Systems and Computers i n Japan, Vol. 21, No. 6 , 1990 Trans la ted from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. 72-DII, No. 12, 1988, pp. 2683-2691.

Estimation for Spread Forms of Spindle Waves in Multichannel EEG

Yoshikazu Ueda and Naohiro I s h i i , Members

Facul ty of Engineering, Nagoya I n s t i t u t e of Technology, Nagoya, Japan 466

SUMMARY

The s p i n d l e wave i s one of t he most i m - p o r t a n t EEG s i g n a l s i n exp lo r ing t h e mechan- i s m of s l e e p . There have been few r e p o r t s , however, t h a t examined t h e s p a t i a l proper- t ies o f t h e s p i n d l e wave. One reason f o r t h i s may be t h a t t h e s p i n d l e wave i s d i f f i - c u l t t o be handled as a s t a t i o n a r y t i m e series s i n c e i t i s a short- term s i g n a l . Another reason i s t h a t a mult idimensional s i g n a l p rocess ing technique h a s no t been e s t a b l i s h e d f o r t h e mult ichannel s i g n a l , and an a n a l y s i s w i t h a tremendous amount of d a t a h a s n o t been t r i e d .

This paper a i m s a t t h e a n a l y s i s of t h e empi r i ca l knowledge of d o c t o r s by computer processing, and proposes a n a lgo r i thm which estimates t h e spread form of t h e s p i n d l e wave i n t h e mult ichannel EEG. The method i s based on t h e frequency-wavenumber spectrum, and has f e a t u r e s i n t h a t t h e two phas i c dominants and t h e d i r e c t i o n of spread are est imated from a smaller number of measure- ments, by u t i l i z i n g t h e phase r e l a t i o n s on t h e scope.

The u s e f u l n e s s of t h e proposed a lgo r - ithm w a s v e r i f i e d by a computer s imulat ion. The proposed e s t i m a t i o n method was app l i ed t o t h e a c t u a l s p i n d l e waves, and t h e follow- ing r e s u l t w a s obtained. The p h a s i c dominant of t h e s p i n d l e e x i s t s around t h e c e n t r a l f i e l d f o r t h e low-frequency components and around t h e p a r i e t a l f i e l d f o r high-frequency components. This i s a q u a n t i t a t i v e r e s u l t , which a g r e e s wi th t h e t r a d i t i o n a l q u a l i t a - t i v e f ind ings . A new obse rva t ion w a s a l s o made where t h e s p i n d l e s spread mostly from t h e p a r i e t a l f i e l d t o t h e f r o n t a l f i e l d .

1. In t roduc t ion

The a n a l y s i s of t h e s p i n d l e wave h a s a long h i s t o r y . With t h e r e c e n t i n t r o d u c t i o n

of computers, high-speed, l a rge - sca l e analy- sis of b i o l o g i c a l s i g n a l s , i nc lud ing EEG a n a l y s i s , has been r e a l i z e d . Thus, t h e r e i s produced a remarkable p rogres s of informa- t i o n processing. However, t h e r e s t i l l e x i s t a l a r g e number of i t e m s i n t h e human EEG f o r which t h e phys io log ica l knowledge is not s a t i s f a c t o r y . Although t h e r e i s a body of knowledge which w a s found empi r i ca l ly by medical r e s e a r c h e r s , i t i s q u a l i t a t i v e and not based on q u a n t i t a t i v e reasoning.

In t h e f i e l d of EEG du r ing s l e e p , v a r i - ous s t u d i e s have been made. There are stud- ies from a macroscopic viewpoint, such as the automatic decision-making of t he s l eep s t a g e and t h e i d e n t i f i c a t i o n of t h e s l e e p model. There are a l s o s t u d i e s from a micro- scopic viewpoint, such as t h e comparative s tudy of t h e e f f e c t s of t h e s l e e p depriva- t i o n , daytime s l e e p , mental load and exer- c i s e on t h e s i g n a l s , as w e l l as the s t u d i e s t h a t fol low t h e p a r t i c u l a r f e a t u r e s of s l e e p EEG. Although those s t u d i e s are performed independently a t p re sen t , they are e s s e n t i a l l y c o r r e l a t e d .

The EEG s i g n a l s considered i n those s t u d i e s are t h e s p i n d l e wave and slow wave [ l , 21. The reason f o r those waves being considered as a common i t e m i s t h a t they can e a s i l y be i d e n t i f i e d by observat ion and are used as t h e d e c i s i o n c r i t e r i o n f o r t h e s l eep s t ag ing [ 3 ] . We have i n v e s t i g a t e d the spin- d l e wave [ 7 - 91. Many o t h e r r e sea rche r s i n v e s t i g a t e d t h e s p i n d l e wave. However, few r e s e a r c h e r s i n v e s t i g a t e d t h e s p a t i a l f e a t u r e s .

The reason f o r t h i s seems t o be t h a t t h e s p i n d l e wave is a short-term s i g n a l and i s d i f f i c u l t t o handle as t h e s t a t i o n a r y time-series. Another reason is t h a t t h e multidimensional s i g n a l processing technique has mostly been developed f o r t h e image

10 1SSN0882-1666/90/0006-0010$7.50/0 @ 1990 S c r i p t a Technica, Inc.

Fig. 1. Assumption f o r spread of spin- d l e waves: ( a ) p a r a l l e l type, (b)

r a d i a l t ype , ( c ) complex r a d i a l type.

2. Est imat ion Algorithm f o r Spread Form of Spindle Wave

2 . 1 Assumptions f o r spread form of s p i n d l e wave

W e have obtained a r e s u l t i n which t h e r e seem t o exist two k inds of s p i n d l e waves. One is dominant i n t h e f r o n t a l f i e l d and has a lower frequency component. The o t h e r i s dominant i n t h e p a r i e t a l f i e l d and h a s h ighe r frequency component [9]. This paper examines t h e foregoing f i n d i n g by ob- se rv ing t h e types of i n d i v i d u a l s p i n d l e waves .

The spread forms of t h e s p i n d l e waves can roughly be divided as fol lows.

0 The s p i n d l e wave is dominant i n the o c c i p i t a l f i e l d o r o r i g i n a t e d from the per- iphery of t h e deep b r a i n , and spreads on the scope s u r f a c e [F ig . l ( a ) ] .

processing, and t h e technique h a s n o t been e s t a b l i s h e d f o r t h e mult ichannel s i g n a l . @ The s p i n d l e wave is dominant i n the

f r o n t a l f i e l d o r p a r i e t a l f i e l d , and o r ig in - a t e d from t h e deep b r a i n , spreading uniform- l y from t h e deep r eg ion t o t h e s u r f a c e [Fig. l ( b ) l *

The s p i n d l e wave o f t e n appears i n t h e sleep s t a g e c a l l e d s t a g e 2 . It i s known t h a t t h e frequency of t h e s p i n d l e wave changes with time from the e a r l y s t a g e of s l e e p t o t h e l a t e r s t a g e . Thus, it is recognized t h a t t h e s p i n d l e wave is r e l a t e d c l o s e l y t o t h e s l e e p mechanism. From such a viewpoint , it i s necessary t o exp lo re where and how t h e s p i n d l e wave i s r e l a t e d t o t h e s l e e p mechan- i s m . Th i s paper examines t h e appearance of t h e s p i n d l e wave i n t h e mult ichannel EEG i n t e r m s of t h e s p a t i a l phas i c r e l a t i o n s .

F i r s t , s e v e r a l assumptions are made con- cerning t h e spread form of t h e s p i n d l e wave. A mult ichannel p rocess ing i s proposed as an e s t i m a t i o n of t h e spread form of t h e s p i n d l e wave, based on frequency-wavenumber s p e c t r a l a n a l y s i s [ 4 - 61. The a lgo r i thm f o r t h i s method is p resen ted .

The wavenumber v e c t o r , which i s t h e im- p o r t a n t d a t a i n t h e wavenumber spectrum, i s handled i n gene ra l as a s t a t i o n a r y parameter over t h e e n t i r e space. I n EEG a n a l y s i s , how- ever , t h e s t a t i o n a r y p rope r ty does n o t e x i s t over t h i s space. Consequently, t h e proposed method determines t h e p a r t i a l wavenumber vec to r f o r t h e subspace, and performs t h e e s t ima t ion by a geometr ical method. The accuracy and t h e u s e f u l n e s s of t h e proposed e s t i m a t i o n method i s v e r i f i e d by a computer s imula t ion . F i n a l l y , t h e proposed estima- t i o n method is app l i ed t o t h e s p i n d l e wave. The r e s u l t i s p resen ted and a d i s c u s s i o n i s made.

@ The s p i n d l e wave with two o r more sources i n t h e deep b r a i n i s r e l a t e d t o t h e dominancy i n t h e f r o n t a l o r o c c i p i t a l f i e l d , which e x h i b i t s a behavior as i f exc i t ed by two o r more sources [Fig. l ( c ) ] .

When t h e s i g n a l with such spread forms are observed on t h e scope, they can be iden- t i f i e d as t h e advance o r r e t a r d of t h e phase, u n l e s s t h e frequency of t h e s i g n a l from t h e source undergoes t h e nonuniform modulation on i t s path. I n t h i s paper, those spread forms are denoted as 0 p a r a l l e l type [Fig. l ( a ) ] , @ r a d i a l type [Fig. l ( b ) ] , and @ complex r a d i a l type [F ig . l ( c ) ] . Those t h r e e types of spread forms are assumed.

I n t h e r a d i a l type, t he p o i n t with the most advanced phase i s c a l l e d t h e phas i c dominant. By our assumption, t h e phas i c dominant i s t h e p o i n t on t h e s k u l l c l o s e s t t o t h e source i n t h e deep b r a i n . Since two k inds of s p i n d l e waves are assumed t o e x i s t i n t h e complex r a d i a l type, t h e phas i c domi- n a n t s are l i m i t e d t o two.

2 . 2 Frequency-wavenumber spectrum

The frequency-wavenumber spectrum is defined as a Four i e r t ransform of t h e space- time c o r r e l a t i o n func t ion . The space-time c o r r e l a t i o n func t ion

11

i s the expected c o r r e l a t i o n between t h e vari- a b l e s(x, t) a t t i m e t i n space x and t h e v a r i a b l e s(x + r, t + T) a t t i m e t + T i n space x + r.

When t h e c o r r e l a t i o n f u n c t i o n is con- s i d e r e d , t h e s t a t i o n a r i t y is assumed i n general . Then t h e c o r r e l a t i o n f u n c t i o n can be r ep resen ted i n terms of t h e d i f f e r e n c e of t i m e and space coord ina te s . I n t h i s paper, t h e s t a t i o n a r i t y i s assumed both i n t h e t i m e and space domain. Then t h e c o r r e l a t i o n func- t i o n i s r ep resen ted as a func t ion of t h e t i m e d i f f e r e n c e and t h e s p a t i a l d i s t a n c e :

H ( Y , 5 ) = E " ( z , b 1 t ( z + r , t + r ) l

Four i e r t ransform of t h e forementioned ex- p re s s ion i n t i m e and i n space i s t h e f r e - quency-wavenumber spectrum:

I E x t r a c t c h a r a c t e r i s t i c f r e q . coniponents I __ 4

I Smooth spec t rum i n s p a t i a l domain I

--- - - -- &-% Yes l e s t spread forrii 3 s p a r a l l e l l=---

~~

Approximately e s t i m a t e p h a s i c dominant I I

I P r e c i s e l y es t i i i i a te p h a s i c doiiiinant I

P G h e t i c a l l y j u d i e e s t i m a t e d forms ]

Fig. 2 . Flowchart f o r e s t ima t ing spread forms of s p i n d l e waves.

To d e r i v e t h e spectrum by a computer i n t i m e i s more important i n t h e s p i n d l e wave than t h e s p a t i a l r e s o l u t i o n . From such a viewpoint, t h e smoothing i n t h e s p a t i a l

p rocess ing , t h e fol lowing procedures are ap p 1 ied .

domain is employed i n s t e a d of procedure @ t o r e t a i n t h e frequency information. 0 Four i e r t ransform i n terms of time

(FFT) i s performed, t o determine t h e cross- spectrum .

qhl m ( f 1 = ( 2 K 1 - y H n m ( r 1exp I - j 2 K f r 1 d r

where Hnm(') i s t h e space-time c o r r e l a t i o n

The f requency-wavenumber spectrum i s determined i n t h i s way, but what i s needed i s t h e wavenumber k maximizing the spectrum f o r t h e given frequency. dimensional p l ane i s considered i n t h i s paper , t h e wavenumber is t h e wavenumber vec-

Since a two-

t o r k = (kx, k ). The wavenumber vec to r max-

imizing t h e spectrum i s a f a c t o r r ep resen t - i n g t h e phase d i f f e r e n c e i n t h e measurement space, i n d i c a t i n g t h e d i r e c t i o n i n which

Y f u n c t i o n between channels n and m; @,(f) i s

t h e determined c r o s s spectrum between chan- n e l s n and rn.

@ Expectat ion is derived f o r t h e esti- t h e phase i s t h e most r e t a rded a t a par- mated c r o s s spectrum: t i c u l a r p o s i t i o n .

&*( f ) = E [ h m ( f 11 I f t h e measurement space i s s u f f i c i e n t - ly s m a l l , t h a t s m a l l subspace can be def ined

where ;d (f) is t h e e s t ima ted (smoothed)

c r o s s spectrum between channels n and m. nm as a p a r t i c u l a r p o s i t i o n . I n p r a c t i c e ,

t h e measurement space occupies a c e r t a i n volume. Then t h e c e n t e r of g r a v i t y of t h a t volume i s def ined as a p a r t i c u l a r p o s i t i o n . The a c t u a l u t i l i z a t i o n of t h e wavenumber is discussed i n t h e next s e c t i o n .

@ Four i e r t ransform i n r ega rd t o space (DFS) is performed:

where N i s t h e number of measurement chan- n e l s ; x and xm are s p a t i a l coo rd ina te s of

channels n and m. n

However, by t h i s procedure, t h e informa- t i o n of t h e s p i n d l e wave i n r ega rd t o t h e frequency i s l o s t i n t h e smoothing i n t h e expec ta t ion i n 0. up t o t h e p r e s e n t , t h e frequency r e s o l u t i o n

According t o our s tudy

The e s t ima t ion algori thm f o r t h e spin- d l e wave spread i n t h i s s tudy i s represented by t h e f lowchart of Fig. 2. I n t h e proposed e s t ima t ion , t h e remarkable components i n the frequency band of t h e s p i n d l e wave are se l ec - ted f i r s t . Then i t is examined whether o r no t t h e s p i n d l e wave spread is of p a r a l l e l type i n regard t o t h a t frequency component. I f t h e spread i s decided as p a r a l l e l type,

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!Gain

Frequency

Fig. 3. E x t r a c t i o n of c h a r a c t e r i s t i c components on frequency domain.

i t i s r e g i s t e r e d . I f no t , t h e spread i s re- garded as r a d i a l type, and t h e processing i s continued.

As a p rep rocess ing , t h e rough p o s i t i o n of t h e phas i c dominant i s determined. Then t h e two-stage e s t ima t ion f o r t h e a c c u r a t e p o s i t i o n i s made. t he phas i c dominant i s r e g i s t e r e d . A f t e r completing t h e foregoing procedure f o r a l l s i g n i f i c a n t frequency components, a compre- hensive d e c i s i o n i s made f o r t h e r e g i s t e r e d r e s u l t , t o be combined i n t o a n a p p r o p r i a t e r e s u l t . The processing i n each s t a g e i s descr ibed i n t h e fol lowing .

The r e s u l t i n g p o s i t i o n of

2 . 3 . 1 . Ext rac t ion of f e a t u r e frequency component

To extract t h e f e a t u r e components from t h e frequency components of t h e o b j e c t of a n a l y s i s , t h e mean spectrum i s c a l c u l a t e d f o r a l l measured p o i n t s . The components w i th l a r g e r ga in are e x t r a c t e d . The thresh- o ld i s determined from t h e maximum g a i n , and t h e components above t h e th re sho ld are ex- t r a c t e d as t h e f e a t u r e f r equenc ie s (Fig. 3).

I n t h i s s tudy , t h e mean of t h e maximum g a i n and t h e mean g a i n i s used as t h e th re sho ld :

where N is t h e t o t a l number of measured chan- n e l s ; Gn(kAf) i s t h e ga in of channel n f o r

frequency kAf; 5 ( k A f ) i s t h e maximum gain maX

Fig. 4 . P a r t i a l f i e l d s f o r t e s t i n g spread form as p a r a l l e l .

among z(kAf); L i s t h e number of frequency components.

When two ad jacen t f e a t u r e f r equenc ie s exist , it is impossible t o dec ide whether it i s due t o a s i n g l e frequency component o r a c t u a l l y two frequency components. This dec i s ion i s made by e s t ima t ing t h e spread of t h e frequency component, and i n t h e follow- ing, they are considered as s e p a r a t e compon- e n t s .

2 .3 .2 . Smoothing i n s p a t i a l domain

It h a s a l r eady been descr ibed t h a t the expec ta t ion p rocess ing i s requ i r ed i n de t e r - mining t h e frequency-wavenumber spectrum. Usually, t h e expec ta t ion i s c a l c u l a t e d by t h e ensemble averaging o r by smoothing. How- ever , s i n c e t h e r e do n o t e x i s t a l a r g e num- be r of s p i n d l e wave d a t a , t h e ensemble aver- age is d i f f i c u l t due t o i t s short-term property. Consequently, smoothing i s t h e only p o s s i b i l i t y .

It i s no t d e s i r a b l e t o degrade t h e r e s o l u t i o n i n t h e frequency domain i n t h e a n a l y s i s of t h e s p i n d l e wave. Consequently, t h e smoothing i n t h e s p a t i a l domain i s used in s t ead . The smoothing is made by t h e aver- aging based on eight-neighbor d a t a , as i n t h e usual image processing.

where Si is t h e spectrum of channel i; xi

and xn are t h e s p a t i a l coo rd ina te s normal-

i zed by t h e d i s t a n c e between measurement p o i n t s of-channels i and n. spectrum Sn(kAf 1 f o r channel n i s given by t h e foregoing expression.

The smoothed

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Fig , 5. Approximate e s t i m a t i o n of pha- s i c dominants.

2 . 3 . 3 . P a r a l l e l i s m check f o r spread

I n t h e fo l lowing , i t i s determined whether t h e s i g n a l spread i s r a d i a l o r pa ra l - l e l type , i n r ega rd t o t h e p a r t i c u l a r f r e - quency component. By t h e assumption i n t h i s paper , t h e phas i c dominant exists i n t h e measurement r e g i o n i f t h e spread i s r a d i a l . Consequently, t h e wavenumber v e c t o r s should have d i f f e r e n t d i r e c t i o n s i n t h e f o u r sub- r eg ions shown by t h e dashed l i n e i n F ig . 4 .

I n o t h e r words, t h e spread is dec ided as of p a r a l l e l t ype , i f a l l wavenumber vec- t o r s i n t h e f o u r r e g i o n s are i n almost t h e same d i r e c t i o n . The co inc idence of t h e wavenumber v e c t o r d i r e c t i o n s i s dec ided as fo l lows . When t h e minimum c o s i n e va lue be- tween v e c t o r s

exceeds t h e t h r e s h o l d (cos IT/^) ) , t h e d i r ec - t i o n s a r e dec ided as t h e same. When t h e spread i s d e c i d e d a s p a r a l l e l b y t h i s procedure, t h e wavenumber v e c t o r s are r e g i s t e r e d with- o u t t h e e s t i m a t i o n procedure f o r t h e phas i c dominant .

2 . 3 . 4 . C a l c u l a t i o n of p a r t i a l wave- number v e c t o r

A s a p rep rocess ing f o r t h e e s t i m a t i o n of t h e p h a s i c dominant, t h e wavenumber vec- t o r i s c a l c u l a t e d f o r t h e subregion . The wavenumber v e c t o r i s determined a c c u r a t e l y i f t h e r e are a l a r g e number of measurement p o i n t s . On t h e o t h e r hand, t h e computational complexity i n c r e a s e s r a p i d l y wi th t h e number of measurement p o i n t s , s i n c e i t is propor- t i o n a l t o t h e number of combinations of t h e measurement p o i n t s . Consequently, it is d e s i r e d t h a t t h e wavenumber v e c t o r should b e

............. A ..... ........................... B : b . . . i :. .:

:. b . .: . j . .: . + t + +

X : x j

........ ................................ . . . . . . . . + +

b b . . . . C D . .........,.

0 .

+ ............. . :. .j 0 0 ;+ x + j . + j x ; + ;.t + ; . :. . j . . i+ + +': . ........... .............

+ 0 . 0 .

Fig . 6 . F i e l d s f o r p r e c i s e e s t ima t ion of phas i c dominants.

determined from t h e least number of measure- ment p o i n t s .

I n t h i s s tudy , t h e wavenumber v e c t o r i s determined more e a s i l y by e s t ima t ing t h e phas i c dominant by two s t a g e s . I n p r a c t i c e , t h e wavenumber i s determined f i r s t us ing n i n e channels , which are composed of t h e p a r t i c u l a r measurement p o i n t and i t s e igh t - ne ighbors , as t h e subregion . The r e s u l t is def ined as t h e p a r t i a l wavenumber v e c t o r a t t h a t p o i n t . The p a r t i a l wavenumber v e c t o r s are determined f o r a l l of t h e measurement p o i n t s .

2 . 3 . 5 . Rough e s t i m a t i o n of phas i c dominant

I n t h e fo l lowing , t h e p h a s i c dominant is e s t ima ted roughly us ing t h e p a r t i a l wave- number v e c t o r a t each measurement p o i n t . S ince t h e wavenumber v e c t o r i n d i c a t e s t h e d i r e c t i o n of t h e most r e t a r d e d phase, t h e v e c t o r ob ta ined by r e v e r s i n g t h e s i g n ind i - cates t h e d i r e c t i o n of t h e most advanced phase. The p a r t i a l wavenumber v e c t o r , how- eve r , c o n t a i n s an error. Consequently, an allowance should be de f ined f o r t h e d i r ec - t i o n , as i n F ig . 5. The v a l u e , which i s pro- p o r t i o n a l t o t h e ga in of t h e measurement p o i n t a t t h e start of t h e v e c t o r , and in- v e r s e l y p r o p o r t i o n a l t o t h e d i s t a n c e from t h e s tar t p o i n t , i s g iven t o t h e subregion w i t h i n t h e allowance. The same procedure i s app l i ed t o a l l of t h e p a r t i a l wavenumber v e c t o r s , r e s u l t i n g i n t h e two-dimensional d i s t r i b u t i o n f o r each reg ion . The peak i n t h i s d i s t r i b u t i o n i n d i c a t e s t h e rough pos i - t i o n of t h e phas i c dominant. I n t h i s s tudy , 220 deg a long the d i r e c t i o n of t h e v e c t o r i s de f ined as t h e a l lowable range .

14

2 . 3 . 6 . Accurate e s t i m a t i o n of phas i c dominant

I n t h e fo l lowing , t h e phas i c dominant is est imated a c c u r a t e l y , using t h e d a t a a t t h e measurement p o i n t con ta in ing t h e p h a s i c dominant ob ta ined by t h e rough e s t ima t ion , t oge the r w i t h t h e d a t a a t i t s per iphery. Four methods are considered i n t h i s s tudy.

Method A (Fig. 6-A) u t i l i z e s 16 measure- ment p o i n t s i n fou r r e g i o n s , each con ta in ing e i g h t p o i n t s . Method B (Fig. 6-B) u t i l i z e s 1 2 measurement p o i n t s i n fou r r eg ions , each con ta in ing f o u r p o i n t s . Method C (Fig. 6-C) u t i l i z e s 1 2 measurement p o i n t s i n t h r e e r eg ions , each con ta in ing fou r p o i n t s , and t h e e s t i m a t i o n is made us ing t h e c e n t r a l r eg ion and t h e p e r i p h e r a l r e g i o n s f o r which the computation i s impossible . Method D (Fig. 6-D) u t i l i z e s 1 2 measurement p o i n t s i n two r eg ions , each con ta in ing s i x p o i n t s , and the e s t i m a t i o n i s made from t h e p a r t i a l wave- number v e c t o r s a t fou r measurement p o i n t s i n t h e r eg ions f o r which t h e computation i s pos- s i b l e and i n t h e considered r eg ion .

The phas i c dominant es t imated by e i t h e r method i s r e g i s t e r e d f o r t h e f i n a l dec i s ion . The fou r methods are desc r ibed i n t h e f o l - lowing.

Method A. The wavenumber v e c t o r s are determined from measurement d a t a a t e i g h t p o i n t s i n f o u r r eg ions . The phas i c dominant is determined as t h e geometr ical i n t e r s e c - t i o n , as t h e v e c t o r a t t h e c e n t e r of g r a v i t y i n each r eg ion . It i s n o t always t r u e that four v e c t o r s i n t e r s e c t a t a p o i n t ; and, con- sequent ly , t h e mean p o s i t i o n is employed. When t h e roughly est imated p o s i t i o n i s i n the outermost r eg ion , t h i s method cannot be app l i ed . t i o n is s h i f t e d t o t h e a d j a c e n t i n t e r n a l r eg ion , and t h e method i s app l i ed .

Then t h e roughly e s t ima ted posi-

Method B. Th i s method u t i l i z e s measure- ment d a t a a t f o u r p o i n t s i n each of fou r r eg ions . The method i s e s s e n t i a l l y t h e same as method A. t h e number of d a t a is reduced t o reduce t h e computation t i m e .

The on ly d i f f e r e n c e i s t h a t

Method C. I n t h e two forementioned methods, an excep t iona l procedure i s re- quired when t h e roughly e s t ima ted p o s i t i o n is i n rhe outermost region. In t h i s method, t h e roughly e s t ima ted p o s i t i o n i s always set as t h e c e n t r a l r eg ion , and t h e phas i c domi- nan t is determined from t h e wavenumber vec- t o r s a t t h e c e n t r a l r eg ion and t h e a d j a c e n t r eg ions f o r which t h e computation is poss ib l e .

Method D. This method i s a modifica- t i o n of method C where t h e r e are fewer d a t a

of wavenumber vec to r s . i s determined from t h e wavenumber v e c t o r s obtained from t h e measurement d a t a a t s i x p o i n t s i n f o u r r e g i o n s and t h e p a r t i a l wave- number v e c t o r s a t f o u r measurement p o i n t s .

The phas i c dominant

2 . 3 . 7 . F i n a l d e c i s i o n of es t imated r e s u l t

For t h e phas i c dominant o r t h e wavenum- ber v e c t o r i n p a r a l l e l spread f o r each fea- t u r e frequency component r e g i s t e r e d by t h e forementioned procedure, t h e d e c i s i o n i s made as t o whether o r no t they can be com- bined, based on t h e frequency d i f f e r e n c e o r t h e s p a t i a l d i s t a n c e . I n t h i s s tudy , t h e frequency r e s o l u t i o n is used as t h e u n i t . When t h e frequency d i f f e r e n c e i s 1 o r less, t h e wavenumber v e c t o r s are defined as t h e same frequency. When t h e d i f f e r e n c e i s 1 o r 2 , they are def ined as c l o s e f r equenc ie s . ‘hen t h e d i f f e r e n c e i s above 2, they are def ined as remote f r equenc ie s . Using t h e d i s t a n c e between t h e measurement p o i n t s as t h e u n i t , t h e d i s t a n c e less than 2 i s decid- ed as c l o s e , and t h e d i s t a n c e above 2 i s decided as remote.

When two est imated p o i n t s o r v e c t o r s s a t i s f y t h e fol lowing cond i t ion , they are merged i n t o one.

The frequency components of two phasic dominants are t h e same o r c l o s e frequencies , and t h e d i s t a n c e between measurement p o i n t s i s c l o s e .

The frequency components of two wave- number v e c t o r s are t h e same o r c l o s e f r e - quencies , and t h e ang le between v e c t o r s i s less than t h e threshold ( ~ r / 6 ) .

In e i t h e r case, when t h e merge t akes p l ace , t h e mean of t h e corresponding d a t a i s adopted as t h e r e s u l t of re-est imat ion.

3 . Computer Simulation

3 . 1 . Condition f o r s imula t ion

I n t h e fol lowing, t h e a b i l i t y of the proposed a lgo r i thm i n e s t ima t ing t h e phasic dominants a t two o r fewer p o i n t s as w e l l as t h e v e c t o r d i r e c t i o n i n t h e p a r a l l e l - t y p e spread, i s examined. When t h e r e e x i s t s only one phas i c dominant, t h e r e is no problem i n whatever way t h e frequency o r s p a t i a l posi- t i o n is defined. On t h e o t h e r hand, when t h e r e exist two o r more phas i c dominants, t h e frequency d i f f e r e n c e and t h e s p a t i a l d i s - tance w i l l a f f e c t t h e r e s u l t . When t h e r e e x i s t two p a r a l l e l - t y p e spreads, t h e f r e - quency d i f f e r e n c e and t h e d i r e c t i o n s w i l l a f f e c t t h e r e s u l t f o r t h e d i r e c t i o n .

15

l""t-1 ....

Fig. 7. Example of s imula t ion t o estimate phas i c dominant by n o i s e l e v e l . ( ' * I denotes set phas i c dominant, '+' de- n o t e s measurement p o i n t , I - ' denotes es t imated phas i c domi-

nant . )

The s imula t ion is made by d i v i d i n g t h e cases . The phas i c dominant i s c l a s s i f i e d as w a s descr ibed ear l ier . The frequency rela- t i o n s a r e divided i n t o t h e cases w i t h t h e same, c l o s e and remote f r equenc ie s . The s p a t i a l d i s t a n c e i s d iv ided i n t o c l o s e and remote d i s t a n c e s . Consequently, t h e r e is a s i n g l e way of s imula t ion f o r one phas i c dominant, and s i x ways of s imula t ions f o r two phas i c dominants. For t h e p a r a l l e l - t y p e spread, t h e r e i s a s i n g l e way of s imula t ion f o r one phas i c dominant, and s i x ways of s imula t ions f o r two phas i c dominants. The number of measurement p o i n t s i s set as 5 x 5 i n accordance wi th t h e a c t u a l EEG r eco rd ing cond i t ion.

The s i g n a l s used i n t h e s imula t ions are as fol lows. I n t h e case of t h e r ad ia l - type spread, a s i n u s o i d a l wave is given a t each phas i c dominant, with amplitude decaying exponen t i a l ly w i t h t h e d i s t a n c e from t h e measurement p o i n t t o t h e phas i c dominant and with t h e phase delay p r o p o r t i o n a l t o t h e d i s t a n c e . With t h e s i g n a l a t t h e phas i c dominant as t h e r e f e r e n c e , a pseudo-Gaussian n o i s e i s added w i t h SN r a t i o of 0, 5, 10, 15, and 20 dB. When t h e r e e x i s t two phas i c dominants, t h e s i g n a l s are superposed.

I n t h e case of p a r a l l e l spread, a sinu- s o i d a l wave is given t o each measurement p o i n t , w i t h t h e same ampli tude and t h e phase delay along t h e propagat ion d i r e c t i o n . With t h e s i g n a l amplitude as t h e r e fe rence , a pseudo-Gaussian n o i s e is added w i t h SN r a t i o of 0, 5, 10, 15 and 20 dB. When t h e r e exist two v e c t o r s , the s i g n a l s are superposed.

3.2. Resu l t of s imula t ion and discus- s i o n s

The r e s u l t of s imula t ion f o r t he r a d i a l - type spread is discussed i n t h e following. Figure 7 shows a s imula t ion example of t h e e s t ima t ion f o r t h e phas i c dominant. The phas i c dominant i s set a t the p o i n t indi- cated by t h e a s t e r i s k . For t h e s i g n a l s w i th n o i s e w i t h v a r i o u s SN r a t i o s , t h e phas i c dominant i s est imated by the proposed algor- ithm, and t h e resu l t i s ind ica t ed by d o t s . The p l u s s i g n i n the f i g u r e i n d i c a t e s t h e d a t a measurement p o i n t . Twenty t i m e s siniu- l a t i o n were performed f o r each s e t of parame- ters.

It i s seen from t h e f i g u r e t h a t under t h e worst cond i t ion of SN r a t i o of 0 dB, the est imated p o i n t s scatter around t h e set p o i n t , b u t they concen t r a t e w i th the improve- ment of SN r a t i o . Even f o r t h e most s ca t - t e r ed case, t h e r e is l i t t l e s c a t t e r i n g com- pared wi th t h e d i s t a n c e between t h e measure- ment p o i n t s .

The s ta t i s t ica l e v a l u a t i o n of t h e re- s u l t of s imula t ion i s shown i n t h e following. Figure 8 compares t h e four methods i n terms of t h e accuracy of t h e e s t ima t ion of t he phasic dominant. F igu re 9 shows t h e e f f e c t of t h e n o i s e l e v e l . The upper f i g u r e s are t h e r a t i o of c o r r e c t e s t i m a t i o n ( i -e . , num- be r of c o r r e c t estimation/number of phasic dominants), and t h e lower f i g u r e s are t h e mean e r r o r of d i s t a n c e with t h e measurement d i s t a n c e as t h e u n i t . The number of simula- t i o n s i s 80 f o r t h e former and 100 f o r t h e l a t te r .

16

I---. A B C D

T

' a r : .6 I L-+-- -, I - -

0 5 10 15 2 O d B r

S/N r a t i o E s t i m a t i o n method

Fig. 8 Fig. 9

Fig. 8. Estimation of phasic dominants by p r e c i s e es t imat ion methods. Upper f i g u r e shows t h e r a t i o of c o r r e c t es t imat ion. Lower f i g u r e shows t h e est imat ion e r r o r of d i s tance . Simulation d a t a : (a ) one phasic dominant; (b) two phasic dominants with i d e n t i c a l frequency and near d i s tance ; (c) two phasic dominants with near frequency and near d i s tance ; (d) two phasic dominants with f a r frequency and near d i s tance ; (e ) two phas ic dominants with i d e n t i c a l frequency and f a r d i s tance ; ( f ) two phasic dominants with near frequency and f a r d i s tance ; and (g) two phas ic dominants with

f a r frequency and f a r d i s tance .

Fig. 9 . Estimation of phas ic dominants by noise level. Upper f i g u r e shows the r a t i o of c o r r e c t es t imat ion. Lower f i g u r e shows the est imat ion e r r o r of d i s tance .

Simulation d a t a are t h e same a s i n Fig. 8.

Comparing t h e r a t i o of c o r r e c t estima- t i o n , i t i s seen t h a t t h e r a t i o decreases with the degradat ion of t h e s imulat ion con- d i t i o n , independently of t h e method. I n methods A and B, t h e r a t i o of c o r r e c t esti- mation is low when t h e s p a t i a l d i s t a n c e is remote. This seems due t o t h e f a c t t h a t t h e est imat ion i s u n s a t i s f a c t o r y when t h e phasic dominant is c l o s e to t h e per iphery. Compar- ing the methods i n terms of t h e mean e r r o r of d i s t a n c e , method A is t h e b e s t , followed by method C , al though t h e r e is a d i f f e r e n c e depending on t h e condi t ions.

Comparing t h e e f f e c t of n o i s e i n terms of t h e r a t i o of c o r r e c t es t imat ion , t h e r e is l i t t l e d i f f e r e n c e due t o t h e noise l e v e l , al though t h e r e is a d i f f e r e n c e due t o t h e simulation condi t ion. Comparing t h e methods i n terms of t h e mean e r r o r of d i s tance , t h e e r r o r decreases with t h e increase of SN r a t i o . Viewed as a whole, method A seems t o be t h e b e s t f o r accura te es t imat ion of the phasic dominant.

The r e s u l t of s imulat ion of p a r a l l e l spread is discussed i n t h e following. Figure 10 shows the e f f e c t of no ise l e v e l . The upper f i g u r e is t h e r a t i o of cor rec t es t imat ion and t h e lower f i g u r e i s the mean e r r o r of angle , where t h e e r r o r i s defined a s t h e angle between the t h e o r e t i c a l and t h e estimated d i r e c t i o n s . The number of simula- t i o n s f o r each measurement d a t a i s 400.

Comparing t h e e f f e c t of the noise l e v e l i n t e r m s of t h e r a t i o of cor rec t es t imat ion, it is seen t h a t t h e c o r r e c t es t imat ion de- c reases with t h e degradation of t h e simula- t i o n condi t ion. The r a t i o increases with t h e increase of SN r a t i o and decrease of the noise l e v e l . Comparing i n terms of the mean e r r o r of angle , t h e e r r o r is l a r g e when two v e c t o r s a t t h e same frequency are d i r e c t e d t o completely d i f f e r e n t d i r e c t i o n s , s ince only one vec tor can be estimated. The e r r o r is s u f f i c i e n t l y small f o r o t h e r condi t ions.

17

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0 5 10 15 20dB S / N r a t i o

F ig . 10. Es t ima t ion of spread d i r e c - t i o n s by n o i s e l e v e l . Upper f i g u r e de- n o t e s t h e r a t i o of c o r r e c t e s t ima t ion . Lower f i g u r e deno tes t h e e s t i m a t i o n e r r o r of a n g l e between set d i r e c t i o n and e s t ima ted one. S imula t ion d a t a : (h) one spread; ( i ) two sp reads w i t h iden- t i c a l frequency and nea r d i r e c t i o n ; ( j ) two sp reads w i t h n e a r frequency and n e a r d i r e c t i o n ; ( j ) two sp reads w i t h nea r frequency and nea r d i r e c t i o n ; (k) two sp reads w i t h f a r frequency and n e a r d i - r e c t i o n ; (1) two sp reads w i t h i d e n t i c a l frequency and f a r d i r e c t i o n ; (m) two sp reads w i t h nea r frequency and f a r d i s - t ance ; and (n) two sp reads w i t h f a r f r e -

quency and f a r d i s t a n c e .

Viewed as a whole, t h e r e s u l t of esti- mation i s degraded w i t h t h e deg rada t ion of t h e s imula t ion c o n d i t i o n . A s t o t h e a c t u a l spread of t h e s p i n d l e wave, one can ha rd ly expec t t h a t t h e c o n d i t i o n i s degraded both i n t e r m s of frequency and t i m e domains. Con- sequen t ly , t h e proposed a lgo r i thm w i l l g i v e a f a i r l y s a t i s f a c t o r y r e s u l t of e s t ima t ion .

4. Es t ima t ion of Spindle Wave Spread

4.1. Data f o r s p i n d l e wave

The s p i n d l e waves cons idered i n t h i s s tudy a r e those which appeared simultaneous- l y i n mul t i channe l s , i n t h e improved wave- form r e c o g n i t i o n method [ 7 ] w e proposed. The measurement p o i n t s i n t h e s l e e p EEG

c-

. .

0

Fig . 11. EEG measure ( l e f t ) and square measure ( r i g h t ) .

r eco rd ing were set a t 19 channels s p e c i f i e d by t h e i n t e r n a t i o n a l 10 - 20 method, a s i s shown i n F ig . 11 ( l e f t ) . To s impl i fy t h e a n a l y s i s , t h e measurement p o i n t s are ad jus t - ed t o 5 x 5 squa re g r i d measurement p o i n t s shown i n F ig . 11 ( r i g h t ) . The va lue a t t h e measurement p o i n t wi thout d a t a i s comple- mented by t h e neighborhood d a t a . I n o t h e r words, t h e a n a l y s i s is made assuming t h a t t h e t o t a l number of channels i s N = 25. The frequency components are assumed as 10 - 16 Hz, con ta in ing t h e band of t h e s p i n d l e wave.

4.2. Resu l t and d i s c u s s i o n s

This s tudy i s concerned wi th t h e e x i s t - ence of two k inds of s p i n d l e waves as w e l l as t h e i r d i s t r i b u t i o n s on t h e scope. The waves are d iv ided i n t o those wi th frequency components below 12.5 Hz and those wi th fre- quency component above 12.5 Hz. F igu re 12 shows t h e phas i c dominant of t h e s p i n d l e wave es t imated by t h e a c c u r a t e e s t i m a t i o n o f method A .

Figure 13 shows t h e r e s u l t where t h e mean p o s i t i o n due t o t h e frequency d i f f e r - ence on t h e scope i s c a l c u l a t e d from t h e d i s t r i b u t i o n of t h e phas i c dominant ob ta ined by t h e a c c u r a t e e s t ima t ion .

I n e i t h e r case , t h e mean p o s i t i o n of t h e p h a s i c dominant w i t h frequency component below 12.5 Hz d e v i a t e s t o t h e f r o n t a l d i r e c - t i o n , compared w i t h t h a t w i t h frequency com- ponent above 12.5 Hz. t h e s i s i s app l i ed t o t h e d i f f e r e n c e of those s p a t i a l p o s i t i o n s , and a r e s u l t was obta ined by t h e T - t e s t t h a t t h e r e is a s i g n i f i c a n t d i f f e r e n c e wi th t h e r e j e c t i o n r a t i o of 1 pe rcen t .

The t e s t i n g of hypo-

The r e s u l t sugges t s t h a t t h e s p i n d l e wave w i t h h ighe r frequency component appears t o t h e o c c i p i t a l s i d e on t h e scope compared wi th t h e one wi th lower frequency component. This ag rees wi th our p rev ious observation'.

18

* \ . . * :*: . *. " :.

\ . . .I. .& .

Fig. 12. S p a t i a l d i s t r i b u t i o n of phas i c dominants.

front under 12.5Hz

-.2

A B C D

Fig. 13. S p a t i a l p o s i t i o n of phas i c dominants by frequency components.

Fig. 14. Spread d i r e c t i o n of sp ind le waves.

F igu re 14 shows t h e spread of t h e pa ra l - l e l - type s p i n d l e wave. I n t h e f i g u r e , t h e wavenumber v e c t o r i s shown with t h e c e n t e r of t h e c i r c l e as t h e start. It is obvious t h a t t h e r e e x i s t s a l a r g e number of v e c t o r s d i r e c t e d t o t h e f r o n t a l f i e l d , and t h e r a t i o of t h e v e c t o r s i n t h e range of k 4 5 deg from t h e f r o n t a l d i r e c t i o n i s 77.4 percen t . Con- s i d e r i n g t h a t t h e wavenumber v e c t o r repre- s e n t s t h e spread of t h e s i g n a l i n d i c a t e s an i n t e r e s t i n g f a c t t h a t t h e r e ex i s t s a l a r g e number of s p i n d l e waves spreading from t h e o c c i p i t a l t o t h e f r o n t a l f i e l d .

5. Conclusions

This paper proposed a n e s t i m a t i o n algor- ithm t o i n d i c a t e t h e spread o f t h e s l e e p s p i n d l e wave. consider ing t h e work and f u n c t i o n assignment of t h e mul t i channe l s i g n a l p rocess ing in t h e frequency and s p a t i a l domains. Another

The algori thm is ob ta ined by

f e a t u r e is t h a t t h e phase information i s ex- t r a c t e d using t h e frequency-wavenumber spec- trum, and t h e phas i c dominant t o two p o i n t s as w e l l as t h e spread d i r e c t i o n are est imated based on a small number of measurement p o i n t s .

It w a s seen as a r e s u l t of a p p l i c a t i o n t o t h e s l e e p s p i n d l e wave t h a t most of t h e p a r a l l e l - t y p e s p i n d l e waves spread from t h e p a r i e t a l t o t h e f r o n t a l f i e l d . This i s an i n t e r e s t i n g p rope r ty and needs f u r t h e r in- v e s t i g a t i o n . When the r a d i a l spread i s c l a s s i f i e d according t o t h e frequency com- ponents, they concen t r a t e t o t h e c e n t r a l f i e l d f o r low f r equenc ie s and t o p a r i e t a l f i e l d f o r h igh frequencies . This ag rees with our p rev ious r e p o r t s and t h e t r a d i t i o n a l f i nd ings .

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Problems l e f t f o r f u r t h e r s tudy are t h e r e l a t i o n between t h e work and f u n c t i o n as- signments of t h e mult ichannel s i g n a l process- i ng t o t h e frequency and s p a t i a l domains. The a lgo r i thm should be improved so t h a t as much information as p o s s i b l e i s r e t a i n e d . The spread types of t h e s p i n d l e waves should be

1.

2.

3.

i n v e s t i g a t e d f u r t h e r .

REFERENCES

Azumi and Shirakawa. Usefulness of s p i n d l e wave as t h e i n d i c a t i o n of s l e e p . Psych. Med., 25, pp. 169-176 (1983). Kobayashi, Endo, T s u j i , and Takahashi. A mathematical model of s l e e p . Tech. Rep. , I. E. C. I . E. , Japan, MBE86-4 (1986). A. Rechtschaffen and A. Kales (Tr. Seino). At las of Sleep EEG. Ishiyaku Publ. Co. (1965).

4.

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9.

M. Hino. S p e c t r a l Analysis . Asakura Publ. Co. (1977). P. L . Nunez. E lec t r ic F i e l d s of t h e Brain. Oxford Univ. Press (1981). L. J . Piason and D . G. Chi lders . Fre- quency-Wavenumber Spectrum Analysis of EEG Mul t i e l ec t rode Array Data. IEEE Trans. Biomed. Eng., BME, 2, 3, pp.

I s h i i , Ueda, I s h i i and Terajima. A study of waveform recogn i t ion i n EEG a n a l y s i s . Jour . MEBE, 24, 7, pp. 524- 529 (1986). Ohta, Terashima, I s h i i , Ueda, I s h i i , Iwata, and Okada. Fea tu re e x t r a c t i o n of s l e e p s p i n d l e wave i n a l l - n i g h t s l eep EEG. C l in . EEG, 28, 8, pp. 527-533 (1986). Ueda, I s h i i and I s h i i . A study of m e a - surement method i n frequency a n a l y s i s of s l e e p s p i n d l e wave. Trans. (D), I . E . C . I . E . , Japan, J70-D, 10, pp. 19W- 1991 (Oct. 1987).

192-206 (1974).

AUTHORS (from l e f t t o r i g h t )

Yoshikazu Ueda graduated i n 1984 from t h e Dept. I n f . Eng., Fac. Eng., Nagoya I n s t i t u t e of Technology, and obtained a Master's degree i n I n f . Eng. from t h e r e i n 1986. P r e s e n t l y , he i s i n t h e d o c t o r a l program a t Nagoya I n s t i t u t e of Technology. H e is engaged i n r e sea rch on bio- l o g i c a l s i g n a l p rocess ing , e s p e c i a l l y t h e time-series of mult ichannel s l e e p EEG.

Naohiro I s h i i graduated i n 1963 from t h e Dept. Electrical Eng., Fac. Eng., Tohoku Uni- v e r s i t y , and obtained a D r . of Eng. degree from t h e r e i n 1968. H e w a s an A s s i s t a n t , Fac. of Med., Assoc. P ro f . i n 1975, and p r e s e n t l y i s a P ro f . of Elect. Comp. Eng., Fac. Eng., Nagoya I n s t i t u t e of Technology. H e i s engaged i n r e sea rch on threshold l o g i c , medical information processing and non l inea r processing.

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