Spiking Neural P Systems with Functional Astrocytes

Spiking Neu al P Sys ems wi h Func ional As ocy es Luis F. Mac´ıas-Ramos, 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, Spain A da. Reina Me cedes s/n. 41012 Se illa, Spain [email p o ec ed], [email p o ec ed] Abs ac . Spiking Neu al P Sys ems (SN P Sys ems, o sho ) is a de eloping ield wi hin he uni e se o P Sys ems. New a ian s a ise cons an ly as he s udy o hei p ope ies, such as compu a ional comple eness and compu a ional e iciency, g ows. Va ian s equen ly inco po a e new ing edien s in o he o iginal model inspi ed by eal neu ophysiological s uc u e o he b ain. A singula elemen p esen wi hin ha s uc u e is he as ocy e. As ocy es, also known collec i ely as as oglia, a e cha ac e is ic s a -shaped glial cells in he b ain and spinal co d. In his pape , a new a ian o Spiking Neu al P Sys ems inco po a ing as ocy es is in oduced. These as ocy es a e modelled as compu ing de ices capable o pe o ming unc ion compu a ion in a single compu a ion s ep. In o de o expe imen ally s udy he ac ion o Spiking Neu al P Sys ems wi h as ocy es, i is necessa y o de elop so wa e p o iding he equi ed simula ion ools. Wi hin his end, P– Lingua o e s a s anda d language o he de ini ion o P Sys ems. Pa o he same so wa e p ojec , pLinguaCo e lib a y p o ides pa icula implemen a ions o pa se s and simula o s o he models speci ied in P–Lingua. Along wi h he new SN P Sys em a ian wi h as ocy es, an ex ension o he P–Lingua language allowing de ini ion o hese sys ems is p esen ed in his pape , as well as an upg ade o pLinguaCo e, including a pa se and a simula o ha suppo s he a o emen ioned a ian . 1 In oduc ion Spiking Neu al P Sys ems we e in oduced in [10] in he amewo k o mem-b ane compu ing [16] as a new class o compu ing de ices which a e inspi ed by he neu ophysiological beha iou o neu ons sending elec ical impulses (spikes) along axons o o he neu ons. A SN P Sys em consis s o a se o neu ons placed as nodes o a di ec ed g aph (called he synapse g aph). Each neu on con ains a numbe o copies o a single objec ype, he spike. Rules a e assigned o neu ons o con ol he way in o ma ion lows be ween connec ed neu ons, i.e. ules assigned o a neu on allow i o send spikes o i s neighbou ing neu ons. SN P Sys ems usually wo k + 2 + 2 in a synch onous mode, whe e a global clock is assumed. In each ime uni , o each neu on, only one o he applicable ules is non-de e minis ically selec ed o be execu ed. Execu ion o ules akes place in pa allel amongs all neu ons o he sys em. Since he in oduc ion o his model, many compu a ional p ope ies o SN P Sys ems ha e been s udied. I has been p o ed ha hey a e Tu ing-comple e when conside ed as numbe compu ing de ices [10], used as language gene - a o s [5,3], o compu ing unc ions [15]. Also, many a ian s ha e come in o scene b inging new ing edien s in o he model (o some imes d opping some o hem), while o he s modi y i s beha iou , ha is, i s seman ics. Mo i a ion o his “ esea ch boom” can be ound in a ques o bo h enhancing exp essi i y and e iciency o he model, as well as explo ing i s compu a ional powe . As a di ec esul o all o his, he e is an ex ensi e (and g owing) bibliog- aphy ela ed o SN P Sys ems. Fo ins ance, i has been shown [4] how usage o p e-compu ed esou ces makes hem able o sol e compu a ionally ha d p ob- lems in cons an ime. Also, s udy o di e en kinds o asynch onous “wo king modes” has been conduc ed [18]. In wha conce ns o he addi ion o new ing e- dien s in o he model, his in ol es (naming only some examples) weigh s [20], an ispikes [12], ex ended ules [18] o budding and di ision ules [13]. A SN P Sys ems a ian wi h as ocy es was i s i n oduced i n [ 2]. As o- cy es a e glial cells connec ed o one o mo e synapses ha can sense he whole spike a ic passing along hei neighbou ing synapses and, e en ually, modi y i . Thei unc ionali ies include biochemical suppo o endo helial cells ha o m he blood-b ain ba ie , p o ision o nu ien s o he ne ous issue, main enance o ex acellula ion balance, and a ole in he epai and sca ing p ocess o he b ain and spinal co d ollowing auma ic inju ies. I has been shown ha as- ocy es p opaga e in e cellula Ca wa es o e long dis ances in esponse o s imula ion, and, simila ly o neu ons, elease ansmi e s (called glio ansmi - e s) in a Ca -dependen manne . Mo eo e , wi hin he do sal ho n o he spinal co d, ac i a ed as ocy es ha e he abili y o espond o almos all neu o ansmi e s [9] and, upon ac i a ion, elease a mul i ude o neu oac i e molecules ha in luences neu onal exci abili y. Synap ic modula ion by as ocy es akes place because o he 3-pa associa ion be ween as ocy es and p esynap ic and pos synap ic e minals o ming he so- called “ ipa i e synapse” [1]. Such disco e ies ha e made as ocy es an impo an a ea o esea ch wi hin he ield o n eu oscience, hus a lso a n i n e es ing e lemen o c onside b inging in o Na u al Compu ing disciplines like Memb ane Compu ing. 2 The model p esen ed in [2], p e y complex, was hen simpli ied i n [ 17], in which only inhibi o y as ocy es we e conside ed. This simpli ica ion w as e- cen ly e ised again in [14], whe e “hyb id” as ocy es we e in oduced. Be- ha iou o an as ocy e o his kind, inhibi o y o exci a o y, elied on he amoun o spikes passing on i s neighbou ing synapses, in ela ion o a gi en h eshold associa ed o i . Thus, o a gi en as ocy e as wi h associa ed h eshold wi h k spikes passing along i s neighbou ing synapses synas a a ce ain ins an , a) i k > , he as ocy e as has an inhibi o y in luence o n he neighbou ing synapses, and he k spikes a e simul aneously supp essed ( ha is, he spikes a e emo ed om he sys em); b) i k < , he as ocy e as has an exci a o y in luence on he neighbou ing synapses, all spikes su i e and pass o hei des- ina ion neu ons, eaching hem simul aneously; c) i k = , he as ocy e as non-de e minis ically chooses an inhibi o y o exci a o y in luence on he neigh- bou ing synapses. I is possible o wo o mo e as ocy es o con ol he same synapse. In his case, only i e e y as ocy e has an exci a o y in luence on he synapse he spikes passing along ha synapse su i e. In his pape , again, a new a ian is in oduced. Based upon he o iginal model de ined in [2], new ing edien s a e in oduced in o de o u n as ocy es in o unc ion compu a ion de ices. B ie ly, a se o pai s ( h eshold, unc ion) is associa ed wi h each as ocy e. Exis ing spike a ic measu ed on dis inguished neighbou ing con ol synapses a ached o he as ocy e is ma ched agains he h esholds un il one o hem is selec ed. Subsequen ly, he associa ed unc ion o he ma ched h eshold is selec ed. A his poin , ha unc ion is compu ed ak- ing as a gumen s he amoun s o spikes measu ed on dis inguished neighbou ing ope and synapses a ached o he as ocy e. Finally, he esul o he unc ion compu a ion is sen h ough a dis inguished ope and synapse. So, by in oducing his new kind o as ocy es, no only co e ing o unc- ionali y o he as ocy es de ined in [2] is achie ed, also any compu able pa ial unc ion be ween na u al numbe s can be compu ed in a single compu a ion s ep. Mo eo e , his new ing edien eases he design o machines ha calcula e unc ions, as as ocy es can be iewed as “mac os”. In addi ion, a P–Lingua based simula o o he p oposed model has been de eloped, which also simula es he model de ined in [14]. The a o emen ioned simula o is an ex ension o he one p esen ed in [11]. P–Lingua is a p og am- ming language in ended o de ine P Sys ems [7,8,19], ha comes oge he wi h a Ja a lib a y p o iding se e al se ices (e.g., pa se s o inpu iles and buil -in simula o s). This pape is s uc u ed as ollows. Sec ion 2 is de o ed o in oduce he o mal speci ica ion o S N P S ys ems w i h Func ional A s ocy es (SNPSFA o sho ). Sec ion 3, is de o ed o show applica ions o he p esen ed model. Sec ion 4 is de o ed o simula ion: A P–Lingua syn ax o de ining SNPSFA is in oduced, along wi h se e al examples. Finally, he simula ion algo i hm is shown. Sec ion 5 co e s conclusions and u u e wo k. 2 Spiking Neu al P Sys ems wi h Func ional As ocy es In his sec ion, we in oduce SN P Sys ems wi h Func ional As ocy es. 2.1 Syn ax ASpiking Neu al P Sys em wi h Func ional As ocy es (SNPSFA o sho ) o deg ee (m, l), m ≥1, l≥1, is a cons uc o he o m Π= (O, σ, syn, as , ou ), whe e: –O={a}is he single on alphabe (ais called spike); –σ={σ1, . . . , σm}is he ini e se o neu ons, o he o m σi= (ni, Ri),1≤ i≤m, whe e: •ni≥0 is he ini ial numbe o spikes con ained in σi; •Riis a ini e se o ex ended ules o he ollowing o m: E/ac→ap whe e Eis a egula exp ession o e a, and c≥1, p≥1 wi h c≥p; –syn ={s1, . . . , sθ} ⊆ {1, . . . , m}×{1, . . . , m}wi h (i, i)6∈ syn is he se o synapses; –as ={as 1, . . . , as l}is he ini e se o as ocy es, wi h as j,(1 ≤j≤l) o he o m as j= (syno j, sync j, ωj, Tj, Fj, pj(0), γj), whe e: •syno j={so j,1, . . . , so j, j} ⊆ syn, j≥1, is he as ocy e ini e se o ope and synapses, o de ed by a lexicog aphical o de imposed on syno j; •sync j={sc j,1, . . . , sc j,qj} ⊆ syn, qj≥0, is he as ocy e ini e se o con ol synapses; •ωj∈ { ue, alse}is he as ocy e con ol-as-ope and lag; •Tj={Tj,1, . . . , Tj,kj}, kj≥1, is he as ocy e ini e se o h esholds, such ha , Tj,α ∈N, (1 ≤α≤kj) and Tj,1< . . . < Tj,kj; •Fj={ j,1, . . . , j,kj}is he as ocy e ini e mul ise (some elemen s in Fj can be he same) o na u al unc ions such ha o each α(1 ≤α≤kj): ∗ j,α is a compu able unc ion be ween na u al numbe s; ∗i ωj= ue hen j,α is a una y unc ion; ∗i ωj= alse and j= 1 hen j,α is a una y cons an unc ion; ∗i ωj= alse and j>1 hen j,α has a i y j−1; •pj(0) ∈Nis he as ocy e ini ial po en ial; •γj∈ { ue, alse}is he as ocy e po en ial upda e lag; –ou ∈σis he ou pu neu on. 2.2 Seman ics In o de o p ecise seman ics o a SNPSFA, le us in o mally in oduce some opological aspec s o he model and he na u e o he i ing p ocess. Gi en a synapse sg= (σg,1, σg,2)∈syn, i an as ocy e is linked o sg, i can be iewed as ha i “makes con ac ” wi h sgin he “space be ween” s1 gand s2 g(i can be said ha he as ocy e is “a ached” o he synapse as well). I he e exis s se e al as ocy es a ached o sg, all o hem make con ac a he same in e media e poin . These as ocy es can simul aneously ead he spike a ic going om σg,1 o σg,2a an ins an and e en ually modi y i . Keeping in mind he in ui i e ideas exp essed abo e, we p oceed now o o - mally speci y he seman ics o SN P Sys ems wi h Func ional As ocy es as an ex ension o he one de ined o he well-known SN P Sys ems model. A global clock is assumed and in e e y compu a ion s ep one and only one ule can be selec ed o a gi en neu on. Le us in oduce he ollowing no a ion as a ma e o con enience: gi en a synapse sy= (σ1 y, σ2 y), we deno e by σ1 y he inpu neu on o syand by σ2 y he ou pu neu on o sy. An as ocy e can sense he spike a ic passing along i s neighbou ing synapses, bo h con ol and ope and ones. Fo an as ocy e as j, i he e a e kspikes passing along he con ol synapses in an ins an and he cu en po- en ial o as ja is p, hen he alue s=k+pis compu ed. A his poin , he numbe hsa is ying ha s∈[Tj,h, Tj,h+1) is compu ed ou o s. Le us no ice ha i s < Tj,1 hen h= 1, and i s > Tj,kj hen h=kj. Following his, by using bo h hand he boolean alue ωj, a numbe s0is compu ed as ollows. I ωj= ue hen s0= j,h(s) di ec ly. O he wise, wo cases a e conside ed: a) i he numbe o ope and synapses jis one, hen s0= j,h(0); and b) i he num- be o ope and synapses is g ea e han one and assuming ha x1, x2, . . . , x j−1 spikes a e passing along he espec i e ope and synapses associa ed o as j , hen s0 = j,h(x1, x2, . . . , x j−1). Finally, he mul ise s o he inpu and ou pu neu- ons associa ed o he ope and and con ol synapses a e upda ed. Fo he ou pu neu ons: a) i hey a e associa ed o con ol synapses, hen hei co esponding mul ise s a e added he spikes passing along he synapses a ins an ; and b) i hey a e associa ed o ope and synapses, hen no change is applied o hei mul ise s, excep o neu on sjo, j , which is added s0 spikes. Simila ly, mul ise s co esponding o inpu neu ons associa ed o bo h ope and and con ol synapses a e sub ac ed he spikes passing along he a o emen ioned synapses a ins an . As a las ema k, i he as ocy e po en ial upda e lagγj = ue hen he as ocy e po en ial in + 1 will be inc emen ed in s uni s. O he wise, he as o- cy e po en ial does no change. 3 Applica ions o Spiking Neu al P Sys ems wi h Func ional As ocy es As men ioned be o e, by in oducing SNPSFA co e ing o unc ionali y o as- ocy es de ined in [2] is achie ed. Also, as ocy es wi hin SNPSFA a e able o compu e any compu able na u al pa ial unc ion :Nm− → Nin a single compu a ion s ep. Le us illus a e his ac by showing how o e-implemen he examples co e ed in [2] wi hin he scope o ou p oposed model. Mo eo e , he co esponding P–Lingua iles o he a o emen ioned examples a e co e ed in Sec ion 4, hus by unning he in oduced simula o agains hese iles, i s wo king p ocess can be checked in ela ion o he seman ics p esen ed abo e. 3.1 Exci a o y and Inhibi o y As ocy es Fi s couple o examples shows how o implemen exci a o y and inhibi o y as- ocy es espec i ely, wi h a gi en h eshold k. Implemen a ion in ol es de ining wo unc ions: (x), which is he iden ically ze o unc ion o a i y one, and g(x). Exci a o y as ocy e, as exc, is depic ed in he Fig. 1 wi h i s o mal speci ica ion being: as exc = ({(p0, q)},{(p, q0)}, ue, {0, k},{ (x), g(x)},0, alse) and i s wo king equa ion, assuming ha αspikes pass h ough synapse (p, q0) a a gi en ins an , being: as exc(α, ) =  (α) = 0 i 0 ≤α < k g(α) i α≥k Fig. 1. Exci a o y as ocy e. Inhibi o y as ocy e, as inh, is s uc u ally iden ical o as exc, wi h i s o mal speci ica ion being: as inh = ({(p0, q)},{(p, q0)}, ue, {0, k + 1},{g(x), (x)},0, alse) and i s wo king equa ion, assuming ha αspikes pass h ough synapse (p, q0) a a gi en ins an , being: as inh(α, ) = g(α) i 0 ≤α≤k (α) = 0 i α≥k+ 1 3.2 Logic Ga es Second couple o examples shows how o implemen logical ga es, conc e ely AND-ga es and NAND-ga es espec i ely. Implemen a ion in ol es de ining wo unc ions, (x) and g(x), bo h o hem una y cons an unc ions, which asso- cia es he 0 and 1 na u al alues espec i ely o e e y x∈N. AND-ga e as ocy e, as and, is depic ed in he Fig. 2 wi h i s o mal speci ica ion being: as and = ({(p, q)},{(A, A0),(B, B0)}, alse, {1,2},{ (x), g(x)},0, alse) and i s wo king equa ion, assuming ha α, 0≤α≤2 spikes in o al pass h ough synapses (A, A0) and (B, B0) a a gi en ins an , being: as and(α, ) =  (0) = 0 i 0 ≤α≤1 g(0) = 1 i α= 2 Fig. 2. AND-ga e as ocy e. NAND-ga e as ocy e, as nand, is s uc u ally iden ical o as and, wi h i s o mal speci ica ion being: as nand = ({(p, q)},{(A, A0),(B, B0)}, alse, {1,2},{g(x), (x)},0, alse) and i s wo king equa ion, assuming ha α, 0≤α≤2 spikes in o al pass h ough synapses (A, A0) and (B, B0) a a gi en ins an , being: as nand(α, ) = g(0) = 1 i 0 ≤α≤1 (0) = 0 i α= 2 3.3 Disc e e Ampli ie Las example shows how o implemen a disc e e ampli ie which, as soon as he spike amoun passing h ough con ol synapse (B, B0) goes beyond a gi en h eshold k, compu es he ampli ica ion unc ion ∗,n(x) = n∗x om he inpu gi en a E, o he wise no ampli ica ion is pe o med. Rules al→albelonging o neu on pa e in e p e ed in he same way as in [2]. Implemen a ion in ol es de ining wo unc ions: g(x) = ∗,n(x) and (x), which associa es x o e e y x∈N. Disc e e ampli ie as ocy e, as amp, is depic ed in he Fig. 3 wi h i s o mal speci ica ion being: as amp = ({(p, p0),(q0, q)},{(B, B0)}, alse, {0, k},{ (x), g(x)},0, alse) and i s wo king equa ion, assuming ha a a gi en ins an α spikes pass h ough synapse (B, B0) and βspikes pass h ough synapse (p, p0), being: as amp(α, β, ) =  (β) = βi 0 ≤α < k g(β) = n∗βi α≥k 4 A P–Lingua Based Simula o o SNPSFA This sec ion in oduces a P–Lingua simula o o SNPSFA, ex ending he one p esen ed in [11]. SNPSFA a e only pa ially simula ed because only ce ain unc- ions can be de ined w i hin P –Lingua amewo k. A lso, l e u s n o ice ha an ex ension o he simula o p esen ed he e in ended o simula e SNPSA as in o- duced in [14] is being de eloped. P–Lingua syn ax o speci ying a o emen ioned SNPSFA is in oduced, along wi h se e al examples. To conclude, he simula ion algo i hm is shown. 266 Fig. 3. Disc e e ampli ie as ocy e. 4.1 P–Lingua Syn ax A se o new ea u es has been inco po a ed in o P–Lingua in o de o suppo SNPSFA. New ins uc ions ha e been included o de ine bo h as ocy es and unc ions, ex ending he P–Lingua model speci ica ion amewo k o Spiking Neu al P Sys ems. Thus, hese ins uc ions can be used only when he sou ce P–Lingua iles de ining he models begin wi h he ollowing sen ence: @model<spiking_psys ems> In wha ollows, P–Lingua syn ax o de ining SNPSFA is in oduced. –As ocy es. The ollowing sen ence can be used o de ine a SNPSFA as ocy e as b j, wi h bs anding o binde , as he as ocy es p esen ed in [2] inspi ed he unc- ional as ocy es p esen ed in his pape : @mas b = ( label-j, ope and-synapses-j,con ol-synapses-j,con ol-ope and- lag-j, se - h esholds-j,se - unc ions-j, po en ial-j,upda e-po en ial-j ); whe e: •label-jis he label o he as ocy e; 2. Binde , A., F eund, R., Oswald, M., Vock, L.: Ex ended spiking neu al P Sys ems wi h exci a o y and inhibi o y as ocy es. 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