Mathematical Appendix

Transcription

1 Mahmacal Appn arl J Frson Th Wllcom Dp. of Cognv Nrology Unvrsy Collg Lonon Qn Sqar Lonon U WCN 3BG Tl Fa mal k.frson@fl.on.cl.ac.k Conns I. Inrocon II. Inp-op mols for sngl rgon III Inp-sa-op mols for mlpl rgons IV Mlvara ARMA mols V Conclson Rfrncs INTRODUCTION A. Ovrvw Ths chapr prsns a horcal rvw of mols ha ar s for ffcv conncvy. In hs scsson w focs on h nar an form of h mols hmslvs an lss on smaon or nfrnc sss. Th am s o rla h varos mols commonly mploy an o mak hr nrlyng assmpons an rqrmns mor ransparn.

2 As w hav sn n h prcng chaprs hr ar a nmbr of mols for smang ffcv conncvy sng nromagng m srs PET fmri EEG an MEG. By fnon ffcv conncvy pns on a mol hrogh whch s fn opraonally Frson al 995. Ths chapr rvws h prncpal mols ha col b aop an how hy rla o ach ohr. W consr ynamc casal mols DCM Gnrals Convolon Mols GCM [b- ]cohrnc srcral qaon mols SEM an mlvara aorgrsson mols MAR. In brf w wll show ha hy ar all spcal cass of ach ohr an ry o mphass hr pons of conac. Howvr som fnamnal sncons ars ha g h slcon of h appropra mols n ffrn saons. Fgr abo hr Sngl or mlpl rgons? Th frs sncon rss pon whhr h mol s s o plan h coplng bwn h nps an h rsponss of on cll assmbly or rgon or whhr h mol ncompasss nracons among h sas of mlpl rgons. In rms of mols hs sncon s bwn np-op mols.g. mlpl-np snglop mols MISO an mlpl-np mlpl-op mols MIMO an plc np-sa-op mols. Usally h np-op approach s concrn wh h nonlnar ransformaon of nps by a rgon o proc s ops. Th mplc sas corrspon o hn sas of a sngl rgon an h ffcv conncvy concrns h vrcal lnk bwn nps an ops s Fgr a. In conrasncon h np-sa-op approach s gnrally concrn wh characrsng h horzonal coplng among varabls ha rprsn h sas of ffrn rgons. Ths sas ar obsrv vcarosly hogh h ops s Fgr b. Eampls of np-op mols ncl h Volrra formlaon of ffcv conncvy an rla cohrnc analyss n h spcral oman. An ampl of a mol ha rs o sma horzonal coplng among hn sas s DCM. A crcal aspc of vrcal np-op mols of ffcv conncvy s ha hy can proc who rfrnc o h hn sas. Convrsly h horzonal nracons rqr nrc accss o h sas or som srong assmpons abo how hy proc ops. In shor analyss of ffcv conncvy can b consr as ryng o characrs h np-op bhavor of a sngl rgon or h

3 coplng among h sas of svral rgons sng an plc np-sa-op mol. Blow w sar by rvwng np-op mols an hn rn o np-saop mols. Drmnsc or sochasc nps? Th scon ky sncon s bwn mols whr h np s known an f.g. DCM an hos n whch s no MAR an SEM. Only h formr class of mols affors rc masrs of ffcv conncvy. Th rmanng mols ar sfl for sablshng h prsnc of coplng nr cran assmpons abo h np sally ha s wh nos ha rvs h sysm. Ths sncon pns on whhr h nps nr as known an rmnsc qans.g. prmnally sgn cass of vok rsponss or whhr w know or can assm somhng abo h nsy fncon of h nps.. s sascs p o scon or hghr orrs. Mos mols of h sochasc vary assm h nps ar Gassan... an saonary. Som sochasc mols.g. cohrnc s local saonary assmpons o sma hgh orr momns from obsrvabl b nosy nps. For ampl polyspcral analyss rprsns an nrma cas n whch h nps ar obsrv b only hr sascs ar s. Howvr h ky sncon s no whhr on has accss o h nps b whhr hos nps hav o b ra as sochasc or no. Saonary assmpons n sochasc mols ar crcal bcas hy prcl fll analyss of vok nronal rsponss or ransns ha by hr nar ar non-saonary. Dsp hs hr ar saons whr h np s no obsrvabl or nr prmnal conrol. Ths saons prcl h smaon of h paramrs of DCMs. Approachs lk MAR an SEM can b s o proc f h nps can b rgar as saonary. Th sncon bwn rmnsc an sochasc nps s crcal n h sns ha wol b nappropra o aop on class of mol n a con ha calls for h ohr. 3 Conncons or sascal pnncs? Th fnal sncon s n rms of wha s bng sma or nfrr. Rcall ha fnconal conncvy s fn by h prsnc of sascal pnncs among rmo nrophysologcal masrmns. Convrsly ffcv conncvy s a paramr of a mol ha spcfs h casal nflncs among bran sysms. I s sfl o sngsh nfrncs abo sascal pnncs an smaon of

4 ffcv conncvy n rms of h sncon bwn fnconal an ffcv conncvy. Eampls of approachs ha ry o sablsh sacally pnncs ncl cohrnc analyss an MAR. Ths s bcas hr hs chnqs o no prsm any mol of how hn sas nrac o proc rsponss. Thy ar nrs only n sablshng [sally lnar] pnncs among ops ovr ffrn frqncs or m lags. Alhogh MAR may mploy som mol o assss pnncs hs s a mol of pnncs among ops. Thr s no assron ha ops cas ops. Convrsly SEM an DCM ry o sma h mol paramrs an cons analyss of ffcv conncvy propr. Gnrals convolon approachs fall no hs class bcas hy rs on h smaon of krnls ha ar an qvaln rprsnaon of som np-sa-op mol paramrs. B Effcv conncvy Effcv conncvy s h nflnc ha on nronal sysm rs ovr anohr a a synapc or nsmbl lvl. Ths shol b conras wh fnconal conncvy whch mpls a sascal pnnc bwn wo nronal sysms ha col b ma n any nmbr of ways. Opraonally ffcv conncvy can b prss as h rspons nc n an nsmbl n or rgon by np from ohrs n rms of paral rvavs of h arg acvy wh rspc o h sorc acvs. Frs E an scon E k orr conncvs ar hn E E k k Frs-orr conncvy mbos h rspons vok by a chang n np a. In ohr wors s a m-pnan masr of rvng ffcacy. Scon-orr conncvy rflcs h molaory nflnc of h np a on h rspons vok a. An so on for hghr orrs. No ha n hs gnral formlaon ffcv conncvy s a fncon of crrn np an nps ovr h rcn pas. In conras fnconal conncvy s mol-fr an smply rflcs h mal nformaon I. In hs papr w ar concrn only wh mols of ffcv conncvy

5 Frhrmor mplc n Eq s h fac ha ffcv conncvy s casal nlss s allow o b ngav. I s sfl o nroc h ynamc qvaln n whch h rspons of h arg s masr n rms of changs n acvy E& & & & E k k whr &. In hs ynamc form all nflncs ar casal an nsananos. Bfor consrng spcfc mols of ffcv conncvy w wll rvw brfly hr bass s Chapr : Effcv Conncvy. C Dynamcal sysms Th mos gnral an plasbl mol of nronal sysms s a nonlnar ynamcal mol ha corrspons o an analyc mlpl-np mlpl-op MIMO sysm. Th sa an op qaons of a analyc ynamcal sysm ar & f θ y λ ε 3 Typcally h nps corrspon o sgn prmnal ffcs.g. smls fncons n fmri or rprsn sochasc rvs or sysm prrbaons. Sochasc obsrvaon rror ε ~ N Σ nrs lnarly n hs mol. For smplcy h prssons blow al sngl-np sngl-op SISO sysms an wll b gnrals lar. Th masr rspons y s som nonlnar fncon of h sas of h sysm. Ths sa varabls ar sally nobsrv or hn.g. h confgraonal sas of all on channls h polarsaon of vry nrc comparmn c.. Th paramrs of h sa qaon mboy ffcv conncvy hr n rms of mang h coplng bwn nps an ops MISO mols of a sngl rgon or hrogh h coplng among sa varabls MIMO mols of mlpl rgons. Th obcv s o sma an mak nfrncs sally Baysan abo hs paramrs gvn h ops an possbly h nps. Somms hs rqrs on o spcfy h form of h sa qaon. A bqos an sfl form s h blnar appromaon o 3; panng aron

6 & A B C y L 4 f A f B f C λ L For smplcy w hav assm an f λ. Ths blnar mol s somms prss n a mor compac form by agmnng h sas wh a consan X& M N X y HX 5 X M f A N C B H [ λ L] s Frson. Hr h mol's paramrs comprs h marcs θ { A B C L}. W wll s h blnar paramrsaon whn alng wh MIMO mols an hr rvavs blow. W wll frs al wh MISO mols wh an who rmnsc nps. II. INPUT-OUTPUT MODELS FOR SINGLE REGIONS A Mols for rmnsc nps - Th Volrra formlaon In hs scon w rvw h Volrra formlaon of ynamcal sysms. Ths formlaon s mporan bcas allows h np-op bhavor of a sysm o b characrs n rms of krnls ha can b sma who knowng h sas of h sysm. Th Flss fnamnal formla Flss al 983 scrbs h casal rlaonshp bwn h ops an h hsory of h nps n 3. Ths rlaonshp conforms o a Volrra srs whch prsss h op y as a gnrals convolon of h np crcally who rfrnc o h sa varabls. Ths srs s smply

7 a fnconal Taylor panson of h ops wh rspc o h nps Bna 99. Th rason s a fnconal panson s ha h nps ar a fncon of m. y h h y κ κ θ ε θ 6 wr κ s h h orr krnl. In Eq 6 h ngrals ar rsrc o h pas or hsory of h nps. Ths rnrs Eq 6 casal. In som saons an acasal formlaon may b appropra.g. n whch h krnls hav non-zro vals for fr nps - s Frson an Büchl. On mporan hng abo 6 s ha s lnar n h nknowns nablng nbas smas of h krnls sng las sqars. In ohr wors 6 can b ra as a gnral lnar obsrvaon mol nablng all h sal smaon an nfrnc procrs s Chapr : Effcv Conncvy for an ampl. Volrra srs ar gnrally hogh of as a hghorr or gnrals nonlnar convolon of h nps o prov an op. To nsr smably of h krnls hy can b pan n rms of som appropra bass fncons q o gv h gnral lnar mol q q h h y β κ ε β 7 Th Volrra formlaon s sfl as a way of characrsng h nflnc of nps on h rsponss of a rgon. Th krnls can b rgar as a r-paramrsaon of h blnar form n Eq4 ha ncos h mpls rspons o np. Th krnls for h sas ar

8 κ X κ κ κ 3 M M N N M X M N M X 8 Th krnls assoca wh h op follow from h chan rl h Hκ h Hκ h 9 s Frson for als. If h sysm s flly nonlnar hn h krnls can b consr local appromaons. If h sysm s blnar hy ar globally ac. I s mporan o rmmbr ha h smaon of h krnls os no assm any form for h sa qaon an complly schws h sas. Ths s h powr an waknss of Volrra-bas analyss. Th Volrra formlaon can b s rcly n h assssmn of ffcv conncvy f w assm h masr rspons of on rgon conss h np o anohr.. y. In hs cas h Volrra krnls hav a spcal nrpraon; hy ar synonymos wh ffcv conncvy. From 6 h frs orr krnls ar y κ E y Ensons of Eq6 o mlpl nps MISO mols ar rval an allow hgh-orr nracons among nps o a sngl rgon o b characrs. Ths approach was s n Frson an Büchl o amn paral molaon of V nps o V5 by smang an makng nfrncs abo h appropra scon orr krnl. Th avanag of h Volrra approach s ha nonlnars can b moll an sma n h con of hghly nonlnar ransformaons whn a rgon an y h smaon an nfrnc proc n a sanar lnar las sqars sng. Howvr on has o assm ha h nps conform o masr rsponss lswhr

9 n h bran. Ths may b nabl for EEG b h hmoynamc rsponss masr by fmri mak hs a mor qsonabl approach. Frhrmor hr s no casal mol of h nracons among aras ha wol ohrws offr sfl consrans on h smaon. Th rc applcaon of Volrra smaon n hs fashon smply amns ach no on a a m assmng h acvs of ohr nos ar vrcal masrmns of h nps o h no n qson. In smmary alhogh h Volrra krnls ar sfl characrsaons of h np-op bhavor of sngl rgons hy ar no consran by any mol of nracons among rgons. Bfor rnng o DCMs ha mboy hs nracons w wll al wh h SISO saon n whch h np s ra as sochasc. B Mols for sochasc nps Cohrnc an Polyspcral analyss In hs scon w al wh sysms n whch h np s sochasc. Th am s o sma h krnls or hr spcral qvalns gvn only sascs abo h on srbon of h nps an ops. Whn h nps ar nknown on gnrally maks assmpon abo hr srbonal proprs an assms [local] saonarnss. Alrnavly h nps may b masrabl b oo nosy o srv as nps n Eq7. In hs cas hy can b s o sma h np an op nss n rms of hghr orr cmlans or polyspcral nsy. Th nh orr cmla of h np s c } { n n whr w hav assm hr an hrogho ha E { }. I can b sn ha cmlans ar a gnralsaon of ao-covaranc fncons. Th scon-orr cmlan s smply h ao-covaranc fncon of lag an smmarss h saonary scon-orr bhavor of h np. Cmlans allow on o formla 6 n rms of h scon orr sascs of np an ops. For ampl

10 a a a a y c y c κ κ Eq says ha h cross-covaranc bwn h op an h np can b compos no componns ha ar form by convolvng h h orr krnl wh h np's h cmlan. Th mporan hng abo hs s ha all cmlans grar han scon orr of Gassan procsss ar zro. Ths mans ha f w can assm h np s Gassan hn κ c c a a y 3 In ohr wors h cross-covaranc bwn h np an op s smply h aocovaranc fncon of h nps convolv wh h frs-orr krnl. Alhogh s possbl o formla h covaranc bwn nps an ops n rms of cmlans h mor convnonal formlaon s n frqncy spac sng polyspcra. Th nh polyspcrm s h Forr ransform of h corrsponng cmlan } { n n n n c g n π 4 Agan polyspcra ar smply a gnralsaon of spcral nss. For ampl h scon polyspcrm s spcral nsy an h hr polyspcrm s bspcral nsy. I can b sn ha hs rlaonshps ar gnralsaons of h Wnr- hnchn horm rlang h ao-covaranc fncon an spcral nsy hrogh h Forr ransform. Inrocng h spcral nsy rprsnaon s 5 w can now rwr h Volrra panson Eq6 as

11 Γ s s h π π π π θ 6 whr h fncons Γ Γ κ κ ar h Forr ransforms of h krnls. Ths fncons ar call gnrals ransfr fncons an ma h prsson of frqncs n h op gvn hos n h np. Crcally h nflnc of hghr orr krnls or qvalnly gnrals ransfr fncons mans ha a gvn frqncy n h np can nc a ffrn frqncy n h op. A smpl ampl of hs wol b sqarng a sn wav np o proc an op of wc h frqncy. In h Volrra approach h krnls wr nf n h m oman sng h nps an ops rcly. In hs scon sysm nfcaon mans smang hr Forr ransforms.. h ransfr fncons sng scon an hghr orr sascs of h nps an ops. Gnrals ransfr fncons ar sally sma hrogh smas of polyspcra. For ampl h spcral form for 3 an s hgh-orr conrpars ar! n n n n y y y g g n g g g g g g M Γ Γ Γ 7 Gvn smas of h rqs [cross]-polyspcra hs qals can b s o prov smas of h ransfr fncons s Fgr. Ths qals hol whn h Volrra panson conans s h nh orr rm an ar a gnralsaon of h classcal rsls for h ransfr fncon of a lnar sysm [frs qaly n Eq7]. Th mporanc of hs rsls n rms of ffcv conncvy s h mplc

12 manng confrr on cohrnc an b-cohrnc analyss. Cohrnc s smply h scon-orr cross spcrm g bwn h np an op an s rla o y frs-orr ffcs.. h frs-orr krnl or ransfr fncon hogh Eq7. Cohrnc s hrfor a srroga markr for frs-orr or lnar conncvy. Bcohrnc or h cross-bspcrm g s h hr-orr cross-polyspcrm y an mpls a non-zro scon-orr krnl or ransfr fncon. Bspcral analyss was s n a smplf form o monsra nonlnar coplng bwn paral an fronal rgons sng MEG n Chapr Volrra krnls an ffcv conncvy. In hs ampl cross-bspcra wr sma n a smpl fashon sng m-frqncy analyss. C Smmary In smmary Volrra krnls gnrals ransfr fncons characrs h npop bhavor of a sysm. Th nh orr krnl s qvaln o nh orr ffcv conncvy whn h nps an ops conform o procsss ha ma nracons among nronal sysms. If h nps an ops ar known or can b masr prcsly h smaon of h krnls s sraghforwar. In saons whr sochasc nps an ops ar lss prcsly obsrv krnls can b sma nrcly hrogh hr gnrals ransfr fncons sng crosspolyspcra. Th robsnss of krnl smaon confrr by panson n rms of mporal bass fncons s rcapla n h frqncy oman by smoohnss consrans rng smaon of h polyspcra. Th spcral approach s lm bcas assms h sysm conans only h krnl of h orr sma an saonarnss. Th non bhn h frs lmaon rlas o h sncon bwn paramr smaon an varanc paronng n sanar rgrsson analyss. Alhogh s prfcly possbl o sma h paramrs of a rgrsson mol gvn a s of non-orhogonal planaory varabls s no possbl o nqly paron varanc n h op cas by hs planaory varabls.

13 III INPUT-STATE-OUTPUT MODELS FOR MULTIPLE REGIONS In hs scon w arss mols for mlpl nrconnc rgons whr on can masr h rsponss of hs rgons o np ha may or may no b known. Alhogh s possbl o n h chnqs of h prvos scons o covr MIMO sysms h nsng nfrncs abo h nflnc of np o on rgon on h rspons of anohr ar no sffcnly spcf o cons an analyss of ffcv conncvy. Ths s bcas hs nflncs may b ma n many ways an ar no paramrs n rms of h ffcv conncvy among h rgons hmslvs. In shor on s no nrs n h vrcal rlaonshp bwn mlpl nps an mlpl ops b n h horzonal nracons among h sa varabls of ach rgon Fgr. A paramrsaon ha ncos hs nrrgonal coplng s hrfor rqr. All h mols scss blow assm som form or mol for h nracons among h sa varabls an amp o sma h paramrs of hs mol somms who acally obsrvng h sas hmslvs. A Mols for known nps Dynamc Casal Mollng. Th mos rc an gnrc approach s o sma rcly h paramrs of Eq3 an s hm o comp ffcv conncvy as scrb n Eq an Eq. Alhogh hr ar many forms on col aop for Eq3 w wll focs on h blnar appromaon whch s possbly h mos parsmonos b sfl nonlnar appromaon avalabl. Frhrmor as shown blow h blnar appromaon r-paramrss h sa qaons of h mol rcly n rms of ffcv conncvy. Dynamc casal mollng os no ncssarly nal h s of a blnar mol. In DCMs can b spcf o any gr of bologcal comply an ralsm sppor by h aa. Howvr blnar appromaons rprsn h smpls form o whch all DCMs can b rc. Ths rcon allows analyc rvaon of krnls an ohr compaons lk ngrang h sa qaon o proc n an ffcn fashon. Each rgon may comprs svral sa varabls whos casal nrpnncs ar smmars by h blnar form n Eq4. Hr h ky conncvy paramrs of h sa qaon ar marcs M an N. For a gvn s of nps or prmnal con h blnar appromaon o any s of sa qaons s

14 X& JX X J J M X N 8 Noc ha hr ar now as many N marcs as hr ar [mlpl] nps. Th blnar form rcs h mol o frs-orr conncons ha can b mola by h nps. In MIMO mols h ffcv conncvy s among h sas sch ha frs-orr ffcv conncvs ar smply X& E& J X X E X J 9 hs ncls conncons wh h consan rm n Eq5. No ha hs ar con-snsv n h sns ha h Jacoban J s a fncon of prmnal con or nps [ ]. A sfl way o hnk abo h blnar paramr m marcs s o rgar hm as h nrnsc or lan ynamc conncvy n h absnc of np an changs nc by ach np s h prvos chapr for a fllr scrpon E& N f E& M C A B Th lan ynamc conncvy among h sas s A. Ofn on s mor nrs n h B as mboyng changs n hs conncvy nc by ffrn cognv s m or rgs. No ha C s ra as h np-pnn componn of h conncon from h consan rm or rv. Clarly wol b possbl o nroc ohr hgh orr rms o mol nracons among h sas b w wll rsrc orslvs o blnar mols for smplcy.

15 Th fnamnal avanag of DCM ovr alrnav srags s ha h casal srcr s ma plc by paramrsng h sa qaon. Th smaon of ffcv conncvy an nsng nfrncs ar sally hrogh posror mo analyss bas on normaly assmpons abo h rrors an som sabl prors on h paramrs. Th paramrs of h blnar form ar θ { A B C L}. If h prors ar also spcf nr Gassan assmpons n rms of hr pcaon η θ an covaranc η θ y C θ Gass-Nwon EM schm can b aop o fn h posror mo s h prvos chapr for als. In ssnc ynamc casal mollng comprss spcfcaon of h sa an op qaons of an nsmbl of rgon-spcfc sa varabls. A blnar appromaon o h sa qaon rcs h mol o frs-orr coplng an blnar rms ha rprsn h molaon of ha coplng by nps. Posror nsy analyss of h mol paramrs hn allows on o sma an mak nfrncs abo nr-rgonal conncons an h ffc of prmnal manplaons on hos conncons. As mnon abov h sa qaons o no hav o conform o h blnar form. Th blnar form can b comp aomacally gvn any sa qaon. Ths s mporan bcas h prors may b spcf mor narally n rms of h orgnal bophyscal paramrs of h DCM as oppos o h blnar form. Th choc of h sa varabls clarly has o accommoa hr rol n mang h ffc of nps on rsponss an h nracons among aras. In h smpls cas h sas varabls col b rc o man nronal acvy pr rgon pls any bophyscal sa varabls n o rmn h op.g. h sas of hmoynamc mols for fmri. Implc n choosng sch sa varabls s h assmpon ha hy mol all h ynamcs o h lvl of al rqr. Man fl mols an nral mass mols ar sfl hr n movang h nmbr of sa varabls an h assoca sa qaons. Consrans on h paramrs of h mol ar mplmn hrogh hr prors. Ths rsrc h paramr smas o plasbl rangs. An mporan consran s ha h sysm s sspav an os no vrg ponnally n h absnc of np. In ohr wors h prors nsr ha h largs gnval of J s lss han zro.

16 Smmary In smmary DCM s h mos gnral an rc approach o nfyng h ffcv conncvy among h sas of MIMO sysms. Th nfcaon of DCMs sally procs sng Baysan schms o sma h posror mo or mos lkly paramrs of h mol gvn h aa. Posror mo analyss rqrs only h sa qaons an prors o b spcf. Th sa qaons can b arbrarly complca an nonlnar. Howvr a Blnar appromaon o h casal nflncs among sa varabls srvs o rc h comply of h mol an paramrss h mol rcly n rms of frs orr conncvy an s changs wh np h blnar rms. In h n scon w al h saons n whch h np s nknown. Ths prcls DCM bcas h lklhoo of h rsponss canno b comp nlss w know wha cas hm. B Mols for sochasc nps SEM an rgrsson mols Whn h nps ar ra as nknown an h sascs of h ops ar only consr o scon orr on s ffcvly rsrc o lnar or frs-orr mols of ffcv conncvy. Alhogh s possbl o al wh scr-m blnar mols wh wh nos nps hy hav h sam covaranc srcr as ARMA aorgrssv movng avrag mols of h sam orr Prsly 988 p66. Ths mans ha n orr o sngsh bwn lnar an nonlnar mols on wol n o sy momns hghr han scon orr c.f. h hr orr cmlans n bcohrnc analyss. Consqnly w wll focs on lnar mols of ffcv conncvy nr wh saonary nps. Ths nps ar h nnovaons nroc n h las chapr. Thr ar wo mporan classs of mol hr: Ths ar srcral qaon mols an ARMA mols. Boh ar fn paramr lnar mols ha ar sngsh by hr pnncy on ynamcs. In SEM h nracons ar assm o b nsananos whras n ARMA h ynamc aspc s ran plcly n h mol. SEM can b rv from DCMs by assmng h nps vary slowly n rlaon o nronal an hmoynamcs. Ths s appropra for PET prmns an possbly som poch-rla fmri sgns b no for vn-rla sgns n ERP or fmri. No ha hs assmpon prans o h nps or prmnal sgn no o h m consans of h ops. In prncpl wol b possbl o apply DCM o a PET sy.

17 Consr a lnar DCM whr w can obsrv h sas prcsly an hr was only on sa varabl pr rgon & f A A y λ Hr w hav scon obsrvaon rror b allow sochasc nps ~ N Q. To mak h conncon o SEMs mor plc w hav pan h conncvy mar no off-agonal conncons an a lang agonal mar mollng n cay A A. For smplcy w hav absorb C no h covaranc srcr of h nps Q. As h nps ar changng slowly rlav o h ynamcs h chang n sas wll b zro a h pon of obsrvaon an w oban h rgrsson mol s by SEM. & A A Ths shol b compar wh Eq7 n Chapr Effcv Conncvy. Th mor convnonal movaon for Eq s o sar wh an nsananos rgrsson qaon A ha s formally ncal o h scon ln abov. Alhogh hs rgrsson mol obscrs h conncon wh ynamc formlaons s mporan o consr bcas s h bass of commonly mploy mhos for smang ffcv conncvy n nromagng o aa. Ths ar smpl rgrsson mols an SEM. Smpl Rgrsson mols A can b ra as a gnral lnar mol by focssng on on rgon a a m for ampl h frs o gv

18 A [ ] n M 3 A n c.f. Eq8 n Chapr Effcv Conncvy Th lmns of A can hn b solv n a las sqars sns by mnmsng h norm of h nknown sochasc nps for ha rgon.. mnmsng h nplan varanc of h arg rgon gvn h sas of h rmanr. Ths approach was propos n Frson al 995 an has h avanag of provng prcs smas of conncvy wh hgh grs of from. Howvr hs las sqar smaors assm rahr mplasbly ha h nps ar orhogonal o h sas an mor mporanly o no nsr h nps o ffrn rgons conform o h known covaranc Q. Frhrmor hr s no parclar rason ha h np varanc shol b mnms s bcas s nknown. Srcral qaon mollng ovrcoms hs lmaons a h cos of grs of from for ffcn smaon Srcral qaon mollng In SEM smas of A mnms h ffrnc L vrgnc bwn h obsrv covaranc among h [obsrvabl] sas an ha mpl by h mol an assmpons abo h nps. T A A T A Q A T T 4 Ths s crcal bcas h conncvy smas mplcly mnms h scrpancy bwn h obsrv an mpl covarancs among h sas nc by sochasc nps. Ths s n conrasncon o h nsananos rgrsson approach abov or ARMA analyss blow n whch h smas smply mnms nplan varanc on a rgon by rgon bass. C Qas-blnar mols PPIs an moraor varabls Thr s a sfl nson o h rgrsson mol mplc n Eq ha ncls blnar rms form from known nps ha ar snc from sochasc nps

19 ncng [co]varanc n h sas. L hs known nps b no by v. Ths sally rprsn som manpla prmnal con sch as cognv s.g. anon or m. Ths rmnsc nps ar also known as moraor varabls n SEM. Th nrlyng qas-blnar DCM for on sch np s & A Bv 5 Agan assmng h sysm has sl a h pon of obsrvaon A & Bv A Bv 6 Ths rgrsson qaon can b s o form las sqars smas as n Eq3 n whch cas h aonal blnar rgrssors v ar known as psychophysologcal nracon PPI rms for obvos rasons. Th corrsponng SEM or pah analyss sally procs by crang ra 'vral' rgons whos ynamcs corrspon o h blnar rms. Ths s mova by rwrng h las prsson n Eq6 as A v B v 7 I s mporan o no ha psychophysologcal nracons an moraor varabls n SEM ar acly h sam hng an boh spak o h mporanc of blnar rms n casal mols. Thr rlav sccss n h nromagng lrar s probably o h fac ha hy mol changs n ffcv conncvy ha ar gnrally mch mor nrsng han h conncon srnghs pr s. Eampls ar changs nc by anonal molaon changs rng procral larnng an changs ma pharmacologcally. In ohr wors blnar componns affor ways of characrsng plascy an as sch play a ky rol n mhos for fnconal ngraon. I s for hs rason w focss on blnar appromaons as a mnmal DCM n h prvos scon.

20 D Smmary In smmary SEM s a smpl an pragmac approach o ffcv conncvy whn ynamcal aspcs can b scon a lnar mol s sffcn an h sa varabls can b masr prcsly an v h np s nknown b sochasc an saonary. Ths assmpons ar mpos by gnoranc abo h nps. Som of hs rprsn rahr svr rsrcons ha lm h ly of SEM n rlaon o DCM or sa-spac mols consr n.. Th mos profon crcsm of smpl rgrsson an SEM n magng nroscnc s ha hy ar mols for nracng bran sysms n h con of nknown np. Th whol pon of sgn prmns s ha h nps ar known an nr prmnal conrol. Ths rnrs h ly of SEM for sgn prmns somwha qsonabl. IV MULTIVARIATE ARMA MODELS ARMA mols can b gnrally rprsn as sa-spac or Markovan mols ha prov a compac scrpon of any fn paramr lnar mol. From hs sa-spac rprsnaon MAR mols can b rv an sma sng a vary of wll-sablsh chnqs. W wll focs on how h sa-spac rprsnaon of lnar mols of ffcv conncvy can b rv from h ynamc formlaon an h assmpons rqr n hs rvaon. As n h prvos scon l s assm a lnar DCM n whch nps comprs saonary wh nos ~ N Q ha ar offr o ach rgon n qal srngh.. C. Ths rnrs Eq3 a lnar sochasc ffrnal qaon SDE & A y L 8 Th val of a som fr lag comprss a rmnsc an a sochasc componn η ha obans by rgarng h ffcs of h np as a cmlaon of local lnar prrbaons

21 η η τ τ τ A A 9 Usng h assmpon ha h np s srally ncorrla Q T h covaranc of h sochasc par s ηη τ τ τ τ Q W A T A A T T A A T T A T 3 I can b sn ha whn h lag s small A an Q W. Eqaon 9 s smply a MAR mol ha col b sbc o h sal analyss procrs. A η τ 3 By ncorporang h op ransformaon an obsrvaon rror w can agmn hs AR mol o a fll sa-spac mol wh sysm mar A F τ np mar W G an obsrvaon mar L. L y Gz F ε 3

22 whr z s som wh nnovaon ha mols ynamcally ransform sochasc np. Ths formlaon wol b appropra f h sa varabls wr no rcly accssbl an obsrvaon nos ε was larg n rlaon o sysm nos z. A frs-orr AR mol s sffcn o complly mol ffcv conncvy f w col obsrv all h sas wh rasonabl prcson. In saons whr only som of h sas ar obsrv s possbl o compnsa for lack of knowlg abo h mssng sas by ncrasng h orr of h mol. y F p p L ε F Gz 33 Smlar vcs ar sng n h rconsrcon of aracor sng mporal mbng a varos lags. No ha ncrasng h orr os no rnr h mol nonlnar smply accommoas h possbly ha ach rgon's ynamcs may b govrn by mor han on sa varabl. Howvr ncrasng mol orr looss any rc conncon wh formal mols of ffcv conncvy bcas s no possbl o ransform an ARp mol no a nq DCM. Havng sa ha ARp mols may b vry sfl n sablshng h prsnc of coplng vn f h ac form of h coplng s no spcf c.f. Volrra characrsaons. In smmary scr-m lnar mols of ffcv conncvy can b rc o mlvara AR or mor gnrally ARMA mols whos coffcns can b sma gvn only h sas or ops by assmng h nps ar wh Gassan an nr wh h sam srngh a ach no. Thy hrfor opra nr h sam assmpons as SEM b ar r m-srs mols. Th problm s ha MAR coffcns n F can only b nrpr as ffcv conncons whn h ynamcs ar lnar an all h sas can b obsrv hogh h obsrvaon mar. In hs cas E F τa 34

23 Compar wh Eq9. Howvr hgh-orr MARp o rprsn a sfl way of sablshng sascal pnncs among h rsponss rrspcv of how hy ar cas. Fgr abo hr V CONCLUSION W hav rvw a srs of mols all of whch can b formla as spcal cass of DCMs. Two fnamnal sncons organs hs mols. Th frs s whhr hy pran o h coplng of nps an ops by h nonlnar ransformaons nac among hn sas of a sngl rgon or whhr on s mollng h laral nracons among h sa varabls of svral sysms ach wh s own nps an ops. Th scon sncon s Fgr s ha bwn mols ha rqr h nps o b f an rmnsc as n sgn prmns an hos whr h np s no nr prmnal conrol b can b assm o b wll bhav sally... Gassan. Gvn only nformaon abo nsy of h nps or h on nsy of h nps an ops mposs lmaons on h mol of ffcv conncvy aop. Unlss on mbracs momns grar han scon orr only lnar mols can b sma. Many mhos for non-lnar sysm nfcaon an casal mollng hav bn vlop n saons whr h sysms' np was no nr prmnal conrol an n h cas of SEM no ncssarly for m-srs-aa. Volrra krnls an DCMs may b spcally sfl n nromagng bcas w al plcly wh m-srs aa gnra by sgn prmns.

24 Rfrncs Bna JS 99 Nonlnar Sysm Analyss an Infcaon from Ranom Daa. John Wly an Sons Nw York USA Flss M Lamnabh M an Lamnabh-Lagarrg F 983 An algbrac approach o nonlnar fnconal pansons. IEEE Transacons on Crcs an Sysms 3: Frson J 995 Fnconal an ffcv conncvy n nromagng: A synhss Hman Bran Mappng ;56-78 Frson J an Büchl C Anonal molaon of V5 n hman Pro Nal Aca. Sc USA 97: Frson J Baysan smaon of ynamcal sysms: An applcaon o fmri. NroImag 6:

25 Lgns for Fgrs Fgr Schmac pcng h ffrnc bwn analyss of ffcv conncvy ha arss h np-op bhavor of a sngl rgon an hos ha rfr plcly o nracon among h sas of mlpl rgons. Fgr Ovrvw of h mols consr n hs chapr. Thy hav bn organs o rflc hr pnnc on whhr h nps ar known or no an whhr h mol s a m-srs mol or no.

26 MISO sysm vrcal conncon hrogh sas MIMO sysm horzonal conncons among sas E y nps nps E& & sas & f θ sas sas & f θ & f θ sas & f θ & 3 f3 θ rspons y λ ε rspons y λ ε rspons y λ ε rspons y λ ε 3 3 3

27 SEM MAR Transfr fncons cohrnc an bcohrnc Volrra krnls s an n orr krnls DCM wh a blnar appromaon Effcv Conncvy Conncvy n rms of a rgon s rspons o nps MISO Conncvy n rms of nracons among rgons MIMO Drmnsc Sochasc Drmnsc Sochasc M κ κ y y A f y f f f & & & E λ A η τ A Dynamc Sac Qas-blnar nsons v Rqr nps No nps b assms sas ar obsrv rcly s orr mols M y y g g g g g Γ Γ