A Cerebral Cortex Model that Self-Organizes Conditional Probability Tables and Executes Belief Propagation

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Griffin Weaver
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1 A Cerebra Core Mode ha Sef-Orgazes Codoa robaby Tabes ad Eeces Beef ropagao ICHISGI Absrac Ths paper descrbes a era eor ode of cerebra core BESOM ode ha acqres codoa probaby abes for a Bayesa eor sg sef-orgazg aps ad esaes saes of rado arabes h a approae beef propagao agorh. The approae agorh s dered fro soe asspos. A era eor ha eeces he dered agorh s good agreee h s-ayer ad co srcres ha represe he aaoca characerscs of a cerebra core ay respecs. Ths ode has scaabe e ad space copees ad s herefore qafed o be a ode of he bra a arge-scae forao processor. I. INTRODCTION Throgh ese eroscece sdes rece years he roes of dda areas he cerebra core ad eoros aos of forao o he aaoca srcre of area coecos hae bee accaed ad orgazed. Hoeer dersadg of he core s o eogh o reprodce he a fcos of he core o a coper. Sef-orgazg ap SOM [] s a ode ha reprodces soe core fcos o a coper. SOM s a ache earg agorh ha s characerzed by copee earg ad eghborhood earg. A SOM ca achee cserg hgh-desoa p der spersed codos ad oe. Ths agrees h oe of he feares of cerebra forao processg. The co srcre see he prary sa core ad oher areas s edece ha sggess ha he core s a sor of SOM. A Bayesa eor[] has aso bee sed as a ode for he core[6]. I s a graphca ode ha represes a dreced acycc graph of casa reaos beee rado arabes. Obseraos of soe rado arabes h he eor ao for esao of he aes of he reag rado arabes based o codoa probaby abes. For hs esao a effce cacao ehod caed a beef propagao agorh s sed. Ths eor has feares sch as ose-ressa paer recogo abgos forao-based pasbe ferece ad rea-e operao ha are aso agreee h feares of cerebra forao processg. The core areas for a bdrecoay ed eor. Ths srcre s sar o ha of he Bayesa eor. Oe sa core ode ha has a echas sar o boh SOM ad a Bayesa eor s he Seece Aeo Mode SAM[]. The SAM a herarchca cobao of ICHISGI s h Naoa Ise of Adaced Idsra Scece ad TechoogyAIST Tsba Cera Tsba Ibara Japa. e-a: y-chsg@as.go.p copee earg odes has bdrecoa sga roes a boo-p sga ha seds he ress of dda odes recogo ad a op-do oe ha seds predcos based o pas epereces ad coes. The SAM reprodces soe of he feares of cerebra sa forao processg sch as ose-ressa recogo ad arge segeao ad s cosdered o be a pasbe ode for a cerebra core. I addo o he SAM here are odes for he core hch a op-do sga represes predcos [4][5][6]. These odes hoeer hae o sef-orgazg echas. Tradoa odes aso hae he sse of scaaby. The core s a arge-scae forao processor ha cosss of 4 bo eros ad shod se a copaoa agorh ha rs a a reasc speed hrogh parae processg. Hoeer he order of e ad space copey s o dscssed radoa odes. I hs paper e propose a BESOM BdrEcoa SOM as a ode for he core ha has boh he feares of he SOM ad he Bayesa eor. The BESOM s a herarchca SOM h bdrecoa coecos. Each SOM s sed o sef-orgaze ad oba a codoa probaby abe for he eor. The BESOM repaces he SAM s earg ad recogo cacag fora h oe based o a beef propagao agorh hereby prodg a foog for frher epaso of he ode ad for effce cacao. Ths paper cosss of fe sbseqe secos. Seco II prodes a oere of he BESOM ode s archecre. Secos III ad IV dea he earg ad recogo seps respecey. I Seco IV scaaby s aso dscssed. Seco V descrbes he agreee beee hs ode ad he aaoca feares of a cerebra core. Seco VI prodes cocsos. II. ARCHITECTRE The BESOM cosss of odes ha are coeced he for of a dreced acycc graph. Each ode cosss of seera s. If o odes are ed by a edge he s cded each ode are coeced a copee bpare graph Fg.. Each coeco has a egh ha ares h earg. Seco IV descrbes he srcre of s ore accraey. I as assed ha a eor srcre does o ary h earg ad s ge as a pror oedge. The BESOM repeas he earg sep ad he recogo sep aeraey. A earg sep pdaes eghs of coecos based o he res of he preos recogo sep. A recogo sep cacaes ops of odes hch represe he sae of he ord based o he obsered aes

2 Node Node Node Node Fg.. Ip ecors o he SOM he earg sep. Each bac correspods o he ae h a poseror probaby. Fg.. BESOM Archecre. See Fg. 6 for srcre ad deaed coecos aog s. ad he crre codoa probaby abes. I he earg sep each ode ors as a SOM s copee ayer. Each ode csers p ecors ha are se fro chd odes. The res of he SOM s earg ca be regarded as a codoa probaby as descrbed he e seco. The pdaed codoa probabes are sed a he e recogo sep. The eor of SOMs for a herarchca srcre sch ha SOMs a he hgher ayers of a eor epress ore absrac forao ha copresses ore p forao ha hose a he oer ayers. I he recogo sep a eor of odes ors as a Bayesa eor. Each ode represes a rado arabe. Each cded oe ode correspods o a possbe ae of he rado arabe. Eera ps fro sesors obsered aes are ge as ops of he oes odes odes ha hae o chd odes. I he recogo sep accordace h eera ps ad each ode s codoa probaby abe a approae beef propagao agorh descrbed Seco IV s sed o cope poseror probabes of dda arabes. The ress are sed he e earg sep. I ers of a ode for a cerebra core he BESOM s erpreed as foos. The BESOM s odes ad s are eqae o he hyper-cos ad cos of he core. I he prary sa core dda s are eqae o he oreao cos. III. EARNING STE I he earg sep each ode ors as a SOM copee ayer ad csers ps fro s chd odes. Asse ha Node has chd odes. I hs sep he SOM recees esaed aes of he dda chd odes as ps. Ge ha a esaed poseror probaby of Node s y s BEy a eee of a p ecor fro s epressed as foos. f BE y a 0 oherse BE y Naey a eee correspodg o a h a a poseror probaby s he oher eees are 0 Fg.. The esaed ae of Node becoes a er for copee earg. I he er a referece ecor a ecor of eghs s brogh cose o a p ecor. Assg ha he egh of he coeco beee Node s er ad Node s y s hs egh s he pdaed h he foog re: α α here α s a earg rae. I hs sep eghborhood earg shod aso be doe sg a proper eghborhood fco. Here ge ha he eghborhood rads s sffcey sa ad eggbe ad a earg rae α s eqa o for -h earg of each a egh s he eqa o a codoa probaby y. For deas see he apped. IV. RECOGNITION STE I he recogo sep a approae beef propagao agorh s eeced o esae aes of he dda rado arabes. Ths seco dcaes he probes rodcg a beef propagao agorh o a era eor ode ad prodes soos o hese probes. The characerscs of he dered approae agorh are he dscssed. A. Approao of codoa probaby abes Bayesa eors hae a probe ha he sze of a codoa probaby abe creases epoeay agas he ber of pare odes. A he sae e he eeco cos of he beef propagao agorh [] Fg. aso creases he sae order. I Fg. he arabe BE s a abbreao of

3 BE ha s a esaed poseror probaby of a ae s of a rado arabe here s s a ber of possbe aes of. The arabe s a abbreao of ha s a essage o a ode fro a pare ode cocerg a ae. Oher arabes are abbreaed he sae aer. I he case of BESOM as a ode of a cerebra core ay be assed ha he dda odes cded he sae ayer ear dffere ses of feares ad ha her dda recogo ress are aos depede of each oher. I addo he ber of p feares reqred by he sas of a cera ode ay be assed o be sa becase of he sparseess of ero frgs. I hs sao f a eas oe of Node s pare odes reqres a feare ca be esaed ha hs feare es. Based o hese asspos a codoa probaby abe for Node ca be approaed h he s of codoa probabes eared by pare s SOMs. Ths asspo s sar o ha of he osy-or ode [] so ca be epeced ha hs sao res a spe beef propagao agorh ad redce he e ad space copey sbsaay. I s ecessary o erfy he esabshe of h a physoogca epere or a coper sao. I hs paper s assed o be ge ad a approae beef propagao agorh s dered fro hs asspo. B. Iforao fro essage receer Each fora for ad he agorh Fgre ecdes forao cae fro he ode ha recee a essages cacaed by he fora. Ths cases copey of he agorh. If a eor has a ree srcre he agorh becoes sper as sed [6]. I cerebra core hoeer geeray oe area has coecos o seera hgher areas so e cao asse ha he eor s srcred a ree for. I hs paper coersey e asse ha oe ode has cosderaby ay pare ad chd odes. By assg hs forao ha sppors aes of a cera ode s rado arabe are oray obaed fro seera odes so cso of forao fro a essage receer ay o greay affec he esao ress. Based o hs asspo e se a approao of cso of forao fro a essage receer. C. Approae beef propagao agorh I addo o he o ds of approaos descrbed aboe s assed ha essages fro pare odes are BE α Fg.. orazed. Orga beef propagao agorh. 4 The a beef propagao agorh ca be approaed as sho Fgre 4. See he apped for a deaed derao. I he approaed beef propagao agorh each ode recees he aes of y ad Z fro s chd odes ad he aes of BE fro s pare odes ad deeres aes of s o BE ad Z. The agorh repeas hese cacaos he aes of he dda arabes coerge e a orga oopy beef propagao agorh. y s he ress of he chd odes recogo h he se of ay boo-p forao. κ s a predco based o forao fro pare odes. BE s a poseror probaby. Z s sed for he orazao of BE ad aso represes he degree of agreee beee he predcos ad obseraos Node. I s eresg ha he er Z appears a fora for. Ips fro a ode h a good agreee beee predcos ad obseraos hae a saer effec o he recogo of pare odes. I oher ords he feares o agreed h predcos are ephaszed o he recogo. Ths effec ress fro Asspo. The aes of dda arabes hch oy arse fro a ery spe cacao ca be sffcey peeed h a era eor. I parcar a er prodc cacao codced sg or κ s sabe for ero eeco. I addo a codoa probaby abe ca be obaed fro Hebba earg by syapses g o hese eros. Noe ha oe codoa probaby s eared by o eros κ ad saeosy. D. Scaaby Becase p ecors he earg seps are sparse he obaed codoa probaby abe ay aso be sparse.

4 Z Z κ ρ BE κ ρ ρ y y Z y BE Fg. 4. Dered approae beef propagao agorh. The ber of o-zero eees of a p ecor s a cosa ber ber of chd odes ad depede of he ber of s s h each chd ode. The resg codoa probaby abe ay eaby hae he sae feares. Tag adaage of hs characersc e ca oer he order of e ad space copees. I he case of core syapses h a egh of 0 ca be eaed. Ths ca he epose crease of syapses reqred. A space copey ha s reqred o epress oe s codoa probaby abe referece ecor s O s hch ca be redced o O by ag adaage of abe s sparseess. A e facor ha reqres a er prodc cacao ade by ad κ reach O by he sae reasog. Therefore a e for a er prodc cacaos parae copao h a ode s O. The e copees parae copao for oher arabes are Oogs or ess. As sho aboe he approae agorh dered hs paper s scaabe. Hece ca be sad o be qafed as a ode for he cerebra core s forao processg agorh ers of e ad space copees. V. CORRESONDING TO NEROSCIENCE FINDINGS A. Aaoca characerscs of a cerebra core The cerebra core has a s-ayer srcre. Areas of he core are bdrecoay coeced. These coecos are o o hae he foog regares[7]. Boo-p coecos are dreced fro ayer III o IV. Soe areas aso hae coecos fro ayer V o IV. Top-do coecos are dreced ay fro ayers V ad VI o ayer I. There are aso a fe coecos fro ayer III o ayer I. Frherore based o a co s aaoca srcre forao p o ayer IV s cosdered o be op fro a-ayer V a ayers II ad III h he co[8]. BE Node κ y Z BE Node Node Fg. 5. Correspodece of approae beef propagao agorh ad he s-ayer srcre of cerebra core. Cosderg hese o fdgs he s-ayer core srcre fors a ery srage cosrco hch he forao ayer III a prosoa res of forao processg h he co s se o a hgher area ad he forao ayer V a fa res s se bac o a oer area. The fcoa eag of hs srcre s o. B. Correspodg o he approae agorh Of he see arabes appearg he approae beef propagao agorh fe arabes reaed o ode cocao are apped o a area coeco re. The res s sho Fgre 5. We seeced a ayer II o I for κ becase he ayer I coas sa ber of ces. ayers V ad VI ere seeced for Z based o a deph reaed o BE ahogh a ayer III ca be cosdered. As sho he Fgre 5 a approae agorh ca be drecy correspoded o he coeco re ad cao be cosdered o be cocdeay correspode o. Top-do coecos fro ayer III o I cao be epaed by hs ode. Fgre 6 shos ha he see ds of arabes are paced a co so as o be cosse h aaoca fdgs. A fo of forao he order of ayers IV IIIII ad V as descrbed aboe correspods h ha a order of arabes of ρ ad BE. Based o he fdgs forao frher fos he order of ayers V VI ad IV[8]. Ths forao roe cao be epaed by hs ode. I addo he foog agreees ca be obsered h he fdgs: aos a forao processg s codced ercay h he co ay horzoa fbers are see ayers I ad IV ad here are ay sa ces ayers II ad IV. Z I II III IV V VI

5 BE BE BE BE BE BE κ κ κ κ I II ρ ρ III Z BE BE Z y y y Z y y y IV V VI Z BE BE Fg. 6. Nera eor hch eeces he approae beef propagao agorh. Ths fgre shos he fo of forao beee arabes h o s ad of Node. Node recees forao fro o pare odes ad ad o chd odes ad. The eor srcre s sar o he co srcre of a cerebra core ay respecs see a e. C. Sae ayer s area coeco probes The cerebra core soees has horzoa coecos beee o areas ha cao be epaed he crre BESOM ode. We h hese coecos are cocao roe h a dffere prpose sch as depede copoe aayss. VI. CONCSION A approae beef propagao agorh as dered for bdrecoay coeced SOM eor ad as deosraed ha he agorh boh correspods e o he s-ayer ad co srcres of he cerebra core ad s scaabe. Ths ode represes frg rae of eros h he core ha gh be obsered physoogca eperes. I order o sae hs ode ad codc arge-scae ess he echas o ae odes h he sae ayer depede each oher s be ecdaed. We are crrey acg hs chaege by rodcg a echas of depede copoe aayss o hs ode. If hs sse s soed ao for saos o be perfored erfyg he adeqacy of a approae epresso a codoa probaby abe. I addo o he basc fcos of he BESOM ode descrbed hs paper e h s ecessary o eed he ode for sace o cde seece aeo ad shor-er eory echass. The BESOM eor has a ery hgh ee of epresse poer. We hae sared o ode a echas for acqrg aco seqeces doe by preoor ad sppeeary oor areas ad aso ode a echas for acqrg a sae raso abe of eera ord sed for aco pag by prefroa area. Boh odes are epressed by BESOM eor cobed h a reforcee earg echas ha s a ode of he basa gaga. I he fre e a o reprodce a bra fcos o copers. AENDI I ca be sho ha eghs obaed earg seps are regarded as codoa probabes as foos. e {0} be a -h eee of a p ecor fro s chd ode ad e be earg res here eas -h earg of a. We do o cosder

6 eghborhood earg here. e be he ber of es ha a becoes he esaed ae of Node. e y α be a earg rao. Ge he ae of eqas he foog. > α α Ths ae s a rao beee he ber of es ha a becoes he esaed ae of Node ad he ber of es ha a becoes he esaed ae of Node. Assg ha he esao ress are rgh hs ae s he codoa probaby. y y A approae beef propagao agorh sed for earg seps s dered as foos. Frs ca be approaed as foos sg. 6 7 Here he foog eqao s esabshed fro 4 asspo for orazao. 8 Hece 7 s coered o he foog epresso. ca be approaed as foos by cdg forao fro a essage receer. The foog eqao cded a fora for ca be approaed as foos sg ad 8. By cdg forao fro a essage receer ad 6 he aboe fora s approaed as foos. By sbso of he aboe eqao ca be approaed as foos. Arragg he aboe ress e oba he agorh sho Fgre 4. REFERENCES [] T. Kohoe Sef-Orgazg Maps. Sprger-Verag 995. [] J. ear robabsc Reasog Iege Syses: Neors of asbe Iferece Morga Kafa 988. [] K. Fsha Nera eor ode for seece aeo sa-paer recogo ad assocae reca AIED OTICS 6 : Dec 987. [4] M. Kaao H. Hayaaa T. I: A forard-erse opcs ode of recproca coecos beee sa areas. Neor: Copao Nera Syses [5] R..N. Rao ad D.H. Baard redce codg he sa core: a fcoa erpreao of soe era-cassca recepe-fed effecs Nare Neroscece Vo. No. pp Ja 999. [6] George D. Has J. A herarchca Bayesa ode of ara paer recogo he sa core proc. of IJCNN 005 o. pp [7] adya D.N. ad eera E.H. Archecre ad coecos of corca assocao areas. I: eers A Joes EG eds. Cerebra Core Vo. 4: Assocao ad Adory Corces. Ne or: e ress [8] Gber C.D. Mcrocrcry of he sa-core Aa ree of eroscece 6:

Chapter 1 - Free Vibration of Multi-Degree-of-Freedom Systems - I

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