1 I n t r o d u c t i o n THE COMPOUND extracellular potentials (CEPs) may be thought of as a linear summation of temporally and spa- tially dispersed SFEAPs. In accordance with the potential theory, the shape of an intracellular action potential (IAP), its duration T~. and wavelength (b = vT,..) as well as the propagation velocity v of the excitation wave along the activated fibre influence SFEAPs in a different manner at different fibre-electrode distances (LoRENTE DE N6, 1947; ROSENFALCK, 1969; PATTLE, 1971; DIMITROVA, 1973; DIMI- TROV and DIMITROVA, 1974a; b; PLONSEY, 1974; DIMITROV, 1987).

Knowledge of the radial changes of the SFEAPs quanti- tative characteristics would shed light on the question whether the information included in SFEAPs is sufficient for estimation of both the activated fibre location in the volume conductor and the main parameters (i.e., T~,, v and b) specifying the activated fibre functional state. This knowledge would also help to elucidate whether the infor- mation included in CEPs is sufficient for estimating the parameters (analogous to the above ones) of the equivalent

First received 24th October 1990 and in final form 25th April 1991

�9 IFMBE: 1992

Medical & Biological Engineering & Computing May 1

generator by means of which CEPs can be described as a first-order approximation.

The aim of this study is to analyse the possibility of solving the inverse problem using the amplitude and inte- gral characteristics of surface-recorded CEPs.

2 M e t h o d Because of their small magnitudes the SFEAPs can be

recorded directly (without any averaging) only near the activated fibre. On the other hand, not only is the SFEAP shape, its value, the ratios between the amplitudes of the individual SFEAP phases (DIMITROV and DIMIXROVA, 1974a; b; PLONSEV, 1974) and their durations (LORENTE DE N6, 1947; ROSENFALCK, 1969; PATTLE, 1971; DIMITROVA, 1973; DtMITROV, 1987) dependent on the fibre-electrode distance, but so too are the sensitivities of the SFEAP amplitude (DIMITROVA, 1973; DIMITROV and DIMITROVA, 1974a, b; 1977), temporal (DIMITROV, 1987) and integral (DIMITROV e t al., 1989) characteristics to changes in the main parameters which specify the activated fibre's func- tional state. Therefore, the results obtained at a certain fibre-electrode distance cannot be applied to another. That is why at an arbitrary fibre-electrode distance (including

992 357

that typical of surface recordings) the SFEAP quantitative characteristics could only be correctly investigated theo- retically, for example by means of mathematical models.

In this investigation the SFEAPs were calculated by the method suggested by DIMITROV (1987). The lAP shape was assumed to be typical for a nonmyelinated nerve fibre without afterpotentials, as used in our previous work (DIMITROVA, 1973; DIMITROV and DIMITROVA, 1974a ; TRAY- ANOVA and DIMITROV, 1982; DIMITROV, 1987). The volume conductor was assumed to be infinite or semi infinite, resistive, homogeneous and isotropic.

The fibre-electrode distance y was varied in the range 0.01-50 mm; the IAP duration T~, in the range 0.625-5 ms; the propagat ion velocity v in the range 2 - 1 6 m m m s - 1 ; and the coefficient of the IAP asymmetry Kay, i.e. the ratio between T~, and the depolarisation phase duration, was varied in the range 2-5-5 (without any practical alteration in T~,).

0 [3 o

0 [3 o

o

0 [3 0

[3 o

0 0

0 0 0

0 5

[3 0 0

0 o [3

0 [3 0

0 [3 O

fl 0 0 0

0 0 0

0 0 O

--1]

1] 0 [3

0 [3

I

0 [3 [3

[3 [3 [3

n O

n 0 0

F i g . 1 Cross-sections of the territory shapes used in the present paper. (a) Territory shaped as a circle; (b) territory shaped as an ellipse with its longer axis oriented trans- versely to the electrode line; (c) territory shaped as an ellipse with its longer axis oriented parallel to the elec- trode line

To assess the effect of the spatial dispersion of the acti- vated fibres generating the CEP, we have summarised SFEAPs generated by a number of activated fibres located in territories with different diameters and cross-sectional shapes. We have used cross-sections shaped as a circle or as an ellipse with its longer axis oriented transversely or in parallel to the skin surface (Fig. I).

The effect of desynehronisation was analysed assuming that the activated fibres are located on the same axis.

3 Simulat ion results

3.1 Radial changes o f amplitude and integral characteristics o f S F E A P s

For a certain lAP shape (for a certain K~ in the present investigation), the fibre-electrode distance y, the wave- length b and the relevant amplitude of the SFEAP nega- tive phase A,p h are related by a comparatively simple relationship (Fig. 2) which can be analytically expressed by

A*A o b"- 2. A.ph - y . ( l )

A* = 0.3972(I - exp ( -ky /b ) ) (2)

a 2 g m r a

A o - ( 3 ) 4o~

where a is the activated fibre radius; V,, is the IAP anapli-

3 " c

2 b

c.l

f i , j g o

,

-3 ~ - , ; , OJ.8 -2.4 - ,0 -1.6 -112 -0.8 -0.4 0 0.4

log (y /b )

Fig. 2 Dependence of the Anphb 2 product on the y/b ratio for different IAP asymmetrieS: (a) K,s = 2.5; (b) Ka~ = 3"5; (c) K,s = 5. Anp h is the amplitude of the SFEAP negative phase, b is the wavelength and y is the fibre-electrode dis- tance

tude; aa and o- e are the conductivities of the axoplasm and of the volume conductor, respectively; n ~< 3 is a coefficient (Table 1) depending on the y/b ratio and on the lAP shape (i.e. on K , J ; and k is a coefficient (Table 2) depending on the y/b ratio.

At a large fibre-electrode distance the extracellular field has a quadrupole character, n = 3. Moreover for y/b > 1.5 the integral of the SFEAP negative phase can be expressed by

A*A o b Inp h - - (4) vy z

A~' = 0.1535(1 -- exp ( - p y / b ) ) (5)

where p is a coefficient (Table 3) depending on the y/b ratio.

Table 1 Values of the coefficient n for different y/b ratios and different K.~

y/b

K,~ 0-001 0-01 01 02 04 1 2 4 10

25 068 188 2-47 269 2-84 2.95 2-98 2.99 3.00 35 0.74 192 2-50 2.70 2.85 2-95 2.98 2.99 3.00 50 0 - 8 0 196 2-53 2.71 2.85 2.95 2.98 2.99 3-00

Table 2 Values of the coefficient k for some fixed values of the y/b ratio

y/b 1 2 3 4 6 10

k 2"25 1.79 1.51 1.36 1.27 1.20

358 Medical & Biological Engineering & Computing May 1992

Table 3 Values of the coefficient p for some fixed values of the y/b ratio

y/b 1 1.5 3 4 6 10

p 3.28 2.68 1.91 1.69 1.55 1.48

Using eqns. 1 and 4 it can be shown that for surface recordings the normalised integral of the negative phase of the S F E A P (in relation to the amplitude of the S F E A P negative phase) is p ropor t iona l to the fibre-electrode distance and inversely propor t iona l to the propaga t ion velocity

I = Inph ~ 0"3864 -y (6) A n p h 13

It should be noted that the linear relationship between the normalised negative integral and the fibre-electrode dis- tance is also valid for recording by bipolar (parallel or transversal) electrodes with a small interpole distance.

3.2 Estimation of the fibre-electrode distance

Estimation of the fibre-electrode distance y does not offer any difficulty, because the normalised integral of the S F E A P negative phase depends linearly on y (cf. eqn. 6) when making surface recordings.

To assess the source locat ion it is necessary to determine the values Ii of the normalised integrals of the negative phases of the potentials recorded simultaneously by means of three surface unipolar electrodes mounted equidistantly (L) along a line oriented transversely to the fibre axis. If the potential recorded by a bipolar electrode with the poles moun ted along the same line is equal to zero (i.e., the poles are symmetrical in relation to the fibre axis) and the distance between the electrode middle point and the first (i = 1) unipolar electrode is 1-5L (Fig. 3) then as shown in the Appendix

I i - fl yi - - (7) CZ

where

and

512 _ 312 _ 212 fl=

1012 - 611 - 413

•/(13 _ f l ) 2 _ ( 1 1 _ f l ) 2

~=- -l-~L 5

When assessment of the source location is carried out some errors can be made due both to possible deviations of the relationship I(y) (eqn. 6) f rom linearity (Table 4, Yo = 0 or 5 mm) and as a result of the relatively small L compared to the depth Yo at which the fibre is located (Table 4, Yo = 20mm). In the latter case Yi are only just distinguished. The errors can be reduced by substituting 11 in eqns. 7 for ct and fl by

I* = ~/(512 212 ) I

and using such an interelectrode distance L that 1.5L

Y o . As the estimation error of the smallest fibre-electrode

distance y~ is the greatest, this error will be presented further on.

If there are some problems with the unipolar recording realisation then bipolar (parallel or transverse) electrodes with a small interpole distance can be used instead of the

Medical & Biological Engineering & Computing

. 1 L I1

L

Fig. 3 Scheme of the arrangement of the electrodes (one bipolar and three unipolar) for estimation of the fibre-electrode distances Yi. L is the interelectrode distance and Yo is the source depth

Table 4 Estimation errors (per cent) of the fibre-electrode dis- tances Yi for different source depths Yo and different interelectrode distances L

Yi mm

L mm Yo mm Ya per cent Y2 per cent Y3 per cent

5 0 + 5.40 + 2.00 + 1 "01 5 5 +4.23 + 1.96 + 1.06 5 10 + 1 "10 +0-60 +0-30 5 20 + 15'04 + 12-89 +9.93

10 20 --4.48 --2"66 + 1.63

using corrected values of 11

5 0 + 1.97 +0.72 +0-37 5 5 + 1.27 +0.60 +0-31 4 10 - 1.00 -0 .50 -0-60 5 20 -2-08 - 1.69 - 1-36

10 20 - 1.71 -0 .99 -0 .65

unipolar electodes as long as the centre points of the bipolar electrodes coincide with the locations of the uni- polar electrodes in Fig. 3.

3.3 Estimation o f the wavelength

Knowing the amplitude of the negative phase A,p h and the radial distance y, the A o b product could be estimated in the case of surface recordings using eqn. 1

A o b - A,ph y3 A* (8)

In view of the fact that for calculation of A* the value of the y/b ratio has to be known, it can initially be assumed that A* = 0.3972. In this case the smaller the value of the y/b ratio, the greater the error (Table 5) of the estimation of the Ao b product . This error can be reduced if A* and, respectively, A o b are determined by means of an iteration procedure. In this way the est imation error can be smaller than 1 per cent for y/b ~ 1.

3.4 Estimation o f the l A P duration

The A o T~, product can be estimated using the estimated values of the A o b product and of the velocity v. Another

M ay 1992 359

Table 5 Estimation errors (per cent) of the wavelength b (assuming A o -- 1)for different y/b ratios when the noniteration procedure is used

y/b 0.5 0-666 0"75 1 1 1"5 2 3 4 6

b (real) 20 15 20 15 10 10 5 5 2.5 2.5 b (estimated) 13-98 11-96 16.6 13-42 8.94 9.5 4.86 4.94 2-489 2.4998 error, per cent -30.1 -20.3 -17"0 -10"5 -10'5 -5 ' 0 -2"8 -1"1 -0"44 -0.01

way for estimation of the A o Ti, product is using the inte- gral of the SFEAP negative phase

Ao Ti, = i,ph y2 (9) A~

In similar ways as in Section 3.3, the estimation of the Ao T~, product can be carried out both by noniterative (A* = 0.1535) or by iterative procedures. In both cases the estimation errors of the Ao T~, product are smaller than the corresponding estimation errors of the Ao b product.

Use of the integral characteristic 1,p n for estimation of the Ao T~, product is preferable in practice not only because of the fact that in this case only one estimated parameter is used. It is a priori evident that the integral is more noise-resisting and immune to the effects of the tem- poral and spatial dispersion than the amplitude. More- over, the power of the estimated parameter (y) is smaller. In eqn. 9 the power equals 2, but is equal to 3 in eqn. 8 by means of which the A o b product can be estimated.

The estimated value of the Ao T~, product which is obtained using the estimated values of A,ph, y and v can be used as a control value.

When the estimation errors are determined further on we shall use the amplitude characteristic A,ph giving greater errors.

The SFEAPs have no DC component (DIMITROV et al., 1989) and that is why the recording of the Anp h a s well as of the lnp h would not offer any difficulty if the magnitudes of the SFEAPs were not so small.

On the surface the quantitative parameters of CEPs gen- erated by a set of fibres activated almost synchronously can be measured. In this case it is possible to estimate the following products:

N A o bA = Acphy3 (10) 2A*

N A o Ti, A 1 - - ICphy2 (11) 2A~"

where N is the number of the activated fibres; ACph and ICph are the amplitude and integral of the CEP negative phase; A and A, are scale factors depending on the temporal and spatial dispersion of the SFEAP; the scale factor 2 is con- nected with surface recordings and it is due to the doub- ling of the potentials at the conductor/insulator boundary.

3.5 Estimation of the propagation velocity The propagation velocity v can be estimated using one

of the noninvasive methods which have already been described (LIYDSTR6M et al., 1970; PARKER and SCOTT, 1973; LYNN 1979; VAN DER VLIET et al., 1980; STULEN and DELUCA, 1981; JASROTIA and PARKER, 1983; NAEIJE and ZORN, 1983; BROMAN et al., 1985; SOLLIE et al., 1985; DAVIES and PARKER, 1987; HUNTER et al., 1987; RABABV et al., 1989; ZWARTS, 1989).

Another possibility is to take advantage of the fact that the ratio between the temporal and spatial derivatives of the potential is equal to v. The signal recorded by a paral- lel oriented bipolar electrode with a small* interpole dis- tance d is proportional to the potential spatial derivative

(DIMITROV and DIMITROVA, 1977). A signal proportional to the potential temporal derivative can be formed by sub- traction of the potential recorded unipolarly (at the bipolar electrode pole located nearer to the site of the excitation origin) with a small time delay ~ from the same potential, but without any delay. The ratio between the amplitude of this signal and the amplitude of the signal recorded by a parallel oriented bipolar electrode is equal to w/d. The errors resulting from the deviations of the above signals from the corresponding derivatives can be eliminated by determination of the delay zl at which the amplitudes of both the signals are equal. Then

v = d / ~ l

In this case only the estimation error is equal to zero. When z = 0-5zl the error is +0.1 per cent, but it is -2 .28 per cent for z = 2Zr Therefore, if the time period for the estimation of v is limited and it is impossible to search iteratively for the delay it is preferable to apply a delay of

75 2 ~ d/Vma x

where V,,a x is the greatest velocity expected. If there are some problems with unipolar recording, then

two bipolar electrodes having a common pole and located on a line along the fibre length can be used for carrying out the procedure. In this case, the temporal derivative can be formed by means of a time delay of the signal recorded by the first electrodes and the spatial derivative by sub- tracting the signals recorded by the two bipolar electrodes.

The degree of desynchronisation in the activation of the fibres as well as the differences in the distances between the recording electrodes and each of the activated fibres do not influence the accuracy of the estimation of v, carried out by the method proposed in the present paper, as long as v is the same for every one of the activated fibres. The two signals described above have the same amplitudes only in the case when the delay in the formation of one signal is equal to the ratio between the interpole distance d of the bipolar electrode and the velocity of the excitation propa- gation along the fibre.

3.6 Assessment of the effect of the desynchronisation To assess the effect of the desynchronisation on A~.ph, on

the accuracy of the estimation of Yi and on the product

c 3 AnPhY (12)

N A ~ Ti" ~ 2A*v

we have assumed that the activated fibres are located on the same axis, but are activated at different moments which are equally distributed in the folowing ranges: from 0 to 2/v (ms); from 0 to 4/v (ms); from 0 to 6/v (ms); and from 0 to 8/v (ms). We have calculated the relationships of the results obtained to the relevant values in the case when the degree of desynchronisation is equal to zero.

The greater the degree of the desynchronisation D, the

* In actual fact one interpole distance may be assumed to be small if the recorded signal amplitude is half the amplitude of the signal recorded by a parallel oriented bipolar electrode with a two fold interpole distance. In the case of surface recordings an interpole distance of 3 5 m m may be considered as a 'small ' one.

360 Medical & Biological Engineering & Computing May 1992

smaller the amplitude A~,ph. On the other hand, the greater the fibre-electrode distance, the weaker the relative effect of the desynchronisation on A~,ph (Table 6). At large radial distances, typical of surface recordings neither the IAP shape (i.e. K J nor the wavelength b change the effect of the desynchronisation on the ACnph .

Table 6 Percentage change in the CEP negative phase amplitude A~,ph (in relation to the amplitude when the desynchronisation is zero) depending on the degree of the desynchronisation D at differ- ent source depths Yo assuming that the wavelength b = 5 mm

O ms

0 2/v 4/v 6/v 8/v Yo mm per cent per cent per cent per cent per cent

5 0 --4.99 --17.43 --33.26 -48.51 10 0 --1-37 --5-42 --11-59 --19-20 20 0 --0-33 --1-42 --3.17 --5.58

The effect of the desynchronisation on the I~,ph is similar but it is less pronounced than that on the A~,ph and as a result the normalised integral of the CEP negative phase increases weakly as the degree of desynchronisation increases.

The degree of the desynchronisation affects the error of

Table 7 Estimation errors (per cent) of the fibre-electrode dis- tance Yl depending on the degree of desynchronisation D for differ- ent source depths Yo and different interelectrode distances L assuming that the wavelength b = 5 mm

D ms

0 2/v 4/v 6Iv 8Iv L mm y 0 m m per cent per cent per cent per cent per cent

5 5 + 1.27 + 2.48 + 6.22 + 12.14 + 20.49 5 10 - 1 . 0 0 +1.20 +3.00 +6.50 +10.90 5 20 - 2 - 0 8 - 0 . 5 2 +0.41 + 1.11 +2-20

10 20 -1 .71 - 3 . 6 6 -3 -16 - 2 . 3 0 -1 .21

the estimation of the equivalent source location. The greater the degree of desynchronisation, the greater the value of the estimated Y0 (Table 7). The magnitude and the sign of the error depend on the interelectrode distance L, as well.

When the desynchronisation is not pathologically great,

Table 8 Estimation errors (per cent) of the NA o T~. product depending on the degree of desynchronisation D for different source depths Yo and different interelectrode distances L assuming that the wavelength b = 5 mm

D ms

0 2/v 4Iv 6/v 8/v L mm y o m m per cent per cent per cent per cent per cent

5 5 --0.22 +2-05 +7.75 +17-14 +30.40 5 10 --3.36 --2.89 +2.39 +8.05 +15.13 5 20 8.98 -4 -83 - 3 . 0 7 -2 -60 - 1.66

10 20 - 7 . 8 3 -13 .48 -12-64 -11 .38 - 9 . 8 2

the CEPs recorded on the surface are insensitive to it, and hence desynchronisation is not a fundamental obstacle for estimating the rest parameters which specify the func- tional state of the fibres. The error of the estimation of the N A o T~, product is also not so great (Table 8) because of the fact that the errors in the estimation of the other parameters are partially compensated for.

3.7 Assessment o f the effect o f the territory shape and diameter

To assess the effects of the shape and diameter of the

Medical & Biological Engineering & Computing May 1

activated fibres' territory on A~,ph, the accuracy of the esti- mation of the distance between the electrode line and the territory geometrical centre and the N A o T~, product we examined three shapes of territory (Fig. 1) with four differ- ent diameters 2r for different fibre-electrode distances. The greater the distance between the electrode and activated fibres, the smaller the ACnph increase (Table 9) compared with the case when all the activated fibres are concentrated in the geometrical centre of the territory. Owing to the nonlinear radial decline of the SFEAPs the A~,ph increase is strongly dependent on the distances from the electrode to the closest activated fibres. As a result the estimated dis- tance y between the electrode line and territory geometri- cal centre is smaller (Table 10) than the actual distance. Thus the electrical and geometrical centres of the territory are different. The estimation errors also depend on the distances from the electrode to the closest activated fibres. Finally, the estimation errors for the N A o T~, product are not so great (Table 11) when the activated fibres' territory is not oriented parallel to the electrode line. Note that the estimation errors due to the temporal (Table 8) and spatial (Table 11) dispersion have opposite signs. Consequently, the actual total estimation errors will possibly be smaller than those shown in Table 8 and Table 11.

4 D i s c u s s i o n The analytical expression obtained for the amplitude of

SFEAP negative phase as a function of the wavelength and fibre-electrode distance is valid for arbitrary fibre-electrode distance. It can be considered as an analytical gener- alisation of the dependencies between the amplitude of SFEAP negative phase and the wavelength for different fibre-electrode distances, which were found by DIMITROVA (1973).

According to Helmholtz' theorem (HELMHOLTZ, 1853) the inverse problem cannot be solved in full using surface- recorded compound action potentials. Indeed, the location and main parameters of each one of the activated fibres cannot be estimated when they are numerous. However, this information is not of interest for clinical practice. In the majority of cases the possibility of estimating the equivalent generator parameters by means of which the functional state of the activated fibres can be assessed is more important.

The results obtained in the present investigation show that the estimation of the distance between the activated fibres' territory centre and surface electrodes as well as the propagation velocity and Na2Vm Tin( tT a/ ff e) product is not so difficult provided that the CEP spikes do not overlap. These are the cases when the self-induced activity is not so great or when the activity is induced by a stimulus. Although the above-stated product includes some indefi- niteness, the assessment of its changes could be very useful if some additional information which takes into account morphological or functional features of the activated fibres is used. For example, assuming that the fibre radius a or aa/a e ratio as well as N do not alter during the experiment, the changes in the product could be interpreted as changes in Vm and/or Tin. However it has never yet been supposed to be possible to assess these parameters by noninvasive methods.

The methods which have been used up to now have enabled the propagation velocity to be estimated from the time delay of a propagating IAP between two recording sites along the activated fibres. In these methods the greater the distance between the recording sites, the greater the estimation accuracy, but due to the excitation bound-

992 361

Table 9 Percentage change in the CEP negative phase amplitude ACnph (as a relationship with the amplitude when all the activated fibres are concentrated in the territory geometrical centre) depending on the shape and diameter 2r of the activated fibres')erritory for different territory centre depths Yo assuming that the wavelength b = 5 mm and desynehronisation D = 0

Territory cross-section shaped as a circle

Territory cross-section shaped as an ellipse oriented transversely

Territory cross-section shaped as an ellipse oriented in parallel

2r mm

Yo mm 2 4 6 10 2 4 6 10 2 4 6 10

10 6-8 9-5 14-0 28-7 6.9 9-6 14.1 31.5 6-6 8.2 10.8 18.6 20 1-8 7-0 8-6 13-6 1.9 7-5 9-6 16.1 1.6 6-3 6.8 8.0

Table 10 Estimation errors (per cent) of the fibre-electrode distance Y a depending on the shape and diameter 2r of the activated fibres' territory for different territory centre depths Yo assuming the wavelength b = 5 mm and desynchronisation D = 0

Territory cross-section shaped as a circle

Territory cross-section shaped as an ellipse oriented transversely

Territory cross-section shaped as an ellipse oriented in parallel

2r mm

Yo mm 2 4 6 10 2 4 6 10 2 4 6 10

10 --0"5 --2.0 --4"0 --11.3 --0.7 --2"2 --4.6 --11-7 --0"3 --0'7 --1.2 --2"8 20 --0.2 --0.7 --1-4 -3"7 --0'2 --1-1 --2.4 --6-3 --0-2 --0-3 0-1 1-1

Table 11 Estimation errors (per cent) of the NA o Ti. product depending on the shape and diameter 2r of the activated fibres" territory for different territory centre depths Yo assuming the wavelength b = 5 mm and desynchronisation D = 0

Territory cross-section shaped as a circle

Territory cross-section shaped as an ellipse oriented transversely

Territory cross-section shaped as an ellipse oriented in parallel

2r mm

yomm 2 4 6 10 2 4 6 10 2 4 6 10

10 5.2 3'0 1'3 --7'9 4'8 2.7 --1.1 -11.4 5-8 5"9 6'8 8"9 20 1-2 4.9 4.2 1.5 1.4 4.1 1"9 --4-2 1-1 5"3 6"8 11.6

ary effects this is not convenient in the case of short fibres. Using the simple method proposed in the present paper, the necessary accuracy can be achieved by applying a very small distance between two recording sites.

The results of the present work were obtained assuming that the conductive medium was isotropic. In actual appli- cation, however, these results can easily be related (cf DIMI- TROV and DIMITROVA, 1974b) to an anisotropic volume conduc to r having conductivi ty cr x along the fibres which differs f rom (in particular is greater than) the conductivi ty ~ry across the fibres. The procedures for estimating the fibre-electrode distance and propaga t ion velocity are valid in the case of an anisotropic conductive medium without any modifications. The estimated values of the N A o b produc t (according to eqn. 10) and the N A o Ti, product (according to eqn. 11) will be reduced x/ax /ay times. More- over, as the medium anisot ropy is equivalent (DIMITROV and DIMITROVA, 1974b) to the ~ times reducing of b and the accuracy of the est imations grows with increasing y/b it follows that the accuracy of the estimations in a real anisotropic volume conduc to r will be better than in an isotropic one.

5 Conclusions

(a) For a large fibre-electrode distance typical of surface recordings the ampli tude of the S F E A P negative phase, wavelength and fibre-electrode distance are related by a comparat ively simple relationship.

(b) Using the ampli tude and the integral characteristics of the surface-recorded potential generated by infinite homogeneous fibres the inverse problem can be solved to a great extent.

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fatigue and action potential conduction velocity changes studied with frequency analysis of EMG signals. Electromyoor., 10, 341-356.

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NAE1JE, M. and ZORN, H. (1983) Estimation of action potential conduction velocity in human skeletal muscle using the surface EMG cross-correlation technique. Electromyoyr. Clin. Neuro- physiol., 23, 73-80.

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Appendix According to eqn. 6

I i = o~y i + fl

and

(13)

I~-- ,8 y i = - (14)

where ~ and /~ are the coefficients of the linear relationship between the normalised integrals I i and the fibre-electrode dis- tances y/.

With reference to Fig. 3 and taking into account the geometri- cal relationships we obtain

y~ = yg + (1 .5L) 2 (15)

yz z = yo 2 + (2.5L) 2 (16)

y~ = yg + (3-5L) 2 (17)

Substituting eqn. 14 into eqns. 15-17 gives

(i t _ ,8)2 = (i ~ _ ,8)2 + (1.5L~)2

(12 _ ,8)2 = (i ~ _/ / )2 + (2.5Lc02

(13 _ ,8)2 = (Io -- ,8)2 + (3.5Lc0Z

Subtracting eqn. 18 from eqn. 19, the result is

(18)

(19)

(20)

( 12 _ ,8)2 _ ( I 1 - - ,8)2 = 4L2c~2 (21)

and similarly eqn. 18 from eqn. 20

(i 3 _ ,8)2 _ ( I , - - ,8)2 = 10L2~2 (22)

From eqns. 21 and 22 we obtain

5 ( 1 2 - ,8) 2 - 3 ( I t - ,8)2 = 2(13 _ ,8)2 (23)

Solving eqn. 23 the coefficient ,8 is found to be

512_31 t 2_2132 ,8 = (24)

10t 2 - 6t 1 - 413

Using eqn. 22 the coefficient ~ is found to be

/ ( 1 3 _ ,8)2 _ (i t _ ,8)2 (25) i -L V

Authors" biographies George V. Dimitrov was born in Sofia, Bul- garia, in 1942. He received the MS degree in Biomedical Engineering from the Institute of Electrical Engineering, Leningrad, USSR, in 1969, and Ph.D. degree in Biophysics from the Institute of Physiology, Bulgarian Academy of Sciences, in 1974. He is an Associate Professor of Biomedical Engineering at the Bulgarian Academy of Sciences, Sofia. His research inter-

ests are in the field of bioetectrical phenomena.

Zoja C. Lateva was born in Bulgaria in 1952. She received the MS degree in Computer System Engineering from the Institute of Elec- trical Engineering, Leningrad, USSR, in 1976, and Ph.D. degree in Biophysics from the Central Laboratory of Bioinstrumentation & Automation, Bulgarian Academy of Sciences, in 1991. In 1980 she joined the Bulgarian Academy of Sciences as a Research Associate.

Her main research interests include computer modelling and analysis in electrophysiology.

Nonna A. Dimitrova was born in Leningrad, USSR, in 1945. She received the MS degree in Biomedical Engineering from the Institute of Electrical Engineering Leningrad, USSR, in 1969, and Ph.D. degree in Biophysics from the Institute of Physiology, Bulgarian Academy of Sciences, in 1975. She is an Associate Pro- fessor of Biophysics at the Bulgarian Academy of Sciences, Sofia. Her research interests are in

the field of mathematical modelling and analysis of bioelectrical phenomena.

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