A First Model for Hebbian Learning with Spiking Neural P Systems

A Fi s Model o Hebbian Lea ning wi h Spiking Neu al P Sys ems Miguel A. Gu i´e ez-Na anjo, Ma io J. P´e ez-Jim´enez Resea ch G oup on Na u al Compu ing Depa men o Compu e Science and A i icial In elligence Uni e si y o Se illa A da. Reina Me cedes s/n, 41012, Se illa, Spain E-mails: {magu ie ,ma pe }@us.es Summa y. Spiking neu al P sys ems and a i icial neu al ne wo ks a e compu a ional de ices which sha e a biological inspi a ion based on he ansmission o in o ma ion among neu ons. In his pape we p esen a i s model o Hebbian lea ning in he ame- wo k o Spiking Neu al P sys ems by using concep s bo owed om neu oscience and a i icial neu al ne wo k heo y. 1 In oduc ion When an axon o cell Ais nea enough o exci e cell Bo epea edly o pe sis en ly akes pa in i ing i , some g ow h p ocess o me abolic change akes place in one o bo h cells such ha A’s e iciency, as one o he cells i ing B, is inc eased. D. O. Hebb (1949) [13] Neu oscience has been a ui ul esea ch a ea since he pionee ing wo k o Ram´on y Cajal in 1909 [22] and a e a cen u y ull o esul s on he man and he mind, many in e es ing ques ions a e oday open p oblems. Two o such p oblems o cu en neu oscience a e he unde s anding o neu al plas ici y and he neu al coding. The i s one, he unde s anding o neu al plas ici y, is ela ed o he changes in he ampli ude o he pos synap ic esponse o an incoming ac ion po en ial. Elec- ophysiological expe imen s show ha he esponse ampli ude is no ixed o e ime. Since he 1970’s a la ge body o expe imen al esul s on synap ic plas ici y has been accumula ed. Many o hese expe imen s a e inspi ed by Hebb’s pos u- la ed (see abo e). In he in eg a e-and- i e o mal spiking neu on model [9] and also in a i icial neu al ne wo ks [12] is usual o conside a ac o was a measu e o he e icacy o he synapse om neu on o ano he . 212 M.A. Gu i´e ez-Na anjo, M.J. P´e ez-Jim´enez The second one, he neu al coding, is ela ed o he way in which one neu on sends in o ma ion o o he ones. I is in e es ed on he in o ma ion con ained in he spa io- empo al pa e n o pulses and on he code used by he neu ons o ansmi in o ma ion. This esea ch a ea wonde s how o he neu ons decode he signal o i he code can be ead by ex e nal obse e s and unde s and he message. A p esen , a de ini e answe o hese ques ions is no known. The elemen a y p ocessing uni s in he cen al ne ous sys em a e neu ons which a e connec ed o each o he in an in ica e pa e n. Co ical neu ons and hei connec ions a e packed in o a dense ne wo k wi h mo e han 104cell bodies pe cubic millime e . A single neu on in a e eb a e co ex o en connec s o mo e han 104pos synap ic neu ons. The neu onal signals consis o sho elec ical pulses (also called ac ion po- en ials o spikes) and can be obse ed by placing a ine elec ode close o he soma o axon o a neu on. The junc ion be ween wo neu ons is a synapse and i is common o e e o he sending neu on as a p esynap ic cell and o he ecei ing neu on as he pos synap ic cell. Since all spikes o a gi en neu on look alike, he o m o he ac ion po en ial does no ca y any in o ma ion. Ra he , i is he numbe and he iming o spikes which ma e . T adi ionally, i has been hough ha mos , i no all, o he ele an in o ma ion was con ained in he mean i ing a e o he neu on. The concep o mean i ing a es has been success ully applied du ing he las 80 yea s (see, e.g., [18] o [14]) om he pionee ing wo k o Ad ian [1, 2]. None heless, mo e and mo e expe imen al e idence has been accumula ed du ing ecen yea s which sugges s ha a s aigh o wa d i ing a e concep based on empo al a e aging may be oo simplis ic o desc ibe b ain ac i i y. One o he main a gumen s is ha eac ion imes in beha io al expe imen a e o en oo sho o allow long empo al a e ages. Humans can ecognize and espond o isual scenes in less han 400ms [24]. Recogni ion and eac ion in ol e se e al p ocessing s eps om he e inal inpu o he inge mo emen a he ou pu . I a each p ocessing s eps, neu ons had o wai and pe o m a empo al a e age in o de o ead he message o he p esynap ic neu ons, he eac ion ime would be much longe . Many o he s udies show he e idence o p ecise empo al co ela ions be ween pulses o di e en neu ons and s imulus-dependen synch oniza ion o he ac i i y in popula ions o neu ons (see, o example, [5, 11, 10, 6, 23]). Mos o hese da a a e inconsis en wi h a concep o coding by mean i ing a es whe e he exac iming o spikes should play no ole. Ins ead o conside ing mean i ing a es, we conside he ealis ic si ua ion in which a neu on ab up ly ecei es an inpu and o each neu on he iming o he i s spike a e he e e ence signal con ains all he in o ma ion abou he new s imulus. Spiking neu al P sys ems (SN P sys ems, o sho ) we e in oduced in [15] wi h he aim o inco po a ing in memb ane compu ing1ideas speci ic o spike-based 1The ounda ions o memb ane compu ing can be ound in [20] and upda ed bibliog a- phy a [25]. A Fi s Model o Hebbian Lea ning wi h SN P Sys ems 213 neu on models. The in ui i e goal was o ha e a di ec ed g aph we e he nodes ep esen he neu ons and he edges ep esen de synap ic connec ions among he neu ons. The low o in o ma ion is ca ied on he ac ion po en ials, which a e encoded by objec s o he same ype, he spikes, which is placed inside he neu ons and can be sen om p esynap ic o pos synap ic neu ons acco ding o speci ic ules and making use o he ime as a suppo o in o ma ion. This pape is a i s answe o he ques ion p oposed by Gh. P˘aun in [21] ela ed o link he s udy o SN P sys ems wi h neu al compu ing and as he sugges s, he s a ing poin has been no only neu al compu ing, bu also ecen disco e ies in neu ology. The pape is o ganized as ollows: i s we discuss abou SN P sys ems wi h inpu and delay and a new compu a ional de ice called Hebbian SN P sys em uni is p esen ed. In sec ion 3 we p esen ou model o lea ning wi h SN P sys ems based on Hebb’s pos ula e. An illus a i e expe imen ca ied ou wi h he co esponding so wa e is shown in sec ion 4. Finally, some conclusions and u he discussion on some opics o he pape a e gi en in he las sec ion. 2 SN P Sys ems wi h Inpu and Decay An SN P sys em consis s o a se o neu ons placed in he nodes o a di ec ed g aph and sending signals (called spikes) along he a cs o he g aph (called synapses). The objec s e ol e acco ding o a se o ules (called spiking ules). The idea is ha a neu on con aining a ce ain amoun o spikes can consume some o hem and p oduce o he ones. The p oduced spikes a e sen (maybe wi h a delay o some s eps) o all neu ons o which a synapse exis s ou going om he neu on whe e he ule was applied. A global clock is assumed and in each ime uni each neu on which can use a ule should do i , bu only (a mos ) one ule is used in each neu on. One o he neu ons is conside ed o be he ou pu neu on, and i s spikes a e also sen o he en i onmen (a de ailed desc ip ion o SN P sys ems can be ound in [21] and he e e ences he ein). In his sec ion we in oduce he Hebbian SN P sys em uni which is an SN P sys em wi h m+ 1 neu ons (mp esynap ic neu ons linked o one pos synap ic neu on) endowed wi h inpu and decay. A he s a ing poin all he neu ons a e inac i e. A es , he memb ane o biological neu ons has a nega i e pola iza ion o abou −65mV , bu we will conside he inac i i y by conside ing he he numbe o spikes inside he neu on is ze o. The dynamics o a Hebbian SN P sys em uni is qui e na u al. A he s a ing poin , all neu ons a e a es and in a ce ain momen he p esynap ic neu ons ecei e spikes enough o ac i a e some ules. The ins an o he a i al o he spikes can be di e en o each p esynap ic neu on. These spikes ac i a e one ule inside he neu ons and he p esynap ic neu ons send spikes o he pos synap ic neu on. In he pos synap ic neu on a new ule can be igge ed o no , depending on he a i al o spikes and i may send a spike o he en i onmen . 214 M.A. Gu i´e ez-Na anjo, M.J. P´e ez-Jim´enez 2.1 The Inpu The basic idea in SN P sys ems aken om biological spiking neu on models is he codi ica ion o he in o ma ion in ime. The in o ma ion in a Hebbian SN P sys em uni is also encoded in he ime in which he spikes a i e o he neu on and he ime in which he new spikes a e emi ed. The inpu will be also encoded in ime. The idea behind his codi ica ion is ha he p esynap ic neu ons may no be ac i a ed a he same momen . I we conside a Hebbian SN P sys em uni as pa o a wide neu al ne wo k, i is qui e na u al o hink ha he spikes will no a i e o he p esynap ic neu ons (and consequen ly, hei ules a e no ac i a ed) a he same ime. In his way, i we conside a Hebbian SN P sys em uni wi h mp esynap ic neu ons {u1, . . . , um}, an inpu will consis o a ec o ~x ={x1, . . . , xm}o non-nega i e in ege s whe e xi ep esen s he ime uni o he global clock in which he neu on uiis ac i a ed2. 2.2 The Decay The e ec o a spike on he pos synap ic neu on can be eco ded wi h an in a- cellula elec ode which measu es he po en ial di e ence be ween he in e io o he cell and i s su oundings. Wi hou any spike inpu , he neu on is a es co - esponding o a cons an memb ane po en ial. A e he a i al o he spike, he po en ial changes and inally decays back o he es ing po en ial. The spikes, ha e an ampli ude o abou 100mV and ypically a du a ion o 1-2 ms. This means ha i he o al change o he po en ial due o he a i al o spikes is no enough o ac i a e he pos synap ic neu on, i decays a e some milliseconds and he neu on comes back o i s es ing po en ial (see Fig. 1). This biological ac is no implemen ed in cu en SN P sys ems, whe e he spikes can be inside he neu on o a long ime i hey a e no consumed by any ule. In he Hebbian SN P sys em uni , we in oduce he decay in he ac ion po en ial o he neu ons. When he impulse sen by a p esynap ic neu on a i es o he pos synap ic neu on, i i is no consumed o igge ing any ule in he pos synap ic neu on i decays and i s con ibu ion o he o al change o po en ial in he pos synap ic neu on dec eases wi h ime. This decayed po en ial is s ill able o con ibu e o he ac i a ion o he pos synap ic ule i o he spikes a i e o he neu on and he addi ion o all he spikes igge any ule. I his one does no occu , he po en ial decays and a e a sho ime he neu on eaches he po en ial a es . Figu e 2 shows a scheme in which wo p esynap ic neu ons send wo spikes each o hem a di e en momen s o a pos synap ic neu on. Figu e 3 shows he changes o po en ial in he pos synap ic neu on ill eaching he h eshold o i ing a esponse. In o de o o malize he idea o decay in he amewo k o SN P sys ems we in oduce a new ype o ex ended ules: he ules wi h decay. They a e ules o he o m 2In Sec ion 5 we discuss abou o he codings o he inpu . A Fi s Model o Hebbian Lea ning wi h SN P Sys ems 215 Fig. 1. Dynamics o one spike Fig. 2. Two p esynap ic and one pos synap ic neu on E/ak→(ap, S); d whe e, Eis a egula exp ession o e {a},kand pa e na u al numbe s wi h k≥p≥0, d≥0 and S= (s1, s2, . . . , s ) is a ini e non-inc easing sequence o na u al numbe s called he decaying sequence whe e s1=kand s = 0 . I E=ak, we will w i e ak→(ap, S); dins ead o ak/ak→(ap, S); d. The in ui ion behind he decaying sequence is he ollowing. When he ule E/ak→(ap, S); dis igge ed a 0we look in S= (s1, . . . , s ) o he g ea es lsuch ha p≥sl. Such slspikes a e sen o he pos synap ic neu ons acco ding wi h he delay din he usual way. No ice ha slcan be equal o p, so a his poin his new ype o ule is a gene aliza ion o he usual ex ended ules. A 0+d+1, he slspikes a i e o he pos synap ic neu ons. The decay o such spikes is de e mined by he decaying sequence. I he spikes a e no consumed by he igge ing o a ule in he pos synap ic neu on, hey decay and a ime 0+d+2 we will conside ha sl−sl+1 spikes ha e disappea ed and we only ha e sl+1 spikes in he pos synap ic neu on. I he spikes a e no consumed in he ollowing s eps 216 M.A. Gu i´e ez-Na anjo, M.J. P´e ez-Jim´enez by he igge ing o a pos synap ic ule, a 0+d+ 1 + −l he numbe o spikes will be dec eased o s = 0 and he spikes a e los . This de ini ion o decay3can be seen as a gene aliza ion o he decaying spikes p esen ed in [7]. In ha pape a decaying spike ais w i en in he o m (a, e), whe e e≥1 is he pe iod. F om he momen a spike (a, e) a i es in a neu on, eis dec emen ed by one in each s ep o compu a ion. As soon as e= 0, he co esponding spike is los and canno be used anymo e. In his way, a ule E/ak→ap;d(k > p) whe e apa e pdecaying spikes (a, e) can be seen wi h ou no a ion as E/ak→(ap, S); dwi h S= (s1, . . . , se+2), s1=k, s2=· · · =se+1 =pand se+2 = 0. 2.3 Hebbian SN P Sys em Uni s Hebbian SN P sys em uni s a e SN P sys ems wi h a ixed opology endowed wi h inpu and decay. They ha e he ollowing common ea u es: •The ini ial numbe o he spikes inside he neu ons is always ze o in all Hebbian SN P sys em uni s, so we do no e e o hem in he desc ip ion o he uni . •All he p esynap ic neu ons a e linked o he pos synap ic neu on and hese a e all he synapses in he SN P sys em, so hey a e no p o ided in he desc ip ion. •The ou pu neu on is he pos synap ic one. Bea ing in mind hese ea u es, we desc ibe a Hebbian SN P sys em uni in he ollowing way. 3Fu he discussion abou he decay can be ound in Sec ion 5. Fig. 3. The po en ial a he pos synap ic neu on A Fi s Model o Hebbian Lea ning wi h SN P Sys ems 217 De ini ion 1. A Hebbian SN P sys em uni o deg ee mis a cons uc HΠ = (O, u1, . . . , um, ), whe e: •O={a}is he alphabe ( he objec ais called spike); •u1, . . . , uma e he p esynap ic neu ons. Each p esynap ic neu on uihas asso- cia ed a se o ules Ri={Ri1, . . . , Rili}whe e o each i∈ {1, . . . , m}and j∈ {1, . . . , li},Rij is a decaying ule o he o m: ak→(anij , S); dij We will call nij he p esynap ic po en ial o he ule and dij is he delay o he ule. No e ha all ules a e igge ed by kspikes. The decaying sequence S will be discussed below. • is he pos synap ic neu on which con ains only one pos synap ic ule E∗ p/ap→ a; 0 whe e E∗ pis he se 4o egula exp essions {n∈N|n≥p}. We will call p he h eshold o he pos synap ic po en ial o he Hebbian SN P sys em uni . By conside ing he decaying sequences we can dis inguish among h ee ypes o Hebbian SN P sys em uni s: •Hebbian SN P sys em uni s wi h uni o m decay. In his case he decaying sequence Sis he same o all he ules in he mp esynap ic neu ons. •Hebbian SN P sys em uni s wi h locally uni o m decay. In his case he decaying sequence Sis he same o all he ules in each p esynap ic neu on. •Hebbian SN P sys em uni s wi h non-uni o m decay. In his case each ule has associa ed a decaying sequence. A Hebbian SN P sys em uni is an abs ac machine whe e a global clock is assumed ( he sys em is synch onized). I akes an inpu and can p o ide an ou pu o no , depending i he po en ial in he pos synap ic neu on eaches o no i s h eshold. The concep o inpu o a Hebbian SN P sys em uni is de ined as ollows: De ini ion 2. An inpu o a Hebbian SN P sys em uni o deg ee mis a ec o ~x = (x1, . . . , xm)o mnon-nega i e in ege s xi. AHebbian SN P sys em uni wi h inpu is a pai (HΠ, ~x)whe e HΠ is Hebbian SN P sys em uni and ~x is an inpu o i . The in ui i e idea behind he inpu is encoding he in o ma ion in ime. Each xi ep esen he momen , acco ding o he global clock, in which one spike is p o ided o each p esynap ic neu on. 4This ule is an adap a ion o he concep o a ule om an ex ended spiking neu al P sys em wi h h esholds aken om [7]. 218 M.A. Gu i´e ez-Na anjo, M.J. P´e ez-Jim´enez 2.4 How i wo ks In his subsec ion we p o ide a desc ip ion o he seman ics o a Hebbian SN P sys em uni . As we saw be o e, each xiin he inpu ~x = (x1, . . . , xm) ep esen s he ime in which kspikes a e p o ided o he neu on ui. A he momen xiin which he spike a i es o he neu on uione ule (ak→(anij , S); dij) is chosen in a non-de e minis ic way among all he ules o he neu on. Applying i means ha kspikes a e consumed and we look in S= (s1, . . . , s ) o he g ea es lsuch ha nij ≥sl. Such slspikes a e sen o he pos synap ic neu ons acco ding o he delay dij in he usual way, i.e., slspike a i e o he pos synap ic neu on a he momen xi+dij + 1. The decay o such spikes is de e mined by he decaying sequence. As we saw abo e, i he spikes a e no consumed by he igge ing o a ule in he pos synap ic neu on, hey decay and a ime xi+dij + 2 we will conside ha sl−sl+1 spikes ha e disappea ed and we only ha e sl+1 spikes in he pos synap ic neu on. I he spikes a e no consumed in he ollowing s eps by he igge ing o a pos synap ic ule, a x0+dij +1+ −l he numbe o spikes will be dec eased o s = 0 and he spikes a e los . The po en ial on he pos synap ic neu on depends on he con ibu ions o he chosen ules in he p esynap ic neu ons. Such ules send spikes ha a i e o he pos synap ic neu on a di e en momen s which depend on he inpu ( he momen in which he p esynap ic neu on is ac i a ed) and he delay o he chosen ule. The con ibu ion o each ule o he pos synap ic neu on also changes along he ime due o he decay. Fo mally, he po en ial o he pos synap ic neu on is a na u al numbe calcu- la ed as a unc ion R∗which depends on he ime , on he inpu ~x and on he ules chosen in each neu on R∗(R1i1, . . . , Rmim, ~x, )∈N. Such a na u al numbe ep esen s he numbe o he spikes a he momen in he pos synap ic neu ons and i is he esul o adding he con ibu ions o he ules R1i1, . . . , Rmim. The Hebbian SN P sys em uni p oduces an ou pu i he ule o he pos sy- nap ic neu on ,E∗ p/ap→ais igge ed, i.e., i a any momen he amoun o spikes in he pos synap ic neu on is g ea e han o equal o he h eshold p, hen he ule is ac i a ed and igge ed. I he e does no exis such , hen he Hebbian SN P sys em uni does no send any spike o he en i onmen . Bea ing in mind he decay o he spikes in he pos synap ic neu on, i any spike has been sen ou by he pos synap ic neu on a e an app op ia e numbe o s eps, any spike will be sen o he en i onmen . F om a p ac ical poin o iew we ha e a bound o he numbe o s eps in which he spike can be expelled, so we ha e a decision me hod o de e mine i he inpu ~x p o ided o he Hebbian SN P sys em uni p oduces o no an ou pu . Example 1. Le us conside he ollowing Hebbian SN P sys em uni HΠ = (O, u1, u2, ) wi h non-uni o m decay, whe e: A Fi s Model o Hebbian Lea ning wi h SN P Sys ems 219 •O={a}is he alphabe ; •u1, u2a e he p esynap ic neu ons. The p esynap ic neu ons u1, u2ha e associ- a ed he se s o ules R1={R11, R12, R13}and R2={R21, R22}, espec i ely, wi h R11 ≡a3→(a2,(3,2,0)); 0 R21 ≡a3→(a2,(3,2,0)); 1 R12 ≡a3→(a, (3,1,0)); 1 R22 ≡a3→(a, (3,1,0)); 0 R13 ≡a3→(a3,(3,0)); 0 • is he pos synap ic neu on which con ains only one pos synap ic ule E∗ 2/a2→a; 0. No ice ha in his example, he ules send all he p esynap ic po en ial o he pos synap ic neu on bu i only las s one ime uni be o e being los . I hey a e no consumed immedia ely, hey disappea . Case 1: Le us conside he inpu ~x = (0,0), i.e., a = 0 h ee spikes a e placed in each p esynap ic neu on. We ep esen he con ibu ion o each ule o ~x = (0,0) in he ollowing able. No ice ha o ≥3 he con ibu ion is ze o o all he ules. R11 R12 R13 R21 R22 = 1 2 0 3 0 1 = 2 0 1 0 2 0 Conside ing he di e en con ibu ions o he ules and bea ing in mind ha in each neu on only one ule is non-de e minis ically chosen, he changes in he pos synap ic po en ial o ~x = (0,0) a e desc ibed in he ollowing able. R11 R21 R12 R21 R13 R21 R11 R22 R12 R22 R13 R22 = 1 203 3 14 = 2 2 32010 No ice ha wi h he inpu ~x = (0,0), he pos synap ic neu on ac i a es he ule a = 1 i he chosen ules a e R11 R21,R13 R21,R11 R22 o R13 R22. I he chosen ules a e R12 R21, hen he ule is ac i a ed a = 2 and i he chosen ules a e R12 R22 hen he pos synap ic ule is no ac i a ed. Case 2: Le us conside now he inpu ~x = (1,0), i.e., a = 0 h ee spikes a e placed in he p esynap ic neu on u2and in = 1 o he h ee spikes a e placed in u1. As abo e, we ep esen he con ibu ion o ~x = (1,0) in he ollowing able. R11 R12 R13 R21 R22 = 1 0 0 0 0 1 = 2 2 0 3 2 0 = 3 0 1 0 0 0 The changes o he po en ial R∗in he pos synap ic po en ial o ~x = (1,0) a e desc ibed in he ollowing able. 226 M.A. Gu i´e ez-Na anjo, M.J. P´e ez-Jim´enez •The a e o lea ning ²= 0.1 S ep 1: Le us conside he inpu ~x = (0,2). The con ibu ion can be sum- ma ised in he ollowing able: R11 R12 R13 R21 R22 P = 1 30 0 30 0 0 60 = 2 15 70 15 0 0 100 = 3 0 30 0 0 30 60 = 4 0 15 0 80 15 110 = 5 0 0 0 70 0 70 = 6 0 0 0 30 0 30 = 7 0 0 0 15 0 15 The e o e, a ime = 2 he po en ial o he pos synap ic neu on eaches a alue g ea e han he h eshold 70, hen (0,2) = 2. We can compu e now he alues (0,2) ij =xi+dij + 1, s(0,2) ij = (0,2) − (0,2) ij and L(s(0,2) ij ) o e e y ule Rij. A e compu ing he alues L(s(0,2) ij ) o e e y ule Rij, he new weigh s a e calcula ed as w0 ij =wij +² L(s(0,2) ij ) These alues a e summa ised in he ollowing able (0,2) ij s(0,2) ij L(s(0,2) ij )wij w0 ij R11 1 1 2 1 1.2 R12 2 0 4 1 1.4 R13 1 1 2 1 1.2 R21 4−2−1 1 0.9 R22 3−1−1 1 0.9 The e o e, a e his is s ep he new weigh s a e w0 11 = 1.2, w0 12 = 1.4, w0 13 = 1.2, w0 21 = 0.9 and w0 22 = 0.9. S ep 2: Le us conside he new ex ended Hebbian SN P sys em uni buil by eplacing he ini ial weigh s by he new w0 ij and le us conside he second inpu ~x2= (0,0). The con ibu ion can be summa ized in he ollowing able. R11 R12 R13 R21 R22 P = 1 30 0 30 0 30 90 = 2 15 80 15 70 15 195 = 3 0 70 0 30 0 100 = 4 0 30 0 15 0 45 = 5 0 15 0 0 0 15 The e o e, a ime = 1 he po en ial o he pos synap ic neu on eaches a alue g ea e han he h eshold 70, hen (0,0) = 1. We can compu e now he alues (0,0) ij =xi+dij + 1, s(0,0) ij = (0,0) − (0,0) ij and L(s(0,0) ij ) o e e y ule Rij. A Fi s Model o Hebbian Lea ning wi h SN P Sys ems 227 A e compu ing he alues L(s(0,0) ij ) o e e y ule Rij, he new weigh s a e calcula ed as w00 ij =w0 ij +² L(s(0,0) ij ) These alues a e summa ized in he ollowing able (0,2) ij s(0,2) ij L(s(0,2) ij )w0 ij w00 ij R11 1 0 4 1.2 1.6 R12 2−1−1 1.4 1.3 R13 1 0 4 1.2 1.6 R21 2−1−1 0.9 0.8 R22 1 0 4 0.9 1.3 The e o e, a e his is s ep he new weigh s a e w0 11 = 1.6, w0 12 = 1.3, w0 13 = 1.6, w0 21 = 0.8 and w0 22 = 1.3. The use o weigh s needs mo e discussion. The weigh s a e de ined as eal numbe s and memb ane compu ing de ices a e disc e e. I we wan o deal wi h disc e e compu a ion in all he s eps o he lea ning p ocess we ha e o choose he pa ame e s ca e ully. The ollowing esul gi es a su icien cons ain o ha ing an in ege numbe o spikes a any momen . Theo em 1. Le abe he g ea es non-nega i e in ege such ha o all p esynap ic po en ial nij he e exis s an in ege zij such ha nij =xij ×10a. Le bbe he smalles non-nega i e in ege such ha o all ini ial weigh wij and o he a e o lea ning ² he e exis he in ege s kij and ksuch ha wij =kij ×10b and ²=k×10b. I a−b≥0, hen o all p esynap ic po en ial nij and all he weigh s wob ained along he lea ning p ocess, nij ×wis an in ege numbe . In o he wo ds, i he e exis s aand bsuch ha all he p esynap ic po en ials nij can be exp essed as nij =xij ×10a o an app op ia e in ege xij and he ini ial weigh s wij and a e o lea ning ²can be exp essed as wij =kij ×10b and ²=k×10b o app op ia e in ege numbe s kij, k and a−b≥0 hen o all p esynap ic po en ial nij and all he weigh s wob ained along he lea ning p ocess, nij ×wis an in ege numbe . P oo . I su ices o conside he ecu si e gene a ion o new weigh s wn+1 =wn+ ²L(sn) and he e o e wn+1 =w0+²(L(s0) + · · · +L(sn)). I we de elop nij ×wn+1 acco ding o he s a emen o he heo em, we ha e nij ×wn+1 =xij ×10a×[k0×10−b+ (k×10−b(L(s0) + ···+L(sn)))] = 10a−b×xij ×[k0+k(L(s0) + · · · +L(sn))] Since xij ×[k0+k(L(s0) + · · · +L(sn))] is an in ege numbe , i a−b≥0 hen nij ×wn+1 is an in ege numbe . 228 M.A. Gu i´e ez-Na anjo, M.J. P´e ez-Jim´enez 4 An Expe imen Le us conside he Hebbian SN P sys em HΠ = (O, u1, u2, ) wi h uni o m decay, whe e: •O={a}is he alphabe ; •u1, u2a e he p esynap ic neu ons. The p esynap ic neu ons u1, u2ha e associ- a ed he se s o ules R1={R11, R12, R13}and R2={R21, R22}, espec i ely, wi h R11 ≡a3000 →(a3000, S); 0 R21 ≡a3000 →a1000; 0 R12 ≡a3000 →(a2000, S); 1 R22 ≡a3000 →a3000; 3 R13 ≡a3000 →(a2000, S); 7 •The decaying sequence is S= (3000,2800,1000,500,0). • is he pos synap ic neu on which con ains only one pos synap ic ule E∗ 1200/a1200 →a; 0. Le EHΠ be he Hebbian SN P sys em uni HΠ ex ended wi h he ini ial weigh s w11 = 0.5, w12 = 0.5, w13 = 0.5, w21 = 0.5 and w22 = 0.5. Le us conside he lea ning p oblem (EHΠ, X, L, ²) whe e •EHΠ is he ex ended Hebbian SN P sys em uni desc ibed abo e, •Xis a se o 200 andom inpu s (x1 i, x2 i) wi h 1 ≤1≤200 and x1 i, x2 i∈ {0,1,...,5} •Lis he lea ning unc ion L:Z→Z L(s) =    3 i s= 0 1 i s= 1 −1 o he wise •The a e o lea ning is ²= 0.001 We ha e p og ammed an app op ia e so wa e o dealing wi h his lea ning p oblems. A e applying he lea ning algo i hm, we ob ain a new ex ended Heb- bian SN P sys em uni simila o EHΠ bu wi h he weigh s w11 = 0.754, w12 = 0.992, w13 = 0.3, w21 = 0.454, w22 = 0.460 Fig 4 shows he e olu ion o he weigh s o he synapses. The lea ning p ocess shows clea ly he di e ences among he ules. •The wo s ule is R13. In a debugging p ocess o he design o an SN P Sys em ne wo k ha ule should be emo ed. The alue o he weigh has dec eased along all he lea ning p ocess. This ac means ha he ule has ne e con- ibu ed o he success o he uni and hen i can be emo ed. The eason is A Fi s Model o Hebbian Lea ning wi h SN P Sys ems 229 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 Weigh s Inpu s Sin1 Sin2 Sin3 Sin4 Sin5 Fig. 4. The e olu ion o he weigh s clea . The ule emi s ou spikes and he pos synap ic ule is ac i a ed wi h wo spikes. E en wi h he decay, he po en ial p o ided by he ule is oo much o igge ing he ule. •On he o he ex eme, he bes ules a e R11 and R21. In mos o he cases, (no all) hese ules ha e been in ol ed in he success o he uni . •The o he wo ules R21 and R22 ha e e en ually con ibu ed o he success o he uni bu no so clea ly as R11 and R21. We can also guess he easons. Fo R11, he p esynap ic po en ial, 1000, has li le in luence in he pos synap ic po en ial and o R22, he p esynap ic po en ial is la ge han he h eshold, bu i has a la ge delay, so he a i al o i s po en ial o he pos synap ic neu on is o en la e han he ac i a ion o he pos synap ic ule. 230 M.A. Gu i´e ez-Na anjo, M.J. P´e ez-Jim´enez 5 Conclusions and Fu u e Wo k The in eg a ion in an unique model o concep s om neu oscience, a i icial neu al ne wo ks and spiking neu al P sys ems is no an easy ask. Each o he h ee ields ha e i s own concep s, languages and ea u es. The wo k o in eg a ion consis s on choosing ing edien s om each ield and ying o compose a compu a ional de ice wi h he di e en pa s. This means ha some o he ing edien s used in he de ices p esen ed in his pape a e no usual in he SN P sys ems amewo k. Al hough he au ho s ha e ied o be as close o he SN P sys em spi i as possible some ema ks should be conside ed. In he pape , he inpu o he de ice is p o ided as a ec o ( 1, . . . , m) o non-nega i e in ege s, whe e i ep esen s he momen in which one ule (non- de e minis ically chosen) o he neu on uiis ac i a ed. Ob iously, his is no he usual way o p o ide he inpu o an SN P sys em. None heless, he in o ma ion encoded in he ec o ( 1, . . . , m) can be p o ided o he inpu neu ons by mspike ains we e all he elemen s a e 0’s and he e is only one 1 in he posi ion i. In his way, he inpu is encoded by mspike ains, which is close o he s anda d inpu s in SN P sys ems. The idea o p o iding he inpu wi h a spike ain o 0’s and only one 1 in he posi ion ica ies ou new p oblems. In he li e a u e o SN P sys ems, in he ins an ionly one spike is supplied o he neu on ui. In ou de ice we wan ha a ule o ype a →ap;dis ac i a ed wi h > 1. A his poin we can conside se e al choices. The i s one is o conside ha a ime i he spike ain p o ides spikes, bu his choice lead us a om he SN P sys em heo y. A second op ion is o conside ha he spike ains ha e consecu i e 1’s and each o hem p o ide one spike. The emaining elemen s in he ain a e ze os. In his way he momen iwill be he ins an in which he spikes ha e been p o ided o he neu on. A d awback o his p oposal can be ha can be a big numbe and his inc ease he numbe o s eps o he de ice. A hi d choice is o conside ampli ie modules as in Figu e 5. The le mos neu on ecei es a spike ain whe e all he elemen s a e 0’s bu he i− h which is 1. A he momen ionly one spike is supplied o he neu on. A i+ 1, one spike a i es o he pos synap ic neu ons, and each o hem sends one spike o he igh mos neu on, so a i+ 2 exac ly spikes a i e simul aneously o he las neu on. These h ee solu ions can be an al e na i e o he use o he ec o ( 1, . . . , m) and dese e o be conside ed o u he esea ch in his opic. Ano he main concep in his pape is he delay. I has s ong biological in u- i ion, bu i is di icul o inse in o he SN P sys ems heo y. The main eason is ha i we conside he spike as he in o ma ion uni i does no make sense o alk abou a hal o a spike o a hi d o a spike. In ha sense, he app oach o decay om [7] is ull o sense since one spike exis s o i is los , bu i s po en ial i is no dec easing in ime. The key poin o he decay in his pape is aken om he de ini ion o ex ended SN P sys ems. In such de ices, a neu on can send a di e en amoun o spikes depending on he chosen ule. So, in such de ices he in o ma ion is no only A Fi s Model o Hebbian Lea ning wi h SN P Sys ems 231 a a;0 a a;0 a a;0 a a;0 a a;0 a an;d ... ... Fig. 5. Ampli ie module encoded in he ime be ween wo consecu i e spikes, bu on he numbe o spikes. This lead us o de ine he decay as a dec emen in he numbe o spikes. In his way, we can conside ha a pulse be ween wo neu ons is composed by a ce ain numbe o spikes which can be pa ially los depending on he ime. In his pape , such a decay has been implemen ed by ex ending he ules wi h a ini e dec easing sequence which can be uni o m, locally-uni o m o non uni o m o he se o ules. O he implemen a ions a e also possible. P obably, he decay can also be implemen ed wi h an ex a neu on as in Figu e 6 which sends o he inal neu on a decaying sequence o spikes. a a;0 p a a ;d1 n1 k ak a ;d n ... wi Inpu neu on Decay neu on Pos synap ic neu on Fig. 6. Including a decay neu on The use o weigh s also dese es o be discussed. In Theo em 1 we p o ide su icien condi ions o handling a e e y momen an in ege numbe o spikes. In his way, he p esen ed de ices keep he p inciple o disc e e compu a ion o SN P sys ems. None heless, u he ques ions should be conside ed. Fo example, he use o nega i e weigh s o weigh s g ea e han one. Should we conside nega i e weigh s and/o a nega i e con ibu ion o he pos synap ic po en ial? On he o he hand, he use weigh s g ea e han one leads us o conside ha he con ibu ion o 232 M.A. Gu i´e ez-Na anjo, M.J. P´e ez-Jim´enez one ule o he pos synap ic po en ial is g ea e han i s own p esynap ic po en ial. Can he e iciency o he synapses ampli y he po en ial beyond he numbe o emi ed spikes? 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