The Role of Random Spikes and Concurrent Input Layers in Spiking Neural Networks

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1 The Role of Random Spkes and Concurren Inpu Layers n Spkng Neural Neworks Jans Zuers Faculy of Compung, Unversy of Lava, Rana bulvars 19, LV-1586 Rga, Lava ans.zuers@lu.lv Absrac. In spkng neural neworks (SNNs), unlke radonal arfcal neural neworks, sgnals are propagaed by a pulse code nsead of a rae code. Ths resuls n ncorporang he me dmenson no he nework and hus heorecally ensures a hgher compuaonal power. The dfferen prncple of operaon makes learnng n SNNs complcaed. In hs paper, a Hebbanlearnng based learnng algorhm, nroduced by he auhor earler, s furher developed hrough applyng random spkes n order o guaranee frequen frngs of neurons for each layer. In addon, he mpac of concurren npu layers o he learnng ably s shown. The nroduced deas have been llusraed wh expermenal resuls. Keywords: Spkng Neural Neworks, Unsupervsed Learnng, Randomness. 1 Inroducon A neural nework s a compuaonal model whch s made up of relavely smple nerconneced processng uns whch are pu no operaon hrough a learnng process. Neural neworks are useful ools for solvng many ypes of problems. These problems can be characersed as mappng (ncludng paern assocaon and paern classfcaon), cluserng, and consran opmsaon. There are several neural nework models whch are avalable for each ype of problem [3]. Typcally, he neurons of a nework communcae va a rae code,.e., by ransmng pure connuous values rrespecve of he mngs of he sgnals. Work wh radonal models s usually arranged n seps durng each sep exacly one paern s classfed or recognzed. Consderng hs, any emporal nformaon, f presen n he problem, or nerconnecedness of paerns s ou of area of responsbly of a neural sysem, so should be processed by a superor sysem. When spkng neural neworks appeared, he neural nework models became ready o be classfed no hree generaons. The frs generaon of arfcal neural neworks conssed of McCulloch-Ps hreshold neurons [9], where a neuron sends a bnary sgnal f he sum of s weghed ncomng sgnals rses above a hreshold value. The second generaon neurons use connuous acvaon funcons nsead of hreshold funcons o compue he oupu sgnals.

2 In conras o classcal models, spkng neural neworks, or he hrd generaon neural neworks, are desgned o make use of he emporal dynamcs of neurons. Here, communcaon beween neurons nvolves a pulse code he exac mng of he ransmed sgnal (.e., he spke) s of key mporance, bu he srengh of he sgnal s no consdered. SNNs are beleved o be very powerful, bu her operaon s more complcaed and her applcaon s no as obvous as s he case wh ordnary neural neworks. Neverheless SNNs have been focused as an area of much research. In [12], he mehod of adapve layer acvaon has been nroduced for Hebbanlke learnng n SNNs. The curren paper furher develops hs approach (a) by addng necons of random spkes o nduce he learnng process, and (b) by exendng nework srucure wh concurren npu layers of neurons n order o enrch presence of emporal emplaes n he npu srucure represened by axonal delays. 2 Spkng Neural Neworks Spkng neural nework s a model of arfcal neural neworks, n whch sgnals are propagaed va spkes nsead of raes (.e., values). SNNs are poenally very powerful, sll complcaed o operae. 2.1 General Descrpon Each neuron n an SNN produces a ran of spkes (.e., fres), and he operaon of he neuron s fully characerzed by he se of frng mes [4] (Fg. 1): F { ( 1), ( n, K ) } =, (1) where (n) s he mos recen spke of neuron. One of he mos frequenly appled spkng neuron models s he negrae-andfre model [1][2]. In hs model, npu sgnals (boh acual and recen) from a neuron are combned ogeher n order o acheve enough acvaon for he neuron o fre. Neuron s sad o fre f s acvaon sae u reaches he hreshold ϑ. A he momen when he hreshold s crossed, a spke wh he frng me (f) s generaed. Thus Eq. (2) can be clarfed wh Eq. (3) [4]: ( f ) { ;1 f n} = { u ( =ϑ} F = ), (2) where he acvaon sae u s gven by he lnear superposon of all conrbuons: u ( ) = η ( ^ ) + Γ ( f ) w F ε ( ( f ) ), (3)

3 where Γ s he se of presynapc neurons o neuron ; ^ s he las spke of neuron ; he funcon η akes care of refracorness afer a spke s emed; he kernels ε model he neurons response o presynapc spkes. s ϑ exp( ); η( s) = τ 0; s> 0 s 0, (4) where τ s a me consan. ax ax s s ax exp( ) exp( ); s > 0 ε ( ) = s τ m τ, s ax 0; s 0 (5) where τ s and τ m are me consans, and ax s he axonal ransmsson delay for presynapc neuron. Axonal ransmsson delay (or smply, axonal delay) s an essenal par of a spkng neuron (Fg. 1). I delays ransmed sgnals for he subsequen neurons, herefore realzes a knd of shor-me memory. Axonal delays are usually nalzed n random wh creaon of an SNN. Eqs. (4) and (5) have been adaped from [4], [8]. In Fg 2, he naure of acvaon of spkng neurons s vsualzed. The algorhm n Fg. 3 gves a general overvew of operaon of a spkng neuron. Incomng spkes Φ n w Processng Weghs Neuron ax Φ ou Axonal delay Oucomng spkes Fg. 1. A sngle spkng neuron. The small flled crcles w denoe synapc weghs. Φ n denoes an ncomng spke ran. The neuron produces he spke ran Φ ou, delayed by axonal delay ax. (Adaped from [7])

4 ϑ (a) w ε ( ( f ) ) Γ ( f ) F ( w ε ( f ) ) ϑ (b) η ( ^ ) u ( ) ^ Fg. 2. Graphc nerpreaon of acvaon sae u (Eq. (3)) a) Acvaon u () s a superposon of responses of ncomng spkes and he weghs w. The kernels ε descrbe he response of u o presynapc spkes. (b) Acvaon u () reaches he hreshold ϑ a me (f). The rese of acvaon s mmedaely performed n order o avod connuaon of spkng over a ceran perod. Ths refracorness s acheved by addng a kernel η ( (f) ). The pure negrae-and-fre model s very smple, bu he fre sequences appear oversmplfed and unable o reproduce rch and rregular naure of spkng of bologcal sysems. In order o avod regulares and deermnsces, numerous exensons of he model, as well as nermedae ones wh oher models have been proposed [6][10]. 2.2 Learnng n Spkng Neural Neworks If supervsed learnng mehods grealy predomnae n non-spkng neural neworks, hen he suaon wh SNNs s que dfferen. Hebban learnng ([5]) s a ypcal learnng mehod wh SNNs. The smples way o descrbe Hebban learnng s ha he wegh modfcaons depend on he level how pre- and possynapc acves mach (Eq. (6)). Operaon of radonal neural neworks s carred ou hrough propagang of values, and he process of propagaon s occurrng whou any consrans he neural acves nfluence value srenghs, whle he propagaon s occurrng regardless hese values. w = c y y. (6)

5 Wh SNNs, spkes are he ones, whch are beng propagaed over he nework. Unlke radonal neural neworks, here are condons requred for he process o occur, snce spkes are generaed only a ceran condons. Ths s a noceable praccal problem. Algorhm run_spkng_neuron () me sep of operaon ϑ hreshold o be reached by acvaon of he neuron ax axonal delay for he neuron Begn Compue acvaon u by processng mngs of ncomng spkes and weghs (Eq. (3)) If u ϑ Then % acvaon reaches hreshold Generae spke a me sep Delay he spke by ax o become ncomng spke for subsequen neurons (See Fg. 1) Fg. 3. Operaon of a spkng neuron a fxed me sep. 2.3 The Learnng Mehod Usng Adapve Layer Acvaon In [12], a Hebban rule-based mehod for learnng n SNNs s proposed. I s bul consderng he followng predefned prncples (no mandaory for SNNs n general): 1) Wegh changes are performed only afer he neuron has us fred. Thus an addonal mechansm s requred o ensure spkes o be generaed; 2) Wegh changes are performed only upwards (as an award for a correc mng of a spke, Eq. (7)). Thus an addonal mechansm s requred o regulae wegh values o avod hem all o become oo bg. w ( f ) = α ( w w ) ε ( ), (7) ( f ) F where α s he learnng rae; w s a consan represenng maxmum allowed wegh value, ε he neurons response o presynapc spkes. Algorhm run_spkng_neuron_adapve (, β) me sep of operaon β acvaon acceleraon facor of he layer ϑ hreshold o be reached by acvaon of he neuron ax axonal delay for he neuron Begn Compue acvaon u by processng mngs of ncomng spkes and weghs (Eq. (3)) If β u ϑ Then % (acceleraed) acvaon reaches hreshold Generae spke a me sep Delay he spke by ax o become ncomng spke for subsequen neurons Fg. 4. Operaon of a spkng neuron wh adapve acceleraon facor. Acvaon acceleraon facor β deermnes sensvy of he neuron. These condons were obeyed hrough nroducng he followng key nnovaons (See he algorhm n Fgs. 4 and 5):

6 1) Adusmen of he average acvaon sae of neurons n each layer n order o ensure a deermned average rhyhm of spkes whn a layer; 2) Manenance of he fxed average volume of synapc wegh n each neuron n order o avod a suaon n whch all he weghs converge o maxmum possble value. Algorhm learnng_adapve_layer_acvaon () me sep of operaon β acvaon acceleraon facor of he layer (n he begnnng s se o 1.0) ϑ hreshold o be reached by acvaon of he neuron run_spkng_neuron_adapve algorhm o operae one neuron wh adapve acceleraon facor Begn Forall neurons n he layer Call run_spkng_neuron_adapve (, β) % run neuron (See he algorhm n Fg. 4) If neuron has us fred Then Forall weghs n he neuron Increase wegh proporonally o he respecve ncomng acvaon (Eq. (7)) Decrease unformly all he weghs o keep a fxed average wegh value of he neuron Updae layer sascs abou neuron acvy n he layer (frequency of spkes) If neurons acvy s srong (accordng o he sascs) Then Decrease β Else Increase β Fg. 5. Learnng wh mehod of adapve layer acvaon for one layer. In addon o neuron weghs, also he acvaon acceleraon facor β undergoes learnng. 3 Random Spkes and Concurren Inpu Layers Alhough spkng neural neworks are beleved o be very powerful, hey are complcaed o operae, and ranng hem s sll a challengng ssue. In hs secon, wo dfferen deas have been proposed o operae a knd of SNNs (bul accordng o he prncples saed n he prevous secon): 1) Inecng random spkes o nduce learnng of an SNN; 2) Exendng nework srucure wh concurren npu layers. The proposed mehods help o realze a way, how an SNN s se o operae. The expermenal resuls show hese mehods o help o successfully overcome varous problems o buld SNNs. 3.1 The Learnng Mehod Usng Random Spkes Applyng effecs of randomness s a mehod already used o exend negrae-and-fre neural neworks [11]. Ths secon nroduces he approach of usng random spkes n learnng wh SNNs. The curren research s connuaon of he prevous work, where he followng prncple n operaon of an SNN was assumed: learnng whn a neuron s performed only afer he neuron has us fred,.e., has us generaed a spke. Snce normally

7 frng s accomplshed only when acvaon of he neuron reaches a ceran hreshold value, he problem o overcome s how o provde a neuron wh condons o frequenly fre n order o undergo learnng. In he prevous research, hs problem was proposed o solve by usng he acvaon acceleraon facor. The man dea of s ha he requremens of frng are beng urned down, f neural acvy n he layer s nsuffcen. Such a suaon s ypcal a he very begnnng of he learnng process. In he new approach, hs prncple s subsued by generang random spkes wh a ceran probably. Ths prncple ensures he mnmum acvy of a neuron, and by hs, also he learnng process o happen. Roughly speakng, he new mehod has he effec smlar o he one of he mehod of adapve layer acvaon, ye s smpler and so much easer o mplemen. The algorhm n Fg. 6 unvels he prncple of random spkes used n operaon of a neuron. The algorhm n Fg. 7 shows he learnng algorhm for a whole layer usng hs prncple. Adusng weghs of neurons remans he same as n he prevous approach. Algorhm run_spkng_neuron_random_spkes (, ρ) me sep of operaon ρ random spke probably (0 ρ << 1) ϑ hreshold o be reached by acvaon of he neuron ax axonal delay for he neuron ge_random_value generaes a random value from nerval 0..1 Begn Compue acvaon u by processng mngs of ncomng spkes and weghs (Eq. (3)) fre False If u ϑ Then % acvaon reaches hreshold fre True Elsef ge_random_value < ρ Then % probably o generae random spke fre True If fre ϑ Then Generae spke a me sep Delay he spke by ax o become ncomng spke for subsequen neurons Fg. 6. Operaon of a spkng neuron wh random spkes. Spkes are generaed also n random o guaranee a ceran frequency. Algorhm learnng_random_spkes (, ρ) me sep of operaon ρ random spke probably (0 ρ << 1) ϑ hreshold o be reached by acvaon of he neuron run_spkng_neuron_random_spkes algorhm o operae one neuron wh random spkes Begn Forall neurons n he layer Call run_spkng_neuron_random_spkes (, ρ) % run neuron (See Fg. 5 for he algorhm) If neuron has us fred Then Forall weghs n he neuron Increase wegh proporonally o he respecve ncomng acvaon (Eq. (7)) Decrease unformly all he weghs o keep a fxed average wegh value of he neuron Fg. 7. Learnng wh he mehod of random spkes for one layer.

8 3.2 Usng Concurren Inpu Layers n Nework Srucure Ths secon proposes an SNN archecure wh concurren npu layers o enhance he ably of a neural nework o deec regulares n an npu. Commonly a neural nework has a lnear feed-forward srucure. Ths also holds for SNNs (Fg. 8). For echncal reasons, npu layers are usually no rue layers: npu values o he whole neural nework are drecly pu o he oupus of he npu layers, so hey don compue anyhng (See he algorhm n Fg. 9). Wh SNNs, he npu layer can addonally have he role of delayng npu sgnals. As axonal delays are predefned n random, npu layers of SNNs provde wh a knd of spaal represenaon of emporal npu nformaon. npu axonal delays npu layer as a se of axonal delays hdden layer oupu layer wh no axonal delays oupu Fg. 8. A smple example of a feed-forward lnear archecure of a spkng neural nework. Connecons beween layers are depced as arrows, and hey denoe all-o-all connecon beween he neurons of boh layers. Algorhm run_neural_nework Begn Do many mes Fech nex npu no X Se X o oupus of he npu layer (n SNNs, drecly generae spkes accordng o X) Process he res of layers n a row Fg. 9. Runnng a neural nework wh lnear srucure and a sngle npu layer. Insead of a sngle npu layer, several concurren npu layers are proposed by he auhor. Snce axonal delays of neurons are predefned random values, bunch of npu layers provde wh several knds of spaal represenaons of npu sgnals. (Fg. 10) The dea s very smple: a each eraon, he npu s duplcaed for all he npu layers (see he algorhm n Fg. 11).

9 By hs, a nework receves several varees of parly delayed npu sgnals, so concurren npu layers serve as a grd of frs level percepon. As he expermenal resuls show, hs srucural feaure leads o faser and beer learnng n an SNN. npu duplcaon of he npu axonal delays concurren npu layers hdden layer oupu layer wh no axonal delays oupu Fg. 10. A spkng neural nework archecure wh concurren npu layers. Algorhm run_neural_nework_concurren_npu_layers Begn Do many mes Fech nex npu no X Forall L n npu layers Do Se X o L Process he res of layers n a row Fg. 11. Runnng a nework wh concurren npus. 4 Expermenal Work To show he nroduced mehods workng, a large expermenal work was accomplshed usng orgnal sofware for ha. Each expermen was run o solve a ask of deecng one knd of emporal paerns n an npu sream. Expermens were carred ou usng several confguraons of SNNs o be able o compare hem n order o mark ou he effec of he wo deas nroduced n hs paper.

10 4.1 Seup of he Expermens The proposed learnng mehod was esed on he deecon of reerave emporal paerns n an npu sream. A raned neural nework should be able o deec such paerns by gvng a spke n oupu afer he paern s encounered. The esng envronmen produces hree bnary sgnals a each dscree me momen and passes hem o he npu of he neural nework. In general, sgnals are generaed a random, bu somemes paerns from a predefned sore are nerwoven no hem. There are sx possble emporal paerns n he sore (Fgure 12). The proporon of 1s and 0s n he random par equals he one n he paerns 20% Fg. 12. The 6 emporal paerns used n expermens: wdh=3 sgnals, lengh=10 mes; black squares denoe 1 or spke, bu whe squares 0 or no spke A each separae expermen, one of he sx paerns was occasonally pu no npu sream (7 paerns of sze 10 whn 200 me seps n average), so he proporon of paerns n he npu sream was =35%. Oupu spkes of raned neworks were analyzed of how hey mach paerns n he npu sream. Paern recognon rae was measured as a rae of responses of a nework o npu paerns a a fxed poson wh regard o he end of a paern. Posons red were of he nerval [-5, +5]. As for nsance, paern recognon rae of 90% means ha we have a poson p, one for whch s rue ha for 90% of paerns of he npu sream, a spke of he neural nework s encounered a he poson p wh regard o paerns. In addon, he res of spkes, false posve ones, also were couned (Fg. 13.). False posve rae refers o he proporon beween false posve spkes and acual paerns n he npu sream. npu paern # oupu false posve spkes a correc h, f he fxed poson s of 2 Fg. 13. An excerp of an npu sream, and examples of responses of he nework o. Here he value of poson can be chosen as 2, so he paern recognon rae s compued as 100% (as he only paern s recognzed), and he false posve rae s 2 (wo false posve spkes agans one paern).

11 To show he nroduced mehods workng, a oal of 15,000 expermens was conduced. Each expermen conssed of 20,000 eraons n he learnng phase, and hen 200 eraons n he recall phase, he resuls of whch were recorded and furher analyzed. Fve dfferen confguraons of SNNs were used (Table 1) accordng wh he general srucure shown n Fg. 10 and he algorhm n Fg 7. Dfferen confguraons of SNNs were used o be able o compare hem n order o mark ou he effec of he wo deas nroduced n hs paper: random spkes and concurren npu layers. The funconaly of SNNs was bul as descrbed n secons 2.1 and 3; wh random spke probably 4%, hreshold 0.3, and axonal delays randomly se from nerval [0, 4]. Along analyzng drec resuls of he expermens, also sde affecs were deermned, hose of false posves and poson of paern deecon. Table 1. Nework confguraons used n he expermenaon. Name of he confguraon Usage of random Nework archecure descrpon (for all spkes ypes, here are 3 npus and 1 oupu) A. Random=N No 4 concurren npus, 12 hdden neurons B. Random=N No 8 concurren npus, 24 hdden neurons C. Random=Y Yes 4 concurren npus, 12 hdden neurons D. Random=Y Yes no concurren npus, 24 hdden neurons E. Random=Y Yes 8 concurren npus, 24 hdden neurons 4.2 Analyss of he Resuls Fg. 14 shows he general overvew of expermenal resuls for he fve confguraons of SNNs. The confguraons C, D, and E show he posve effec of usng he random spkes o recognon. The confguraon E shows he mpac of addonal concurren npu layers, whle D and E show he effec of boh concurren npu layers and a suffcenly szed hdden layer. The confguraon D (n conras o C) mgh sgnal a large hdden layer o brng he same effec as concurren npu layers do, however suffers from nsably of deecon (see Fg. 16). The bes resuls were obaned by obeyng all he followng condons: 1) Usng random spkes; 2) Usng more concurren npu layers; 3) Usng a hdden layer wh a suffcen sze. The man resuls demonsrae SNNs o be successfully raned o deec emporal paerns. Durng expermens, here also were undesrable sde effecs, whch should be reduced durng furher research.

12 120% Paern recognon rae 100% 80% 60% 40% 20% 0% 22% A. Random=N 4x % B. Random=N 8x % C. Random=Y 4x % D. Random=Y 1x % E. Random=Y 8x Confguraon of SNNs Fg. 14. Paern recognon rae obaned for varous confguraons of SNNs. For he bes resuls, random spkes and concurren npu layers should be used. The mos noceable one s ha of false posves (Fg. 15). The resuls show ha, along successful deecon of npu paerns, a lo of redundan spkes are addonally generaed by a neural nework. 'False posve' rae A. Random=N 4x B. Random=N 8x C. Random=Y 4x D. Random=Y 1x E. Random=Y 8x Confguraon of SNNs Fg. 15. Hgh false posve raes sgnal he algorhms o be furher mproved and/or beer confgured/seup. Fg. 16 shows how he archecure of he nework nfluences he lengh of deeced paerns. The neworks of he confguraons C and D deec mos paerns a earler posons (See also Fg. 13), whle he sronger neworks (E) ypcally deec paerns almos a he end poson (-1), moreover he poson of deecon was much more deermned (50% for he posons of hghes occurrence). The confguraon D s especally unsable, so has such a good recognon rae (Fg. 14) us because of he mehodology used.

13 60.0% Occurrence rae 50.0% 40.0% 30.0% 20.0% 10.0% 0.0% Poson wh regard o end of paern C. 4x D. 1x E. 8x Fg. 16. Occurrence dsrbuon of posons (wh regard o he ends of paerns) a whch he bes paern recognon rae was obaned (only for he hree confguraons wh random spkes). The confguraon E shows he mos sable resuls. 5 Concluson Spkng neural neworks are beleved o be very powerful; for all ha hey are comparavely complcaed, especally n erms of learnng. In hs paper, wo deas have been nroduced o enhance learnng and operaon of a varey of SNNs (characerzed by he prncple ha learnng whn a neuron s performed only afer he neuron has us fred): 1) Usng random spkes o addonally smulae he neuron; 2) Usng several concurren npu layers o beer preprocess he npu. The prncple of random spkes ensures he mnmum acvy of a neuron, and by hs, also he learnng process o happen. Expermenal resuls show essenal for a good recognon rae. Alhough he expermens confrm he sgnfcance of hdden layers wh suffcen szes, ye for he bes resuls, concurren npu layers are very mporan. The proposed approach furnshes an effec o learnng of SNNs smlar o one n he prevous research; neverheless he new mehods are less complcaed and have demonsraed o be more sable. References 1. Burk, A.N.: A revew of he negrae-and-fre neuron model: I. Homogeneous synapc npu. In: Bologcal Cybernecs, Volume 95, Number 1, 2006, pp Sprnger (2006) 2. Burk, A.N.: A revew of he negrae-and-fre neuron model: I. Inhomogeneous synapc npu and nework properes. In: Bologcal Cybernecs, Volume 95, Number 2, 2006, pp Sprnger (2006) 3. Fause, L.: Fundamenals of Neural Neworks. Prence-Hall, Inc. (1993)

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