We analyze the phenomenon of anticipating synchronization of two excitable
systems with unidirectional delayed coupling which are subject to the same
external forcing. We demonstrate for different paradigms of excitable system
that, due to the coupling, the excitability threshold for the slave system is
always lower than that for the master. As a consequence the two systems respond
to a common external forcing with different response times. This allows to
explain in a simple way the mechanism behind the phenomenon of anticipating
synchronization.Comment: 4 pages including 7 figures. Submitted for publicatio

A. Hodgkin

A. Pikovsky

A. A. Andronov

Francesco Marino

J. Nagumo

Marzena Ciszak

R. FitzHugh

R. Toral

Raúl Toral

Salvador Balle

English

arXiv

4 pages, 7 figures.-- PACS nr.: 05.45.Xt.-- ArXiv pre-print available: http://fr.arxiv.org/abs/cond-mat/0401176.-- PMID: 15447342 [PubMed].We acknowledge financial support from MCYT (Spain) and FEDER through projects no. TIC2002-04255-C04, BFM2000-1108, and BFM2001-0341-C02-01. S. B. acknowledges financial support from MEC through sabbatical grant no. PR2002-0329.Peer reviewe

Ciszak, Marzena

Marino, Francesco

Toral, Raúl

Balle, Salvador

Digital.CSIC

VOLUME 93, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R Sweek ending10 SEPTEMBER 2004Dynamical Mechanism of Anticipating Synchronization in Excitable SystemsMarzena Ciszak,1 Francesco Marino,2 Rau´l Toral,1,2 and Salvador Balle1,21Departament de Fı´sica, Universitat de les Illes Balears, Edificio Mateu Orfila, Campus UIB, 07122 Palma de Mallorca, Spain2Instituto Mediterra´neo de Estudios Avanzados (IMEDEA), CSIC-UIB, C. Miquel Marques 21, 07190, Esporles, Spain*(Received 9 January 2004; published 7 September 2004)114102-1We analyze the phenomenon of anticipating synchronization of two excitable systems with unidirec-tional delayed coupling which are subject to the same external forcing. We demonstrate for differentparadigms of excitable system that, due to the coupling, the excitability threshold for the slave system isalways lower than that for the master. As a consequence the two systems respond to a common externalforcing with different response times. This allows us to explain in a simple way the mechanism behindthe phenomenon of anticipating synchronization in excitable systems.DOI: 10.1103/PhysRevLett.93.114102 PACS numbers: 05.45.XtThe synchronization of nonlinear dynamical systemsis a phenomenon common to many fields of science [1],and it has been an active research subject since the workby Huygens in 1665. Recently, the synchronization ofchaotic systems in a unidirectional coupling configurationhas attracted great interest due to its potential applica-tions to secure communication systems [2]. Particularattention has been paid to the so-called anticipatingsynchronization regime [3], where two identical chaoticsystems can be synchronized by unidirectional delayedcoupling in such a manner that the ‘‘slave’’ (the systemwith coupling) anticipates the ‘‘master’’ (the one withoutcoupling). More specifically, the coupling scheme pro-posed in [3] for the dynamics of the master, xt, andslave, yt is the following:_x  Fx (1)_y  Fy Kxy (2)where x and y are vectors, F is a vector function,  is adelay time, y  yt  and K is a positive definedmatrix. For appropriate values of the delay time  andcoupling strength K, the basic result is that yt  xt, i.e., the slave ‘‘anticipates’’ by an amount  the outputof the master.This regime has been theoretically studied in severalsystems [4–6], and experimentally demonstrated in elec-tronic circuits [7] and chaotic semiconductor lasers [8].This same phenomenon has recently been shown tooccur also when the dynamics, instead of chaotic, isexcitable [9]. It was shown that, when both systems areexcited by the same noise, and for a certain range ofcoupling parameters, the randomly distributed pulses ofthe master are preceded by those of the slave. This allowsfor predicting the occurrence of excitable pulses in themaster. Since many biological systems (as neurons andheart cells) exhibit excitable behavior and often operatein a feedback regime in a noisy environment, the study of0031-9007=04=93(11)=114102(4)$22.50 the delayed coupling effects in a presence of noise iscertainly of wide concern.The anticipating synchronization regime has beenoften described as a rather counterintuitive phenomenonbecause of the possibility of the slave system anticipatingthe evolution of the master [3,5,7]. The aim of this paperis to provide a simple clear physical mechanism for thisregime in delayed coupled excitable systems subject tocommon forcing. When the stimuli trigger excitablepulses, the apparent anticipation of the slave over themaster is due to a reduction of its excitability thresholdinduced by the coupling term. As a consequence, themaster and the slave respond to the common externalforcing with different response times, defined as thetime needed to reach a reference level on the excitablepulse. The proposed dynamical picture allows us to ex-plain all the general features of the phenomenon as wellas to determine in a natural way the maximum permittedanticipation time.A dynamical system commonly used to study excitablebehavior is Adler’s equation for a scalar variable x, [12]_x   cosx; (3)where x is an angular variable (modulo 2) and  thecontrol parameter. For jj< 1, there are two fixed pointsat x	  	 arccos, one being a stable focus (x) and theother (x) an unstable saddle point. If jj> 1, there areno fixed points, and the flow consists in an oscillation of x.This limit cycle develops through an Andronov bifurca-tion at c  	1 [13,14], where the two fixed pointscollide and annihilate. For jj< 1, the system displaysexcitable behavior: after a large enough perturbation, thesystem will recover its initial state (modulo 2) throughan orbit that closely follows the heteroclinic connectionof the saddle and the node. During this orbit, the system isbarely sensitive to external perturbations.In order to study anticipating synchronization, we con-sider two identical Adler’s systems with delayed unidirec-tional coupling under the effect of an externalperturbation It acting simultaneously on both systems,2004 The American Physical Society 114102-1FIG. 1. Time series of the master system x (solid line) andslave system y (dashed line) subjected to white Gaussian noiseof zero-mean and correlations htt0i  Dt t0, ob-tained by numerical simulation of Eqs. (4) and (5). Otherparameters are:   0:95, K  0:01, and   1. The noiseintensity is D  0:017.FIG. 2. (a) Response time tr versus  for the Adler system xperturbed by p0t with p0  2 from Eq. (6). (b) Time seriesfor xt for   0:95 (solid line) and   0:97 (dashed line).Both systems have been perturbed at t0  10 by a pulse ofconstant amplitude p  1:7 and duration t  0:4.FIG. 3. Response of the master (solid line) and slave (dashedline) for three different amplitudes of the singular perturbationof duration t  0:4 at time t0  10: (a) p  1:7, (b) p  1:65,and (c) p  1:61. Other parameters are   0:95,   5, andK  0:01.VOLUME 93, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R Sweek ending10 SEPTEMBER 2004_x   cosx  It (4)_y   cosy  Kx y  It (5)When It  t is zero-mean Gaussian noise, antici-pating synchronization occurs as shown in Fig. 1, wherewe plot the master and slave outputs for a particular valueof K > 0 and . Note that the slave system anticipates thefiring of a pulse in the master by a time interval approxi-mately equal to . If we increase the coupling constant Kor the delay time  beyond some values, anticipatingsynchronization is degraded; i.e., the slave system canemit pulses which do not have a corresponding pulse inthe master’s output, although the reverse case never oc-curs. Upon further increasing K or , the anticipationphenomenon disappears. The results are analogous tothose obtained in [9] for the FitzHugh-Nagumo model[10].In order to understand the mechanism of the observedphenomenon, we first analyze the behavior of the masteralone under the effect of a single perturbation It p0t t0. The effect of this perturbation appears onlyas a discontinuity of the xt variable, xt0   xt0   p0.Assuming xt0   x, an excitable pulse is fired ifxt0 > x, i.e., p0 > 2 arccos. The maximum of thepulse corresponds to xr  =2, which happens at timetr R=2xt0 dxcosx which yieldstr  112p ln2666641 bb1 tanxt0 2  11 bb1 tanxt0 2  1377775 (6)where b 11q. The response time tr is plot in Fig. 2(left panel) as a function of  for a given value of p0 Notethat below the excitability threshold, p0 < 2 arccos, tr114102-2does not exist. For > cosp0=2 tr is a decreasing func-tion of  which approaches zero as ! 1. This showsthat the response time to an above-threshold externalperturbation progressively decreases as the Andronovbifurcation point (jj  1) is approached, in agreementwith the numerical result shown in Fig. 2 (right panel).The decrease of tr as the excitability threshold lowers,and the fact that in the coupled system the slave can emitpulses that are not followed by a pulse in the master,suggest that the mechanism for anticipation in themaster-slave configuration is that the slave has a lowerexcitability threshold than the master. This is supportedby the following qualitative argument: Imagine that att  t0 both systems, master and slave, are in the rest statext0   yt0   x. The effect of the perturbationchanges both values to xt0   xt0   p0, yt0  yt0   p0. Because of the coupling, the slave can beconsidered to have at this time an effective efft0Kxt0 yt0 Kp0. Since efft> also114102-2FIG. 4. The ratio between the slave and the master excitabil-ity threshold as a function of K for 1  0:05, 2  0:2, 3 0:35, and 4  0:5. Considered systems have parameter  0:95. Perturbation is applied at time t0  10 with magnitudep  1:635 and duration t  0:4. The dashed line correspondsto the constant excitability threshold of the master.FIG. 5. Two coupled systems (master and slave) with a cou-pling parameter K  0:01 and delay time (a)   1, (b)   5,and (c)   50. Both systems have   0:95 and are perturbedat time t0  60 with a pulse of magnitude p  1:7 and durationt  0:4.FIG. 6. Time series for the variable x1 of the FitzHugh-Nagumo system for a  1:01 (dashed line) and a  1:08(solid line). In both cases it is   0:09. As indicated by thevertical dotted line, the system is perturbed at a time t0  200by a pulse of amplitude p  0:4 and duration t  1.VOLUME 93, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R Sweek ending10 SEPTEMBER 2004for all times t such that t0  t < t0  , the excitabilitythreshold of the slave has been reduced and the responsetime decreases.To give a more rigorous evidence for this explanation,we consider now two coupled systems, Eqs. (4) and (5),where It is a pulse of constant amplitude p and durationt acting at time t0 in which both systems are in the reststate xt0   yt0   x. For a sufficiently large pertur-bation, the master and the slave respond with an excitablespike and the slave pulse anticipates the master pulse[Fig. 3(a)]. For small perturbations, no pulses are gener-ated and both systems respond proportionally to theapplied stimulus [Fig. 3(c)]. Note that in this case theslave responds later than the master, the retardation beingof the order of . However, an intermediate amplitude ofthe perturbation triggers the emission of an excitablepulse by the slave system while the master respondslinearly [Fig. 3(b)]. This confirms a lowering of theexcitability threshold of the slave as compared to themaster (see Fig. 4), which is systematically found for allcoupling parameters that yield anticipating synchroniza-tion. Therefore the effect of this particular couplingscheme on the slave system is to lower its excitabilitythreshold in such way that the difference between theresponse time of the master and the slave to an externalperturbation equals approximately the delay in the cou-pling term, . It is worth noting that when K or  tendto zero, not surprisingly the thresholds for the slaveand the master tend to be equal, while for large valuesof  the difference between the two thresholds is verylarge. A further confirmation of this mechanism is that,for   0, the slave anticipates the master wheneverslave > master.Clearly, the same reasoning can be followed if theperturbation applied to both systems is a white noiseprocess. This allows us to explain why the erroneoussynchronization events correspond to the slave systemfiring a pulse that is not followed by a pulse in the master:for a particular noise level the master response is propor-114102-3tional to the perturbation while the slave emits an excit-able pulse. By increasing the noise level both master andslave emit excitable pulses, each pulse of the slave antici-pating that of the master.Since, as we have shown, master and slave systemsrespond to external perturbations with different responsetimes, a question which arises is whether it is possible tochose the parameters such that the anticipation time isarbitrarily large, in particular, larger than the masterresponse time,  > tr, a result that would violate thecausality principle. In order to answer this question, weplot in Fig. 5 the results of integrating Eqs. (4) and (5)under the effects of a single perturbation for three differ-ent values of the parameter . When  < tr [Fig. 5(a)] or  tr [Fig. 5(b)], the anticipation time is approximatelyequal to . However, when  tr, the anticipation timegreatly differs from the delay time, such that the slaveanticipates the master by a time interval always lower114102-3FIG. 7. Response of the master (x1, solid line) and slave (y1,dashed line) for two coupled FitzHugh-Nagumo systems witha  1:01,   0:09,   4, and K  0:1, after perturbation att0  200 by a pulse of amplitude p and duration t  1. (a) Forlarge amplitude p  0:4 both systems pulse, whereas (b) forthe smaller amplitude p  0:3 there is only pulse in the slavevariable.VOLUME 93, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R Sweek ending10 SEPTEMBER 2004than tr [Fig. 5(c)]. This is a reasonable limit to theanticipation time: the pulse cannot anticipate the pertur-bation which created it. In other words, master and slaveare both ‘‘slaves’’ of the external perturbation, althoughthe presence of the master signal into the coupling termcontributes to lowering the excitability threshold of theslave leading to the anticipation phenomenon.In order to assess the generality of our hypothesis, wehave also considered two delayed coupled scalarFitzHugh-Nagumo systems: _x1; _x2 x2  x1  x313; a x1; (7) _y1; _y2 y2 y1 y313Kx1 y1;; a y1: (8)In the excitable regime, which occurs when jaj> 1, thesystem possesses a single steady state. As the criticalvalue jacj  1 is approached, the excitability thresholdis lowered [15]. In this sense, the control parameter aplays the same role as the parameter  in Adler’s equa-tion. In fact, we have checked that also in this case theresponse time of the system to an external perturbationdecreases as the critical value ac is approached (seeFig. 6).For the unidirectionally delayed coupled system wefind, as before, that the excitability threshold for the slaveis lower than that of the master (see Fig. 7) and that themaximum anticipation time is limited by the responsetime of the master. The same phenomenology has alsobeen observed for two delayed coupled Hodgkin-Huxleysystems [11].114102-4The ubiquity of this effect is, in our opinion, an in-dication that the lowering of the excitability threshold ofthe slave in a delayed coupling scheme is a generalmechanism for anticipating synchronization in excitablesystems. This mechanism allows us to explain the phe-nomenon in such dynamical systems and it evidences itscausality: the master and slave systems follow the appliedexternal perturbations, although the response time of theslave system is shorter due to the effects of the coupling.Moreover, we have shown that the anticipation time islimited by the response time of the master system. Therelevance of this type of mechanism for the synchroni-zation of coupled chaotic systems is an open question thatwill be studied in the near future.We acknowledge financial support from MCYT (Spain)and FEDER through projects no. TIC2002-04255-C04,BFM2000-1108, and BFM2001-0341-C02-01. S. B. ac-knowledges financial support from MEC through sab-batical grant no. PR2002-0329.*URL: http://www.imedea.uib.es[1] A. Pikovsky, M. Rosemblum, and J. Kurths,Synchronization: A universal concept in nonlinear sci-ences, (Cambridge University Press, Cambridge,England, 2001).[2] L. Pecora, T. Carrol, G. Johnson, and D. 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Dynamical mechanism of anticipating synchronization in excitable systems

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Dynamical Mechanism of Anticipating Synchronization in Excitable Systems

Dynamical mechanism of anticipating synchronization in excitable systems

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