• 検索結果がありません。

Coherence Resonance in Propagating Spikes in the FitzHugh-Nagumo Model-香川大学学術情報リポジトリ

N/A
N/A
Protected

Academic year: 2021

シェア "Coherence Resonance in Propagating Spikes in the FitzHugh-Nagumo Model-香川大学学術情報リポジトリ"

Copied!
4
0
0

読み込み中.... (全文を見る)

全文

(1)

IEICETRANS・FUNDAMENTALS,VOL.E84−A,NO.6JUNE2001 1593

LETTER

CoherenceResonanceinPropagatingSpikesinthe

FitzHugh−NagumoModel

YoHORIKAWA†a),Reg加ゎγ〟emわer

First,the resul七s ofcomputer simulation on the One−dimensionalFitzHughrNagumomodelareshown.

血/dま=∂2γ/∂諾2一項−α)(℃−1)一触+が(£)

dぴ/離=ど(り−γu)

(α=0・2,ど=0・003,7=0・5,0≦ご≦30)(1) where Gaussian white noise n with zero mean and

strengtho−(E(n(t)n(壬′))=CT26(L−t′))isaddedtov

at七heoneend(x=0)・Spikesaregeneratedbythe

?Oiseandpropagatetotheotherend・Equation(1)

1S numericallycalculatedbythesimpleEulermethod

with△x=1.O and△i,=0.2.The time series of v

atthenoise−addedpoint(x=0)andafterpropa−

tion(x=25)witho・=0.2,0.4andO.6areshownln

Fig.1.Thespikes ofpropershape areobtained after

propagation(x=25)owingtothewaveshapingaction

Ofexcitablemedia[17],Whiletherearelargevariations

due to the noise at x=0.Figure2plots the mean

andstandarddeviation(S・D・)oflOOOOinterspikein−

tervals ofthe propagating spikes atニご=25against the noise strength o..Both take minimum values at O−⊆≧0・38.Thismeansthefrequencyandregularityof

the spikes are absoIutely highest at the intermediate

noisestrength・Thecoefncientofvariation(theratioof

thestandarddeviationtothemean)oftheinterspike

intervalsisalsominimum at thesame noisestrength

O’望0.38,thoughnotshownhere.Notethatthevalues

Ofthenoisestrengtharesmallerthantheamplitudeof

thespikesandareconsideredtobephysicallyrelevant.

This coherence resonance canbe obtainedinwide

ParameterrangeS.Figure3showschangesintheop−

timalnoisestrengthcTopt andthemeaninterspikein−

terva17二plbyaddingconstantinputZatx=Otothe

right−handsideofdv/dtinEq・(1).Theoptimalnoise

Strengthandthemeaninterspikeintervalaresmallat

O・3≦Z≦1・3,Wherethespikesareperiodicallygener−

atedwithoutnoiseinthesingleelement.

Next,eXperimentonNagumo’sactivetransmission

line[14],ananalogcircuitfortheFitzHugh−Nagumo

modelisdone.TheN−Shapednonlinearcurrentdevice

(T.D.)andinductorareconstructedwithoperational

amplifiers[18],aSShowninFig.4.TheNagumo,sac−

tivelineismadebycoupling20elementswithresistors.

Thevalueoflんissettobe4.9Vsothattheelements are mono,Stable.White noise sourceis added to V−1 SUMMARY CoherenceresonanCeinpropagatingspikesgen−

eratedby noiseinspatiallydistributedexcitablemediaisstud− ied with computer simulation and circuit experiment on the

FitzHugh−Nagumomodel・Whitenoiseisaddedtotheoneend

Ofthe media to generate spikes,Which propagate to the other end・Themean andstandard deviationoftheinterspikeinter− Valsofthespikesafterpropagationtakeminimumvaluesatthe

intermediatestrength ofthe added noise.Thisshowsstronger COherencethanobtainedint,hepreviousstudies. たeyぴ0γdβご βねcんα5t五cγeβ0γもα几Ce,COんeγe†lCeγe50γ乙α乃Ce,e:rC正αらge med盲α,ダ豆まzガ㍊gん−Ⅳ叩祝mOmO(ヱeら叩豆如pγOp叩α舶0れ Stochasticresonanceinexcitablemediaisofwideinter−

estsinceitmayberelatedtosensorysignalprocessing

innervoussystems[1]・Recently,itwasshownthatco−

herenceresonance(stochasticresonancewithoutinput

Signals),Whichwasfoundinsomelimitcyclemodels

[2],OCCurSalsoinexcitablemedia[3][12].Thatis,the

COherenceorregularityofthespikesgeneratedbyaddi−

tivenoiseinexcitablemediaisoptimalatintermediate

noisestrength.

Inthislettercoherenceresonanceinspatiallydis−

tributedexcitablemediaisstudiedwithcomputersimu−

1ationandcircuitexperimentontheFitzHugh−Nagumo

model,aSimplemodelofanervefiber[13],[14=]・White

noiseisaddedtotheoneendofthemedia,bywhich

Spikes aregenerated and propagatetotheotherend.

It is shown that the mean and standard deviation of

theinterspikeintervalsofthespikesafterpropagation

take minimum values atintermediate noise strength・

Thisis stronger coherence than that obtainedin the

previousstudiesonexcitablemedia,inwhichsomerel−

ative measures ofcoherence,e・g.,the product ofthe

heightofthepeakandthequalityfactorofthepower

SpeCtrum,the coefBcient ofvariation andthe correla−

tiontimewereused.Althoughsimilarnon−mOnOtOnic

relationsofthefiringfrequencytothenumbersofion

channels have been obtained in computer simulation

OnStOChasticversionsoftheHodgkin−Huxleymodel,in

Whichfluctuationsinionchanneldynamics aretaken

intoaccount[15=16],themechanismcausingthisco−

herence resonanceis differentfrom them,

ManuSCriptreceivedNovember16,2000. ManuscriptrevisedFebruary23,2001.

†TheauthoriswiththeFacultyofEngineerlng,Kagawa

University,Takamatsu−Shi,761−0396Japan. a)E−mail:horikawa@eng.kagawa−u.aC.jp

OLIVE 香川大学学術情報リポジトリ

(2)

IEICETRANS・FUNDAMENTALS,VOL.E84−A,NO.6JUNE2001 1594 (a)打=0.2 0.4 0.3 こ. 言0・2 0.1 0.0 0 0 0 + 1 0 1 ︵○︶ゝ 0

0・5 / 1

+ 5 0 5 0 0 0 ︵SN︶ゝ 1500 ≠1000 艮 500 0 5000 10000 f (b)グ=0.4 0 0t5 / 1 1.5 Fig・3 0ptimalnoisestrengthJ。Ptandmeaninterspikeinter− Val7㌔p士VS・COnStantinputZ(theFitzHugh−Nagumomodel). 5000 10000 t (c)J=0.6 2.0 1.0 望 0.0 −1.0 −2.0 1.0 5 0 5 0 0 0 ︵∽N︶ゝ 5000 10000 t Fig・1 Timeseriesofvatthenoise−addedpoint(x=0)and afterpropagation(x=25)withJ=0.2(a),0.4(b)andO.6(c) (theFitzHughーNagumomodel)・

Fig・4 Nagumo’s active transmissionline(a)and an analog CircuitforoneelementwithOPamps(b), 0 0 0 0 0 ︵∪ 5 4 0 nY O O O O O O O O 3 2 1 u双≦

duringlOOsecareplottedagainstthenoisestrengtho・

inFig.6.ThemeanandS.D.oftheinterspikeintervals

afterpropagationtakeminimumvaluesattheinterme−

diatevalueso.;¥2・0山2.5Vofthenoisestrength.

Itwasshownthatthemeanandstandarddeviation

Oftheinterspikeintervalsofthepropagatingspikesgen−

eratedbypointstimulusinexcitablemediatakemini−

mumvaluesattheintermediatelevelsofnoisestrength.

ThepreviousStudiesoncoherentresonanCeinthesin−

gle element of excitable media have shown that the

meaninterspikeintervaldecreasesmonotonicallyasthe

noisestrengthincreases[4=8],[9].Theresultsobtained

inthisstudyshowtheexistenceofstrongercoherence

thanthoseinthesestudies,andarealsodifferentfrom

thepreviousresultsoncoherenceresonanceandnoise−

SuStainedpatternsinspatia11ydistributedexcitableme−

dia閏,[10],[11],[19].

The mechanism of this coherence resonance is a

COmbinationofthedecreaseintheinterspikeintervals

Ofthe spikes duetothe noise,Whichis attributedto

themeanfirstpassagetimefortheOrnstein−Uhlenbeck

0.0 0.6 0.8 0.0 0.2 0.4 (7 0.6 0.8 Fig.2 Meanandstandarddeviation(S.D.)oflOOOOinterspike intervalsofthepropagatingspikesatニ℃=25vs.noisestrengtho・ (theFitzHugh−Nagumomodel)・

inthefirstelement,bywhichspikesaregeneratedand

propagateintheline.Figure5showsthetimeseriesof

thevoltageinthelstand20thelementwiththenoise

StrengthJ=0.5,2.O and3.5V.The mean and S.D.

Oftheinterspikeintervalsinthe20thelementrecorded

(3)

LETTER 1595 SPikeswiththeintervaltothepreviousonesmallerthan therefractoryperiodfailtopropagate.Thisfailureof thespikepropagationmakesthemeanandS.D.ofthe interspikeintervalslarger.Itisdi侃culttoseethespike failureduetotherefractoryperiodinthesingleelement modelsincethenoiseglVeSlargevariationsatthestim− uluspoint,aSCanbeseeninFigs.1and5.Thewave Shaping actionaccordingtospikepropagationinspa− tiallydistributedexcitablemediamakesthiscoherence resonance clear. Refbrences [1]L.Gammaitoni,P.H云.nggi,P.Jung,and F.Marchesoni, “Stochastic resonance,”Rev.Mod.Phys.,VOl.70,nO.1, pp.223287,Jan,1998. [2]H.Gang,T.Ditzinger,C.Z.Ning,andH.Haken,“Stochas− ticresonancewithout,eXternalperiodic force,”Phys.Rev. Lett.,VOl.71,nO.6,pp,807p810,Aug.1993.W.J.Rappeland S・H・Strogatz,”Stochasticresonanceinanautonomoussys− temwith anonuniformlimitcycle,”Phys.Rev.E,VOl.50, no.4,pp.3249L3250,Oct,.1994.L.IandJ.M.Liu,“Experq 主ment,alobservationofstochasticresonancelikebehaviorof autonomousmOtioninweaklyionizedrfmagnetoplasmas,” Phys.Rev.Lett.,VOl.74,nO.16,Pp.3161−3164,April1995. A.Neiman,P.Ⅰ.Saparin,and L.Stone,“Coherence reso−

nance at noISy preCurSOrS Ofbifurca七ionsin nonlinear dy−

namicalsystems,”Phys.Rev.E,VOl.56,nO.1,pp.270−273, July1997.

[3]A.S.Pikovsky andJ.Kurths,“Coherence resonancein a noise−driven excitable system,”Phys.Rev.Lett.,VOl.78,

no.5,pp.775−778,Feb.1997.

[4]A.Longtin,“Autonomousstochasticresonanceinbursting

neurons,”Phys.Rev.E,VOl.55,nO.1,pp.868876,Jan.1997. [5]J・M・Casado,“Noise−inducedcoherenceinanexcitablesys−

tem,”Phys.Lett.A,VOl.235,pp.489−492,Nov.1997. [6]S.G.Lee,A.Neiman,and S.Kim,“Coherence resonance

in aHodgkin−Huxleyneuron,”Phys.Rev.E,VOl.57,nO.3,

pp.3292−3297,March1998.

[7]D.E.Postnov,S.K.Han,T.G.Yim,and O.V.Sosnovt−

SeVa,“Experimentalobservationofcoherence resonancein

CaSCaded excitable systems,”Phys.Rev.E,VOl.59,nO.4,

Pp.R3791TR3794,April1999.S.K.Han,T.G.Yim,D.E. Postnov,andO.V.Sosnovtseva,“Interactingcoherenceres− OnanCeOSCillators,”Phys.Rev.Lett.,VOl・83,nO.9,pp.1771p 1774,Aug.1999. [8]J.R Pradines,G.V.Osipov,andJ.J.Collins,“Coherence resonanceinexcitableandoscillatorysystems:Theessen− tialroleofslow andfast dynamics,”Phys.Rev.E,VOl.60, no.6,pp.6407−6410,Dec.1999. [9]B−LindnerandLSchimansky−Geier,“Analyticalapproach tothestochasticFitzHugh−Nagumosystem andcoherence resonance,”Phys.Rev.E,VOl.60,nO.6,pp.72707276,Dec. 1999. [10]Y.Wang,D.T.W.Chik,andZ.D.Wang,“Coherenceres− OnanCe and noise−inducedsynchronizationinglobal1ycou−

pledHodgkin−Huxleyneurons,”Phys.Rev.E,VOl.61,nO.1,

pp.740−746,Jan.2000.

[11]B・HuandC,Zhou,”Phasesynchronizationincouplednon− identicalexcitable systems and array−enhanced coherence

resonance,”Phys.Rev.E,VOl.61,nO.2,pp.RlOOl−RlOO4, Fbb.2000. [12]G.Giacomelli,M.Giudici,S.Ba11e,andJ.R.Tredicce, “Experimentalevidenceofcoherenceresonanceinanop七i− Calsystem,”Phys.Rev.Lett.,VOl.84,nO.15,pp.3298−3301, (a)打=0.5V 0 0 00 ︵>︶︵こ> ︵U O ハリ O OOOO T−︵>〓○号T 0 00 800 1000 200 40号(mse3 (b)J=2.0V 0 0 0 ∩︶ ︵>︶︵こゝ ■ ︵>︶︵OZ︶ゝ ■ 0 0 ︵U O O O O O 0 200 40号(mse300 800 1000 (c)打=3.5V 0 0 0 0 0 0︵UOOO ︵>︶︵モ Tl ︵>︶︵○号T 0 200 400 600 800 1000 f(msec) Fig・5 Timeseriesofthevoltageinthenoise−added(1st)ele− mentand20thelementwitho.=0・5V(a),2.OV(b)and3.5V (c)(Nagumo’sactiveline). ∩︶ ∩︶ 0 0 0 0 5 0 5 0 54433 ︵。心S∈︶ u票≦ 1.0 2.O J(∨) 3.0 4.0 0.0 0 0 ∩︶ 0 0 0 5 0 5 0 32211 ︵0心S∈︶.凸.S 0.0 1.0 2.O J(∨) 3.0 4.0 Fig.6 MeanandS.D.oftheinterspikeintervalsofthepropa− gatingspikesinthe20thelementvs・nOisestrengtho.(Nagumo’s activeline).

processandadouble−Wellpotentialsystem[20】,andthe

failureofspikepropagationduetotherefractoryperiod

[17]・Thatis,aSthenoisestrengthincreases,theinter−

Spikeintervals at thestimulus point decrease,but the

(4)

IEICETRANS.FUNDAMENTALS,VOL.E84LA,NO.6JUNE2001

1596

Scott,Neurophysics,JohnWiley&Sons,NewYork,1977・ [18]J.P.Keener,“Analog circuitryfor the vander Poland

FitzHugh−Nagumoequations,”IEEETrans・Syst・,Man・&

Cybern.,VOl.SMC−13,nO.5,pp,1010−1014,Sept./Oct.1983. [19】P.Jung,A・Cornell−Be11,F・Moss,S・Kadar,J・Wang,and

K.Showalter,“Noisesustainedwavesinsubexcitable me− dia:From chemicalwaves to brain waves,”Chaos,VOl.8,

no.3,pp.567LT575,Sept.1998.H.Hempel,L・Schimansky− Geier,andJ.Garcfa−Ojalvo,“Noise−SuStained pulsating

patterns and globalosci11ationsin$ubexcitable media,” Phys.Rev.Lett.,VOl.82,nO.18,pp・3713−3716,May1999・ [20]N.S.Goeland N・Richter−Dyn,Stochastic Modelsin Bi−

ology,Academic Press,New York,1974・H・C・Tuckwell, St,OChasticProcessesintheNeurosciences,SIAM,Philadel− phia,1989.P.H嵐nggi,P.Talkner,and M・Borkovec, “Reaction−rate theory,”Rev.Mod.Phys・,VOl・62,nO.2, pp.251−341,April1990, Apri12000. [13】R・FitzH11gh,“Impulsesandphysiologicalstatesintheoret− icalmodelsofnervemembrane,”Biophys.].,VOl.1,pp.445− 466,1961. [14]J・Nagm,S・Arimoto,andS・Yoshizawa,“Anactivepulse transmlSS10nlinesimulatingnerveaxon,”Proc.IRE,VOl.50, no.10,pP.2061−2070,Oct.1962. [15]E・SkaugenandLWall@e,“Firingbehaviourinastochas− ticnervemembranemodelbasedupontheHodgkin−Huxley equations,”Acta Physiol.Scand.,VOl.107,Pp・343−363, 1979.

[16]A.F.Strassberg and L.].DeFelice,“Limi七ations of the Hodgkin−Huxleyformalism:Effectsofsinglechannelkinet− icsontransmembranevoltagedynamics,”NeuralCompu− tation,VOl.5,pp.843−855,1993.

[17]R.FitzHugh,“Mathematicalmodels of excitation and PrOPagationinnerve,”in BiologlCalEngineering,ed・H・P・

Schwan,pp.1−85,McGraw−Hill,New York,1969.A.C.

OLIVE 香川大学学術情報リポジトリ

参照

関連したドキュメント

Analogs of this theorem were proved by Roitberg for nonregular elliptic boundary- value problems and for general elliptic systems of differential equations, the mod- ified scale of

Then it follows immediately from a suitable version of “Hensel’s Lemma” [cf., e.g., the argument of [4], Lemma 2.1] that S may be obtained, as the notation suggests, as the m A

Spikes, Some results on the on the asymptotic behavior of the solutions of a second order nonlinear neutral delay differential equations, Contemporary Mathematics 129 (1992),

Correspondingly, the limiting sequence of metric spaces has a surpris- ingly simple description as a collection of random real trees (given below) in which certain pairs of

[Mag3] , Painlev´ e-type differential equations for the recurrence coefficients of semi- classical orthogonal polynomials, J. Zaslavsky , Asymptotic expansions of ratios of

[5] Fonda A., Mawhin J., Quadratic forms, weighted eigenfunctions and boundary value prob- lems for nonlinear second order ordinary differential equations, Proc.. Edinburgh 112A

[r]

As a result of the Time Transient Response Analysis utilizing the Design Basis Ground Motion (Ss), the shear strain generated in the seismic wall that remained on and below the