CMOS current-mode chaotic neurons

499 CMOS Cu en -Mode Chao ic Neu ons Manuel Delgado-Res i u o and Angel Rod iguez-V&quez Depa amen o Analog Design Cen o Nacional de Mic oelec onica, cma lia sn, 41 012-Se illa, Spain This pape p esen s wo nonlinea CMOS cu en -mode ci cui s ha implemen neu on soma equa ions o chao ic neu al ne wo ks, and ano he ci cui o ealize p og ammable cu en -mode synapse using CMOS-compa ible BJT's. They ha e been ab ica ed in a double-me al, single-poly 1.6pm CMOS echnology and hei measu ed pe o mance eached he expec ed unc ion and speci ica ions. The neu on soma ci cui s use a no el, highly accu a e CMOS ci cui s a egy o ealize piecewise-linea cha ac e is ics in cu en -mode domain. Thei p o o ypes ob ain educed a ea and low ol age powe supply (down o 3 ) wi h clock equency o 5OOkHz. As ega d o he synapse ci cui , i ob ains la ge linea i y and con inuous, linea , weigh adjus men by explo a ion o he exponen ial-law ope a ion o CMOS-BJT's. The ull acco dance obse ed be ween heo y and measu emen s suppo s he de elopmen o u u e analog VLSI chao ic neu al ne wo ks o emula e biological sys ems and ad anced compu a ion. INTRODUCTION Recen s udies on eal ne e memb anes in neu ophysiological expe imen s ha e shown ha he dynamical beha io o biological neu ons a e much mo e complex (including chao ic esponse) han ha exhibi ed by con en ional models used in a i icial ne wo ks, ypically ep esen ed by simple h eshold o sigmoid elemen s [l], [2]. Consequen ly, in he las yea s, new schemes o a i icial neu al ne wo ks ha e eme ged whose pu pose is o mo e ealis ically emula e he chao ic esponses expe imen ally obse ed om biological sys ems. One o he simples chao ic neu al ne wo ks has been epo ed in [3], and cons i u es a modi ica ion o he Nagumo-Sa 0 model [4], whe e ins ead o ollowing an all-o -none law o he ac ion po en ial, modelled by an uni s ep ou pu unc ion U(*), he neu on shows a con inuously g aded s imulus- esponse cu e, ep esen ed by a non-linea unc ion Am). The model in [3], illus a ed in Fig. 1, is de ined by he ollowing ini e-di e ence equa ions: ni(n + 1) = kxi(n) - a&(n)) +Ai@) , n = 0, 1, ... (1) yi(n + 1) =&(n + I)) whe ex,(n+l) and y,(n+l) a e he in e nal s a e and he ou pu o he i h chao ic neu on a he disc e e ime n+l, espec- i ely; a and k a e he scaling and damping ac o s o e ac- o iness ( esidual e ec o a neu on once i ed), espec i ely; and A,(n) is he inpu exci a ion a he ins an n, gi en by Figu e 1 Analog compu e concep o he chao ic neu on ci cui . The nonlinea block is gi en by g(x) = k - ax), acco ding o (1). M N Ai(n) = Wjjyj(n) + Vj,Ij(n) - Bi (2) j= 1 j= 1 whe e he i s e m compu es he in luence o he A4 neu ons d i ing he i h neu on; he second, he exci a ion om he N ex e nal inpu s, 4; and 0, is he h eshold o he i h neu on. Las ly,A-) cons i u es he neu on ou pu unc ion ep esen ed by he piecewise-linea model whe e E is a posi i e numbe de ining he s eepness o he unc ion. Many s udies on chao ic neu al ne wo ks in gene al, and using he p e ious model in pa icula , e eal ha such ne wo ks no only se e as an expe imen al ehicle in he s udy o senso y ne e sys ems, bu also p o ide he means o impo an enginee ing applica ions. In his sense, chao ic neu al ne wo ks ha e been p oposed o sol e di icul op imiza ion p oblems [5], [6] ; o dynamical associa i e pa em classi ica ion [7]; and o signal de ec ion and classgca ion in noisy en i onmen s [8], and i is o eseeable ha new applica ions will a ise in he nea u u e. In spi e o he s ong economical in e es in ol ing hese applica ions, ew up- o-da e physical implemen a ions o chao ic neu ons ha e been p oposed. Thus, i is ad isable o gi e ci cui ealiza ions o hese models. Fu he mo e, due o he echnological end owa ds sys em in eg a ion, hese ci cui s mus be well-sui ed o VLSI, and, i possible, compa ible wi h s anda d low cos CMOS echnologies. The pu pose o his communica ion is o p o ide monoli hic implemen a ions o he disc e e- ime chao ic 500 neu on desc ibed in (l), as well as o he Nagumo-Sa o model ha , p e iously s a ed, subs i u es he nonlinea unc ion in (3) by a uni s ep U(*). Also, a synapse ci cui wi h con inuous, linea p og ammabili y which employs CMOS-compa ible BJT's is epo ed. CURRENT MODE IMPLEMENTATIONS Cu en -mode echniques ha e been employed o implemen bo h neu ons, ollowing he concep ual diag am shown in Fig.1. Summa ion is easily ealized exploi ing KCL. The delay ope a ion can be ealized as a cascade o wo ack-and-hold swi ched-cu en s ages, as p oposed by Hughes e al. [9]. Fig. 2 shows he schema ics o his block. Nonlinea i ies ha e been achie ed using a no el, highly accu a e CMOS ci cui s a egy o ealize PL cha ac e is ics in cu en -mode domain. I is based on he ec i ying cha ac e is ics o he cu en swi ch [lo], which p o ides e y high esolu ion and i ually ze o cu en o se , no 9 I : 1+1 : B Figu e 2 Cu en -mode ack and hold ci cui . + -a I (a) (b) Nagumo-Sa o Model (4 Aiha a Mode Figu e 3 Piecewise linea mapping :I ci cui s o he neu ons. in luenced by ansis o misma ches (indeed, minimum size ansis o s we e used in bo h p o o ypes). Fig. 3 shows he co esponding schema ics o bo h PL unc ions. Cu en ampli ie s can be implemen ed by p ope ly a ioed bila e al cu en mi o s. Fi s ly, le us conside implemen a ion o he nonlinea block, g(-), o he Nagumo-Sa 0 model, acco ding o he concep o Fig.1. Fig.3(a), which consis s o wo cu en sou ces ( ealized in p ac ice by cu en mi o ou pu b anches), ou ansis o s and wo digi al in e e s, shows a concep ual schema ic o he ealiza ion o he PL cha ac e is ics o Fig.3(b). T ansis o s MCSn and M,, in Fig.3(a) ope a e as a cu en -con olled-cu en -swi ch, wkle ansis o s MVSn and MVSp ope a e as ol age-con olled cu en swi ches. Any posi i e inpu cu en inc eases he inpu ol age, u ning Qs de ice ON, and since bo h de ices in he cu en swi ci ha e he same ga e ol age, MCSn OFF. Simul aneously, he ol age a he second in e e ou pu e ol es o he high logic s a e, u ning M s,, ON and M s OFF. Thus, a cu en h(n) -a (ob ained by KCL a noBe NI) is di ec ed o he ou pu node h ough he ansis o M Sn -- he igh -hand piece o Fig.3(b) is implemen ed in his manne . Simila ly, nega i e inpu cu en s u n MCSn and MVsp ON, so ha a cu en kx(n) + a ci cula es h ough M sp o he ou pu node. The ou pu esponse o he neu on can be aken om he ol age a he ou pu o he in e e IN2 which swings om ail o ail depending upon he sign o he inpu cu en , hus esembling he equi ed uni s ep unc ion in ol age mode. Ob iously his bina y signal mus be combined wi h analog swi ches and cu en sou ces o p ope ly d i ing o he neu ons in he ne wo k. Since cu en disc imina ion in he p oposed ci cui elies on in eg a ion unc ion pe o med a he inpu node, esolu ion is e y high, no in luenced by ansis o misma ches (measu emen s om he CMOS p o o ype display esolu ion o 12pA's). Ope a ion speed is also e y high, limi ed mainly by nonlinea ansien s in he ansis o s ha implemen he cu en sou ces used o d i e nodes NI and N2. Also, he eedback c ea ed by in e e IN1 yields signi ican educ ion o he dead-zone exhibi ed by he d i ing poin cha ac e is ics measu ed a he inpu node, ha is p opo ional o (VTn + IVTp() / Ai",, whe e VTn and VTp a e he h eshold ol ages o he ansis o s and A,, is he in e e DC gain. This is an appealing ea u e ha enables educ ion o in e s age loading e o s caused by ini e equi alen MOS ansis o s Ea ly ol ages. Conce ning he implemen a ion o he PL cha ac e is ic shown in Fig. 3(d) o he Aiha a model, Fig. 3(c) shows he co esponding schema ic. The cu en swi ches, oge he wi h he cu en sou ces, a e used o disc imina e he inpu cu en in o h ee pa hs acco ding i x(n) is lowe han -E, g ea e han E, o is comp ised be ween bo h alues. In he wo i s cases, he pa hs a e ou ed o he same node and ampli ied by k, while i x(n)E [-E, E] he cu en is ampli ied by -y, whe e y = WE - k, acco ding o (1) and (3). The ou pu o he neu on can be easily aken in his case by eplica ion and scaling he ou pu cu en o he ampli ie wi h gain -y. 50 1 PROGRAMMABILITY ISSUES FOR CURRENT MODE BLOCKS In cu en -mode domain wo di e en app oaches o p og ammabili y can be conside ed depending on he na u e o he associa ed con olling signals: disc e e p og ammabili y, whe e con olling signals a e digi al, and con inuous p og ammabili y, whe e con olling signals a e analog. Disc e e p og ammabili y can be inco po a ed in a e y simple way, by analog mul iplexing o cu en con ibu ions om di e en mi o s. These mi o s can ei he implemen ixed empla es (wi h applica ion, o ins ance, in cases whe e well-de ined asks mus be sequen ially pe o med), o be bina y-weigh ed ( o mo e gene al applica ion). Disc e e p og ammabili y p o ides ease o con ollabili y and accu a e esul s, a he cos o a s ong a ea penal y. Fo educed a ea and con inuous weigh adjus men , analog p og ammabili y should be conside ed. A simple way o achie e analog p og ammabili y is using unable ansconduc o s, as he one shown in Fig. 4(a). Fig. 4(b) shows a p og ammable cu en mi o using his ansconduc o . Two di e en si ua ions a ise depending on whe he ansis o s ope a e in weak o in s ong in e sion. Analysis o bo h ope a ing condi ions shows he ollowing, - iq = 1s 21 = 2 (4) i. In s ong PllBl 'In weak 'El As can be seen, he dependence is linea o weak in e sion; hence, his la e case p o ides la ge weigh adjus men anges. I is illus a ed in Fig. 5, showing he cu en weigh as a unc ion o IB2/IB o di e en alues o IB,/I,, whe e IB is a no maliza ion ac o o alue lOnA o weak in e sion (Fig. 5(a)) and 50pA o s ong in e sion (Fig. 5(b)). Also, nonlinea i y cancella ion is exac in weak in e sion due o he exponen ial na u e o cu en o ol age cha ac e is ics, while i is only app oxima e o s ong in e sion: nonlinea i y in he weak in e sion case is less han I% up o io=IB2, while he co esponding alue o s ong in e sion is i,=O.13lB2. D awbacks o weak in e sion a e low accu acy, due o misma ch, and educed speed. These can be o e come by using CMOS compa ible la e al BJTs [ 1 I], which exhibi exponen ial ea u e o la ge cu en anges, and wi h excellen ma ching p ope ies [ 121. EXPERIMENTAL RESULTS Bo h neu ons ha e been ab ica ed in a double-me al, single-poly 1.6pm CMOS echnology. Fig. 6 shows he co esponding mic opho og aphs. Some ex a miscellaneous ci cui y has been added o bo h ci cui s o enable es ing he ou pu cu en and he possibili y o ei he open o close he eedback loop. Dummy swi ches we e also added o educe he in luence o clock eed h ough. All cu en ampli ie s we e bina y-weigh ed o he issue o p og ammabili y. Bias cu en I, o he delay s ages was se (4 (b) Figu e 4 (a)Tunable ansconduc o . (b) P og ammable cu en mi o . i - P Figu e 5 Weigh (iJih) a ia ion wi h IBZIIB o di e en alues o IB~IlB in Fig. 4(b). (a) T ansconduc o s ope a ing in weak in e sion I - 10nA. (b) 'kansconduc o s ope a ing in s ong in e sio;, 50pA. o 50pA. To al a ea occupa ion is 0.096mmL o he Nagumo-Sa 0 neu on, and 0.225mm2 o he Aiha a neu on. Fig. 7 shows he cha ac e is ics measu ed in open loop o bo h ci cui s, using he HP41 45 semiconduc o analyze , wi h a ail- o ail powe supply o only 3 . Fo he Nagumo-Sa 0 neu on (Fig.7(a)), a = 20pA, k = 1 and he inpu anges om -20pA o 20pA. Fo he Aiha a neu on (Fig.7(b)), E = 2pA, y= 10 , k = 1 and he inpu sweeps om -1OpA o IOW. In bo h p o o ypes de ia ion om linea i y is less han 0.2%, and he measu ed cu en o se amoun s o ew PA'S. (a) Nagumo-Sa o Model (b) Aiha a Model Figu e 6 Mic opho og aphs o he p o o ypes. ah) 6w 20.0 4.0 Idi 0.0 -20.0 -20.0 0.0 20.0 I 2.Wdl (U) (a) Nagumo-Sa o Model -20.0 0.0 20.0 I 2.Wdi (pA) (b) Aiha a Model Figu e 7 Measu ed open-loop cha ac e is ics. Fig. 8 shows he expe imen al bi u ca ion ees o bo h neu ons, when he damping ac o k = 0.5 and he neu on exci a ion A, aken as he bi u ca ion pa ame e , anes om -20pA o 20~A. All o he pa ame e s we e ixed o he alues p e iously ci ed. Clock equency was se o 500kHz. All hese measu emen s a e in ull acco dance wi h he heo e ical esul s. Also, gi en he educed a ea, as well as ope a ing wi h only 3 powe supply, we belie e bo h neu on implemen a ions a e e y app op ia e o high densi y chao ic neu al ne wo ks on a single chip. RERERENCES G. Ma sumo o, K. Aiha a, Y. Hanyu, N. Takahashi, S. Yoshi- zawa and J. Nagumo: “Chaos and Phase Locking in No mal Squid Axons”. Physics Le e s A, Vol. 123 pp. 162-166, Aug. 1987. C. A. Ska da and W. J. F eeman: “How B ains make chaos in o de o make sense o he wo ld’. B ain and Beha io al Science, Vol. 10, pp. 161-195, 1987. K. Aiha a, T. Takabe and M. Toyoda: “Chao ic Neu al Ne wo - ks”. Phys. Le . 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