3350449513

V* in ctapmion -«*> łstubllshed spatul flltering techniques

♦ r«z, I

•V*c!row*phalograpy. eagnetoencephalogiapy. spatial fllt*ring, lnv*rse probl«os

•    • ’ i- ’    • •- i - - - - - - - - - - - - -

IM* sak* th# assutption that the approilMt* posltlons and oflłntatlons of    • •

n*uronal sourc*s of lnt*r*st hav* b»*n id*ntiti*d. Tn* posltlons of th* sources nay be obtaln*d, for exa*iple. by using Ing Ing j Mthogs such as LORETA ItttfMttUm). spatlal fllt*rlng bas*d r*ural actlvlty lndlces ldtt{łtoiSMV2tU, Piotrowskl2014a, I |Pioirouski2914<). or wlih r*f*renc« to publlshed r*uioici*nce stodlłi that hav* ldentlfied cortlcal reglons of Interest. On the Otber hand. \t«xthf(>ri«atltioa» for EEG and SBparately for HEC with nic* reference* herc.;    i

♦    rwtlon) •tesl

♦

♦cai late ♦ 1S |T*X

tio {    )    - - -|

j\l !-sr(tlC' ('. ■ Itlo-i)

,Assuae s\«s to be a v«ctor of rea\-v«lu*d randen variabl*s s> _;.\dots.x .1, each wltb a flnlte varianc*. Th* nean waloe of S\x$ I ls d**ot«d by SVMtht*(E)(\x]S and th* covarlance *atrlx of $\x$ ls denoiłd by tVnsthcal{OH\xJt. SI .$ stands for Identlty ratru of siz* $ks, whil* sl ,-»S stands for a dlagcnal Mtrli of size SkS ln whlch th* flrst SpS entrics of th* nain dlagonal |ar* «qual to 1 and all oth*r «ntrles areVqual to-0. let S\signa <,\dots,\slgM »$ be the slngular valu«s of natrlx sAUn\Mthbb(R)'<> '*»»»» "li organized ln nonincreasing order, where |M,i Let naw $AUn\Mttibt){R}*{n \liw n)$ be o symetrie I “»trlx. $\lent>da(A)S denotes th* s-ector of eigen-ralues of SAS organized ln non-increaslrg order. Let SA-M\Slgra,H‘tS be an eigerwalu# dccoaposltlon of SA.s By SP    we denote th* orthogonal projectlon fdtrlx onto the subspac* spanned by

tha eigenvectors corr*spondlng to the SsS largest elg«nvalu*s of-SAS. where Sl\leq s\leq n.S for a glven positlve (s«ol)d*flnlt«l rsstrl. >A\in\i*athbb{R}'<>> \ti»«* we denot* by $A'<s :i\ln\rsłthbb(R)-(™ »)$ the wniqu* *atrlx such that <A'Ii/.'IA’0/2NA$ and LSA'U .'is ls also posltlyę Ise*lldeflnit* uit^HornlWS). ~

Tf* array r*spons* (leadfleld) aatrlx deflning th* relatlonshlp between-sls dipole sources and SrS sensors ls constructed as- [

i^S    • Idots.M{\th*ta t))Vln\Mthbb(R}*(* uim lis where tor Sl=l.\dots.lS. SH(\theto ,)\lnViathb6lR)'«S ls the    |

,l*adfl*ld ęatrix of tn* Sis-Ih sourc*. s\r-(\th*t* iAdot$.\theta OS ls such that S\th*ta ,-\{r i.u_i\)$, where Sr ,$ ls the    I

sourc* wmion, and i-i ,s ls th* orlłntatlon unit v*ctor for the SiS-th sourc*. Me assine that the sources¹ posltlons and onentations ar* flx*d during the Masurewnt period. For nadablllty. w* will drop the explldt depcndenc* of SHS oo-S\thS belcw.

♦    larsgus

♦

•x/l«te «ndard La» 4*/SlZ* ♦naih/«i ♦he o ♦nath/an ♦nath/ans* ♦*ath/a*s

♦    rvsth/ar •phlcs ♦phlcs/grt ♦ptocs ♦phlcs ♦phlcs/tr ♦exconf

♦    phlcs/dv

♦    fonts/ar, ♦fonts/ar ♦apltx ♦yent/co*

I - :    (ćŁ■- ME!, forward Model)

w» consid*r sls dicole sources ot bram *l*ctrical actluity and EEG / MEG neasurcoents obtalned with St^lS sensors at a spcdtied ti*e int*rval Iwithout loss of głnerallty. let thls int*rval t>* slaply S10.11SI. Then. th* Sn \tlies 1S randa* vector SlyttlS co*pos*d of t»* aeasureaents at a glv*n tli* Instant StS can b* iodel*d as \cite(VanV**nl997. Kosherl999. S«klhara2088}:    i

(-••• •(-■• ’•>:•) • .

(aodel)

•(tl-H j<t)*H ₍\q_c(t)a\n<t).

f <    i*» }    .    -

•here rardoa v«ctor s\q(t)\ln\wthbb{R)“-S represents electric / łagnetlc dipole rownts ot sources of interest.

.SHUn\Mthbb{R)‘(*\tUH 11$ thelr array response (leadfleld) wtrlx. rardon vector S\q clt|\ln\rwthbb(R)*H represents interferlng I activity correlated with actlvtty of interest S\qS. SH Aln\iMthbb(R)*(«iti»« Us thelr leadfleld *atrlx. and randen v*ctor ł\'-iti\ln\nathtb(R)*.s expr*sses rols* *«asur*d at the sensors. Me assuie that the leadfleld aatrlces SHs and SK _CS havc fuli colsan rank. Also. we assuie that ł\(\q(t>.\ t\ln(B,l|\)S. S\(\q <(t>A t\ln(0.1|\)j. and S\(\n(t)A t\in|8.1}\)Ś are zero-eean weakly statlonary stochastlc processes and that S\»athcaUO>l\q(t)l=<S. S\isMhcal{0)I\q «(t))*OS« and S\nathcal{0}|\n(t))=KS ara c«sitive deflntte. Me eooel noise S\n$ as |    u )

>H 9 l_a(t)*\n_,.

| r-3{wqu4tlo*)

^■er* s\q _ts represents th* background actlvlty of the braln and-An ,s a Gaussian aeasurcient nois* uncorrelated at the sensors. |

♦ e/dte.S ♦atlng/ ♦e/lfthe «figur*

»

♦figur ♦Is/tn.sta ♦fss/ot ♦fonts/u* ♦fonts/un ♦apUx/ur ♦at Uncs ♦ra-phy

2.2. KEC/ MM; fnrwutd'1«W

We conudcr ■' dipole ««irre» nf bum dccirwal xiivii> md M:G / MMi incnurnacnb chłiuicd »ilh >n I iClHorw w a >pccM>cd inne uativaJ (wilhoul Iih» of promili). Irt liii' nvr>jl he Minpl) [l> 11> Ihcn.llie ri * 1 nnikini veclnr * tl ciMiipoKd «f ik- mtatureincab m o gne* limę in»i»m I can lv n.Hi ltd a. |h «)

y(l) - //qii W.q,(f) • nu).    <l>

where uniom «co lor q{f) •- R' n|>rełom» cłolrtc / niw nclicdipolemomenwe<tourne.ofiMcml II R"'¹ Iheir :«n> fopauo lleadfkUlrandom łcotiw q,(l) • S' rcfreonu mcerfeniiy ailmty coirrtiled wiMiaclinlycd inlcr-cm q. II R"'¹ Ihm leadlirkl matm. wd random \vcKa n|C| • R” e<|nc»-*-' nonę mcaiurtil 41 ilw wnwn. Meli-sumę diat tfie leadlield raairkes H and II, have (uli column rank Ako. weKwae(hai (q(4) i O l|) |q, (ri. i -li V).aml |nli;. l • |ll.I;) a«r rero-inean wcaklt sianon-aryiiośhatiu |w (te sse> and diai Pq l|j f.T>q. fi O. and T>n •:    V acpenalisedefmilc Weinodrl noise n a»

n - «sq_tif| ł n,„    (2)

where q, reprc«nlł tlić batkymj ni acimly of ile liraan ani n,„ a Gaussian iucasuvn»:nl rausc unumeialed :i llic sensors

Mos^l III cncompasses resimji siak- ani eseni-rrlak-d ntmiiayi a. folk™> we issume iliai tfe reeauareincr* inay h: distdtd i«o rwo stayes (such ls pir- and pui-siuwilus penods m cu<nl-rvl»Kd rccmdinysi. In ihj fnj siaęe oni) hockpcund acliviiy

-V KEUUCEO-KANK Ml.LING SPYTIAI. III.TER 4. M:\IEKICAI. l:\AMPI.K

|4| T 1’iodoł prtc»»:x.l di ks f<« I xf.rr*. pp 4

|S) k \ Ihw*

hrklye U«K

|6] 11 D Van A. Su/oki.

Incwl) ma; sol. 44, pd

|7) 1 C Moshrr. M MMi foruardl no. ? pp 24S-2

18) K Stkih»J and

nrs for £Je« łomot A Ikrlm. 2WI8.

{■oćel}) encoepesses resting stat* and eeent■related recordlngs a* follows: w* as o iwo st*g*s (such as pre- and post-stinulus perlods ln event-r*lated recordlngs)

♦ŻMUlt ♦8

♦■•filie \o

(»*»-<*d rar* aulling Spatial Fllter)

s*mon{lluHrual Caaapl*}

♦fll lWi

♦    r*z, I

♦

5. KEFEKENCES

|l | k D PaKual-M.nquL "Rrvvu of meihods for sohmc il>: EEG msirse f<obiciu' Asrernoii"mil łaurnul<•(lino elre/mmApwnMi. sol 1. na I. pp 7S-86. I9W

|2| A. Mo»eev. .' M Gatpar. I A ScttreidcT. aml A. T. Hcrdoun. “Applitation of mulo-souKe minimum swv a ret beuu&amiers fot n\omiru:i>jn i< owrelalol neural aiiissh " Nrumlmitt, sol S8.no 2,pp 481-496.Scpc. 2011.

«4

♦eu-ral a

, V*<lEEEblb)

bl»llogr*Ohy{r*T*r*rc*s)

? :(:oc«*n )

♦at Une

♦    rez. I

♦

•rnatlon) ♦tes)

«

♦cas Q

|31 T- pK«rmxii. Ił Guterrv/- I Yamala. and 1 /y-

peressiu. Ra!usvd-rani neural ailmly indr.u fur KŁGiMEG uuilb-source locali/alion." m Prot IEEE ICASSP. Bcecnce, Ilaly.Mas 201-!. pp 4708-4712.

’««d savlng)