BioMedical Engineering OnLine

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1 BoMedl Engneerng Onne BoMed Centrl eserh Effet of neurl onnetvty on utoovrne nd ross ovrne estmtes Mrk M Steker* Oen Aess Address: Dertment of Neurology, Gesnger Medl Center, Dnvlle, PA 78 USA Eml: Mrk M Steker* - mrk_steker@yhoo.om * Corresondng uthor Publshed: 6 Jnury 7 BoMedl Engneerng Onne 7, 6: do:.86/475-95x-6- eeved: Otober 6 Aeted: 6 Jnury 7 Ths rtle s vlble from: htt:// 7 Steker; lensee BoMed Centrl td. Ths s n Oen Aess rtle dstrbuted under the terms of the Cretve Commons Attrbuton ense (htt://retveommons.org/lenses/by/., whh ermts unrestrted use, dstrbuton, nd reroduton n ny medum, rovded the orgnl work s roerly ted. Abstrt Bkground: Mesurements of uto nd ross ovrne funtons re frequently used to nvestgte neurl systems. In nterretng ths dt, t s ommonly ssumed tht the lrgest ontrbuton to the reordngs omes from soures ner the eletrode. However, the otentl reorded t n eletrode reresents the suermoston of the otentls generted by lrge numbers of tve neurl strutures. Ths retes stutons under whh the mesured uto nd ross ovrne funtons re domnted by the tvty n strutures fr from the eletrode nd n whh the dstne deendene of the ross-ovrne funton dffers sgnfntly from tht desrbng the tvty n the tul neurl strutures. Methods: Dret lton of eletrostts to lulte the theoretl uto nd ross ovrne funtons tht would be reorded from eletrodes mmersed n lrge volume flled wth tve neurl strutures wth sef sttstl roertes. esults: It s demonstrted tht the otentls reorded from monoolr eletrode surrounded by dole soures n unform medum re redomnntly due to tvty n neurl strutures fr from the eletrode when neuronl orreltons dro more slowly thn /r or when the sze of the neurl system s muh smller thn known orrelton dstne. eordngs from qudruolr soures re strongly deendent on dstnt neurons when orreltons dro more slowly thn /r or the sze of the system s muh smller thn the orrelton dstne. Dfferenes between bolr nd monoolr reordngs re dsussed. It s lso demonstrted tht the ross ovrne of the reorded n two stlly serted eletrodes delnes s ower-lw funton of the dstne between them even when the eletrl tvty from dfferent neuronl strutures s unorrelted. Conluson: When etrellulr eletrohysolog reordngs re mde from systems ontnng lrge numbers of neurl strutures, t s mortnt to nterret mesured uto nd ross ovrne funtons utously n lght of the long rnge nture of the eletr felds. Usng reordng eletrodes tht re bolr or qudruolr mnmzes or elmntes these effets nd hene these eletrodes re referred when eletrl reordngs re mde for the urose of uto nd ross orrelton nlyss of lol eletrl tvty. Pge of (ge number not for tton uroses

2 BoMedl Engneerng Onne 7, 6: htt:// Bkground eordngs of sontneous or evoked eletrl tvty from lrge grous of neurons re often used to robe the funton of neurl systems. One fundmentl roblem tht rses n nterretng suh dt s fndng the loton the genertors of sef eletrl tterns from reordngs mde t dstne from the soures. Ths "nverse roblem" does not hve unque soluton [,] sne there re lolzed hrge dstrbutons suh s the losed dole sheet tht do not rodue eletr felds outsde the regon ontnng the soures [] nd hene do not hnge the mesured otentl. However, f ddtonl restrtons n be led on the tyes of soure dstrbutons tht re relst, unque soluton my be found [4] n ertn ses. The most ommon set of smlfyng ssumtons s tht the soure onssts of sngle or very few doles but ths ssumton my not be resonble f there re strong orreltons between dstnt neurons. Beuse of the dffultes nherent n solvng the "nverse roblem", t s very mortnt to obtn nsght nto ths roblem through n understndng of the "forwrd roblem" of redtng the eletr felds generted by vrous known genertors. The rmry gol of ths er s to onsder the "forwrd roblem" of omutng the felds generted by lrge numbers of orrelted neurl strutures wth the sef ntent of understndng those fetures of the genertor, reordng eletrode, nd neuronl orrelton funton whh determne whether reordngs of eletrl tvty re domnted by neurons ner the eletrode or neurons fr from the eletrode. From the outset, t should be noted tht ths roblem s very dfferent from tht of determnng the dstrbuton of felds generted by ndvdul soures. Even n tht smle roblem there re "ner feld otentls" suh s the qudruolr felds generted by trvellng ton otentls [5] whh re hghly eked ner the soure nd "fr feld otentls" whh n vry more slowly thn /r t lrge dstnes from the soure [5]. In ths er, only soures ssoted wth trdtonl "ner feld otentls" wll be onsdered. The seondry gol wll be to demonstrte the effet tht long rnge nture of eletrostt felds hs on the ross ovrne between the sgnls reorded from eletrodes serted by gven dstne. Seflly, n ths er, the uto nd ross ovrne funtons for sgnls reorded from eletrodes mmersed n lrge ontnuous medum of neurl elements wll be onsdered. Sne the gols of ths er re qulttve rther thn quntttve, very smlst model system wll be studed n whh grou of very smll reordng eletrodes s led t the enter of very lrge homogeneous sherl regon ontnng dentl neurl strutures. The deendene of the feld reorded from these eletrodes on the rdus of the sherl regon wll be tken s n ndtor s to whether there s sgnfnt ontrbuton to the reorded otentl from neurons fr from the eletrode. The mesured ross ovrne funton wll be omred wth the tul ross ovrne of tvty n stlly serte neurl strutures.. Methods/esults. The rdo Consder the stuton n whh smll eletrode of rdus s led n homogeneous sherl regon of rdus flled wth unformly dstrbuted neurl genertors. One gol of ths er wll be to omute the ross ovrne funton: ν (, s, τ whh reresents the ross ovrne between sgnls reorded from two eletrodes one loted t oston nd nother loted dstne s wy from the frst s funton of,, τ(tme dfferene nd the ssumtons bout the orrelton between neurl genertors throughout the medum. Also of nterest wll be study of ν (τ ν (,, τ whh s the utoovrne funton for sgnls reorded from the enter of the sherl medum s funton of. Stutons n whh the vlue of lm ν( τ does not est suggest tht under these ondtons, reordngs wll be domnted by ontrbutons from dstnt neurons nd wll be strongly deendent on the sze nd she of the regon n whh the eletrode s mmersed. The ore of the roblem n be esly llustrted n smle rgument remnsent of tht whh leds to Olber's rdo. If the otentl from set of neurl soures dstne r from n eletrode flls s the net ontrbuton to the otentl from ll neurl r m soures n sherl shell loted t dstnes between nd + Δ from the eletrode wll be roortonl to 4π Δ. If m s less thn or equl to eh suessvely more dstnt shell of soures ontrbutes ether m greter or equl mount to the totl otentl nd hene the reorded otentl s very senstve to the et she nd sze of the neurl regon. When m, the ontrbuton from eh suessvely more dstnt shell dmnshes s but the totl otentl dverges logrthmlly nd so, even n ths se, t s eeted tht the reorded otentl wll be strongly deendent on the she nd sze of the volume n whh the reordng eletrode s led. Only when m > does the mesured otentl reh fnte lmt for lrge vlues of nd hene most of the Pge of (ge number not for tton uroses

3 BoMedl Engneerng Onne 7, 6: htt:// eletrl tvty reorded n be onsdered to ome from neurons ner the eletrode. Ths result suggests tht when reordng from hghly orrelted dolr or qudruolr soures most of the eletrl tvty reorded from n eletrode omes from neurons fr from the eletrode. Ths rdol result rses out of the mlt ssumton tht neuronl tvty s hghly orrelted even t lrge dstnes. The followng dsusson wll demonstrte how the bove result deends on the detls of the orrelton funton desrbng tvty n stlly serted neurons.. The multole enson The generl eresson for the eletr otentl rodued t the loton of the eletrode s: ρ(, t ϕ( t, d ( where s the sherl volume of rdus nd ρ( ', t s the hrge densty t the ont ' nd tme t. Consder the ontrbuton to the totl otentl mde by the genertors n smll element of se Δ entered round the ont. Ths s gven by: ρ(, ϕ(,, t tδ d + Ths eresson n be rerrnged to dsly the otentl n terms of the multole enson of the hrge densty n the regon by usng the relton []: where the ngles (θ", φ" desrbe lol oordntes (.e. orentton of " wthn Δ, (θ, φ " desrbe the orentton of the regon Δ n relton to the orgn of oordntes, nd t s ssumed tht the regon of nterest s smller n ts mmum dmeter thn the dstne between the orgn of oordntes nd the regon. The Y lm (θ, φ re sherl hrmons nd Y lm (θ, φ re ther omle onjugtes. Substtutng ( nto ( demonstrtes tht the totl ontrbuton to the otentl from soures n the regon Δ n be wrtten s: where Δ ( l l 4π Y ( l l lm θ, φ Y l m l + + lm ( θ, φ ( ϕ( t,, l Δ 4π Y ( θ, φ Q ( l l lm lm l m l + +,Δ t ( 4 l Qlm(, t Ylm(, (, t d Δ θ φ ρ + 5 Δ Q lm s the verge multole moment er unt volume t nd tme t (e the multole moment densty. It s mortnt to understnd the rorte hoe of the volume. It should be hosen to be lrge enough tht eh volume onssts of very smlr olleton of neurl elements but smll enough tht ts ledng multole moments re smlr eh tme the regon s tvted. Ths restrton n be reled gretly wthout lterng the underlyng onlusons of ths study f one onsders the felds generted by ouled neurl strutures of dfferent tyes. The totl otentl wll smly be the sum of the otentls generted by the strutures of eh tye. Integrtng (4 over ll the volume elements n the shere system yelds the followng eresson for the totl reorded otentl s: where (θ, φ re the ngles desrbng the loton of the element under study reltve to the enter of the oordnte system. The dvntge of ths enson over ( s tht n mny neurl systems the felds generted re domnted by ether qudruolr (l or dolr (l omonents nd so only theses omonents of the bove sum re of rtl mortne.. The ovrne funton One of the most ommonly used desrtors of sontneous eletrl tvty s the ross-ovrne funton. The gol of ths seton s to relte the tterns of multole moment tvton to the ovrne between the sgnl reorded by n eletrode of rdus nd smlr eletrode dstne s from the frst eletrode embedded n sherl volume of rdus. One stndrd defnton of the ross ovrne funton s: where the followng notton s used: ( l ϕ( t, π 4 Y ( θ, φ Q (, t l l lm lm l m l + + d 6 ( ν ( s,, τ ϕ ( t, ϕ ( t, ϕ ( st, τ ϕ ( st, τ ( ( ( 7 T ft ( lm T ftdt ( ( 8 T T for ny funton of tme f(t. Substtutng (6 nto (7 yelds: Pge of (ge number not for tton uroses

4 BoMedl Engneerng Onne 7, 6: htt:// ν( s,, τ d d l m l l+ l + + l l ml where s now tken s the regon bounded by the surfe of both eletrodes nd the shere of rdus whh bounds the regon flled wth neurl strutures. One very mortnt smlfton ours when t n be ssumed tht the struture of eh neurl regon remns onstnt over tme nd tht the flututons n neurl tvty over tme n eh regon just modfy the mgntude of the multole moments over tme. In rtulr, ths mens tht t s ossble to wrte: Q (, t ( σ(, t lm where lm ( gves the mgntude of the mml multole moment generted when eh regon s fully tvted nd σ(, t s slr funton desrbng the degree of tvton of neurl elements t oston t tme t. Note tht n ths reresentton, the orentton of sef multole moments s sefed by the reltve weghts of the moment for the dfferent vlues of m for gven vlue of nd so ths theory lso llows for rbtrry vrtons of the multole moments from loton to loton. Ths mens tht: Substtutng nto ( yelds: where: Ylm Yl ( θ, φ ( θ, φ m lm lm l l Q (, t Q (, t Q (, t + τ Q (, t + τ ( 9 m m + s ( lm Qlm(, t Qlm(, t Ql m (, t Ql m ( + τ, t + τ lm( l m ( σ(, t σ(, t σ(, t + τ σ( (, t + τ ν( s,, τ Y d lm( θ, φ Yl m ( θ, φ d lm( l m ( l m lm l l+ l + l+ g (,, l τ l + s + ( g(,, (, t (, t (, t ( τ σ σ σ + τ σ, t + τ ( s the red ovrne between the level of tvty n dfferent regons of the neurl network. It should be noted tht the ovrne hs been ssumed to deend only on the dstne between the regons nd the loton of one of the regons. Before ssng onto the more generl se dsussed n Aend B, t s nstrutve to onsder the se n whh the level of tvton t n eh regon s totlly unorreltted: g (,, τ δ( h (, τ ( 4 for ny funton h(, τ. Ths mles tht: ν( s,, τ d l l l Y l + lm( θ Y, φ l m ( θ, φ lm( ( ( lm h, τ ( 5 l l mlm l s In Aend D, the stuton where s s not zero s dsussed, but n the se where both s : ν( τ d l l l m lm l l+ l Y lm( θ, φ Yl m ( θ, φ lm( l m ( h(, τ ( 6 l The net smlfton omes when dvdng the volume ntegrl nto ts rdl nd ngulr omonents: d d dω π π dω dφ sn( θ dθ Defnng: t s ossble to wrte: Now, the ngulr ntegrl n (7 s over fnte regon nd sne the nd h(, τ re bounded then so s q ll' (, τ. et: then: Ω Note tht sne the soures re ether dolr or qudruolr then: q ll' (, τ unless l l' or Ω l l qll (, τ dω Ylm Y mlm l l+ l + ( θ, φ lm ( θ, φ lm( lm ( h(, τ ( 7 Ω ν( τ d q τ l l ll (, ( l l + 8 q, τ q ( τ; ( < ( ll ll 9 ν( τ qll ( τ d q l l l+ l ll ( τ l l l+ l l+ l l l + ( so tht l+l'-l >. Ths mens tht s nreses wthout lmt the ovrne funton remns bounded by the quntty: lm ν( τ < qll ( τ l+ l l l l+ l Pge 4 of (ge number not for tton uroses

5 BoMedl Engneerng Onne 7, 6: htt:// whh s fnte. Thus, ontrbutons to the utoovrne funton from neurl strutures fr from the eletrode re smll when the tvty n dfferent regons s not orrelted nd s eeted from the romnent erne of the rdus of the eletrode,, s mnly determned by neurl strutures ner the eletrode. Ths onluson s not nfluened by the orentton or the dstrbuton of multoles wthn the volume s long s the multole densty remns fnte. In Aend A, the sgnl utoovrne funton ν (τ s omuted for more generl ovrne funton for nteger vlues of : g (,, τ m mn h(, τ Although ths mthemtl rgument s more omle, the onluson s tht qudruolr (l soures fr from n eletrode wll not sgnfntly ontrbute to the sgnl ovrne funton s long s mn > nd m s fnte. For dolr soures (l the neuronl orrelton funton, the utoovrne funton n generl does not hve defned lmt s for orrelton funtons of the form ( wth m < s requred for onvergene of the ntegrls requred for omutton of the utoorrelton funton. It s lso ossble to estmte the deendene of the ross ovrne funton on the dstne between the reordng eletrodes. In Aend C t s demonstrted tht, n the se where the tvty n vrous neurl strutures s unorrelted (.e. soure orrelton funton of the tye (4, f the soures re dolr (l, l' then ν(,, s τ. If the soures re qudruolr, then l, l' s ν(,, s τ nd. The rtl observton s tht s even when the tvty n the neurl strutures s unorrelted, the mesured ross ovrne funtons delne s ower-lw funton of the dstne between the eletrodes nd do not dsly the eeted delt funton behvor. Furthermore, s shown n Aend D, when the neuronl orrelton funton s of the ower lw form nd the neuronl strutures rodue redomnntly dolr feld, the mesured orrelton delnes more slowly wth dstne thn the neuronl orrelton funton. In the settng of qudruolr soures, the mesured ross ovrne funton hs the sme deendene on the dstne ( between the eletrodes s the neuronl orrelton funton..4 More omle reordng eletrodes The rguments resented bove nd n Aend A formlly refer to reordngs from monoolr eletrode. However, s demonstrted n Aend B, smlr onlusons n be drwn for omle eletrode whose otentl s the lner ombnton of the otentls t number of other eletrodes. In rtulr, n Aend B, t s demonstrted tht bolr reordng of dolr soures s equvlent to monoolr reordng of qudruolr soures. Thus, lthough monoolr reordngs from hghly orrelted dole soures (s hrterzed by < mn n equton ( re domnted by soures dstne from the eletrode, bolr reordngs from suh hghly orrelted dole soures reeve the lrgest ontrbuton from soures lose to the eletrode. Thus, the reordngs from dfferent eletrodes n be vstly dfferent when they re reordng from etended neurl strutures wth long-rnge orreltons..5 The qulttve rgument In order to obtn qulttve understndng of these mthemtl rguments, t s helful to return to the orgnl rgument of seton.. From the dsusson bove, t s ler tht tht the key element left out of the orgnl rgument ws the fnte rnge of neuronl orreltons. It s nstrutve to onsder very rude qulttve rgument whh tkes nto ount these fnte rnge effets. Assume tht the neurl system n be dvded u nto dsrete elements of volume v nd lbel eh element wth the nde, then the reorded otentl n be wrtten: N m σ r where s the moment ssoted wth eh element, r s the dstne of the ' th element from the eletrode, σ reflets tht stte of tvton of the 'th element nd N s the totl number of elements: N 4 π where s the rdus of the neurl system. In order to estmte the otentl, t s resonble to note tht most of the dstnes, r, re on the order of so tht: N m σ The sgnl vrne s gven by: Pge 5 of (ge number not for tton uroses

6 BoMedl Engneerng Onne 7, 6: htt:// ( Defnng: ( σ σ σ the term: N where N s the number of other elements wth whh the gven element s orrelted. If the orrelton dstne s C, then: N sne the number of elements orrelted wth gven element n never be lrger thn the totl number of elements n the system. Thus: so tht: m Ths qulttve rgument suggests tht reordngs from dole soures m re strongly deendent on the sze of the neurl system when the sze of the system s muh smller thn the orrelton dstne but s stble to hnges n when the system s muh lrger thn the orrelton dstne. The reorded sgnl from qudruolr soures m s ndeendent of only for the lrge systems whle those from hgher order moments m 4 re lwys ndeendent of.. Dsusson The frst result of ths er s tht when there re strong long-rnge orreltons between neurl strutures, the N σ σ σ NN 4 π Mn 4 π, σ σ 4 π Mn (, ( m ( ; << m6 ( ; >> m ( otentl reorded t monoolr eletrode n be domnted by the tvty n neurl strutures fr from the eletrode even f the otentl from eh grou of neurons s of the "ner feld" tye (delnng s /r or fster. Seflly, when the orreltons hve fnte rnge C, domnnt ontrbutons ome from the dstnt neurons when the sze of the system,, s muh smller thn C. In the se where the underlyng neuronl orrelton funtons hve ower lw deendene on the dstne, t ws found tht when the orrelton between eletrl tvty n dfferent neuronl strutures delnes more slowly thn /r, the utoovrne funton of the reorded sgnls from monoolr eletrode s domnted by tvty from dstnt dolr genertors. eordngs from qudruolr genertors re domnted by neurons ner the eletrode s long s the orrelton between the tvty n neuronl strutures dereses s / r ε wth ε >. Another mortnt observton s tht reordngs of dolr soures from bolr eletrodes hve smlr deendenes on the sze of the neurl system s do monoolr reordngs from qudruolr soures. Thus, when there re long rnge orreltons between the tvty n neurons, t s ossble tht there my be mjor qulttve dfferenes between the reordngs mde from ure monoolr nd bolr eletrodes. Ths observton my hve rtl mortne for the seleton of the best eletrodes for reordng events ssoted wth long-rnge orreltons suh s sezures or n fndng the best eletrodes wth whh to erform oherene nlyss [9]. The seond result of the nlyss erformed n ths er s the ft the deendene of the ross ovrne of the eletr otentl reorded from hyslly serted eletrodes hs very dfferent dstne deendene thn the ross orrelton funton desrbng the tvty n dfferent neurl strutures. Ths s true for both dolr nd qudruolr soures f the tvty n the dfferent neurl strutures s unorrelted. However, when the underlyng soure orrelton funton hs ower lw struture, the mesured ross ovrne funton s smlr to the underlyng neuronl orrelton funton when the soures re qudruolr but dey more slowly wth dstne thn the neuronl orrelton funton when the soures re dolr. Ths, n onjunton wth bove dsusson of the effet of dfferent eletrode tyes, suggests tht reordng wth bolr eletrodes for studes of ross ovrne funtons wll rovde better estmte of the underlyng neuronl ross orrelton funton thn reordngs wth monoolr eletrodes. The rnles set forth n ths er re of nterest only f there re tully long rnge orreltons n rel neurl systems. Cler dt on the rnge of neuronl orreltons n humns s lmted but oherenes between wdely se- Pge 6 of (ge number not for tton uroses

7 BoMedl Engneerng Onne 7, 6: htt:// rted rts of orte hve been demonstrted [7] usng monoolr eletrodes. In ddton, the "globl wve" theory of ortl oslltons roosed by Nunez [6] s bsed on estmtons of length of orto-ortl fbres n humns etendng u to m. Ths estmte s smly the rnge of the onnetons between neurons. The tul stl etent of orreltons n be muh longer. Nekelmnn [] hs demonstrted orreltons between neurons n t brn derese very slowly s funton of dstne, etendng beyond m nd tht the stl etent of ths orrelton nreses durng sezures. Also, t should be noted tht n smle sttstl models of systems ner rtl onts suh s the Isng model wth only nerest neghbor oulngs, the orrelton between sns serted by dstne vres (n dmensons s ner the rtl ont where η.8 [8]. Ths rovdes ddtonl evdene for the lkelhood tht neurl +η elements old be synhronzed over lrge dstnes eselly when there re lrge sle oherent oslltons suh s durng sezure. It should be noted tht, n tul ltons, rw unroessed dt from eletrodes s rrely used nd number of tehnques re used to etrt dt tht s onsdered most relevnt for the rtulr lton. One of the most ommon sgnl roessng tehnques s hgh ss flterng of the dt. As desrbed n revous er [5], the otentls from qudruolr soures movng t onstnt veloty suh s those omnyng n ton otentl hve dstne deendent ower setrum wth the sgnl ontnng rogressvely lower frequenes s the dstne between the soure nd reordng eletrode nreses. Thus, f fed hgh ss flter s used, the tul ontrbuton from movng qudruoles (but not qudruolr soures wth fed stl loton nd vryng ntensty my dro off fster thn redted by the model used n ths er nd hene there s redued lkelhood of ontrbutons from dstnt strutures to the reorded otentl. However, dolr soures tylly our t synses, bends n ons, or regons where there s hnge n on dmeter [5] nd do not rogte. Hene, n neurl systems dole soures re generlly n fed ostons nd the result obtned n ths er s more lkely to ly. Of ourse, there re gret number of other sgnl roessng tehnques tht n be used to etrt sef elements of the reorded sgnl nd my be used to enhne the ontrbutons from vrous strutures f there s ror nformton tht dstngushes the reordngs from these dfferent strutures. 4. Conluson eordngs of eletrl tvty from etended neurl strutures suh s brn re ommonly used to understnd the bs mehnsms underlyng vrous brn funtons. One rtl queston s whether the eletrl tvty tht s reorded from n eletrode omes from genertors ner the eletrode or genertors fr from the eletrode. The dstnton between "ner feld" nd "fr feld" otentls ws ntrodued to desrbe stutons n whh lolzed genertor my rodue resonses tht ether delne very qukly or very slowly wth dstne from the genertor. However, even f sngle lolzed genertor ontrbutes mnmlly to the reorded eletrl tvty, the resultnt effet of mny orrelted neurons dstrbuted over lrge regon of se my be sgnfnt. In ths er, t s demonstrted tht for dole soures n unform medum, the reorded otentl s strongly deendent on the neurons fr from the eletrode when neuronl orreltons dro more slowly thn /r or when the sze of the neurl struture s muh smller thn known orrelton dstne. eordngs from qudruolr soures re strongly deendent on dstnt neurons when orreltons dro more slowly thn /r or the sze of the system s muh smller thn the orrelton dstne. Bolr reordngs from dolr soures rodue resonses tht hve the sme roertes of qudruolr soures nd bolr reordngs of qudruolr soures re lwys domnted by lol genertors. In ddton, t s demonstrted tht the ross ovrne funtons omuted from reordngs of eletr otentl n etended neurl systems do not reflet the underlyng neuronl orrelton funton when orreltons between neurl genertors re very short rnge. However, when the orreltons between dfferent neurl regons delne s ower lw funton of dstne, the mesured ross ovrne funton delnes more slowly wth dstne when the soures re dolr. When the soures re qudruolr, the reorded ross ovrne funton hs smlr dstne deendene s the neuronl orrelton funton. Ths suggests tht bolr eletrodes should be used for reordng ross ovrne funtons n the settng of rmrly dolr soures. Aend A-Power lw neuronl orrelton funtons The urose of ths end s to derve eressons for the utoovrne funton of the reorded sgnl for generl lss of neuronl orrelton funtons. As derved n the mn tet, the generl reltonsh between the neuronl ovrne funton g( - ',, τ nd the sgnl utoovrne funton ν (τ s (: Pge 7 of (ge number not for tton uroses

8 BoMedl Engneerng Onne 7, 6: htt:// ν( τ l l Y d lm( θ, φ Yl m ( θ, φ lm( l m ( d l l m lm l l+ l + l+ g (,, τ ( Seflly, the deendene of ν (τ on wll be studed s funton of mn for ovrne funtons of the form: ν( τ m + G (, l, l,..., τ d d l l ; < + l l ; + ( 8 g (,, τ for some ntegers m nd mn >. It s mortnt to relze tht the degree to whh the stl etent of the ovrne s strongly deendent on the vlue of mn. If mn s zero, the orrelton etends over ll ses nd dereses more qukly wth dstne s the vlue of mn nreses. It s lso mortnt to note tht m must be less thn or else the ntegrls to be dsussed below do not onverge. Although more generl eresson n terms of Gegenbuer olynomls s ossble for nonnteger vlues of, ttenton wll be foused on the se where s restrted to nteger vlues. In ths se, equton ( n be used to wrte: Defnng: m mn h(, τ the eresson for the utoovrne funton beomes: The frst ste s to look t the uer bound for the utoovrne funton. There must be funton G (, l, l',,..., τ stsfyng: sne the ntegrl defnng G s over bounded regon nd the ntegrnd s fnte everywhere s long s s fnte. Ths mles tht ( 4 ; < m h(, τ + m g (,, τ h (, τ 4π, mn M + mn YM( θ, φ YM( θ, φ ; + ; < m + h(, τ YM( θ, φ YM( θ, φ ( 5 mn + M M ; + ( G, l, l,...,,, τ dωdω h(, τ l+ l + l l Q lm ( lm ( Ylm(, Ylm (, Q θ φ θ φ YM( θ, φ YM( θ, φ mlm l M M + ( 7 ν( τ m mn + ; < + d d l l Gl (,, l,...,,, τ 7 l l ; + ( ( ( G, l, l,...,,, G τ, l, l,..., τ ;, n order to smlfy the lultons let r, ' r' nd defne. It n be seen tht the deendene of the ovrne funton on wll be determned by the ntegrls: r ; r < r + r r + l r r drdr r l r l dr dr + dr + + r l r r + + l r ; r r r + Evlutng ths ntegrl s strghtforwrd but tedous wth the result: r ;( l + r l r r + ( + r dr dr dr + + l r r ln * ;( l l r + + l l * ( + l ( + l ( l l+ 4 ( l l+ 4 + l ;( l +,( l l+ 4,( l+ ( l l+ 4 ( l + ( l + ( l+ * ln ( + l ( + l + ;( l,( l l+ 4,( l+ ( l + ( l + ( l+ + ( l l+ 4 ( l l+ 4 l ln ;( l +,( l l+ 4,( l+ ( l l+ 4( l + ( l + * ln ;( l +,( l ( ( ln l ( l + ;( l +,( l ( l ( l And nlogously + + l ( ( + + l l r r r + r ;( l l l dr dr + + r r dr ( l + l * r ln ;( l r ( + l ( + l ( + l ( l 4 ( l 4 ;( l,( l +,( l l + 4 ( l ( l + ( l l + 4( l ( + l ( l + It s resonble to ssume tht ll soures re ether dolr or qudruolr n order to smlfy the lultons. Beuse of ths, t s ossble to set l l'. The bove ntegrls suggest tht the resonses for lrge re onvergent f: 4--l < ( ( + l ( l + ln + ( l + l l ;(,,( + ln ;( l,( l + ( + l ( l+ + + ( l ln l ;(,( l +,( l l + 4 ( l ( l + ( l ( ln ( ( + + l l 4 l 4 ;( l,( l +,( l l + 4 ( l ( l l + 4( l ( 9 ( ( Pge 8 of (ge number not for tton uroses

9 BoMedl Engneerng Onne 7, 6: htt:// Ths mles tht, for dolr soures for > nd for qudruolr soures > gurntee onvergent resonses for lrge or n other words tht the reorded sgnl s domnted by the ontrbuton of nerby neurl strutures. Sne must be less thn or else the ntegrls do not onverge, ths demonstrtes tht, lthough there my be sef eetons, n generl wth dolr soures nd ower lw orreltons, the utoovrne funton s strong funton. It should be noted tht ths rgument rovdes only mmum bound on the utoovrne funton nd hene does not rove tht the utoovrne must be etremely senstve to the vlue of outsde the rnge of rmeters dsussed bove. It s ossble, however, n one sel se to demonstrte the senstvty to the vlue of outsde the bove restrtons on. Consder the sef se, l, l' where n ddton: lm( lm h(, τ h( τ G(,,,,,, τ h ( τ l l d d m Y ( θ, φ Y 9 Ω Ω m m m ( θ, φ Y M ( θ Y (, φ M ( θ, φ mlm l M + Orthogonlty of the sherl hrmons mles: G,,,,, τ so tht: h ( τ δ m ( 7 M M M ( ( 4 < h ( m ν( τ ( ( π τ ; Q Q 4 d d M M 7 M ; h ( τ m Q Q ( ( M M 7 M ( 5 whh dverges for lrge. Ths soluton s et rovng tht for unform dole denstes nd the utoovrne funton does strongly deend on, the sze of the neurl system. Aend B esonses reorded from multolr eletrodes It s esy to etend the results to the se of bolr or more omle reordng eletrodes. Consder multolr eletrode n whh the reorded otentl ϕ s (, t s lner ombnton of the otentls t number of lotons: ϕs( t, εϕ( + ξ, t ( 6 where the ξ re the vetors ontng from the geometr enter of the eletrode to eh of the lotons where reordngs re mde ξ. The ε re the weghts determnng how muh the otentl from eh loton ontrbutes to the totl reorded otentl, reorded otentl, Substtutng the result (6 nto ( gves: ρ(, t ρ (, ϕs( t, ε d t d + ξ ρ (, t ερ( ξ, t so tht: l ϕs( t, 4π Y θ φ l lm(, Qlm(, t l m l l + + d l Qlm (, t Ylm(, ( θ φ ρ +, t d Δ Δ l ε Ylm( θ, φ ρ ( + ξ Δ, t d Δ ( 7 ( 8 Ths demonstrtes tht the rguments used bove ly eqully to reordng from omle eletrodes s well s to smle sngle eletrodes by use of the modfed multole moments Q lm '(, t. The seond mortnt queston s how does the use of omle eletrode hnge the reorded otentls from eletr multoles. Frst, t s well known tht the lowest order non-vnshng multole moment of ny hrge dstrbuton s ndeendent of the orgn of oordntes. Now, n (9 t n be seen tht the effet of reordng from dfferent eletrodes ξ s essentlly equvlent to hnge n the orgn of oordntes n the omutton of the multole moment nd so does not ffet the lowest order moment. Ths mens tht s long s ε the lowest order multole moment dsers. Thus, for emle, bolr reordng from dolr soures re domnted by the l terms n the multole enson nd bolr reordngs of the qudruolr soures re domnnted by the l terms. Aend C Evluton of the ross ovrne funton In order to determne the ross ovrne, t s neessry to evlute: Pge 9 of (ge number not for tton uroses

10 BoMedl Engneerng Onne 7, 6: htt:// ν( s,, τ d l m lm l l+ l + + l l + s Consder the se : ν(,, s τ d l m lm l l+ l + + l l s Note tht: nd so: when s >. If we onsder r->nfnty nd sne l, l' > we get nd - > : Now let: then we hve: Y l lm( Yl m lm +, (, ( θ φ θ φ Q Q lm ( h(, τ ( 9 + Ylm( θ, φ Yl ( θ, φ Qlm ( Ql m ( ( h, 4 l m s ; s < l + + YM( θ s, φ s YM( θ, φ s + M M ; s l s + + ν(,, s τ s ; s < l l l + + d dω l m l l + l + l l ml M M ; s l s + + s s YM( θ, φ ( θ, φ ( θ, φ ( θ, φ YM Ylm Y l m Qlm( Ql m ( h(, τ + s l l + l s l + l r l dω + + l m l l + l + s + l Ω l l ml M M l s l YM s s Y + ( θ, φ M ( θ, Y φ lm( θ, φ Yl m ( θ, φ Qlm( Ql m ( h(, τ τ ( ( 4 ( l + l l dω l l m lm l l l l l Ω 4 + ( + + π + s ( 4 l + M M l+ l + YM( s s Y, M(, + θ φ θ φ Y Y h lm ( θ, φ l m ( θ, φ Q lm ( Q l m ( (, τ + + l + l l l s s fll (,, θ, φ dω l+ l + Ω mlm l M M l+ s s YM + ( θ, φ YM( θ, φ Ylm( θ, φ Yl m ( θ, φ ( 44 ν (,, s τ f l l θ s φ s,,, l+ l lml m h( τ l l s ( ( 45 So tht for nstne f the soures re dolr (l, l' then ν(,, s τ. If the soures re qudruolr, then l, l' s ν(,, s τ nd so. Ths ndtes tht even s when the neuronl ross ovrne s delt funton, the mesured ross ovrne deends nversely on the dstne between the eletrodes. Aend D-Qulttve estmte of the dstne deendene of the ross ovrne funton The urose of ths end s to resent smle qulttve rgument llustrtng the reltonsh between the ross ovrne funton of smle slr soure dstrbuton nd the ross ovrne funton of the reorded eletrl otentls. Consder the stuton n whh the reltonsh between the otentl nd the soures s gven by: ϕ( d f( ( ρ ( 46 where f( - ' s the funton tht determnes how the resene of n tve genertor t oston ' nfluenes the otentl t. In the ensung lultons, t wll be muh smler to del wth soures tht re unformly dstrbuted thn to be onerned bout the detls of how soures re dstorted by the resene of n eletrode of fnte sze. In wht follows, the vew wll be tken tht ner the eletrode ρ( s onstnt but: f( ; 47 < ( If the ovrne funton desrbng tvty n dfferent neurl strutures s gven by: ρ( ρ( α( ( 48 t s ossble to use (47 nd (48, to wrte the ross ovrne of the reorded sgnl s: ϕ( ϕ( d d f ( f ( α( s s s s g(, s ϕ ϕ + d d f f + α ( s + s g(, s d d f ( f ( α ( s d d f ( f ( α ( s ( 49 Pge of (ge number not for tton uroses

11 BoMedl Engneerng Onne 7, 6: htt:// sne f - ( ± f(. Consder the slng behvor s funton of s: g(,εs d d f ( f ( α( εs d d f ( f ( α ε s ε ε ε ε d g(, εs ε d d f ( ε f ( ε α ε s where d. If: αε ( εα( (e the orreltons between neurl strutures delnes s P for some number, t s ossble to wrte: Beuse of the restrton (47, there n be no et relton of the form: However, f: for some funton h nd number q: Thus: ( 5 ( [ ] If s >> then α([ " - " - s ] α( s sne the lrgest ontrbutons wll ome from vlues of ", " ner nd so: It s ossble to estmte the lst ntegrl s: ( 5 d g (, + s d ε ε d f( ε f( ε α([ s ] 5 q f( ε ε f( f( h(( θ > q h( ε ε h( ( 5 ( d+ + q g(,εs ε d d h( θ > h ( θ > ([ s] ε ε d+ + q ε d+ + q + ε d d h( h( θ > ε > θ θ θ α ([ s ] ε d d h( h( θ( > θ( > α( [ s ] ( 54 d+ + q g(, εs ε g(, s d+ + q + ε d d h( h ( > > ε θ θ θ ( ε > α ([ s] d+ + q g(, εs ε g(, s d+ + q + ε α( s d d h( h ( > > ( > θ ε θ θ θ( > ε ( 55 ( 56 h q+ d ( q+ d h ε ; q+ d + + d q d q d ε [ h ln ε ] ; q+ d f h( h q ; <. Thus: d q d q h q d g (, s gs (, ( ε ε + ε α s ε q+ d q+ d ( In the se d nd q - (dole soures then: ( 57 h g (, s gs (, ( + + ε ε + ε α s ( ε 58 ( Although ths equton s omle, t s ler tht f only the frst term s onsdered, then + + h g (, εs ε gs (, + ε α( s ( ε ( 59 gs (, s α( s If only the seond term s studed for lrge vlues of ε, then: + + h g (, εs ε gs (, + ε α( s ( ε 4 h gs (, s ( s α ( 6 In ether se, the reorded ross ovrne funton deys slower wth s for lrge dstnes between the eletrodes thn the neuronl orrelton funton α( s. For q - (qudruole soures: g (, εs ε gs (, ε α( s[ h ln ε] + ( 6 Sne the logrthm of ε s very slowly vryng funton, the mesured ross orrelton funton sles very nerly s the neuronl orrelton funton. eferenes. Helmholtz HF: Ueber enge Gesetze der erthelung elektrsher Strome n korerlhen etern mt Anwendung uf de thersh-elektrshen ersuhe. Ann Physk un Cheme 85, 89:-.. Mlmvuo J, Plonsey : Boeletromgnetsm New York: Oford; Jkson JD: Clssl eletrodynms New York: Wley; Cuffn BN: A method for lolzng EEG soures n relst hed models. IEEE Trns Bomed Eng 995, 4: herson SJ, Ingrm M, Perry D, Steker MM: Clssfton of the etrellulr felds rodued by tvted neurl strutures. BoMedl Engneerng Onne 5, 4:5. 6. Nunez P: Neoortl dynms nd humn EEG rhythms New York: Oford; Towle, Crder K, Khorsn, ndberg D: Eletroortogrh oherene tterns. J Cln Neurohysology 999, 6: Pfeuty P, Toulouse G: Introduton to the renormlzton grou nd to rtl henomen. New York: Wley; gerlund TD, Shrbrough FW, Busher NE, Cor KM: Intereletrode oherenes from nerest-neghbor nd sherl hrmon enson omutton of ln nd sl otentl. Eletroeneh Cln Neurohys 995, 95: Pge of (ge number not for tton uroses

12 BoMedl Engneerng Onne 7, 6: htt:// Nekelmnn D, Amz F, Sterde M: Ske-wve omlees nd fst omonents of ortlly generted sezures III. synhronzng mehnsms. J Neurohysol 998, 8: Publsh wth BoMed Centrl nd every sentst n red your work free of hrge "BoMed Centrl wll be the most sgnfnt develoment for dssemntng the results of bomedl reserh n our lfetme." Sr Pul Nurse, Cner eserh UK Your reserh ers wll be: vlble free of hrge to the entre bomedl ommunty eer revewed nd ublshed mmedtely uon etne ted n PubMed nd rhved on PubMed Centrl yours you kee the oyrght BoMedentrl Submt your mnusrt here: htt:// Pge of (ge number not for tton uroses