Table Of ContentDynamicalmechanism ofanticipating synchronization inexcitablesystems
Marzena Ciszak,1 Francesco Marino,2 Rau´l Toral,1,2 and Salvador Balle1,2
1Departament de F´ısica, Universitat de les Illes Balears
2Instituto Mediterra´neo de Estudios Avanzados (IMEDEA), CSIC-UIB,
Ed. Mateu Orfila, Campus UIB, 07122 Palma de Mallorca, Spain
(Dated:February2,2008)
We analyze the phenomenon of anticipating synchronization of two excitable systems with unidirectional
delayed coupling whicharesubject tothesameexternal forcing. Wedemonstrate fordifferent paradigms of
excitablesystemthat,duetothecoupling, theexcitabilitythresholdfortheslavesystemisalwayslowerthan
that for the master. As aconsequence the twosystems respond toacommon external forcing withdifferent
responsetimes. Thisallowstoexplaininasimplewaythemechanismbehindthephenomenonofanticipating
synchronization.
4
0
0
2 The synchronization of nonlinear dynamical systems is a andtheyoftenoperateinfeedbackregimeinanoisyenviron-
phenomenoncommontomanyfieldsofsciencerangingfrom ment,thestudyofthedelayedcouplingeffectsinapresence
n
biologytophysics[1],andithasbeenanactiveresearchsub- ofnoiseiscertainlyofwideconcern.
a
J ject since the work by Huygensin 1665. Recently, the syn-
Theanticipatingsynchronizationregimehasbeenoftende-
2 chronization of chaotic systems in a unidirectional coupling scribed as a rather counterintuitive phenomenon because of
1 configurationhasattractedagreatinterestduetoitspotential
the possibility of the slave system anticipating the unpre-
applications to secure communication systems [2]. Particu- dictable evolution of the master [3, 5, 7]. The aim of this
1
larattentionhasbeenpayedtotheso-calledanticipatingsyn-
v paperistoprovideasimpleclearphysicalmechanismforthis
chronizationregime,anideafirstproposedbyVossin[3]. He
6 regimeindelayedcoupledexcitablesystems,showingthatthe
7 showedthat,insomeparameterregions,twoidenticalchaotic anticipationoftheslaveisduetoareductionofitsexcitability
1 systems can be synchronized by unidirectionaldelayed cou-
threshold induced by the delayed coupling term. As a con-
1 plinginsuchamannerthatthe”slave”(thesystemwithcou-
sequence,themasterandtheslave respondtoacommonex-
0
pling) anticipates the ”master” (the one without coupling).
4 ternal forcing with different response times. The proposed
Morespecifically,thecouplingschemeproposedin[3]forthe
0 dynamicalpictureallowsustoexplainallthegeneralfeatures
/ dynamicsofthemaster,x(t),andslave,y(t)isthefollowing: of the phenomenonas well as to determine in a natural way
t
a themaximumpermittedanticipationtime.Theresultsaresus-
m x˙ = F(x) (1)
tainedbynumericalintegrationofthedynamicalequationsas
- y˙ = F(y)+K(x−yτ) (2) wellasbysimpleanalyticalcalculations.
d
n where yτ ≡ y(t − τ). For appropriate values of the delay Adynamicalsystemcommonlyusedtostudyexcitablebe-
o
haviorisAdler’sequation,[12]
c timeτ andcouplingstrengthK,thebasicresultisthaty(t)≈
: x(t+τ),i.e.theslave“anticipates”byanamountτ theoutput
v
i ofthemaster. x˙ =µ−cosx, (3)
X
Thisregimeanditsstabilityhasbeentheoreticallystudied
r inseveralsystems,fromthesimplestonesdescribedbylinear
a where x is an angular variable (modulo 2π) and µ the con-
differential equations and maps where the mathematical de-
trol parameter. For |µ| < 1, there are two fixed points at
tails can be fully worked out[4, 5], to the more complicated
onessuchassemiconductorlasers[6]operatinginthechaotic x± =±arccosµ,onebeingastablefocus(x−)andtheother
(x ) an unstablesaddlepoint. If |µ| > 1, thereare nofixed
regime. Experimentalevidence of anticipating synchroniza- +
points,andtheflowconsistsinanoscillationofthevariablex.
tionhasbeenshowninChuacircuits[7]andinsemiconductor
ThislimitcycledevelopsthroughanAndronovbifurcationat
laserswithopticalfeedback[8].
µ = ±1[13,14],wherethetwofixedpointscollideandan-
This same phenomenonhas recently been shown to occur c
nihilate. For|µ| < 1,thesystemdisplaysexcitablebehavior:
also when the dynamics, instead of chaotic, is excitable. In
ifwekickthesystemoutofitsstablestatewithalargeenough
refs. [9] the effects of unidirectional delayed coupling be-
perturbation,thetrajectorywillreturntotheinitialstate(mod-
tween two identical excitable systems was studied for both
ulo2π)throughanorbitthatcloselyfollowstheheteroclinic
the FitzHugh-Nagumo [10] and Hodgkin-Huxley [11] mod-
connectionofthesaddleandthenode. Duringthisorbit,the
els. Itwasshownthat,whenbothsystemsareexcitedbythe
systemisbarelysensitivetoexternalperturbationsofmoder-
same noise, and for a certain range of coupling parameters,
ateamplitude.
therandomlydistributedpulsesofthemasterareprecededby
those of the slave. This allowsfor predictingthe occurrence Inordertostudyanticipatingsynchronization,weconsider
ofexcitablepulsesinthemaster. Sincemanybiologicalsys- twoidenticalAdler’ssystemswithdelayedunidirectionalcou-
tems (as neurons and heart cells) exhibit excitable behavior pling underthe effectof an externalperturbationI(t) acting
2
perturbation amplitude p . Note that below the excitability
0
threshold,p <2arccos(µ)(equivalentlyµ<cos(p /2)),t
0 0 r
doesnotexist.Forµ>cos(p /2)theresponsetimet isade-
0 r
creasingfunctionofµwhichapproacheszeroasµ→1. This
resultshowsthatthe responsetime ofan Adlersystem to an
above-thresholdexternalperturbationprogressivelydecreases
as the Andronov bifurcation point (|µ| = 1) is approached,
inagreementwiththenumericalresultshowninFig. 2(right
panel).
FIG.1: Timeseriesofthemastersystemx(solidline)andslavesys-
temy(dashedline)subjectedtowhiteGaussiannoiseofzeromean
andcorrelationsh(ξ(t)ξ(t′)i = Dδ(t−t′),obtainedbynumerical
simulationofeqs.(4,5).Otherparametersare:µ=0.95,K =0.01
τ =1.ThenoiseintensityisD=0.017.
simultaneouslyonbothsystems,
x˙ = µ−cos(x)+I(t) (4)
y˙ = µ−cos(y)+K(x−yτ)+I(t) (5) FIG.2: Leftpanel: Responsetimetr versusµfortheAdlersystem
xperturbedbyp0δ(t)withp0 = 2fromEq. 6. Rightpanel: time
seriesforx(t)forµ=0.95(solidline)andµ=0.97(dashedline).
WhenI(t) = ξ(t)iszero-meanGaussiannoise,anticipat- Bothsystemshavebeenperturbedatt0 =10byapulseofconstant
ingsynchronizationoccursasshowninFig. 1,whereweplot amplitudep = 1.7andduration∆t = 0.4. Notethat,inagreement
the master and slave outputsfor a particularvalue of K and withtheleftpanel,thesystemwiththelargervalueofµpulsesbefore
τ. Notethattheslavesystemanticipatesthefiringofapulse theonewiththesmallervalue.
in themaster bya time intervalapproximatelyequalto τ. If
The fact that the response time decreases with lower ex-
we increasethe couplingconstantK orthe delaytime τ be-
citability threshold, and that in the coupledsystem the slave
yond some values, anticipating synchronizationis degraded,
can emit pulses that are not followed by a pulse in the mas-
i.e.,theslavesystemcanemitpulseswhichdonothaveacor-
ter,suggestthatthemechanismforanticipationinthemaster-
respondingpulseinthemaster’soutput,althoughthereverse
slave configuration is that the slave has a lower excitability
caseneveroccurs. UponfurtherincreasingK orτ,theantic-
thresholdthanthemaster. Thisissupportedbythefollowing
ipationphenomenondisappears. Theresultsareanalogousto
thoseobtainedin[9]fortheFitzHugh-Nagumomodel. qualitative argument: Imagine that at t = t0 both systems,
− −
Inordertounderstandthemechanismoftheobservedphe- masterandslave, are inthe reststate x(t0) = y(t0) = x−.
nomenon,weanalyzethebehaviorofthemasteraloneunder Theeffectoftheperturbationchangesbothvaluestox(t+0)=
the effectof a single perturbationI(t) = p0δ(t−t0) acting x(t−0) + p0, y(t+0) = y(t−0) + p0. Due to the coupling,
at a certain time t . The effect of this perturbation appears the slave can be considered to have at this time an effective
0
onlyasadiscontinuityofthe,say,x(t)variableattimet0 as µeff(t0) = µ+K[x(t+0)−y(t+0 −τ)] = µ+Kp0. Since
x(t+0) = x(t−0)+p0. The condition for the perturbation to µeff(t) > µalsoforalltimestsuchthatt0 ≤ t < t0+τ,the
belargerthantheexcitabilitythreshold,isthatx(t+) > x . excitability threshold of the slave has been reduced and the
0 +
responsetimedecreases.
¿From now on, we set the initial condition to be in the rest
state,x(t−0)=x−,suchtheminimumvaluefortheamplitude To givea more rigorousevidencefor this explanation,we
in order to excite a pulse is p > 2arccosµ and the system considernowtwocoupledsystems,Eqs.(4-5),inthepresence
0
ofasingleperturbationwhichwechoosetobeapulseofcon-
develops a pulse after a certain response time t . This time
r
can be precisely defined as the time it takes x(t) to reach a stantamplitudepandduration∆t actingattimet0 inwhich
− −
givenreferencevalue,e.g. xr = π/2. FromEq. (2)wehave bothsystemsareinthereststatex(t0) = y(t0) = x−. The
t = π/2 dx whichyields resultsarereportedinfig3.
r Rx(t+0) µ−cosx For a sufficiently large perturbation, the master and the
slaverespondwithanexcitablespikeandtheslavepulsean-
1 (1−b)(1+b−1tanx(t+0)) ticipates the master pulse (fig. 3a). For small perturbation
t = ln 2 (6)
r p1−µ2 (1+b)(1−b−1tanx(t2+0)) apmropploitrutidoenanlolyptuolstehseaarpepglieendersattiemdualunsd(bfiogth4csy).stHemowserevsepr,onand
intermediateamplitudeof the perturbationtriggersthe emis-
whereb =q11+−µµ. InFig. 2(leftpanel)weplottheresponse sionofanexcitablepulsebytheslavesystemwhilethemas-
timeasafunctionoftheparameterµforagivenvalueofthe terrespondslinearly(fig3b). Thisconfirmsaloweringofthe
3
FIG. 4: The ratio between the slave and the master excitability
thresholdasafunctionof K forτ1 = 0.05, τ2 = 0.2, τ3 = 0.35
andτ4 = 0.5. Consideredsystemhaveparameterµ = 0.95. Per-
turbationisappliedattimet0 = 10withmagnitudep = 1.635and
duration ∆t = 0.4. The dashed line corresponds to the constant
excitabilitythresholdofthemaster.
FIG.3: Responseofthemaster(solidline)andslave(dashedline)
forthreedifferentamplitudesofthesingularperturbationofduration
the anticipation time is approximately equal to τ. However,
∆t = 0.4 at time t0 = 10: (a) p = 1.7, (b) p = 1.65 and (c)
p=1.61.Otherparametersareµ=0.95,τ =5andK =0.01. when τ ≫ tr, the anticipation time greatly differs from the
delaytime,suchthattheslaveanticipatesthemasterbyatime
intervalalwayslowerthant (5c). Thisisa reasonablelimit
r
excitabilitythresholdoftheslave ascomparedto themaster, totheanticipationtime:thepulsecannotanticipatethepertur-
whichissystematicallyfoundforallcouplingparametersthat bationwhichcreatedit. Inotherwords,masterandslaveare
yieldanticipatingsynchronization. InFig.4weplottheratio both”slaves” oftheexternalperturbation,althoughthepres-
Rbetweentheminimumamplitudeoftheperturbationwhich ence of the master signal into the coupling term contributes
generatesanexcitablepulseintheslaveandtheminimumam- to lower the excitabilitythresholdof the slave leadingto the
plitudethatgeneratesapulsein themaster. Asshowninthe anticipationphenomenon.
figures, the effect of this particular coupling scheme on the
slavesystemistoloweritsexcitabilitythresholdinsuchway
that the difference between the response time of the master
andtheslavetoanexternalperturbationequalsapproximately
thedelayinthecouplingterm,τ. Itisworthnotingthatwhen
K or τ tend to zero, not surprisingly the thresholds for the
slave and the master tend to be equal, while for largevalues
ofτ thedifferencebetweenthetwothresholdsisverylarge.
Clearly, the same reasoning can be followed if the pertur-
bationappliedtobothsystemsisawhitenoiseprocess. This
allowsustoexplainwhytheerroneoussynchronizationevents
correspond to the slave system firing a pulse that is not fol-
lowedbyapulseinthemaster:foraparticularnoiselevelthe
master response is proportionalto the perturbationwhile the
slave emits an excitablepulse. By increasing the noise level
bothmasterandslaveemitexcitablepulses,eachpulseofthe
slavebeinganticipatedrespecttothatofthemaster.
Since,aswehaveshown,masterandslavesystemsrespond
toexternalperturbationswithdifferentresponsetimes,aques- FIG. 5: Two coupled systems (master and slave) with a coupling
tionwhicharisesiswhetheritispossibletochosetheparam- parameterK = 0.01anddelaytime(a)τ = 1, (b)τ = 5and(c)
τ = 50. Both systems have µ = 0.95 and are perturbed at time
eterssuchthattheanticipationtimeisarbitrarilylarge,inpar-
t0=60withapulseofmagnitudep=1.7andduration∆t=0.4.
ticular,largerthanthemasterresponsetime,τ > t , aresult
r
thatwouldviolatethecausalityprinciple. Inordertoanswer
thisquestion,weplotinFig.(5)theresultsofintegratingEqs.
4,5undertheeffectsofasingleperturbationforthreedifferent Inordertoassessthegeneralityofourhypothesis,wehave
valuesoftheparameterτ. Whenτ < t (5a)orτ ≈ t (5b), alsoconsideredtwodelayedcoupledFitzHugh-Nagumosys-
r r
4
tems:
x3
(x˙ ,x˙ ) = (x +x − 1,ǫ(a−x )) (7)
1 2 2 1 1
3
y3
(y˙ ,y˙ ) = (y +y − 1 +K(x −yτ),ǫ(a−y )) (8)
1 2 2 1 3 1 1 1
In the excitable regime, which occurs when |a| > 1, the
system possesses a single steady state. As the critical value
|a | = 1 is approached,the excitabilitythreshold is lowered
c
[15]. In this sense, the control parameter a plays the same
roleastheparameterµinAdler’sequation. Infact,we have
checkedthatalsointhiscasetheresponsetimeofthesystem
toanexternalperturbationdecreasesasthecriticalvaluea is
c
approached(seeFig.6).
FIG.7: Responseofthemaster(x1,solidline)andslave(y1,dashed
line)fortwocoupledFitzHugh-Nagumosystemswitha=1.01,ǫ=
0.09,τ = 4,K = 0.1,afterperturbationatt0 = 200byapulseof
amplitudepandduration∆t=1.Forlargeamplitude,p=0.4,case
(a),bothsystemspulsewhereasforthesmalleramplitude,p= 0.3,
thereisonlypulseintheslavevariable.
FEDER through projects TIC2002-04255-C04, BFM2000-
1108 and BFM2001-0341-C02-01. S.B. acknowledges fi-
nancialsupportfromMECthroughsabbaticalgrantPR2002-
FIG. 6: Time series for the variable x1 of the FitzHugh-Nagumo 0329.
system for a = 1.01 (dashed line) and a = 1.08 (solid line). In
bothcases itisǫ = 0.09. Asindicated bytheverticaldotted line,
thesystemisperturbedatatimet0 = 200byapulseofamplitude
p=0.4andduration∆t=1.Notethattheresponsetimedecreases
withincreasinga.
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WeacknowledgefinancialsupportfromMCYT(Spain)and E,68,036209(2003).
