1、AttractionofSpiralWavesbyLocalizedInhomogeneitieswithSmall-WorldConnectioninExcitableMediaWangXiaonanDepartmentofphysics,PekingUniversity,Beiing100871,P.R.ChinaAbstractThetrappingoruntrappingofspiralwavesinatwo-dimensionalhomogeneousexcitablemediawithlocalsmall-worldconnectionsisstudiedbynumericalsi
2、mulation.Thespiralwaveismeanderingwithcompletelylocalregularconnection.Whenchangingaregionofthesystemfromregularconnectionstosmall-worldconnections,thetipofthespiralwavesisobservedtodrifttowardsthesmall-worldregionwheretheaveragepathlengthisshorterthanthatoftheregularconnectionregion.Theanchoringphe
3、nomenonalsooccureswhenincreasingthediffusioncoefficientintheregionandreservingtheregularconnection.Theaboveresultscouldbeexplainedbythechangingofthespeedofthetipininhomogeneousmedia.1.IntroductionSpiralwavesarecharacteristicstructuresofexcitablemediathathavebeenobservedinsystemsasdifferentastheBelou
4、sov-Zhabotinskychemicalreaction1,aggregatingcoloniesofslimemold2,andhearttissuewheretheyaresuspectedtoplayanessentialroleincardiacarrhythmiaandfibrillation3.Suddencardiacdeathresultingfromventricularfibrillationisgeneratedfromthefragmentingorbreakupofspiralwaves4.Spiralwavesarepronetoavarietyofinsta
5、bilities,oneofwhichismeander,andtheycanbemadetodriftandcontrolledbyexternalinfluences5orlocalizedinhomogeneitiesofdefects6J.Fortheexcitablemediacanberegardedasanetworkconsistingofanumberofsitesconnectedwithcertaintopology,wecandrawintoanotherkindoflocalizedinhomogeneitiesbychangingthetopologyofthene
6、twork.Aftertheconceptofsmall-worldconnectionswasproposedbyWattsandStrogatz,ithasquicklyattractedmuchattentionbecausethiskindofconnectionexistscommenlyinthereallife,suchasinsocialsystem7,neuralnetwork8Jandepidemicproblem9.Meanwhile,thelittlechangeofthenetworkcanessentiallychangethefeaturesofthegivenm
7、edia,andplaysaveryimportantroleindeterminingthesystembehaviors.Whythesinusinheartalwayscontrolstherhythmoftheheart?Isitbecausethecharacterandthestructureofsinuscellsaredifferentfromothercardiacmusclecells?Couldasmall-worldnetworkdescribethecharacterofsinusaswellastheabovesystems?Toanswerthesequestio
8、ns,inourwork,Wechangedthewildlyusedregularnetworkinspiralwavestudytosmall-worldnetworkinpartofthesystemtoinvestigatetheeffectsofit.2.Theeffectofsmall-worldnetworkWefocusonmodelingwiththetwo-variablereaction-diffusionmodel:FitzHugh-Nagumomodel110withlocalnearest-neighborcouplingsinaregionwhosesizeisN
9、i*Na：=(a-uij)(uij-l)uij-vij+DwV2uij(1)K+*fj*fJ*9J*911Jdvo-=(buij-vij)+DvVvij(2)V%ij12+wu.1+%J+1-4/j).VRijN1/(匕Tj+匕+j+匕A+%+一4Vij)(4)wherei=l,2,.,N,j=l,2,.,N2,3/isadimensionlessfunctiondescribingthetransmenbranepotentialinabiolobicalexcitablecell,VLjisadimensionlessfunctionsimilartoaslaverrecoverypota
10、ssiumcurrentandDuDvisthediffusioncoefficientofthetwovariables.When0a=0,DvDu,l,theequationexpressesaexcitablemediaandcouldbetakenasasimplifiedmodelforcardiactissue.Inthefollowingdiscussionweuseno-fluxboundaryconditionandthesamesystemparametersthatN=N2=256,a=0.1,b=1.0,=0.005,Du=1/3,Dv=O.Inaddition,Wef
11、ixtheVerticalGradientDistributioninitialcondition:0.055(1-)+0.25心%=F、(5)-0.01(N2-)+0.047y、R(6)Inthiscondition,aspiralwavesisformedinthemiddleofthesystem,andwiththeevolutionofitstiptravellingalongasemi-periodicaloutertrolleylineunderregularconnectioninthewholesystem(SeeFigl(a)(b).Itshowsthatthespiral
12、isameanderingspiralwhichisnotastablestateundertheseparametersinuniformmedia.Then,weproduceasmall-worldnetworkinaNs*Ns(NsN)regionintheplacewherethespiralisformedatthebeginning.Withtheprobabilityps(0ps1),wereconnecteveryedgeinthisregionfromoneofitsoriginalvertextoavertexchosenrandomlyinregionC,inwhich
13、duplicatingedgesisforbidden1l(seeFig1(c).Inthisregion,thechangeofconnectionsleadstothechangeofdiffusionmode.Supposingnodeijisconnectedwithknodes,theyarenodexlyl,nodex2y2,.nodexkyk,thenitsdiffusionitemscouldbeapproximatedas:VUij=h(W1v1+“2.y2+,+uxk.yk卜，,j)%JN1/(匕0+匕2,y2+%,W_匕匕J)(8)Thenusingtheparamete
14、randinitialconditionabove,aspiralwavesisformedaswellasundertheregularconnection,butthetip,smotioniseffectedgreatlybythesmall-worldregion.Bydoingextensivesimulations,wefindthatforsomesmall-worldconnection,whilepassthroughthesmall-worldregion,thetipofthespiralwavescouldbeattractedbytheregionandthenrot
15、atesarounditsboundary(seeFig1(d).Thisphenomenonisjustthecaseweareinterestedin,buttheresultissomewhatrandomthattheattractiononlyoccursundersomeparametersrangeandevenwiththesameparametertheattractiondoesn,toccurallthetime.Itisbecausethatthetopologystructureofthelocalsmall-worldnetworkisoccurredrandoml
16、ybycomputerundertheaboverule.Soweneedadditionalparameterstodescribethecharacterofsuchnetworksandtrytofindthemostrelevantfactor.FigLaIlabovepicturesNl=N2=Ns=10(a)p=0regularnetwork,Ns=10,redverticesisinsmall-worldregion,greenisinoutsideregularnetwork.(b)spiralanditsmotion(tiptrack)whenp=0.(c)p=0.1(d)s
17、piralandtipwhenp=0.1Tocharacterizetheattactabilityofthenetworkwithsameparameters,Wedefineaparameterpawhichequalstotheattractiontimedividedbythetotaltimerepeatedinthesamecondition.Inoursimulationthetotaltimeofoneconditionis50.Withourstudywefindthatpaisrelevanttomanyparametersofthesmall-worldnetworksu
18、chasitssizeNs,thesmall-worldcreationprobabilitypsandtheaveragelengthofthesmall-worldnetwork1.First,paincreaseswithNswiththesameps(seeFig2(a).Second,paalsoincreaseswiththesmall-worldcreationprobabilityps.Whenps=0(correspondingtoregularconnection),thetipcouldn,tbeattracted,whileps=1(correspondingtoran
19、domconnection),thetipcouldmostlybeattracted.During0psl,thereisatransitionthattheattractabilityincreasesrapidlyfrom0to1.Definethetransitionpointpc:whenpa=0.5,thevalueofpsisthetransitionpointpc.WithdifferentsizeNs,pcisdifferent,buttheslopeintransitioncourseissimilar(seeFig2(b),WhenweusepcNs2astheabsci
20、ssa,thetransitioncurveswithdifferentNsareclosedtoasuperposition.ThereforepcNs2isaconstantwhichisacharacteristicparameterofattraction.(seeFig2(c).Third,althoughpscouldexplaintheattractioninpart,westillneedotheressentialcharacterstofurtherdescribethecharacterofnetworksuchascomparativeaveragepathlength
21、1.1=L/L(),Listheaveragepathlengthofsmall-worldnetworkinC12andLoistheaveragepathlengthinwhenps=0(equaltotheregularnetworkaveragelength).Becausesmall-worldconnectioncoulddeclinetheaveragepathlength,1isintherangefrom0to1.Fromthenumericalsimulations,wefindthatthesmallerthe1is,theeasierthetipcanbeattract
22、ed.Because1descendswithps,sopadescendswith1.Meanwhilelc(transitionpoint)increaseswithNs(seeFig2(d).Whywethink1isamoreessentialparameterthanps?Wedeemthatdiffusionspeeddecideswheretobethesource.Itiswellknownthat,thediffusionspeediscorrespondto1,thesmallerthe1is,thefasterthediffusionspeedis.Soweguessth
23、atthemainreasonwhythesmall-worldregioniseasiertobethesourceisthatthediffusionspeedisfasterthere!Fig2(a)paincreaseswithNsps=0.075(b)paincreaseswithps,thethreelinesareNs=10,12,14(c)Thecurveofps*Ns2-pa,thethreelineswithdifferentNsseemstobeasuperposition(d)padeclineswiththeaveragelength13.DiscussionWech
24、eckoutthissuspectbyincreasingthediffusioncoefficientDuinregion(useregularconnection).Asweknow,thediffusionspeedwillincreasewiththeincreasmentofthediffusioncoefficientaswellasthefalloftheaveragelength1.ThistimeWealsofindthatthespiralcouldbeattractedwhenDuislargeenough.Dcisthetransitionvaluefromuntrap
25、pedtotrappedandthecurveofDc-RisshownintheFig3.Thisresulttellsusthatourassumptionseemssuitable.Fig3showthetransitionpointofDufromuntrappingtotrappingwithdifferentsizeofthespecialregionInaddition,Wehavefurtherexplanationsbasedonthedispersionrelation,theeikonalrelationofspiralsandthedefinitionofthetipo
26、fspirals.ThedispersionrelationorLutherequation13is:C=c(yf)4kDu(9)Cisthespeedoftheplanewave,c(vf)isaconstant,kisthechemicalreactionspeedconstant,Duisdiffusioncoefficient.Theeikonalrelation13is:N=C-Pwc(10).Nisthenormalspeedofthewavefront,isthecurvatureofthewavefront.AndthetipistheplacewhereN=014.Then,
27、WechangetheDuofacircleregioninthemiddleofthesystem.ItiscommonthatthetipofspiralistraveledalongwithacirclewhoseradiusisRoandthenthenormalspeedofthetipiszero.IfthetiphavetotravelalongacirclewitharadiusofRwhichisbiggerthanR(),wecanfindthatofthetipisreducedinthesenseofboththenumericalsimulationandtheore
28、ticalreasoning,thusNofthetipisbiggerthanzerointhenormalregion.Ontheotherhand,fromequation(9)and(10),wecanseethatNisdescendedwiththeincreaseofDuwiththesame.Thus,ifKiSconsecutiveintheboundaryoftworegionwithdifferentDu,NwillbesmallerthanzerointheregionwhoseDuisbigenough.Therefore,thetipcouldn,tstayboth
29、inthetworegionswhenitpassesthroughthespecialregionofbigDubuttravelalongtheboundaryoftheregioninstead.ThinkthisanalysiscouldexplainwhytheregionofbigDucouldattractthetipofspiral.Insum,wefindthatlocalchangeoftopologystructurecouldchangetheabilitytotrapthetipofspiral.Withsmall-worldcreationprobabilityPs
30、increasingandaveragelength1descending,thediffusionspeedincreasesanditfinallycausestheimprovementofattractability.Sothesmall-worldregioniseasiertobethesourceofspiralwaves.Basedontheabovework,Weneedtocarryoutmoreacademicanalysisandexplanationabouttheattractingmechanismandtrytofindpracticalmethodtotest
31、ifyourresultsbyexperiment.Reference1A.N.ZakinandA.M.Zhabotinsky,Nature225,535(1970)3 J.M.Davidenko,A.M.Pertsov,R.Salomonz,W.Baxter,andJJaIife,Nature(London)335,349(1992)4 J.N.Weiss,A.Garfinkel,H.S.Karagueuzian,Z.Qu,andP.S.Chen,Circulation99,2819(1999);M.L.Riccio,M.L.Koller,andR.EGilmour,Jr,Circ.Res.
32、84,955(1999)5 K.LAgladzeandRDeKepper,J.Phys.Chem96,5239(1992)6 S.Nettesheim5A.vonOertzen,H.H.Rotermund,andGErtl,J.Chern.Phys.98,9977(1993)7 JJ.CollinsandC.C.Chow,Nature(London)393,409(1998)8 JJ.HopfieldandA.V.M.Herz,Proc.NatI.Acad.Sci.U.S.A.92,6655(1995)9 S.A.PanditandR.E.Amritkar,Phys.Rev.E60,R1119
33、1999)10 R.A.FitzHugh,BiophysJ.1,445(1966)11 D.J.WattsandS.H.Strogatz,Nature(London)393,440(1998)12 R.AIbertandA.L.Barabasi,R.M.Phy74,47(2002)13 Q.Ouyang,2000,PatternFormationinReaction-DiffusionSystems,(PekingUniversity,Beijing)14 V.HakimandA.Karma,P.R.E60,5073(1999)致谢真诚的感谢导师欧阳斤贞教授在这一年半的时间里对我的严格要求，
34、悉心教导。欧阳老师严谨的治学科研态度，求实创新的科学精神，敏锐的科学洞察力和渊博的知识，将使我受益终身。同时，感谢“政基金”对我的资助，正是在它的促动下才使我有了亲身参与科学研究的机会。另外，还要感谢王宏利，蒋闽熙，吕莹等老师同学与我在学术上的交流和对我的热情帮助，正是这种学术氛围使我在知识水平上有所提高，并更加热爱目前所从事的工作。作者简介：王晓楠，女，1981年2月22日生于辽宁省鞍山市，2000年从北京人大附中保送至北京大学物理系。在校期间各方面表现优秀。从2002年7月开始在北京大学物理系介观与人工微结构国家重点实验室下属的非线性科学及生物技术实验室工作。本篇文章正准备投稿。感悟与寄语
35、通过一年多亲身参与科学研究工作，在教训与成绩中我深深的体会到，要完成一项出色的科研工作首先要有广泛的背景知识，这样才能站得高望得远，有更强的方向感；其次便要有严谨求实的治学态度，科学来不得半点马虎，只有在严密的思考和逻辑推理中才可能得到完整正确的认识；另外，科学领域也同其他领域一样，欲取得成绩必须依靠坚持不懈的努力，正是勤奋造就天才。一年多的时间转眼便过去了，但其间的收获与体会却将伴随我继续明天的航程。指导教师简介：欧阳顽,男，教授，博士生导师，长江学者。1989年于法国波尔多第一大学获博士学位,后一直从事非线性科学的基础理论与实验研究。主要研究方向是非线性动力学中的斑图自组织行为。近十年来在
36、该领域取得了一系列重大成果，被国际同行公认为斑图动力学领域的实验科学带头人之一。迄今为止在各类科学杂志上共发表论文近四十篇，其中包括自然杂志三篇，科学杂志两篇，物理通讯快报三篇；八次应邀在不同国际会议上作专题报告，其中包括美国物理学会，加拿大化学学会，美国工业与应用数学学会；十余次在各个大学和研究单位作专题讲座。1996年受聘于日本电器公司（NEC）在美国的研究中心，从事生物计算机的研究开发和其他一些生物基因工程问题。1997年在科学上发表了DNA计算机的论文。文章引起了国际新闻界的广泛注意。包括英国新科学家，美国新闻周刊，日本教育电视台在内的数家新闻媒介作了报道。1998年6月到北京大学物理系从事非线性科学与生物芯片技术开发工作。
