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!!ACTABIOPHYSICASINICAVol.24No.1Feb.2008:2007-08-23:(3030010730270432):(029)853102696@0[1,2][3~11][3~5]interspikeintervalISIpeakISIISIISI[3~5][6]Kaplan[6]FHN[1,7~10,13][8,9]WangKaplan[7,10,13]Wang[11]Wang[8,13]Wang[8]WangISIISI[7,8]FitzHugh-Nagumo,,,,710062:1;,1,;,,,:;;;;;:Q424E-mailrenwei194vip.sina.com1FitzHugh-Nagumo1[6]FitzHugh-Nagumo(FHN)dvdt=-v(v-1)(v-a)-w+I(1)dwdt=c(v-bw)(2)vwIa=0.139b=2.54c=0.008FHN(1)!(t)(2)FHN,dvdt=-v(v-1)(v-a)-w+I+!(t)(3)dwdt=c(v-bw)(4)!(t)!(t)=0;!(t),!(t),!(t')=2D(t-t')D()Diract-Mannella[12]FHN10-3vISI2[7]ISINISI(ISI1,ISI2,,ISIN)myn=(ISIn-m+1,ISIn-m+2,,ISIn)yn#Nynj=(ISIn-m+1j,ISIn-m+2j,,ISInj)(j=1,2,,#N)ISInhfn+h=1#N#Nj=1!ISIjn+hISInhISIn+h(nor-malpredictionerror,NPE)NPE=[(fn+h-ISIn+h)2]1/2[(ISI-ISIn+h)2]1/2(5)ISIISINPE1NPE1ISIKasperSchusterLempel-ZivISI01(k-0.5)TISI(k+0.5)TTISIk-101ISI3001SQSQSQvSQQSQvQSQvQQSQvSQ101101)1;2)S=1,Q=0,SQ=10,SQv=1,Q#SQv103)S=10,Q=1,SQ=101,SQv=10,QSQv1014)S=10,Q=11,SQ=1011,SQv=101,Q#SQv1011232008Fig.1Thebifurcationofequilibriumpointandlimitcycle(v)inthedeterministicFHNmodelwithrespecttocurrentI.Thethicksolidlinedenotesstableequilibriumpoints,buttheshortdotlineindicatesunstableequilibriumpoints.Theupper(lower)thinsolidlinepresentsthevmax(vmin)ofstablelimitcycles,andtheupper(lower)dashdotlinedepictsthevmax(vmin)ofunstablelimitcycles0.80.40.0-0.40.0300.0320.0340.0360.0380.040IV(5)S=1011,Q=0,SQ=10110,SQv=1011,QSQv1011010110c(n)=4LempelZivs(i)(i=1~n)[0,1]c(n)limnc(n)=b(n)=nlog2n(6)b(n)c(n)C(n)=c(n)/b(n)C(n)1(0)p0.5p0.5hlimnc(n)=hb(n)(7)h=-[plog2p+(1-p)log2(1-p)]1p=0.5h1100133.1FHNIFHNI0.03512HopfI0.034425FHNII0.034425I0.0344250.03512I0.034425FHN11vmaxvmin11II=0.03513147.933.2I160D10-8~10-4IT=2501.8850.1FHND=510-72A250ISI2BISIISIISIyISIx,logy=3.18868-0.102xlogyxR=-0.99694ISIISI2C241FitzHugh-NagumoNPE1.20.80.40.0369Predictionstep302520151050400080001200000.80.40.0-0.4TVI1000100100123456789NormalizedISINumberofISI250020001500100050000500100015002000ISInISIn+110001001002468NormalizedISINumberofISI(A)(B)(C)(D)(E)Fig.2TheintegermultiplespikingfromstochasticFHNmodelstimulatedbypulsecurrent.(A)Spiketrains,thelowsubthresholdoscillationiscausedbytheexternalperiodicpulsestimulus;(B)ISIhistogram,thepeaksdecreaseexponentially;(C)ISIreturnmap;(D)TheNPEofISIseriesversusthepredictionstep;(E)ISIhistogram,thepeaksdecreaseexponentially,too.Parameters:(A)~(D)periodofpulsecurrentT=250,pulsedurationis0.1,pulsedensityI=1.885,D=510-7;(E)T=300,otherparametersremainunchangedISINPE2DISI1.033.3IT=29.5I=2.680.1I=029.4FHN3A2A29.5ISI3Bmulti-peak123peakpeak4ISI4peak6ISI3CISINPE1NPE13DISI0.7410~1[6]29.4730.260.099010.1040.5430.260.0990.104252008NPE1.21.00.80.6369Predictionstep353025201510501.20.80.40.0-0.4TVI100101061218NormalizedISINumberofISI(D)(E)(A)(B)(C)4008001200016002000120010008006004002000ISInISIn+105001000150025012507501000100101NormalizedISINumberofISI012416820Fig.3Themulti-modespikingfromdeterministicFHNmodelstimulatedbypulsecurrent.(A)Spiketrains,thelowsubthresholdpulseiscausedbytheexternalperiodicpulsestimulusandnoise.(B)ISIhistogram,thepeaksaredifferenttoFig.2(B).Theydonotdecreaseexponentially.(C)ISIreturnmap.(D)TheNPEofISIseriesversusthepredictionstep.(E)ISIhistogram,thepeaksdonotdecreaseexponentially.Parameters:(A)~(D)periodofpulsecurrentT=29.5,pulseduration0.1,pulsedensityI=2.68;(E)T=30.26,otherparametersremainunchanged4[7~10,13][3,5],[14][15]ISI3ISIISI1ISIISI1FHNFHNWang26121FitzHugh-Nagumo[7,10,13]:[1]BraunHA,WissingH,SchaferK,HirschMC.Oscillationandnoisedeterminesignaltransductioninsharkmultimodalsensorycells.Nature,1994,367(20):270~273[2]YangMH,AnSC,GuHG,LiuZQ,RenW.Understandingofphysiologicalneuralfiringthroughdynamicalbifurcationmachineries.NeuroReport,2006,17(10):995~999[3]LongtinA,BulsaraA,MossF.Time-intervalsequencesinbistablesystemsandnoise-inducedtransmissionofinforma-tionbysensoryneurons.PhysRevLett,1991,67(5):656~659[4]SiegelRM.Nonlineardynamicalsystemtheoryandprimaryvisualcorticalprocessing.PhysicaD,1990,42:385~395[5]LongtinA,BulsaraA,MossF.Bistabilityandthedynamicsofperiodicallyforcedsensoryneurons.BiolCybern,1994,70:569~578[6]KaplanDT,ClayJR,ManningT,GlassL,GuevaraMR,ShrierA.Subthresholddynamicsinperiodicallystimulatedsquidgiantaxons.PhysRevLett,1996,76(21):4074~4077[7]..1999,15(3):482~488[8]..2003,19(3):272~278[9]..2001,17(4):637~644[10]GongPL,XuJX,HuSJ,LongKP.Chaoticinterspikeinter-valswithmultipeakedhistograminneurons.IntJBifurcChaos,2002,12:319~328[11]XieY,XuJX,JiangJ.DeterministicintegermultiplefiringdependingoninitialstateinWangmodel.Chaos,Solitons&Fractals,2006,30(5):1250~1259[12]MannellaR,PalleschiV.Fastandprecisealgorithmforcom-putesimulationofstochasticdifferentialequations.PhysRevA,1990,40:3381~3386[13]..2000,16(2):250~258[14]MascioMD,GiovanniGD,MatteoVD,EspositoE.Reducedchaosofinterspikeintervalofmidbraindopaminergicneu-ronsinagedrats.Neuroscience,1999,89(4):1003~1008[15]Rabinovich,AbarbanelHDI.Theroleofchaosinneuralsys-tem.Neuroscience,1998,87(1):5~14272008ThisworkwassupportedbygrantsfromTheNationalNaturalSciencesFoundationofChina(30