Floydcramerberry@gmail.com 750 Cellularneuralnetworks Theory Text

Cellular Neural Networks: Theory LEON 0. CHUA. rattow. mis 0 erect NALOG CIRCUTTS have pasă a ery important ră n the developmen of modern cero th ‘ok Even in our pl computer a, alg aus ‘diate such fas communications, power, ao: mate contol sao and video cto base of sr veac signal procesa apes. Comentat rattan mets have un eto a frou pee ote doe fo the seral mater. To venea thi problem, + new computtion mal ulei cur wets” has been proposed, wich ură cn tome specs of curstingy aad adapted to ine ‘rests Pa) The hey fears of neal networks are ‘ynchronos. parallel pre, contour time J me and pal ineracton of emir ements Some hoon ot impresie sppstons of neural net Set hin en pope fr re ih ch în înca and woniner programming sete Tm pars reception ad compet von 5) 13 „nth paper, me wil presenta new cet atc called lar neal meson hich poses sone of ‘he de features of eral networks and ich as în portant poten appleaton ia Sash area as mage rin abd pate neaga This architect resented in Sean I A depth ana of cellar Bowral never thon follows fa arcu, th pats guta of mame rea i deed Seton I! aad 2 ‘Sree oc EEE Ct i dl LIN YANG, stot sata. te Les toy amas presented în Secon IV. Compute ‘nwlatonr and typ dyin Deka of «singe ‘ule sect sete il De Bica Sen FA ‘tthe rly though Ben on Tr ing lar ella ‘val wewonks i piere masts ruta 0 miră cellar marl setts în Scion VI. Fs ‘ly feo miar mathe modes are compre wih ‘ir clr eur erei în Secon VI IL. Ancummerena oF CaLLAR ‘The base cit uit of callie neural penala called «ail î ante nar and poke cet ge nem wach bt ae Heer capac, ner a toric and nina eostroled sources, abd indepen ‘hn source The sate of cle neural etme 1 ‘Str wo that found cla uma ney al Ima celular neal network în conned ony 108 ce el, The adjacent sels can tera dec with DE ace Cale oe dna conse together ma) Sia each other incins Peas oft propapaton A ofthe ein ime Sara of ee eval Serul A ekampic of întorcea neta motan shown in Fig. 1 There we ca dete a ‘lla nea etek of any cen, but în this oper, we mill concentrate onthe Iwordmensonal cae Eine Se nil vor our at on age proces 0054 8100125780100 01968 EEE Dooaa eaccono oanag Bagagoo a0005 pese Dooaa EacBesa Boege eesasaa Pa 2 Memento Cindy) ber} 72a = ra AGG Ce Ce eee the cll onthe om and the fh cums cl (a ‘devotee Cc nia FA. Now let ws eine what we ‘ean by a neg of CU Defsiion |: rmchterhed ‘The repeat cal Ci în a cer era neta tind by Nisa) = (CCE Dax == er ve eusisieN) here n poe integer unter: Fig. 2 oma 3 negiborooe ol te see cel cated athe ema? and shown shade) wih rel 2 aod 3 nec Uma, we cll he pol naphibornod 3 SSE ionica te 72 acghborhood 4 33 ethherhood and he r=} agit 2-77 Imaghhrod” Is ex o sow that te magnet ‘stem dined bone eb toner propery nthe lose that fi) = Nk then Clb Ibe Ns or ret.) and CU) acra ner network A typed example fo cll CU, ta dr next ewok shown ia Fig 3, where tbe ies and 7 ‘ou he ir se, ind ap rece The noe Sg tof Cle) el he eof he all ae w „To mot te i me et nial condo sessed to have «msi le han call 1 The nde lage, ead e put of (Cin) an asm tote estan with magie te tan of egal 10 1 The node ve cll the spat “Sacre trom Fig 3 that ech cll CU.) eta one inpendet wage sree Fone dependent cate ‘Sorte Zoe nar captor‘ wo linea esos Ry INR, snd at mon leur vltagecomted aren ret sch ae valea i magia eels vin the onrating np Sta va ad the feedback too the up olage yf cach eho el COLD thom i cal sobe namber of aio cl Parca, fd) and Ij kl) a ie a er conrad carat sued with the shares RD A De and Tall bel) = ijk, fa co E it Te any sent fr dement în exc cl 3 pieces nea vlan trea arent ate J, = (07 )tt,) wh care tile 7) a shows a i „All of he leer and pisceiseiear cond ces că or clr noua emer can be eval feed ‘Sing ozn npr (op amps) [15 [LA simple “Lan cp anp aicea of cll creat hot ow a generat, te cel Applying KCL and KVL e cit equations a e are Cry ere a flows St eat ar) îi EN III) e E DUPA Dati 1tei1s/sN. (a) Opa equation: 1 afet tata) -Eptaiemirejex 05) Contra condos lufo)]ei. Lstemte sen a) mujst. Mgt M Te se. Qe) Parameter exemption: Mic iba D)= Abb De It MISS. 20 €>0,R,>0. cy Remake 0) Al ie cl of car sura network hae the sume cet urarea and cement tale. The I cal esa whch bas +1) por eal ‘tre ris dfn în (AT ote cel ae aed rar cll cana acral network come ely characterized by the set of all sone Sire eats mit de ca (©) Each cal of a clar aer network us i mos three sods. (Sometines we wil chose B= if BCL tet) =O forall ele or eal nat ‘fork In this case, ere at al 00 nae in el Cru Since all calls have te same datum ode nd ace ll creat coments are vaga coated, tr calla sera pet a tel sited for ‘oda ana Moreover sce the mesa clamp es an ea tend cre sparse for re Seats (The Gară of el neural network has th ouput feedback and inst contol mechanisms. The ‘tpt feedback elt depends the nteacve fier wank) Ick apere and Bk) at cn pea. ‘The asumptons in i) are resonable because of lhe symmetry property ofthe seabrksed seo. (4) The takes of the cil sores can be shine convent în practice. Ry and A, deerme the power diipated inthe crows sd ae wo Shonen to be etncen In and i MD CA, the ‘ime constant ofthe nam of the eu and ‘aly bose to he 10 7-10” TM, Dig Rawat or Cee Nina Nerworss Before we design a physical ela eu eter. is ccs) to know i dynamic range oder to parte that itil en, our assumptions on the dam soma stilted inthe preceding scion, The alow {ny theorem pois the foundation for or eign There 1 All suna e, în a celt nut networ re Bounded focal tine 120 andthe Pound ra an computed 9 {he folowing formal or ay ea neural en asa, netat. 4-0) Cc) Pro it, et rc e cl dynamic eta as a) ; FO Aznar) merele, tu) Curta pe Drm usta mila je (4) Ei Due vereme yen (e) E) Ci where = cs incat an Mier cae tun input velar gato (4) a fstonde edinary teren equation ant sons a ÎN sat) = ey EMRE + [eee ae flan. 09) felon at Ieoj eu e] sheet aut) slo. + eecoeelir pole leat) leit 1] lea me slrecetile romeas true ale ea, 00 ste Fenalibl 1 DE) E 6 male] 0To pone tis simply replace the erat Sep er res toc secon expression în e Lyepun anton EL) în ayy EEL roe (ex) snd then ering E( ey wo oua ae) be uP aa Eas] <0 ow “hs Lypunoe fence one wed by ‘Hepa an can e interpreted thea coon ton fant ln scl neat Tae Art im, “hm Thorman e in easy prove the fl Tew 4 For any aie input e and any inal ste «of a cel oral work ve Am E(0)~ conan ta) ao in E aw) Prof From Throne 2 and 3. Eta) a bounded nete decreasing funia of ine 1. Hence EI) com ‘ergs to mat ands deve emerges to 02 Canary [tr the transit of a celular nem ceace hat să oar, we alvays obtain a constant opus In ‘ther word, we hve fim y(t) ~emuant, gfe Mie yeN (le) tm ult) Leta eveigt pe the stad tate eh ofc lar sural nemo floes rm e pot el Than 4 hat under the onion (GE(1)/s)~0, hte are ee imate canes forth state. a end ony -0. LereMele sem. (218) w E aa) 0 and Judi 20) i) 0 ant : PO oo and Juli 6) cae ofthe characterise th pene ine utp fection 2, “Ths wl be clea we cose Fig 4. When, (11 1/R,: for conven ce and witout lo Ienei, et Ai fh) = 2. Reni, apd C=, in he flowing anal Then Jr) Ins te characteristic shown in Fi 3 ‘Consider net e eae ut fa eli ear ‘ural network as shown i Fig. Thee a oly thee ‘Set clement lina apa with n pontine capac {ace , » picoise near Nae contd esr with {tv rng pont characte =f) J) the tae anetion atin Fig. 3). anda Timevarying independent ‘ren sure whose apel gen Dy RU. The two ‘Seats in Fig Sand e egean because they are oth deeb by (2a) which we rewrite for implici îs [re a a Fo ta) ~0, he epava pots and the danie re SL 4) ofthe aaa creat are shown ia Fig în ‘There ae des equiva pons in th eat ome of thm 0, Gena bya ce sbi he thet (ened and = 2 ae sae, and ate dented y slid ‘Points The wstabi equim point i eer observed in veal econ cuts. case of wavelet al acne Therefore air he amen, ad depen on {he inal tat the cava wl slays approach one of ble ei pases and tay thre tec. Foe ‘ample if he nia state ot the cet 203. then the ea) ste wl be cere atthe sabe ultra pante = bat heal tate thee 205, then the sind) ste wil be oară a the stable ep iia pane i Tg) = coma! 0. there ate si ifr ae ofthe dynamic behavior of the atena creat a orn în Pe 1)-(@ For the ese in Fig. 10) ad (here oe th ee eplva pit neo therm abe, nae {he ote wo ar stale Fo he exes n Fie 1) and (0, here are 1 egean pen: cae Is waa and te ‘the a, To the dam ote ie Fig) an (8 ‘here i ah one spr poor he oat, a ‘ile Oe tat all ofthe ate atât ptt erripoding othe sven danie rer sce wih the equivalent cc asl a car paral etn share he common pope > Tet ret now he base cal et fer eta cual neve ince (7) 8 fonction of oly the utp ft) and te IS a the nighorhoad ‘tthe elt fon rom the ret of Thro hat all ‘ofthe steady sate cupa of out ella nur network Se comaant Hence. aftr the inal tacts our ab ewpion (0) = conan iva forthe oy of the catea havior f cular muri ewok Lets ‘mmarize cu taverne ll Whe cit parameter ss Hola) ste) = ea (28) 1 ee Mi [E] then ex alo ur celula ner network mun te at st ea pat fer te ron hs dea to tere. Moteove, tbe matale of all able ela Dama i peter than IIa oe moră We have the folowing proper „AI, Asem ese (25) ln eth vereitejea 8) Renata: 0) The atone theorem în sign ican or ear neal neted became implies hat the Great wll Do. ‘Sela or become chaos (5) (16) 0) Tiran 3 sarme îm co cla moral et mori have brary outputs, Tas proper e cl for sling clica problemas prcesina spion 0 ăn be ay shown by the same ehnigue that micul consrit of (25) od ae (mă ese (Ghean orase bt ease) amt Tas piesa ‘ema (a) te ven without the coniton (23) co See (o’s peace fom ‘galas mina about of poe eich in order parazit sendy at output of ech ‘al eter 21 or Note that thi ceata i Sua voie na Hopfield neural metork ce te ‘iagona coupling coefficients re all sued tobe ‘0 2). To guarants a m Maury outpat {he Hopfield mode ts necessary to choose an în. ate pe e inc ex o e sons Jin Fig. In cour the conesponding tyra clk tol oat ir ară bere 1 one V.. Coana Sonarions of 4 Sours ‘Crusciat Neva Sero 1 this secon we wil prea avery sil example to ntre how the aller new momen decd în ‘Secon 1 work. This example il ali help to provide ‘eter understanding ofthe Gear proved i he reed „The celular neural network fortis example isthe ame ‘that shown în Fig. ta he network sacs 4 ‘The are cement primate ofthe ell CUL) ae hove flow, ge put eal Cnt it input comin 23) For any CUI) © (ij) si r=, ssipotion ste (n Fi 2) et Micfit-LeJ D0 asia, ONO, si-o poor fr any 33 poeme Michi -D 0, Mi fie puri Mi fet je =0 Cote: Retea; 150; Bi fsb) =O, erc(ă.1) e NU.) ace i, fh!) =0, the 33 coats A, 1) lone drain the taint boars the clit ru ctor We wl often spy thee cotit ln (he form of square aay as Sbown in Fig. Mo) Ben forth called the domine template wich species the d some nde ofthe sl neal newer: “The dynamic euon ct a cur pearl etek cerreponding othe above partes ae pen By 20 corfu Foy wd 40) anu apt) (2) elt) =03(leaf-1l- nau) foelciedi eye (27) 1 n consint to rcs the ightchand sie of (24) ino the mdf At) (3) with he ip ofthe emerson ont ape» ‘fina teow: Defntion s Foray cours template T (ac a the example sown in fig Ha) whe Stine te dye ro oe cal ‘Scat we define he conven operator * by Tene Tk where Tim.) denotes the entry in the m ow and th imn of the cin tempat, a= „1.01 and = Cal, respec Noe that inthe bore definea A, 1) assumed to be independent of | and fortis cella ner amor” Ths property iid be apa marin, wich Imps hat 40 Jk) canbe expressed a (d= = eu) 17). Unless stated ters cela eur networks are ‘Samed be the pac arat property Ths op” fy allows us to spl the darie rae of ear Scara twos by ting honing template „To stedy the wast bea of (7) lt n apply an inal voltage (0) scron the capastor of ach ell Gi Em tit aap may ed ay ae Seon ună nila in 2 “The crit smalator we vedo obtain car ansint responses PWLSPICE [7 whch 8 a modified version of SPICES [1] fe pice car circuit ana. The put tpt fs ne the sae a hve for SPICES „Toe anna bebe of he sve cer neural ne. work with she il soon seed în the ah Fig a) has bows sate. The sate vit ct the ‘Great at 5 pS are tows io Fig 0) The mac tum abut valu a he ste variable at #5 i Qual o 6, approxi. The upper bound of ‘computed rem equation (9 of Thre 1 is sual 7. ic sey clove 06 „The corrponding dlp, 6 i 125 pS areshown ia Fig XC) Observe that all utp varabls sune Binary alos ster I or 1. a pried by Theorem 3. (lee {he cruce AC, ji) >17R, state) Since A mould tbe to ‘ch pce 0 pay the rane ofthe eae cel, we nly display the tase behavior a oe ell (2.2) in Fig 3) The nia ‘mains constant at 10, predicted fom Fi A Bele we tvegate Row any tact ales dat a can au inthe ody ste, combi fin the ‘lowing Defaizon 4 A ae el qr ste, eo pale ot cellar neural Stoa 1 defined este vale, ‘fel CU which sates suo : =O camine 00 a e min ier at ‘om Defntion 4 elds foray asamed combination of nas = Fund therfore may a epee a Stel con sea tan equim state af he oer et For cur carent example, eguralenty, the sable cell uta fate cal cet CU) ae the Selaons eof the e ell cit cin otaned by „Pe me ad pie be eta a [= E Ya re iti an a replacing al capacitor by open cca; namely, are Bay pur ap Bur ude the conta: 1» Gh) wt sce ie gt Soman meu ape sere) Fem poy) atenta 03) Punbeenere, since ses, )=seate,) fom (8) în telor at 4) 286) = 90064) Bue) ouale. 09) Omen that thet und idea (9) can en asume five posable vas mame 2,0 dama 4. feos thatthe coespndg vale ht can be sued în the ste vanable are 6, = (~ 202) 4and e flows fom the are ana that each fone sal ici for out present example can have onl st pone ‘bie cll eqiibum ster samele 6, 422 4, Ei „Te cual sale ell equim sate aid by cc cal cleat depends om ial sat a well as se of Se nego el Hence sell ma eventul approach fy one of sabe equim sate eve if nial Ste Tema changed For cumple consider the ai bi © six iinet of inl cnd în Fg 10 ta al the Sse the nit state of cal COL) ae de same dha i st 210: Aer i ransient has deaed sie at 202,132, ~396, and 394 Tes ar dt all ta vina sabe cl eee sate of C22) wil be ‘ined inthe sed sate Te transient Pav f the {il C222) forthe iia ceda în Fg 10 are ‘Shown in Fig 12 Oare at eventhough hey al set From the sume inal pout, the steady sates approach lee pot nile ots Observed be atthe Trane respense is tet mcosarie monotonic the ne Spon in Fig 6) i ese po „Another ineesing phenomenon can e cbiered by hing he 4 int tee fia coon Fi 13 i follows fom the shove definition that cele eral network sts tone a i table ssn eg © Tene îm se cepe e Beta ema n o wap oo] [ae] = ” LI „BASES ‘o * E Îsi center at he sable Sen equim pois. Toia that lor fal tee in Pp 13 are eat within the Basin of the sme able rye equim ‘ott, which shown n Fp 18). “Our inal gel hi seston lo ake gmp a the ‘les cn the eo of a dynam ele fore caller ur network. Leu oer he inal conden owe Pig 16 Ps, kt ws we the same name ule de că byte cloning tempt la Fi i) The ial state în 723 48 and ts corepending output of the cela eal network starting fom ts il ate ar shown in Figs 10 and IM) especie, Next et ws change the va re by wn the new cloning template shows in Fig), The ial te e 5p and it erezie ‘xtpt staring fom the sune inal sae of Fig. 16 ae ‘Sown in Figs 170) and 18), spectively. Alou the iy iflerence between the wo out shown in Fg 1B cuts in cell C2.2) which hav oppose aes. will ‘mein ÎI) that thee two dynam eles perform Sry nt functions wen app to mage proce. ‘General ea. celular teu network process sigan by mapping ica oon gal apes into wnt Seema tct see S_etnw anise a ee e {he inal state space at [o 10.10” and the output ‘pace at [OLE the the dynamical pF, can be proces. This propery cn be ented în be dei of ntre emiri arcing co and fad mira stems Th era teint st complex note system is ery tile if at imponite, to determin, cera Viza or numerical Ag. pews Enea e x tis pomible ond al de slatons by wing eer trate force lg [19 some more efficent one 20, [BIL is never very tine consaming fo lage A tem. Fer our clu neural networks ln view ot the are siberian property, we can salve fr all ‘atm equim pout by ft detemsnng the able all equim sata and then singe sighber itr Setters o ind the corespnding system equim pot ‘AS present above, te dynamic behavo of cal i= {23-25} Both of tem have te paral sira processing fapablty and ae based on the Reset nebo nec tive dynamic rule, The main ference Benea lla ‘eral network and cele stomata machine in her ‘snare been The fre 3 en me ale the later isa icre tie mama tem: Bese the ‘wo systems have many mite, we can wie ear Sloman theory to tay the steed sate bata of Sellar seul ttwors,Anober rari duncan tetwecn thm i tut while the eu nenul networks Say sete to stable equim pots inthe sed Slates ur automa machine usualy ied ith ‘much sicher opună ehaor, sch at prot, aut nd evn mom compler phenomena, Of iama. we have {ned ur celular perl etwors by cooing © mend Sonnet. I we noe some other semi fr the monica cements mat) ove comple phenomena il th coer in calu neal mater. Te to models ‘il be compared in more dedu in Sesion FL VI. Mouraaven Cricean Nrvnat Nemo We cam generalize the spine ceia pearl et. ‘ork inrodoced în Section 1110 4 maker calu rai netwoek. Instead of only esate vale in the Single-ayer case thee may be several sale vals în ach cel of mulayer cellar neta rework The oop of mullyerig emphases he interactions ofthe ate vaibls on the fame Ier To svcd ter, i is Content to we the cometa cpt ts lined în Definition 3, i the folowing. Using the como operate we an ewe a) a) Felt) Ata) Beth usile ex. 65) ‘Then, for multilayer clar neal setworts, te cal mar equation an be expres în the following come a ee ce ©) a ° : ° me of : pn 3 om : ma E Ic ee A EI Jo» oo nd where m dente the suber ofthe aibe în the sine cll et. Here, the comoluton operat + mea mata nds velr isto be decoded Me ait ‘mulphcaten bot withthe operator inert between teh entry ofthe marc and a the ete ‘Observe tat C and A e diagonal maces, whereas A and re lock angular mates. „Remar (6 For mahiayer cellar neural networks al ofthe ‘els prsatd inthe previous wens stl bold ‘ith some mint miieatons The stb es be Proved fom te bt ayer Gaye o e upper ‘oes y ning te beck angular recurs of he ind 8 mate (Since there are Several state variables in el i cut me can chore multiple dynamic rls coca ata for the diferent state va Thi propery ale te aetncek curent ele and allows wo {0 dea ith more completed image procese rien ston to ung mil Amar rue a ma one n) we et ec erent tne conn toe the differnt sat vanat of the el cat [Asa sng case, we can chose Cy 0 fr Sone ‘eae variable, thereby obtaining 3 st of AF. Fecal and alba equations Tos pope) vs © = se even mere Mei în the design of clar seră networks for pata pro VII. Ratanow 0 Patra Dama tions av CHLLULAn AUTOMAT De) der the ratan betwee ur cellar Bea net md ‘torts td thre wo mutate! ‘Cone a paral diferent eqestion Ket The well oan ent equation fam Pays ua, Pulte ae ae o here «x constant called th thermal conduci. The Toler, (sy) of the heat easton isa comnos ‘anc ofthe ime, andthe spe variables and te foncon a(x, )) bs spposimated by a St of functons (7) which i defined as 140) =H dy!) o) tere A and A, re th spice interval inthe andy sasi then th paral deste of 3.) ih eta andj can be reparat by Bale yt), Palsy.) aap on Cu: frat E) “Thus the heat ouation cn be approciatly by setat atena Va aut 0 tape) ad)