Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.protein.bio.msu.ru/~akula/JourVND6_09KulaichevLO.pdf Дата изменения: Mon Nov 2 11:08:42 2009 Дата индексирования: Mon Oct 1 19:50:30 2012 Кодировка:

, 2009, 59, 6, . 766­775


612.821.6

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© 2009 . . .
. .. , e mail: akula @mail.ru 23.12.2008 . 09.02.2009 . , . . : , , , .

On the Informative Value of Coherence Analysis in EEG Studies
A. P. Kulaichev
Department of the Higher Nervous Activity, Lomonosov State University, Moscow e mail: akula @mail.ru Numerous methodological and computing errors characteristic of coherence analysis as applied to EEG studies are discussed. The comprehensive review of basic disadvantages of coherence function does not allow the author to consider it as a reliable and effective index of EEG synchronism. Key words: coherency, spectral analysis, EEG nonstationarity, amplitude modulation.

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) . , [5]. . [7, . 138]: " 1960 . [15] , [14] ". , [15] , , .. , 1955; .. , 1958; .. .. , 1959 . , ( ) [14] [17]. , . , x(t) y(t) ( ) :
(f) =
2

G xy ( f ) , Gx ( d ) Gy ( f )

2

(1)

|Gxy( f )| ­ ; Gx( f ), Gy( f ) ­ x(t) y(t). [1, . 146; 7, . 138; 10, . 342; 13, . 36; 17, . 172] (1), , . [11, . 271] , ( , ). , (1). , [1] . 146 (1) , ( ) . 517.


[2]: 2 . 138, .. (1) . 112, 1 . 294, . 289 . [9, . 463] , . 3, .4, . 43 , (1) , . [10, . 330] (1) , ( [10, . 322]), . , "" . , , . . . f
x ( t ) = a cos 2 ft + b sin 2 ft ; y ( t ) = c cos 2 ft + d sin 2 ft . (2)

, a, b, c, d ( X( f ), Y( f )) () f
X(f) = a + ib; Y(f) = c + id, (3)

i ­ . X( f ) Y( f ), ..
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Ax( f ), Ay( f )
Gx ( f ) = Ax ( f ) = a + b ; Gy ( f ) = Ay ( f ) = c + d , x ( f ) = arctg ( b / a ) ; y ( f ) = arctg ( d / c ) .
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(4)


(5)

, , X( f ), , |X( f )|, ­ x( f ), a b . x(t), y(t)
G xy ( f ) = e + i g , (6)

e = ac + bd, g = bc ­ ad.
G xy ( f ) = =
2

e +g =
2

2

2

(7)

( ac + bd ) + ( bc ­ ad ) =

Gx ( f ) Gy ( f ) ,


x ( f ) = arctg ( g / e ) = x ( f ) ­ y ( f ) . (8)

E[...] ( ).
(f) =
2

E [ G xy ( f ) ] , E [ Gx ( f ) ] E [ Gy ( f ) ]

2

(9)
2



|E[Gxy( f )]|

=

E[e] + E[g]

2

=

= E [ a c + b d ] 2 + E [ b d ­ a d ] 2 , E[Gx( f )] = E[a2+ + b2]. , (9) . , [16] ( "" ), (1),

[17], . . . ­ ­ / . [1, . 179; 11, . 271]: 1) , ­ ; 2) ; 3) , , . . ; 4) , . ­ . , , : , ; , , .. , ( ). , , , . , . (
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), , . , , , , , ­ . . , , , , . , . . . : , , , , , .. , , , [2, . 112; 10, . 342; 11, . 270; 13, . 36; 17, . 172]. , . , , , : ) (1) , ; ) ­1 + 1.


. (9) . (1) [6, . 196], (7)
' ( f ) =
2

E [ G xy ( f ) ] E [ G xy ( f ) ] = , E [ G x ( f ) G y ( f ) ] E [ G xy ( f ) 2 ]

2

2

(10)

2 (, [11, . 270] ). 2: ) 2 = 0­1 0.95; ) 0.074. '2 [11, . 270]: , ­ . , . , . / (8) . 2 , , , . (9) (10) (7), . ( . 1) (9) (10) , . (. 1) x(t) y(t) ( 1­3) . 2 90 . ( 1 x(t) 0 ., 2 ­ 90 .,
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1. Table 1. Coherency dependence on spectral amplitude phase ratio for two monoharmonic signals 1 1 2 E[...] 1 2 E[...] 1 2 E[...] a 4 0 4 0 4 0 b 0 4 0 2 0 1 Gx(f ) 16 16 16 16 4 10 16 1 8.5 c 0 0 0 0 0 0 d 4 4 4 2 4 1 Gy(f ) 16 16 16 16 4 10 16 1 8.5 e = ac + g = ad ­ + bd ­ bc 0 16 8 0 4 2 0 1 0.5 16 0 8 16 0 8 16 0 8 |E[Gxy(f )]|2 = 128 E[Gx(f )]E[Gy(f )] =256, 2 = 0.5 E[Gx(f )Gy(f )] = 256, '2 = 0.5 |E[Gxy(f )]|2 = 68 E[Gx(f )]E[Gy(f )] = 100, 2 = 0.68 E[Gx(f )Gy(f )] = 136, '2 = 0.5 |E[Gxy(f )]|2 = 64.25 E[Gx(f )]E[Gy(f )] = 72.25, 2 = 0.889 E[Gx(f )Gy(f )] = 128.5, '2 = 0.5

2

3

y(t) 90 .). ( 1), (9), (10) 2 = '2 = 0.5, 90 . . 2 ( 2) 2 = = 0.68, ( 3) ­ 2 = 0.889 ( '2 = 0.5). 2 , . , . , , 2 (, '2 2). . () .

, . [6, . 200] , : , . , . [6, . 200]: , . , . , , , . [6, . 187] , 40% , ( , ).
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. 1. . ­ ( 9 ), 128 , 0.5­32 , 64 ; ­ ( 8 ); ­ ( 8 ); ­ ( 8 ). Fig. 1. Typical EEG spectrums: ­ occipital EEG signals with high portion of alpha rhythm: sampling rate = 128 Hz, analog filters with bandwidth of 0.5­32 Hz, records length = 64 s; ­ amplitude cross spectrum (sin gle 8 s epoch); ­ phase cross spectrum (single 8 s epoch); ­ coherency spectrum (over 8 epoch averaged).

( ) , .


. . 1,
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. 2. (. 1, ) 9 (. 1, ) 32 2 . Fig. 2. Cross spectrum phase values of alpha rhythm dominating frequency 9Hz (fig. 1, ) for 32 epochs of 2 s length.
1.0 0.8 0.6 2 0.4 1 0.2 0 0 20 40 60 80 100 %

) ±160 . , [8, c. 68­82; 12, . 29; 13, . 134­137 .]. (. ). , , ­ . , , , ­ . . , . . : . . . , . X( f ), Y( f ) x(t) y(t) : ) r r = 01 (, 0 .); ) 1r , 0­
1

1

. 3. (1) (2) . Fig. 3. Coherency (1) and correlation coefficient (2) dependencies on the percentage of noise in a signal.

. . 1, 8 , 9 . . . 1, , , . . 2 9 32 . , (

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360 . 30 , 1000 2. 2 (0­r)% . 3, : 1) S , ; 2) 60% , 2 ; 3) 2 : 20­60%; 4) 30­40% , 2 > 0.7. [1, . 308] n 2 ±0.1 2 (. 2). 10, , 2 > 0.8. , 0.1­0.8, . . . ? (, , , , )


0.5

4

8

13



. 4. , , 4 , 16 . Fig. 4. Coherency spectrums calculated by three EEG analysers (epoch length = 4 s, over 8 epoch averaged).

100% . ( EDF ), ( . 4): , , .
2. 2 n, 2 ±0.1 Table 2. True values of coherency 2 and number of aver aged ensembles n necessary for estimation of true value 2 with error less than ±0.1 2 n
59

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6

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. 6. ( 16 ): 1 ­ = 4 ; 2 ­ = 2 ; 3 ­ = 2 + . Fig. 6. Variability of averaged coherency values in EEG frequency domains depending on the length of analysed epoch and used correcting window (over 16 epoch averaged): 1 ­ T = 4 s; 2 ­ T = 2 s; 3 ­ T = 2 s with Hann's window.

. 5. , , . . 6 , . 5. , ­ 28­36% ( , 1 ). , , . [8, . 71, 74], . , , ( , ) , , ,
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. 5. ( 16 ), ­ : = 4 ; = 2 ; = 2 + . Fig. 5. Coherency dependence on the length of analysed epochs T and used correcting window (over 16 epoch averaged), from above to down wards: T = 4s; T = 2s; T = 2s with Hann's window.

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1. ., . . .: "", 1971. 540 . 2. ., . . . 1.2. .: "", 1971. 316 . 285 . 3. .. : . . . . 1998. 29(3): 3555. 4. .., .., .., .. . . . . 2002. 4: 8495. 5. .. . . . . . 1997. 47 (5): 918926. 6. .. . .: , 2007. 640 . 7. .. . .: "", 1989. 400 . 8. .., .., .., ., .., .., .., .., .., .., .. . .: "", 1988. 207 . 9. . .. . .: "", 1990. 584 . 10. ., . . .: "", 1978. 428 . 11. .. . : , 1989. 389 . 12. .., .., .., .. . .: "," 1987. 256 . 13. .., .. . .: "", 1978. 238 . 14. Adey W.R., Walter D.O. Application of phase detec tion and averaging techniques in computer analysis of EEG records in the cat. Exper. Neurol. 1963. 7: 186­209. 15. Goodman N.R. Measuring amplitude and phase. J. Franklin Inst. 1960. 270: 437­450. 16. Nolte G., Bai O., Wheaton L., Mari Z., Vorbach S., Hallett M. Identifying true brain interaction from EEG data using the imaginary part of coherency. Clin. Neurophysiol. 2004. 115: 2292­2307. 17. Walter D.O. Spectral analysis for electroencephalo grams: Mathematical determination of neurophysi ological relationships from records of limited dura tion. Exper. Neurol. 1963. 8: 155­181.
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