and DRTon the PPR behaviour was also addressed in Section 3.1. As in thecase of Cas, we observed that if the DRT is kept constant [Fig. 5(aand c)], changes in Tdep do not translate into changes in the PPR-rrelationship [Fig. 5(b and d)]. Thus, the duration of the IAP risingphase can hardly be responsible for the small changes of PPR foundin SFAPs recorded at different radial distances. From the above dis-cussion, we conclude that the DRT is the only parameter thatshows a consistent effect on the dependence of PPR on radial dis-tance: an increase in the DRT is invariably followed by a decreasein the sensitivity of PPR to changes in electrode position. The nextsection provides a biophysical explanation for the role of the IAPspike width on the variation of PPR with radial distance.

4.2. Biophysical explanation for the influence of the IAP spike width onthe PPR-r relationship

Since the formation of an electrical field around a fibre is aninherently spatial matter (Dimitrova, 1973; Dimitrova and Dimit-

rov, 2006), the analysis of how the amplitude characteristics ofSFAP (and therefore the PPR) vary with radial distance should becarried out considering the IAP in the spatial domain. At short ra-dial distances, the SFAP PPR is not determined only by the IAP ris-ing portion but also by the degree to which the field produced bythe IAP repolarization phase influences that generated by the depo-larization one. The degree of interaction between the fields pro-duced by the two IAP phases depends on the spatial distancebetween them as well as on the fibre-to-electrode distance. Letus see how this interaction changes with radial distance dependingon the width of the IAP spike.

Following the studies of Dimitrova and Dimitrov (2002), weconsider a simplified model of the IAP spatial profile, P(x), in whichthe phases of de- and re-polarization are schematically repre-sented by two oppositely-directed lumped dipoles [Fig. 7(a andb)]. These are the dipoles determined by the centres of gravity ofthe first and second phases of the first derivative of the IAP that,for the particular case of the D–D IAP, coincide approximately withthe points of steepest rise (Ai) and decay (Bi) of the spike, respec-tively. The spatial distance between Ai and Bi, hereafter referredto as the de-repolarization distance (DRD), can be calculated asthe product of DRT and v. The strength and orientation of each ofthe dipoles are determined by the spatial derivative of the poten-tial profile along the fibre [dP(x)/dx].

To better conceive how the electrical field develops, inFig. 7(c and d) each dipole is substituted by a solid and a dashedline representing the positions of the positive and negativemaxima, respectively, of the field produced by this dipole atdifferent distances from the dipole axis (see Dimitrova andDimitrov, 2006). Under these maxima lines, we show the poten-tials produced by dipoles A1, B1, A2 and B2 in the space domainat different radial distances: r1 [Fig. 7(e and f)], r2 [Fig. 7(g andh)], and r3 [Fig. 7(i and j)]. Note that the potentials correspond-ing to dipoles Bi are smaller than those corresponding to dipolesAi, as the repolarization phase of the IAP is much longer than thedepolarization one.

4.2.1. Variation of PPR with radial distance for single-fibre potentialsFor simplicity, let us consider only the diagrams corresponding

to the short IAP, PS(x) (the left panel of Fig. 7). In Fig. 7(c) radial dis-tances are represented in the y-axis. In this figure it can be seenthat interactions between the negative and positive maxima ofthe fields of dipoles A1 and B1, (and therefore the values of V1, V2

and PPR of the corresponding potential) depend on the radial dis-tance considered:

- For small values of r, such as r1, interaction between the nega-tive maxima of the fields of A1 and B1 is small. This can beappreciated in Fig. 7(e), where the potentials produced by A1

and B1 overlap only slightly. Since, at this radial distance, thestrength of A1 is much stronger than that of B1, the amplitudesV1 and V2 of the summated potential would be practically thesame as those of the potential produced by A1.

- For the IAP of Fig. 7(a), an optimal radial distance exists (r2), atwhich the negative maxima of A1 and B1 coincide [point p2 in

+ – +–

0 2 4 60

50

100

0 2 4 60

50

100

Rad

ial d

ista

nce

(a.u

.)

r2

r3

DRD = 2 mm DRD = 4 mm

(a)

(c)

+ –

p2

p3

Volta

ge(m

V)

Volta

ge(m

V)

+–A1 B1 A2 B2

Spatial length (mm) Spatial length (mm)

(g)

– +–+– +

–+

(i)Interaction at r3 Interaction at r3

(e)Interaction at r1 Interaction at r1

Interaction at r2 Interaction at r2

r1

A1B1 A2

B2(b)

(d)

(h)

(j)

(f)

x

x

x

x

x

x

x

Short IAP, Ps ,PAIgnoL)x( PL(x)

Fig. 7. (a and b) Spatial profiles of two IAPs, PS(x) and PL(x), simulated by the Dimitrov and Dimitrova model. The IAP phases of de- and repolarization are schematicallyrepresented by two dipoles, Ai and Bi, located at the centres of gravity of the first and second phases of the IAP first derivative, respectively. In each IAP, the spatial distancebetween dipoles Ai and Bi is referred to as the de- and repolarization distance (DRD). (c and d) Presentation of the interactions between the dipole fields produced by thedipoles Ai and Bi at different radial distances ri (dashed–dotted lines) of the electrode. Solid and dashed lines represent the positions of the positive and negative maxima ofthe individual dipole fields, respectively. Interactions between the potentials generated by dipoles A1 and B1 at r1, r2 and r3, are shown in (e, g and i), respectively. Interactionsbetween the potentials generated by dipoles A2 and B2 at r1, r2 and r3, are shown in (f, h and j), respectively. Potentials in different subplots (e and j) are not to scale.

J. Rodríguez et al. / Journal of Electromyography and Kinesiology 22 (2012) 88–97 95

Fig. 7(c)]. At such a radial distance, the positions of the negativepeaks of the potentials produced by A1 and B1 are the same,whereas the negative phase of the B1 potential hardly affectsthe positive phase of the A1 potential [Fig. 7(g)]. As a result, inthe summated potential, a large increase in V2 relative to V1

occurs, which in turn makes PPR reach a value close to its max-imum. This radial distance can be calculated approximately asDRD/

p2 mm (Dimitrova, 1973). This means that r2 would be

close to 2/p

2 � 1.41 mm (Rodríguez et al., 2011a).- For radial distances greater than r2, the negative peaks of the A1

and B1 potentials do not coincide and the PPR of the summatedpotential cannot be expected to be as high as in the case of r2

[see Fig. 7(i)].

From the above it can be concluded that, up to a radial distanceof approximately r2 � 1.41 mm, the radial decline of the SFAP V1

will be faster (i.e., with a steeper slope) than that correspondingto the SFAP V2, and consequently the PPR of an SFAP is expected

to increase monotonically with radial distance up to nearly1.41 mm (note that this distance clearly exceeds the 0.3-mm ra-dius within which SFAPs are usually registered).

4.2.2. The variation of PPR with radial distance depends on the widthof the IAP spike

The weight (influence) of the field produced by the fallingphase of the IAP on that produced by the rising phase does notdepend only on radial distance, but also on the spatial extensionof the IAP profile, i.e., on the de- and re-polarization distance. Toillustrate this, Let us imagine two IAPs with different spatiallengths, one short, PS(x) (DRD = 2 mm), and one long, PL(x)(DRD = 4 mm), such as those depicted in Fig. 7(a and b), respec-tively. For the short IAP, the summation of the negative maximaof the fields of A1 and B1 is optimal at r2 � 2/

p2 mm [Fig. 7(c),

point p2]. For the long IAP, however, we would have to movethe needle further way from the fibre (to the position r2 � 4/p

2 mm) in order to obtain the optimal summation of the

96 J. Rodríguez et al. / Journal of Electromyography and Kinesiology 22 (2012) 88–97

negative maxima of A2 and B2 [Fig. 7(d), point p3]. It follows fromthis that the increase of the SFAP V2 consequent upon a certainincrease in radial distance is expected to be more pronouncedin the case of the shorter IAP. This explains why the sensitivityof PPR to changes in electrode position increases as the spikewidth of the IAP shortens [as shown in Fig. 6(b and d)].

4.3. Applicability and limitations of the results

4.3.1. Using the PPR dependence on radial distance to obtaininformation about the IAP profile

From the above presentation it is clear that the contribution ofthe field produced by the IAP falling phase to that generated by therising phase increases with radial distance and that such an in-crease can be slowed down by making the repolarization portionlonger. This explains why, if the DRT of the IAP spike is made suf-ficiently long (over approximately 0.75 ms), the SFAP PPR becomeslittle influenced by positional changes of the electrode [Fig. 6(b andd)], as found recently in needle movements recorded from the tib-ialis anterior (Rodríguez et al., 2011a). Such a behaviour has beenshown to be true for IAPs with Tdep values ranging from 0.2 to0.4 ms, in good concordance with the data from real intracellularpotentials (Albuquerque and Thesleff, 1968; Ludin, 1973; Hanson,1974; Akaike, 1978; Wallinga et al., 1985) recorded in mammals.With the DRT over 0.75 ms and the Tdep under 0.4 ms, the resultingIAPs of mammals would exhibit spikes with a pronounced asym-metry [Cas greater than 3.68, see data in the table on the left ofFig. 6(c)].

A noteworthy advantage of the IAP description proposed byDimitrov and Dimitrova is that it allows for widening of thespike whilst leaving the depolarization portion unchanged. Thisflexibility enables synthesis of IAPs with a wide range of DRTand Tdep values. This would be useful to model IAPs whosecharacteristics slightly differ from those of mammal IAPs. Forexample, the IAPs recorded in frogs and toads are known tohave depolarization times lying within the range 0.5–1.0 ms(Ishiko and Sato, 1957; Hanson and Persson, 1971; Tayloret al., 1969; Lännergren and Westerblad, 1987). The resultsand biophysical explanation of the present study allow topredict that, even for IAPs with Tdep values within the range0.5–1.0 ms, the SFAP PPR would be relatively unaffected bychanges in radial distance as long as the IAP DRT is sufficientlyhigh (longer than 0.75 ms).

Are the values of IAP DRT and asymmetry necessary to generatePPRs largely unaffected by radial distance found in real (measured)IAPs? In the case of our recent study, potentials were recordedfrom the tibialis anterior (a muscle that, according to Johnsonet al. (1973), contains 73% of type I fibres) of five normal humansubjects at a low contraction level. Therefore, it is highly likely thatthe vast majority of the activated fibres had slow contraction prop-erties. A review of the action potentials recorded from slow-twitchfibres strongly suggests that the DRT of this type of fibre is over0.75 ms. For the soleus muscle, for instance, Wallinga et al.(1985) and Hanson (1974) measured DRT values of about0.74 ms, whereas for the same muscle, Albuquerque and Thesleff(1968) found a DRT of 0.76 ms and Mcardle et al. (1980) a DRT of1.02 ms. In the intercostal muscle Ludin (1973) found a DRT of0.78 ms. As for the shapes of the IAPs, the recordings of Ludin(1973) and Radicheva et al. (1986) evidenced spikes with markedasymmetry, with Cas values of around 4.2 and 3.3, respectively.Thus, the values of DRT and asymmetry derived from real IAPs re-corded in mammals broadly coincide with those values of the D–DIAP necessary to make the PPRs of SFAPs largely independent of ra-dial distance. This suggests that the analytical expression of D–DIAP is appropriate for the modelling of IAPs from human musclefibres.

4.5. Limitations

The inverse approach above described to infer characteristics ofreal IAPs is grounded on the hypothesis that the core-conductormodel (on which the D–D SFAP model is based) is valid to accu-rately describe the generation of extracellular potentials. However,some of the approximations of the core-conductor model, such asthe assumption that the excitation source is distributed along theaxis of the fibre and the negation of the influence of the extracellu-lar potential field when calculating the transmembrane current,are questionable (Henriquez and Plonsey, 1988; Van Veen et al.,1993). Therefore, it should be mentioned that the conclusions de-rived here about the IAP model and characteristics of the IAP spikeare speculative and should be treated with caution. Future studiesinvolving simultaneous recording, from the same fibre, of the IAPand the transmembrane current are needed to establish the extentto which the assumptions of the core-conductor model are correct.

5. Conclusions

It has been shown that that the sensitivity of the SFAP PPR tochanges in radial distance decrease with the decreasing influenceof the field produced by the IAP repolarization phase on that gen-erated by the depolarization one. The duration of the IAP spike,measured as the DRT, is inversely proportional to the degree ofinteraction between the fields of the two IAP phases and thereforethe PPR sensitivity decreases as the DRT increases. The theoreticalbasis for this finding can be understood from a biophysical presen-tation of the IAP in the spatial domain.

It has been tentatively suggested that intracellular actionpotentials recorded in mammals should have a DRT longer thanabout 0.75 ms. Since the duration of the rising phase of these intra-cellular potentials is about 0.3 ms, the shape of the spike should bestrongly asymmetric. The IAP analytical function proposed by Dim-itrov and Dimitrova in 1998 fully accommodates theserequirements.

Acknowledgements

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

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Javier Rodríguez Falces was born in Pamplona in1979. He graduated in 2003, and obtained the Ph.D. in2007 in Telecommunication Engineering from thePublic University of Navarra, Pamplona, Spain. Heworked for the Higher Scientific Investigation Councilof Spain 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 teachingseveral subjects related to digital signal processing,image processing and biomedical engineering. Hisresearch focuses on signal processing applied tobiomedical signals, modeling of biological systems,electromyography, neurosciences and muscle fatigue.

Javier Navallas Irujo was born in Pamplona in 1976.He graduated in 2002, and he obtained the Ph.D. in2008 in Telecommunication Engineering from thePublic University of Navarra, Pamplona, Spain. He hasalso worked as a software engineer. He is presentlyAssistant Professor of the Electrical and ElectronicsEngineering Department of this University. Hisresearch interests are modeling of biological systemsand neurosciences.

Luis Gila Useros received his MD degree from theComplutense University, Madrid, Spain in 1983. In1988 he completed his specialization in Neurology atthe ‘‘Ramón y Cajal’’ Hospital, Madrid. Between 1989and 1998 he worked as a neurologist at the ‘‘SanMillán’’ Hospital, Logroño, Spain. From 1998 to2001 he carried out his specialization training inClinical Neurophysiology at the ‘‘Virgen del Camino’’Hospital, Pamplona, Spain, where at the present timehe is a staff member at the Department of ClinicalNeurophysiology. His research interests includequantitative electromyography and the automaticanalysis of electromyographic signals.

Ibán Latasa Zudaire was born in Pamplona in 1978.In 2002 he obtained his degree in Telecommunica-tion Engineering from the Public University ofNavarre, Pamplona, Spain. Since 2003 he has workedas an engineer in the development of software andhardware for communications systems. Since 2010he has been involved in his Ph.D. studies. His area ofresearch is the development of software and algo-rithmic methods for the processing of EMG signals.

Armando Malanda Trigueros was born in Madrid,Spain, in 1967. In 1992 he graduated in Telecom-munication Engineering at the Madrid PolytechnicUniversity. In 1999 he received his Ph.D. degree fromthe Carlos III University, Madrid. In 1992 he joinedthe School of Telecommunication and IndustrialEngineering of the Public University of Navarra. In2003 he became Associate Professor in the Electricaland Electronics Engineering Department of thisUniversity. During all this period he has beenteaching several subjects related to digital signalprocessing, image processing and biomedical engi-neering. His areas of interest comprise the analysis,modeling and simulation of bioelectrical signals,particularly EEG and EMG.