1 I n t r o d u c t i o n ONE OF the main goals in electrophysiological investiga- tions is discovering any possibilities of estimating some physiologically oriented parameters which could be used for specifying the functional state of the activated fibres. According to Helmholtz' theorem (HELMHOLTZ, 1853) it is not possible to solve the inverse problem in full. In our previous paper (DIMITROV et al., 1992) it was shown that the inverse problem in electroneurography can be solved to some extent using amplitude and integral characteristics of the surface-recorded compound extracellular potentials (CEPs) generated by a number of fibres activated almost synchronously. However, the proposed procedures are limited to the cases of stimulus-induced or relatively weak self-initiated activities, when CEP individual pulses are not overlapped.

LINDSTROM and PETERSEN (1981; 1983) have shown that the total signal power spectrum has the same shape as that of the individual signals even when the CEP individual pulses are overlapped. This feature is very attractive when efforts are exerted to estimate the activated fibres' func- tional state (including maximum activity) by means of spectral characteristics of surface-recorded CEPs.

The aim of the present study is to search for ways of solving the inverse problem in electroneurography using spectral characteristics of surface-recorded CEPs and analysis of the revealed opportunities.

First received 1 l th June and in final form 6th November 1991

�9 IFMBE: 1992

Medical & Biological Engineering & Computing

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, the sensitivities of the single-fibre extracellular action potential (SFEAP) spectral characteristics to changes in the main parameters specifying the activated fibre functional state depend on the fibre-electrode distance (DtMtTROV et al., 1990; LATEVA et al., 1990). Therefore, the results obtained at one fibre- electrode distance cannot be applied to another.

Analytical expressions of the power spectra of the SFEAPs in an anisotropic frequency-dependent volume conductor were obtained on the basis of the properties of the Fourier transform (FT) and the regularities of the current distribution in the volume conductor. The theoreti- cal conclusions drawn for SFEAPs were tested for CEPs by numerical experiments. CEPs were calculated as a linear summation of a number of temporally and spatially dispersed SFEAPs. The desynchronisation in the fibres' activation and differences in the distances from the record- ing electrodes to each activated fibre were simulated in the same manner as in our recent paper (DIMITROV et al., 1992). The descriptions of the method used for calculation of SFEAPs and of the shape of the intracellular action potential (IAP) as well as definitions of the investigated parameters are given in our previous papers (DIMITROV, 1987; LATEVA, 1988; DIMITROV et al., 1990; 1992; LATEVA et al., 1990). In addition, in the present study we took into consideration differences in the tissue electrical conductivi- ties in longitudinal a x and transversal ay directions with

July 1992 399

respect to the fibre direction (BURGER and VAN DONGEN, 1960; RUSH et al., 1963; GEDDES and BAKER, 1967), which are due to the large lengths of the fibres compared with their diameters 2a (ROSENEALCK, 1969). Moreover, we took into consideration the frequency dependence of the trx, try and anisotropy ratio K, , = a~/ar which is a result of the capacitive effects of the fibre membranes (EISENBERG, 1983; GIELEN, 1983; GIELEN et al., 1984; 1986; ZHENG et al., 1984; ALBERS et al., 1986; ROTH et al., 1988).

Because of the fibre/electrode distance dependence of the sensitivity of the SFEAP spectral characteristics to changes in the main electrophysiological parameters, a pre-estimation of the average activated fibre-electrode dis- tance ~ was needed. In addition, consideration of an anisotropic volume conductor with two different conduc- tivities a~ and ay is equivalent to transformation of the actual distance ~r into an equivalent distance Y ~ x / ~ , (DIMITROV and DIMITROVA, 1974).

3 Results

3.1 Power spectra of SFEAPs in an anisotropic frequency-dependent volume conductor

According to KRAKAU (1957) the SFEAP generated by an infinite homogeneous fibre can be perceived as a con- volution of two parts: one (called a weighting function) depending on the propagation velocity v and recording geometry and the other depending on the IAP time course, Taking into account (cf. DIMITROV et al., 1990; LATEVA et al., 1990) the Borel's convolution theorem, differentiation, shift, similarity theorems and linearity of the FT as well as the regularities of the current distribution in an anisotropic volume conductor (cf. DIMITROV and DIMITROVA, 1974), the power spectra of SFEAP in an anisotropic volume con- ductor can be expressed, respectively, as:

PS[ ~uni(t, Xe, y~)] = A..2 V~2 T~. j C( Ti. f)12

for recording by a unipolar electrode at point (x~, y~) of the volume conductor;

PS[ObipIl( t, Xe, Ye, d)]

2 2 z 2 T2nIC(TI.f)]2 = 4d ~ A . . V m

for recording by a bipolar parallel electrode with a small interpole distance d; and

PS[4)~,~_(t, Xe, Ye, d)]

= A,, 2 V 2 TEl C(T~. f )12

xf~4 ]K1(27~ Ye ~N~anf) 12 ( 3 )

for recording by a bipolar radial electrode with a small interpole distance, where

7z2a2 gan t7 a

V~ and T/, = I A P amplitude and duration, respec- tively

C ( f ) - - - F T of the lAP whose amplitude and duration are unity

K o and K~ -- modified Bessel functions of the second kind and orders zero and one, respectively

4 0 0

aa = intracellular electrical conductivity tr,, = ax /~a r = extracellular electrical conductivity of

the anisotropic volume conductor.

These equations are valid whether the conductivities tr x and ar as well as the anisotropy ratio Ka, are frequency- dependent or not.

3.2 Power spectra of surface CEPs At a great radial distance (typical of surface recording)

the time course of the CEP is similar to that of the SFEAP generated by a fibre located on the electrical axis of the activated fibres' territory, i.e. the axis of the equivalent line source. The effects of the differences in the distances from the recording electrode to each activated fibre on the CEP (normalised by the number N of the activated fibres) and on the CEP power spectrum are relatively weak (Fig. 1) if there is no activated fibre located in the proximity of the recording electrode. The contribution of the proximate

7 o:I x

O

"5

4

O 3

"o

-6 2 E

r 1

0

2

i

0"08 0"16 0"24 0.~2

7

6] 'o p x

5

r 4

"o o .m -6 2 E L O c 1

0 0"08 0.16 0-24 0.32

f requency, k Hz

b

Fig. 1 The power spectra o f the CEP (normalised by the number N of the activatedfibres).]'or different territory shapes (I, 2, 3 and 4) and diameters (a) and (b). The radial dis- tance between the territory geometrical axis and the recording electrode Yo = 15 ram. (a) Territory diameter 4ram; (b) territory diameter 6ram. 1: territory cross- section shaped as a circle, N = 32; 2: territory cross- section shaped as an ellipse with its longer axis oriented transversely to the surface, N - - 2 0 ; 3: territory cross- section shaped as an ellipse with its longer axis oriented in parallel to the surface, N = 20; 4: a f b r e located along the territory geometrical axis

Medical & Biological Engineering & Computing July 1992

fibres to the CEP power spectra formation can predomi- nate due to the nonlinearity of the radial decline of the power spectra of the weighting function second derivative. This nonlinearity corresponds to exponent 4 for unipolar recording (eqn. 1) or recording by bipolar radial electrodes (eqn. 3) and to exponent 6 for recording by bipolar parallel electrodes (eqn. 2).

In contrast, the effect of desynchronisation in the fibres' activation on the power spectra can be pronounced. As a result of desynchronisation dips can exist (Fig. 2) in the power spectra even in the case of unipolar recording of CEP. The frequencies of the dips are proportional to the degree of desynchronisation but they do not depend on the propagation velocity of the excitation waves. For an object to explain this let us simulate the CEP formation as a summation of the action potentials generated by N fibres located at equal radial distances from the point of observa- tion and activated at equal time intervals At. If the propa- gation velocities of the individual fibres are the same, the obtained CEP is similar to the integral of the action poten- tial recorded by a bipolar parallel electrode with interpole distance A x = N Atv. The frequency at which the first dip occurs is

2.4

2.2

o 2.0 1

x 1.8

~, 1.6 CL

1"4

1"2

1"0 (IJ _~ 0'8

0"6 0 c 0"4

0'2

0 0 ' 0 4 0'08 0"12 0"16

2'4

2"2

~, 2.0 o

• 1'8 o "~ 1"6

1'4

~, 1.2 0 cL 1'0

0'8

0"6

0"4

0"2

Fig. 2

\

0 ~ , 0"04 0'08 0'12 0'16

frequency, kHz

b

The effects of (a) the desynchronisation in the fibres' acti- vation and (b) of the propagation velocity on the power spectra of the CEP. The radial distance Yo = 15 mm. (a) The frequencies of the dips are proportional to the degree of desynchronisation 1 : 10 ms; 2: 20 ms, The propagation velocity v = 4 mm ms-1. (b) The frequencies of the dips do not depend on the propagation velocity: 2:4 mmms-1; 3: 8 mm ms- 1. The degree ofdesynchronisation is 20 ms

Medica l & Biological Engineering & Comput ing July 1

v v 1 feiv -- A x - N Atv - N At

The above considered dips are well pronounced in case of uniform desynchronisation distribution. Similar results have been obtained by LINDSTR6M (1970) for the dips exist- ing in the power spectra due to the uniform propagation velocity in the case of bipolar parallel recording. Actually, a smooth distribution of the desynchronisation and propa- gation velocity is more likely. As a result, both kinds of dips can be poorly visible. Moreover, in case dips are visible in the power spectra, it is not evident to which of the two causes (desynchronisation or propagation velocity) they are due. This is an extra difficulty in applying the dip method, suggested by LINDSTR6M (1970), for estimation of the propagation velocity.

3.3 Estimation o f the average propagation velocity

The ratio between the power spectra of the CEPs recorded by a unipolar electrode and a bipolar parallel electrode with a small interpole distance, namely

PS[CEP""I(N' Xe' Y~)] = ~2/4d2Tc2f 2 (4) PS[CEPb,pII(N, x~, Ye, d)]

can be used for estimation of the average propagation velocity ~.

Another possible way for estimation of the average pro- pagation velocity is by using of the ratio between the spec- tral amplitudes of the first temporal and spatial derivatives (numerical differences) of the potential. One bipolar parallel-oriented electrode can be used for obtaining both temporal and spatial differences. The temporal difference can be formed by subtraction of the CEP recorded uni- polarly by the bipolar parallel electrode pole located nearer to the excitation origin site and time delayed by a small delay z, from the same potential but without any delay. The CEP recorded bipolarly by the electrode is the spatial difference needed. The ratio between the spectral amplitudes is equal to ~z/d for any frequency component at which there are no dips. To eliminate errors resulting from possible deviations of the above-described signals from the corresponding temporal and spatial derivatives, the delay z has to be obtained in such a way that the ratio between the spectral amplitudes of the two spectra is unity, i.e. ~ = d/z.

As it was shown in our previous paper (DIMITROV et al., 1992), if the computation time is a critical factor it is better to use z = d/~m,x, where ~,,,x is the greatest ~ expected.

The frequency dependence of the tissue conductivities does not influence the accuracy of the average velocity estimation because the ratio between the temporal and spatial derivatives of the signal does not depend on its frequency and corresponding tissue conductivities despite the fact that each of the derivatives depends on them.

The temporal desynchronisation and spatial dispersion of the activated fibres do not affect the accuracy of the estimation of the propagation velocity if the latter is uniform. But in the real case, when the propagation velo- cities differ, the spatial dispersion can strongly affect the accuracy of the estimation of the average propagation velocity, If there are activated fibres at substantially smaller distances to the recording electrode compared with the rest of the activated fibres, their velocities predominate in the result of the average velocity estimation. Hence, the proposed procedure fits the case of recording at a great radial distance from the activated fibres to the electrode, i.e. when the spatial dispersion is of no concern.

992 401

3.4 Estimation of the equivalent average distance (Ye ~Xfan) At a great radial distance for which the CEP spectrum

maximum frequency F,,ax can be within the ultra-low-fre- quency range, where the spectral power of the IAP without after-potentials is practically unchanged (DIMITROV et al., 1990; LATEVA et al., 1990), the weighting function maximum frequency F~a ~ forms Fma x:

uni Fmax ~ VW,x = 247 - - (5)

for unipolar recording, where 247 Hz is the frequency at which the function f2Ko(2nf) has its maximum (PATTLE, 1971), and

Fbipll = 403 (6)

for recording by a bipolar parallel electrode with a small interpole distance, where 403 Hz is the frequency of the maximum of functionf3Ko(2nf).

The estimated equivalent average distance from the recording electrode to the electrical axis of the territory is always smaller than the distance from the electrode to the territory geometrical axis (Table 1). The underlying reason is the nonlinear radial decline of the weighting function spectrum and as a consequence greater contribution of the activated fibres located closer to the recording electrode. The difference between the territory geometrical axis and the estimated ~ is greater under recording by a bipolar parallel electrode than by a unipolar electrode. This is as a result of the stronger radial decline of the weighting func- tion spectrum in the former case. The smaller the relative difference between the radial distances from the electrode to each one of the activated fibres, the smaller the differ- ence between the electrical and geometrical axes of the territory. Note that the estimation error of ~ grows signifi- cantly when the distance from the territory geometrical axis to the recording electrode is commensurable with the territory diameter. For this reason, the method described above is applicable at great radial distances from the terri- tory axis when the spatial dispersion is of no concern, i.e. in the case of surface recordings, but it is not suitable for intraterritory recordings.

The error due to IAP spectral power changes in the frequency range to which F~,~ belongs is negligible com- pared with the error formerly described. The error owing to the temporal desynchronisation in the fibres' activation is negligible too.

At a great distance FW~x belongs to a frequency range in which the tissue conductivities do not change (EISENBERG, 1983; GIELEN, 1983; ROTH et al., 1988) and the effect of the tissue frequency dependence on the estimation of the equivalent average distance is of no concern.

3.5 Estimation of the actual average distance (Ye) and tissue anisotropy ratio (Ka, = ax/ay)

The estimation of the average of the actual distance from the electrode to each of the activated fibres, namely the distance from the electrode to the electrical axis of the activated fibres, could be carried out using one of the two following procedures. The first procedure described in our recent paper (D~MITROV et al., 1992) utilises the normalised integrals of the negative phases of the potentials recorded at different points, as informative parameters I i .

The second procedure is similar to the first one but some spectral characteristics of the potentials are used as informative parameters I i, namely (cf. LATEVA, 1988) the total power (TP) normalised by the squared value of the total spectral amplitude (TSA) for CEP recorded either unipolarly (TP,,i/TSA2,i) or by a bipolar parallel electrode with a small interpole distance (TPbipl I/TSA21plI). Note that in a case of cylindrical symmetry of the electrical proper- ties of the tissue the actual average distance Ye can be estimated using these procedures regardless of the actual anisotropy ratio of the volume conductor. The estimation error when spectral characteristics are used as informative parameters I i is of the same order as it is when integral characteristics are used (cf. DIMITROV et al., 1992).

On estimating the equivalent and actual average dis- tances, the tissue anisotropy ratio Ka. can be estimated through the ratio between these two distances.

As in the procedures suggested for estimating Ye inter- relationships between the informative parameters Ii at dif- ferent volume conductor points are used, these procedures can be applied only under the conditions of poor activity and an activated fibres' territory which is small with respect to ~e. For this reason )3 e and the tissue anisotropy ratio should be determined by a preliminary experiment under the condition of poor activity. Moreover, carrying out a preliminary experiment the frequency dependencies of a x, try and trx/ay should also be measured. The necessity of this measurement will be elucidated in the next section, Section 3.6. On estimating the tissue anisotropy ratio, ~ can be estimated even under maximum activity applying the estimated value of the equivalent average distance (;e ~ K a . ) �9

3.6 Estimation of the variations in the desynchronisation in the fibres" activation and in the risetime and duration of the l A P

After the values of the average propagation velocity and equivalent average distance were estimated, the calculation of the power spectrum of the hypothetical 'compound' intracellular action potential (CIAP) multiplied by the coefficient Aa, can be performed using the following relationships :

Table 1 Estimation errors (per cent) (in conformity with eqns. 5 and 6) of the equivalent distance from the territory geometrical axis to the recording electrode for different shapes and sizes (diameters) of the territory and different electrode configurations. Ye ~ = 15 mm

Territory shape

Territory Territory cross-section shaped Territory cross-section shaped Territory cross-section shaped diameter, as a circle, as an ellipse oriented as an ellipse oriented in

mm per cent transversely, per cent parallel, per cent

Recording by a unipolar electrode

8 - 3 - 4 - 2 12 --18 -2 2 - 3

Recording by a bipolar parallel electrode

8 - 6 - 7 - 3 12 -25 -27 - 4

402 Medical & Biological Engineering & Computing July 1992

AZa, PSECIAP] = P S E C E P , j (7) f4

for unipolar recordings or

A 2,, PS[ C I AP] = PS[ C EPb~e IJ] (8)

4 d 2 n 2 ~ K o ( # e ~ f ) 2

for recordings by a bipolar parallel electrode with a small interpole distance. As a result of this numerical calculation of the ClAP power spectrum, oscillations could appear in the spectrum high-frequency range. They are due to rela- tively very small values of the divisor (practically vanishing for high frequencies). For this reason the changes in the ClAP power spectrum can be investigated within a limited frequency range.

The ClAP spectrum calculated in this way depends on the desynchronisation in the fibres' activation (Fig. 3a) as well as on the amplitude, duration and risetime of the lAP. However, in conformity with the similarity theorem of the Fourier transform, changes in the IAP duration lead to alterations both in the magnitude and distribution of the spectral power (Fig. 3b), while changes in the desynchroni-

0

rn tn

c,

o

~j tn

0 C

1.0

0.9

08

0-7

0"6

0"5

0.4

0-3

0.2

0.1

0

1

0"12 0"24 0"36 0-48

G

4.0

35

0 3.0

3 Ck

~ 2-5

o ~ 2"0

~ 1"5 E o 1"0 c

0.5 1

0 A , L ~'-m,-~ -'2

0.12 0.24 0"36 0"48

frequency, kHz b

Fig. 3 The normalised power spectra of the 'compound' intracel- lular action potential for (a) different degrees of desynch- ronisation in .fibres' activation and (b) for different IAP durations. (a) desynchronisation; 1: Ores; 2: lms, 3: 2ms. The IAP duration Ti, = 1.25ms. (b) The IAP dura- tion T~,: 1; 1.25ms, 2; 2.50ms, 3; 5.00ms

Medical & Biological Engineering & Computing

sation have an effect on the position and steepness of the high-frequency slope without any alteration in the spectral powers in the ultra-low-frequency range. Note that the spectrum calculated according eqns. 7 or 8 is the same as the spectrum of the hypothetical 'compound' lAP obtained as a sum of N temporally dispersed IAPs.

A result similar to that shown in Fig. 3a is obtained in a case when the rate of the IAP rise is changed but the lAP duration remains unchanged. The greater the lAP risetime, the steeper the high-frequency slope. Therefore, it is diffi- cult to dissociate a change in the lAP rise from a change in the desynchronisation. In practice it can turn out that it is also difficult to separate a change in the IAP duration from a change in the desynchronisation or in lAP risetime, if simultaneously with them some other parameters such as N, Aan or V m are also changed.

Some additional complications arise from the frequency dependence of the tissue conductivities. On the one hand the ratio K~,/a, , = al/2/a 3/2 is not a constant but it is a - - x t - - y

frequency-dependent function. Starting from some fre- quency the tissue conductivities increase with increasing signal frequency (EPSTEIN and FOSTER, 1983). In addition the a r increasing starts at a higher frequency than the a x increasing (GIELEN, 1983; GIELEN et al., 1984; ALBERS et al., 1986; ALBERS, 1987; ROTH et al., 1988). According to Gielen (cf. GmLEN, 1983 or ROTH et al., 1988, Fig. 2) the a r increasing starts approximately at 20 kHz, which is practi- cally outside the frequency range of the lAP. As a result, the ratio K, , /a , , and consequently A~, increase with increasing signal frequency, which should lead to a decrease in the steepness of the high-frequency slope of the CEP spectrum. On the other hand, however, the increase in the anisotropy ratio K~, with a x increasing leads to a more pronounced increase in the steepness of the high- frequency slope of the weighting function spectrum and, hence, of the CEP spectrum. Owing to the higher order of the exponent of the frequency, which leads to shifting of the weighting function spectrum to higher frequencies, the increase in the high-frequency slope steepness is more pro- nounced for recording by a bipolar parallel electrode with a small interpole distance than for recording by a unipolar or by bipolar radial electrodes. In the case of a frequency- dependent volume conductor, if the change in the weigh- ting function spectrum is not taken into account under calculation of the ClAP spectrum, the steepness of the calculated spectrum high-frequency slope will be smaller than the real one. Therefore, in solving the inverse problem, the frequency dependence of the tissue conduc- tivities has an effect not only at small distances as in the case of solving the forward problem (GIELEN, 1983; ALBERS et al., 1986; ALBERS, 1987; ROTH et al., 1988), but also at great distances between activated fibres and recording elec- trodes. However, the correction of the calculated CIAP spectrum is not so complicated if the frequency depen- dencies of the ratio K, , /a , . and the anisotropy ratio K~, have previously been determined.

As a consequence of the reasons stated above it follows that there is some indefiniteness regarding the contribu- tions of the temporal dispersion of the excitation waves, the IAP duration and rise, considering them separately, to changes in the CIAP spectrum. Nevertheless, the investiga- tion of the CIAP spectrum could be useful as it is sensitive to alterations in these three parameters even under record- ings by surface electrodes. Moreover, it is possible to arrange the physiological experiment in a manner provid- ing a minimum possibility for some parameters to b e altered during the experiment. For example, under nerve stimulation it may be assumed that at the stimulation site the desynchronisation is equal to zero, but along the nerve

July 1992 403

the temporal dispersion of the excitation waves increases as a result of the differences existing between the propaga- tion velocities along different nerve fibres. Under normal conditions, it can be presumed that N, A , , , I'm, the lAP duration and rise do not alter along the nerve. Thus, the differences in the propagation velocities along different nerve fibres can be estimated.

4 D i s c u s s i o n Recording of the biopotentials by surface electrodes is

fully harmless for patients and individuals investigated. Moreover, the measuring system has minimum effect on the biological object. Therefore, the utilisation of surface electrodes allows greater freedom in the arrangement of the electrophysiological experiments in comparison with cases when needle elctrodes are used. For these major reasons, the present study focuses on analysing opportunities of solving the inverse problem using spectral parameters of potentials recorded at great distances typical of surface recordings.

The anisotropy is a result of the existence of a great number of fibres oriented practically in the same way in the volume conductor whose transverse conductivity is less (because of the cell membranes) than the extracellular medium conductivity. For this reason not only are the conductivities along the fibres a x and especially across the fibres a r less than the extracellular medium conductivity, but so is the common conductivity of the anisotropic volume conductor %, = ~ a y . On one hand, %, decreasing combined with K~, = a~,/ay increasing lead to SFEAP magnitude increasing. On the other hand, the radial decline of the weighting function, and in conse- quence of the SFEAP, is greater for an increased anisot- ropy ratio (DIMITROV and DIMITROVA, 1974).

GIELEN (1983), ALBERS et al. (1986), ALBERS (1987) and RoTrI et al. (1988) revealed that the effect of the frequency dependence of the tissue conductivities on the SFEAP amplitudes depends on the fibre-electrode distance. Taking into account the range of the SFEAP spectral amplitudes' distribution (DIMITROV et al., 1988; 1990; LATEVA et al., 1990), the actual shape of the frequency dependence of the tissue conductivities (GIELEN, 1983; ROTH et al., 1988) as well as the fact that the SFEAP amplitude is proportional to the TSA, it can be clarified why under solving the forward problem the frequency dependence of the tissue conductivities has an influence at small distances from the activated fibre only.

In fact, at great radial distances the SFEAP spectrum is concentrated within the low-frequency range because of the filtering effect of the weighting function. In this fre- quency range ax and tr r and, hence, aan and K, , do not depend on the signal frequency. The effect of the existence of a great number of fibres oriented practically in the same way in the volume conductor (changing the volume con- ductor conductivity) on the SFEAP amplitudes is reduced to a scale factor reflecting the difference between the ratio Ka,/aan and the extracellular medium resistance 1/a e which would occur in the case of there existing no other fibres but the activated one. With fibre-electrode distance decreasing the weighting function spectrum shifts to higher frequencies and its filtering effect in the high-frequency range becomes weaker. Thus, the effects of the frequency dependence of the ratio K~,/%, and of the anisotropy ratio Kan on the weighting function spectrum grow. As a result, the alterations in the TSA and, respectively, in the SFEAP amplitude owing to the frequency dependence of the tissue conductivities become greater. On the other hand, it is obvious that the relative changes in the TSA (i.e. SFEAP

amplitude) also depend on the low-frequency components' contribution to the TSA, which is affected by the weighting function filtering effect in the ultra-low-frequency range (cf. DIMITROV et al., 1990).

With increasing frequency, an increase in o- x is followed by a predominant increase in try. Thus, the increase in the anisotropy ratio Ka, is followed by its decrease (GIELEN et al., 1984; ALBERS et al., 1986; ALBERS, 1987; ROTH et al., 1988). In contrast to GIELEN (1983), ALBERS (1987) reported that the predominant increasing of tr r occurs between l kHz and 10kHz, i.e. in the frequency range where the lAP spectral power has still significance values. In this case the effect of the frequency dependence of the tissue conduc- tivities can be more complicated than the above discussed. The changes in the weighting function spectral powers as a result of the shifting of the functions Ko or K 1 along the frequency axis depending on the actual values of the o- x and ay, are different for different recording electrode con- figurations, which is due to the differences of the exponent of the frequency (cf. eqns. 1-3). The changes in the SFEAP spectral powers are determined by the inter-relation of the corresponding changes in the weighting function spectral powers and in the frequency-dependent function (Kan/aan) z = G x a ; 3.

Under recording by a bipolar parallel electrode changes of a r by a factor of N cause the function K 1 to shift by a factor of x/N and, hence, the weighting function spectral powers to change by a factor of N 3. This is neutralised by the corresponding change of the function a x a~- 3 and so the SFEAP spectral powers do not change. Under recording by a unipolar or a bipolar radial electrode changes of ay by a factor of N cause the weighting function spectral powers to change by a factor of N 2, which is N times smaller than the changes introduced by the function axar -3. Thus, the SFEAP spectral powers change by a factor of N - 1.

Carrying out the above consideration but referring to changes in a x for the two recording electrode configu- rations, a conclusion can be made that the higher the a x the lower the SFEAP spectral powers.

The calculation of the SFEAP and CEP by the network model used by GIELEN (1983) and ALBERS (1987) is very computer time-consuming. The computer time needed can be substantially shortened if the SFEAP (or CEP) is calcu- lated by the inverse Fourier transform of the product of the three frequency-dependent functions, namely the ratio Kan/aan and the Fourier transforms of the lAP (or CLAP) and of the weighting function, which was formerly calcu- lated in the frequency domain.

The CEP spectrum is a product of the three frequency- dependent functions under the condition of a uniform firing rate of the individual fibres. In the case of a variable firing rate the spectrum of the interference signal is a product of four frequency-dependent functions. The addi- tional fourth function (cf. BLtNOWSKA et al., 1979) reflects the parameters of the point process (AGARWAL and GOTT- LIEB, 1975; BLINOWSKA et al., 1979; CHRISTAKOS and LAL, 1980; PAN et al., 1989) related to the moments of the impulses' appearance.

5 C o n c l u s i o n s (a) The formation of the SFEAP in a volume conductor

can be reduced to a relatively simple filtering problem. (b) In contrast to the forward problem, for the inverse

problem solving the frequency dependence of the tissue conductivities is an important feature not only close to the fibre, but also at distance points.

404 Medical & Biological Engineering & Computing July 1992

(c) As a result of desynchronisat ion in the fibres' activa- tion, dips can exist in the CUP power spectra at fre- quencies depending on the degree of desynchronisat ion but not depending on the p ropaga t ion velocity.

References AGARWAL, G. C. and GOTTLIEB, G. L. (1975) An analysis of the

electromyogram by Fourier, simulation and experimental tech- niques, IEEE Trans., BME-22, 225-229.

ALBERS, B. A., RUTTEN, W. L. C., WALLINGA-DE JONGE, W. and BOOK, H. B. K. (1986) A model study on the influence of struc- ture and membrane capacitance on volume conduction in skeletal muscle tissue. Ibid., BME-33, 681-689.

ALBERS, B. A. (1987) Microscopic volume conduction of myoelec- trical activity. Ph.D. Thesis, Twente University of Technology, The Netherlands.

BLINOWSKA, A., VERROUST, J. and CANNET, G. (1979) The deter- mination of motor unit characteristics from the tow-frequency elctromyographic power spectrum. Electromyogr. Clin. Neuro- physiol., 19, 281-290.

BURGER, H. C. and VAN DONGEN, R. (1960) Specific electric resist- ance of body tissues. Phys. in Med. & Biol., 5, 431-447.

CHRISTAKOS, C. N. and LAL, S. (1980) Lumped and population stochastic models of skeletal muscle: implications and predic- tions. Bio. Cybern., 36, 73-85.

DIMITROV, G. V. and DIMITROVA, N. A. (1974) Extracellular potential field of an excitable fibre immersed in anisotropic volume conductor. Electromyogr. Clin. Neurophysiol., 14, 437- 540.

DIMITROV, G. V. (1987) Changes in the extracellular potentials produced by unmyelinated nerve fiber resulting from alter- ations in the propagation velocity or the duration of the action potential. Ibid., 27, 243-249.

DIMITROV, G. V., LATEVA, Z. C. and DIMnXOVA, N. A. (1988) Effects of changes in asymmetry, duration and propagation velocity of the intracellular potential on the power spectrum of extracellular potentials produced by an excitable fibre. Ibid., 28, 93-100.

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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 bioelectrical 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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