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’sactivetransmissionline[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.jpOLIVE 香川大学学術情報リポジトリ
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.oftheinterspikeintervalsafterpropagationtakeminimumvaluesattheinterme−
diatevalueso.;¥2・0山2.5Vofthenoisestrength.
ItwasshownthatthemeanandstandarddeviationOftheinterspikeintervalsofthepropagatingspikesgen−
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
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