Adaptive Fuzzy Spiking Neural P Systems for Fuzzy Inference and Learning

Adap i e Fuzzy Spiking Neu al P Sys ems o Fuzzy In e ence and Lea ning Jun Wang1and Hong Peng2 1School o Elec ical and In o ma ion Enginee ing, Xihua Uni e si y, Chengdu, Sichuan, 610039, China 2School o Ma hema ics and Compu e Enginee ing, Xihua Uni e si y, Chengdu, Sichuan, 610039, China [email p o ec ed] Summa y. Spiking neu al P sys ems (in sho , SN P sys ems) and hei a ian s, in- cluding uzzy spiking neu al P sys ems (in sho , FSN P sys ems), gene ally lack lea ning abili y so a . Aiming a his p oblem, a class o modi ied FSN P sys ems a e p oposed in his pape , called adap i e uzzy spiking neu al P sys ems (in sho , AFSN P sys ems). The AFSN P sys ems no only can model weigh ed uzzy p oduc ion ules in uzzy knowl- edge base bu also can pe o m dynamically uzzy easoning. I is mo e impo an ha he AFSN P sys ems ha e lea ning abili y like neu al ne wo ks. Based on neu on’s i ing mechanisms, a uzzy easoning algo i hm and a lea ning algo i hm a e de eloped. An example is included o illus a e he lea ning abili y o he AFSN P sys ems. Key wo ds: Spiking neu al P sys ems, Fuzzy spiking neu al P sys ems, Adap i e uzzy spiking neu al P sys ems, Fuzzy easoning, Lea ning p onlem 1 In oduc ion Spiking neu al P sys ems (in sho , SN P sys ems) fi s ly in oduced by Ionescu e al. in 2006 [1], a e a class o dis ibu ed pa allel compu ing models, which a e inco po a ed in o memb ane compu ing om he way ha biological neu- ons communica e h ough elec ical impulses o iden ical o m (spikes) [2]. Since hen, a la ge numbe o SN P sys ems and hei a ian s ha e been p oposed [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. F om he iewpoin o eal-wo ld applica ions, SN P sys ems ha e a ac i e due o he ollowing ea u es: (i) pa allel compu ing ad an age, (ii) high unde s andabili y (due o hei di ec ed g aph s uc u e), (iii) dynamic ea u e (neu ons fi ing and spiking mechanisms make hem sui able o model dynamic beha io s o a sys em), (i ) synch oniza ion ( ha makes hem sui able o desc ibe concu en e en s o ac i i ies), ( ) non-linea ly ( ha makes hem sui able o p ocess non-linea si ua ion), and so on. Recen ly, in o de o ake ull he ad an age o SN P sys ems, a class o ex ended SN P sys ems we e 236 J. Wang, H. Peng p oposed by in oducing uzzy logic, which we e called uzzy spiking neu al P sys ems (in sho , FSN P sys ems) [12, 13, 14, 15]. The mo i a ion o p oposing hese FSN P sys ems is o deal wi h he ep esen a ion o uzzy knowledge and model uzzy easoning in some eal-wo ld applica ions, such as p ocess con ol, expe sys em, aul diagnosing, e c. As we know, since knowledge in eal-wo ld applica ions men ioned abo e is equen ly upda ed, hey a e essen ially dynamic sys ems. This equi es ha he FSN P sys ems should be adap i e, ha a , FSN P sys ems mus ha e abili y o adjus hemsel es. Howe e , he FSN P sys ems migh ails o cope wi h po en ial changes o ac ual sys ems due o hei lack o adap i e o lea ning mechanism. Besides, a ew o adap i e SN P sys ems ha e been add essed in ecen yea s [16, 17]. In his pape , we p opose a class o modified FSN P sys ems, which a e called adap i e uzzy spiking neu al P sys ems (in sho , AFSN P sys ems). The p ac ical mo i a ion is o build a no el way o deal wi h he lea ning p oblem o dynamical uzzy knowledge in some eal-wo ld applica ions unde he amewo k o SN P sys ems. Fo his pu pose, based on neu on’s fi ing mechanisms, a uzzy easoning algo i hm and a lea ning algo i hm a e de eloped in his pape . The es o his pape is o ganized as ollows. In Sec ion 2, we fi s ly p esen he AFSN P sys ems, and hen desc ibe a way o model weigh ed uzzy p oduc ion ules by he AFSN P sys ems, finally mo e on o gi e he de eloped uzzy easoning algo i hm and lea ning algo i hm. Simula ion example is p o ided in Sec ion 3. Finally, Sec ion 4 d aws he conclusions. 2 AFSN P Sys ems 2.1 De ini ion o AFSN P Sys ems Cu en ly, uzzy spiking neu al P sys ems (FSN P Sys ems, in sho ) ha e been discussed [12, 13, 14, 15]. Howe e , hey can no adjus hemsel es and lack lea n- ing abili y. In his pape , we will in oduce “adap i e” mechanism in o he FSN P sys ems o p opose a class o adap i e FSN P sys ems, called AFSN P sys ems. De ini ion 1. An AFSN P sys ems ( o deg ee m≥1) is a cons uc o he o m Π= (A, Np, N , syn, I, O) whe e 1) A={a}is he single on alphabe ( he objec ais called spike); 2) Np={σp1, σp2, . . . , σpm}is called p oposi ion neu on se , whe e σpi is i s i- h p oposi ion neu on associa ed wi h a uzzy p oposi ion in weigh ed uzzy p oduc ion ules, 1≤i≤m. Each p oposi ion neu on σpi has he o m σpi = (αi,ωi, λi, i), whe e: a) αi∈[0,1] and i is called he (po en ial) alue o pulse con ained in p opo- si ion neu on σpi.αiis used o exp ess uzzy u h alue o he p oposi ion associa ed wi h p oposi ion neu on σpi. Adap i e Fuzzy Spiking Neu al P Sys ems 237 b) ωi= (ωi1, ωi2, . . . , ωisi)is called he ou pu weigh ec o o he neu on σpi, whe e componen ωij ∈[0,1] is he weigh on j- h ou pu synapse (a c) o he neu on, 1≤j≤si, and siis he numbe o all ou pu synapses (a c) o he neu on. c) iis a i ing/spiking ule, o he o m E/aα→aα, whe e α∈[0,1].E= {α≥λi}is called he i ing condi ion, i.e., i α≥λi, hen he i ing ule will be enabled, whe e λi∈[0,1) is called he i ing h eshold. 3) N ={σ 1, σ 2, . . . , σ n}is called ule neu on se , whe e σ i is i s i- h ule neu on associa ed wi h a weigh ed uzzy p oduc ion ule, 1≤i≤n. Each ule neu on σ i has he o m σ i = (αi, γi, τi, i), whe e a) αi∈[0,1] is called he (po en ial) alue o pulse con ained in ule neu on σ i. b) γi∈[0,1] is called he ce ain ac o . I ep esen s he s eng h o belie o he weigh ed uzzy p oduc ion ule associa ed wi h ule neu on σ i. A he same ime, γiis also he weigh on ou pu synapse (a c) o he neu on. c) iis a i ing/spiking ule, o he o m E/aα→aβ, whe e α, β ∈[0,1]. E={α≥τi}is called he i ing condi ion, i.e., i α≥τi, hen he i ing ule will be enabled, whe e τi∈[0,1) is called he i ing h eshold. 4) syn ⊆(Np×N )∪(N ×Np)indica es synapses be ween bo h p oposi ion neu- ons and ule neu ons. No e ha he e a e no synapse connec ions be ween any wo p oposi ion neu ons o be ween any wo ule neu ons; 5) I, O ⊆Npa e inpu neu on se and ou pu neu on se , espec i ely. In he AFSN P sys ems, he e a e wo ypes o neu ons: p oposi ion neu ons and ule neu ons. In his pape , we deno e p oposi ion neu ons and ule neu ons by ci cles and ec angles espec i ely, shown in Fig. 1. (a) (b) Fig. 1. Two ypes o neu ons: (a) a p oposi ion neu on; (b) a ule neu on. Fo a p oposi ion neu on, i s con en is used o exp ess he uzzy u h alue o he uzzy p oposi ion associa ed wi h i . When i s fi ing condi ion E={α≥λi} is sa isfied, he neu on fi es and i s fi ing/spiking ule E/aα→aαcan be applied. Applying he fi ing/spiking ule E/aα→aαmeans ha he spike con ained in he neu on is consumed, and hen i p oduces a spike wi h alue α, which will be weigh ed by he co esponding weigh ac o . Thus, i s ou pu s a e α·ωi(i= 1,2, . . . , s). 238 J. Wang, H. Peng No e ha each ule neu on is assigned only an ou pu weigh ν. Suppose ha a ule neu on has kp edecesso p oposi ion neu ons. When i ecei es kspikes om i s all p edecesso p oposi ion neu ons and i s fi ing condi ion E={α≥τi} is sa isfied, hen i fi es and i s fi ing/spiking ule E/aα→aβcan be applied. The alue o he ecei ed kspikes is calcula ed as i s con en α:α=x1+x2+. . . +xk. Applying he fi ing/spiking ule E/aα→aβmeans ha he spike con ained in he neu on is consumed, and hen i p oduces a spike wi h alue βwhe e β=α·γ. Thus, i s all ou pu s a e α·γ. Suppose ha a p oposi ion neu on has kp edecesso ule neu ons and i e- cei es kspikes om hem. Le ou pu weigh s o he kp edecesso ule neu ons be γ1, γ2, . . . , γk espec i ely. I (po en ial) alues o he ecei ed kspikes a e x1, x2, . . . , xk espec i ely, hen i s new con en is compu ed by α= (x1+x2+ . . . +xk)/(γ1+γ2+. . . +γk). 2.2 Modeling Weigh ed Fuzzy P oduc ion Rules by AFSN P Sys ems In many eal-wo ld applica ions such as expe sys em, aul diagnosing and p o- cess con ol, uzzy p oduc ion ules a e used o desc ibe he uzzy ela ion be ween wo p oposi ions. In o de o conside he deg ee o impo ance o each p oposi ion in he an eceden con ibu ing o he consequen , weigh ed uzzy p oduc ion ule has been in oduced, and a mo e de ained desc ip ion can be ound in [18, 19, 20]. Howe e , we will discuss he ollowing h ee ypes o weigh ed uzzy p oduc ion ules in o de o s udy AFSN P sys ems in his pape . Type 1: A simple uzzy p oduc ion ule R: IF p1THEN p2(CF =γ), τ, ω Type 2: A composi e conjunc i e ule R: IF p1AND p2AND ··· AND pnTHEN pn+1 (CF =γ), τ, ω1, ω2, . . . , ωn Type 3: A composi e disjunc i e ule R: IF p1OR p2OR · · · OR pnTHEN pn+1 (CF =γ), τ, ω1, ω2, . . . , ωn Abo e h ee ypes o weigh ed uzzy p oduc ion ules can be modeled by he p oposed AFSN P sys ems acco ding o he idea ha each uzzy p oposi ion is mapped in o one p oposi ion neu on and each uzzy p oduc ion ule is mapped in o one ule neu on o se e al ule neu ons. Thus, he h ee ypes o weigh ed uzzy p oduc ion ules a e ep esen ed by he ollowing h ee AFSN P sys ems, Π1,Π2and Π3, espec i ely: •Π1= (A, {σp1, σp2},{σ 1}, syn, I, O) whe e: (1) A={a} (2) Fo each j(j= 1,2), σpj = (αj,ω, λ, j) is a p oposi ion neu on associa ed wi h p oposi ion pj, and jis a spiking ule o he o m E/aα→aα. Adap i e Fuzzy Spiking Neu al P Sys ems 239 (3) σ 1= (α3, γ, τ, 3) is a ule neu on associa ed wi h ule R, and 3is a spiking ule o he o m E/aα→aβ. (4) syn ={(σp1, σ 1),(σ 1, σp2)}. (5) I={σp1},O={σp2}. Fig. 2(a) shows he AFSN P sys em model o Type 1:Π1. •Π2= (A, {σp1, σp2, . . . , σpn, σp(n+1)},{σ 1}, syn, I, O) whe e: (1) A={a} (2) Fo each j(j= 1, . . . , n, n + 1), σpj = (αj,ωj, λj, j) is a p oposi ion neu on associa ed wi h p oposi ion pj, and jis a spiking ule o he o m E/aα→aα. (3) σ 1= (αn+2, γ, τ, n+2) is a ule neu on associa ed wi h ule R, and n+2 is a spiking ule o he o m E/aα→aβ. (4) syn ={(σp1, σ 1),(σp2, σ 1), . . . , (σpn, σ 1),(σ 1, σp(n+1))}. (5) I={σp1, σp2, . . . , σpn},O={σp(n+1)}. Fig. 2(b) shows he AFSN P sys em model o Type 2:Π2(in he case o n= 2). •Π3= (A, {σp1, σp2, . . . , σpn, σp(n+1)},{σ 1, σ 2, . . . , σ n}, syn, I, O) whe e: (1) A={a} (2) Fo each j(j= 1, . . . , n, n+1), σj= (αj,ωj, λj, j) is a p oposi ion neu on associa ed wi h p oposi ion pj, and jis a spiking ule o he o m E/aα→ aα. (3) Fo each j(j= 1, . . . , n), σ j = (αn+j+1, γj, τj, n+j+1) is a ule neu on associa ed wi h ule R, and n+j+1 is a spiking ule o he o m E/aα→aβ. (4) syn ={(σp1, σ 1),(σp2, σ 2), . . . , (σpn, σ n),(σ 1, σp(n+1)),(σ 2, σp(n+1)), . . . , (σ n, σp(n+1))}. (5) I={σp1, σp2, . . . , σpn},O={σp(n+1)}. Fig. 2(c) shows he AFSN P sys em model o Type 3:Π3(in he case o n= 2). 2.3 Fuzzy Reasoning Based on AFSN P sys ems Since he p esen ed AFSN P sys ems mainly ocus on he weigh ed uzzy easoning, we assume ha fi ing h eshold o e e y p oposi ion neu on is λ= 0. This means ha once a p oposi ion neu on con ains a spike wi h α > 0 i will fi es. Acco ding o fi ing mechanism o AFSN P sys ems, uzzy easoning p ocesses o abo e h ee ypes o weigh ed uzzy p oduc ion ules can be desc ibed as ollows: •Fo Type 1, we can se ω= 1 since he e is only one p oposi ion in an eceden o he ule R. Ini ially, assume ha neu on σp1con ains a spike wi h α1>0. A fi s s ep, neu on σp1fi es and emi s a spike wi h α1. A second s ep, neu on σ 1 ecei es he spike. I α1≥τ, hen neu on σ 1fi es and emi s a spike wi h 240 J. Wang, H. Peng (a) s p1 s 1 s p2 w 1 (b) (c) l 1 g s p1 s 1 s p3 w 1 l 1 g s p2 w 2 l 2 s p1 s 1 s p3 1 w 1 l 1 g1 s p2 s 2 2 w 2 l 2 g2 Fig. 2. AFSN P sys ems o weigh ed uzzy p oduc ion ules o h ee ypes: (a) Type 1 ; (b) Type 2 ; (c) Type 3. α1·γ. Neu on σp2will ecei e he spike a nex s ep. Thus, α2can be exp essed by α2={α1·γ, i α1≥τ 0,i α1< τ (1) •Fo Type 2, assume ha neu ons σp1, σp2, . . . , σpn con ain a spike wi h α1> 0, α2>0, . . . , αn>0, espec i ely. A fi s s ep, he nneu on fi e simul a- neously, and emi a spike wi h α1, α2, . . . , αn, espec i ely. A second s ep, neu on σ 1 ecei es he nspikes and i s con en is upda ed as ∑n i=1 αi·ωi. I (∑n i=1 αi·ωi)≥τ, hen neu on σ 1fi es and emi s a spike wi h (∑n i=1 αi·ωi)·γ. Neu on σp(n+1) will ecei e he spike a nex s ep. Thus, αn+1 can be exp essed by Adap i e Fuzzy Spiking Neu al P Sys ems 241 αn+1 =       (n ∑ i=1 αi·ωi)·γ, i (n ∑ i=1 αi·ωi)≥τ 0,i (n ∑ i=1 αi·ωi)< τ (2) •Fo Type 3, we can se ω1=ω2= 1. Assume ha neu ons σp1, σp2, . . . , σpn con ain a spike wi h α1>0, α2>0, . . . , αn>0, espec i ely. A fi s s ep, he nneu on fi e simul aneously, and emi a spike wi h α1, α2, . . . , αn, espec i ely. A second s ep, each neu on σ i ecei es a spike sen by σpi, whose alue is αi, i= 1,2, . . . , n. Le J={j|αj≥τj, j = 1,2, . . . , n}. Then neu ons σ j (j∈J) fi e and each neu on o hem emi s a spike. Neu on σp(n+1) will ecei e he spikes a nex s ep. Thus, αn+1 can be exp essed by αn+1 =   (∑ j∈J αj·γj)/(∑ j∈J γj),i αj≥τj, j ∈J 0,i αj< τj, j = 1,2. . . , n (3) F om uzzy easoning p ocess desc ibed abo e, we can see ha uzzy easoning based on AFSN P sys ems a e easily implemen ed. Thus, h ough fi ing mecha- nism o AFSN P sys ems, ce ain y ac o s can be easoned om a se o known an eceden p oposi ions o a se o consequen p oposi ions s ep by s ep. Le Pcu en ={σpi |σpi ∈Np, αi>0}be a se o cu en enabled p oposi ion neu ons. I a neu on σpi ∈Pcu en , hen i will fi e. Le Rcu en ={σ j |σ j ∈ N , αj> τj}be a se o cu en enabled ule neu ons. Likewise, i a neu on σ j ∈ Rcu en , hen i will fi e. The e o e, uzzy easoning algo i hm based on AFSN P sys ems can be summa ized as ollows. p og am Fuzzy_ easoning_algo i hm inpu Ce ain y ac o s o a se o an eceden p oposi ions, which a e co esponding o I o AFSN P sys ems; ou pu Ce ain y ac o s o a se o consequence p oposi ions, which a e co esponding o O o AFSN P sys ems; begin Pcu en := I; Rcu en := {} P := Np; R := N ; epea Compu e he ou pu s o cu en enabled p oposi ion neu ons in Pcu en ; Find cu en enabled ule neu ons Rcu en o m R; Compu e he ou pu s o cu en enabled p oposi ion neu ons in Rcu en ; P := P - Pcu en ; 242 J. Wang, H. Peng R := R - Rcu en ; Find cu en enabled p oposi ion neu ons Pcu en o m P; un il P = {} and R = {} end. 2.4 Lea ning o AFSN P sys ems In o de o deal wi h he lea ning p oblem o AFSN P sys ems, we assume ha 1) AFSN P sys em model Πhas been de eloped; 2) In he AFSN P sys em model, weigh s and h esholds o all ule neu ons a e known; 3) Ce ain y ac o alues o all neu ons in Iand Oa e gi en. F om he discussion abo e, we know ha he p esen ed AFSN P sys ems a e mainly used o model weigh ed uzzy p oduc ion ules and hese ules consis o h ee ypes. So, an AFSN P sys em model can be di ided in o h ee ypes o sub-s uc u es, which a e shown in Fig.2(a)-(c). The e o e, he lea ning o en i e sys em can be decomposed o se e al simple lea ning p ocedu es o he sub-ne s. This means ha he complexi y o he lea ning algo i hm can be g ea ly educed. Acco ding o abo e assump ion, ce ain y ac o s o he p oposi ion neu ons associ- a ed wi h an eceden p oposi ions a e known, howe e , hei weigh s a e unknown. The e o e, hese weigh s need o be lea ned. No e ha o AFSN P sys em Π1o Type 1, we ha e ω1= 1, while we ha e ω1=ω2= 1 o AFSN P sys em Π3o Type 3. So, only weigh s o AFSN P sys em Π2o Type 2 need o be lea ned. In o de o ca y ou he weigh lea ning, he AFSN P sys em Π2o Type 2 can be con e ed o a single-laye neu al ne wo k, shown in Fig.3. So, Wid ow-Hoff lea ning law (Leas Mean Squa e) can be applied in his pape . a 1 a n a n+1 w n w 1 a 2 w 2 Fig. 3. The single-laye neu al ne wo k con e ed by he AFSN P sys em Π2o Type 2. We can summa ize he lea ning algo i hm o AFSN P sys ems as ollows Adap i e Fuzzy Spiking Neu al P Sys ems 243 p og am Weigh _lea ning_algo i hm inpu T aining da a se D; m = |D|; Lea ning a e del a; ou pu The weigh s (w1, w2,..., wn); begin Selec a se o ini ial weigh s; i=1; epea Compu e he ou pu s e o o i- h aining sample; Upda e he weigh s (w1, w2,..., wn) using Wid ow-Ho lea ning law wi h lea ning a e del a; i=i+1 un il i>m end. 3 Simula ion In his sec ion, a ypical example is selec ed o illus a e he lea ning abili y. Example 1. Le p1,p2,p3,p4,p5and p6a e ela ed p oposi ions o a knowledge base o aul diagnosis. The e a e he ollowing weigh ed uzzy p oduc ion ules: R1: IF p1THEN p4(γ1,τ1) R2: IF p2AND p4THEN p5(ω2,ω4,γ2,τ2) R3: IF p3AND p5THEN p6(γ3,γ4,τ3,τ4) This example includes h ee ypes o ules: R1is a simple ule and R2is a com- posi e conjunc i e ule, while R3is a composi e disjunc i e ule. These weigh ed uzzy p oduc ion ules can be modeled by he ollowing AFSN P sys em Π: •Π= (A, {σp1, σp2, σp3, σp4, σp5, σp6},{σ 1, σ 2, σ 3, σ 4}, syn, I, O) whe e: (1) A={a} (2) Fo each j(j= 1,2,3,4,5,6), σpj = (αj,ωj, λj, j) is a p oposi ion neu on associa ed wi h p oposi ion pj, and jis a spiking ule o he o m E/aα→ aα. He e, λj(j= 1,2, . . . , 6) = 0, and ω1=ω3=ω5= 1. (3) Fo each j(j= 1,2,3,4), σ j = (αk+j, γj, τj, k+j) is ule neu on. σ 1and σ 2a e associa ed wi h ule R1and R2 espec i ely, while σ 3and σ 4a e associa ed wi h ule R3. k+j(j= 1,2,3,4) a e spiking ule o he o m E/aα→aβ. (4) syn ={(σp1, σ 1),(σp2, σ 2),(σp3, σ 3),(σp4, σ 2),(σp5, σ 4),(σ 1, σp4), (σ 2, σp5),(σ 3, σp6),(σ 4, σp6)}. (5) I={σp1, σp2, σp3},O={σp4, σp5, σp6}.