We consider a system of globally coupled active rotators near the excitable regime. The system displays a transition to a state of collective firing induced by disorder. We show that this transition is found generically for any diversity distribution with well defined moments. Singularly, for the Lorentzian distribution (widely used in Kuramoto-like systems) the transition is not present. This warns about the use of Lorentzian distributions to understand the generic properties of coupled oscillators.We acknowledge ﬁnancial support by the MICINN (Spain) and FEDER (EU) through

project FIS2007-60327. L.F.L. is supported by the

JAEPredoc program of CSIC.Peer reviewe

Lafuerza, Luis F.

Colet, Pere

Toral, Raúl

We consider a system of globally coupled active rotators near the excitable regime. The system displays a transition to a state of collective firing induced by disorder. We show that this transition is found generically for any diversity distribution with well defined moments. Singularly, for the Lorentzian distribution (widely used in Kuramoto-like systems) the transition is not present. This warns about the use of Lorentzian distributions to understand the generic properties of coupled oscillators.We acknowledge ﬁnancial support by the MICINN (Spain) and FEDER (EU) through
project FIS2007-60327. L.F.L. is supported by the
JAEPredoc program of CSIC.Peer reviewe

Digital.CSIC

Nonuniversal Results Induced by Diversity Distribution in Coupled Excitable SystemsLuis F. Lafuerza, Pere Colet, and Raul ToralIFISC, Instituto de Fı´sica Interdisciplinar y Sistemas Complejos, CSIC-UIB, Campus UIB, E-07122 Palma de Mallorca, Spain(Received 6 May 2010; published 20 August 2010)We consider a system of globally coupled active rotators near the excitable regime. The system displaysa transition to a state of collective firing induced by disorder. We show that this transition is foundgenerically for any diversity distribution with well-defined moments. Singularly, for the Lorentziandistribution (widely used in Kuramoto-like systems) the transition is not present. This warns about the useof Lorentzian distributions to understand the generic properties of coupled oscillators.DOI: 10.1103/PhysRevLett.105.084101 PACS numbers: 05.45.Xt, 02.50.rSynchronization phenomena play a prominent role inmany branches of science [1]. They have been analyzed interms of phase models which successfully describe sys-tems of weakly coupled limit cycle oscillators. In particu-lar, the Kuramoto model [2] has become a paradigm for thestudy of synchronization (for reviews, see [1,3,4]). It showshow synchronized behavior can appear when the competi-tive effects of coupling and diversity among the individualunits are present.Among other possible sources, diversity in the oscilla-tors is usually introduced by taking their natural frequen-cies from a probability distribution. Although, on generalgrounds (central limit theorem), this distribution should bewell approximated by a Gaussian form, theoretical studiesusually consider a Lorentzian form since it allows for aneasier analytical treatment. It is generally believed that themain results concerning the global synchronization prop-erties are qualitatively independent of the precise form ofthe distribution as long as it is symmetric and unimodal. Inthis Letter, however, we show that a variant of theKuramoto model displays or not a reentrant diversity-induced transition into a state of collective firing, depend-ing on the type of distribution used. This transition ispresent in all the (symmetric and unimodal) distributionsstudied, but not in the case of the Lorentzian. The nonge-neric behavior of the system with a Lorentzian distributionof natural frequencies warns about the indiscriminate useof some recently proposed methods [5] in order to under-stand generic properties of coupled oscillators.We consider the following variant of the Kuramotomodel which describes the dynamics of an ensemble ofglobally coupled active rotators jðtÞ, j ¼ 1; . . . ; N [6]:_ j ¼ !j  sinj þ KNXNl¼1sinðl jÞ: (1)A natural frequency !j < 1 (respectively, !j > 1) corre-sponds to an excitable (respectively, oscillatory) behaviorof the rotator j when it is uncoupled. K is the couplingintensity. Diversity is introduced by considering that the!j’s are distributed according to a probability densityfunction gð!Þ, with mean value ! and variance 2. Themodel is equivalent to the Kuramoto model with zeroaverage frequency and an external periodic driving offrequency  !, as it can be easily seen with the changeof variables j ! j  !t. Throughout the Letter, be-sides the well-known Gaussian and uniform distributions,we will be considering a general family of distributionsLmn ð!Þ, for n > 0, mn> 1, defined asLmn ð!Þ ¼ nðmÞ2ðm 1=nÞð1=nÞnm1ðj! !jn þ nÞm : (2)The variance is finite only for mn> 3 and it is given by2 ¼ 2 ðm3=nÞð3=nÞðm1=nÞð1=nÞ . The Lorentzian distribution corre-sponds to L12ð!Þ and has, hence, an infinite variance,although we still will use  as a measure of diversity.To characterize the collective behavior we use the time-dependent complex variable rðtÞ [2,6]:rðtÞ ¼ 1NXNj¼1eijðtÞ: (3)The Kuramoto order parameter   hjrðtÞji, where h  idenotes time average, is known to be a good measure ofcollective synchronization in coupled oscillators systems,i.e.,  ’ 1 when the oscillators synchronize (j ’ l, 8j,l), and  ’ 0 for desynchronized behavior.For ! & 1 the system displays three different regimes:(i) for small diversity, almost all units are at rest at similarfixed points [rðtÞ does not depend on time and   1];(ii) increasing diversity one enters a dynamical state inwhich a macroscopic fraction of units fire at (roughly) thesame time [rðtÞ is time dependent while  is still close to1]; (iii) for even larger diversity, the system reenters adesynchronized state [ small and rðtÞ time independent].To discriminate between static entrainment and collectivefiring, regimes (i) and (ii), we use the order parameterintroduced by Shinomoto and Kuramoto [7]: ¼ hjrðtÞ  hrðtÞiji; (4)which differs from zero only for collective firing.PRL 105, 084101 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending20 AUGUST 20100031-9007=10=105(8)=084101(4) 084101-1  2010 The American Physical SocietyAn approximate theory to describe these three regimeswas developed in [8]. The theory was independent of theform of the natural frequencies distribution and was alsoapplicable to identical units subject to noise. A recentmethod developed by Ott and Antonsen [5,9] allows usto solve this model exactly (and a large family of relatedones) in the infinite number of oscillators limit and in anumber of cases that include the Lorentzian gð!Þ. Childsand Strogatz [10] used this method to obtain the fullbifurcation diagram of the complex variable rðtÞ for theLorentzian distribution. Contrarily to the results of [8],their exact solution implies that there is no transition tocollective firing increasing the diversity for !< 1. Thenonexistence of the transition can be derived from thebifurcation diagram in the ! space obtained usingthe ideas of [10], see Fig. 1. Regime (ii) takes place for theparameter region located to the right of the SNIC (saddlenode on the invariant circle) bifurcation line and below theHopf line. For !< 1 increasing  one never enters in thisregion, so there is no diversity-induced transition to col-lective firing. This situation is generic for all values of K,since it can be shown that the SNIC bifurcation alwaysstarts at ! ¼ 1,  ¼ 0 with positive slope. The model wasalso studied for the Lorentzian case with a different ap-proach in [11] and the same results were found.We will give now the main sketches of the Ott-Antonsenmethod. Quite generally, we will show that, in fact, themethod can be successfully applied to any nonsingulardistribution gð!Þ. Let fð!;; tÞ be the density of oscilla-tors with frequency ! and phase  at time t. This functionobeys the continuity equation (conservation of the numberof oscillators):@fð!;; tÞ@tþ @@½ _ð!;; rÞf ¼ 0; (5)with _ð!;; rÞ ¼ ! sinþ<ðreiÞ. If the coeffi-cients of the Fourier expansionfð!;; tÞ ¼ gð!Þ21þ X1m¼1½fmð!; tÞeim þ c:c:(6)(c:c: denotes complex conjugate), satisfy the ansatzfnð!; tÞ ¼ ð!; tÞn; (7)then ð!; tÞ and rðtÞ ¼ Rd!R deifð!;; tÞ obey theintegrodifferential equations@@tþ i!þ 12f½Krþ 12  Kr  1g ¼ 0; (8)rðtÞ ¼Zd!ð!; tÞgð!Þ: (9)The manifold defined by (7) is invariant under the systemevolution, so if that condition is fulfilled by the initialcondition, it is fulfilled afterwards. Moreover, in [9] it isshown that for a Lorentzian gð!Þ the long time evolution ofrðtÞ is always described by this reduced manifold.If gð!Þ has a finite set of poles !^1, !^2; . . . outside thereal axis [as is the case for Lmn ð!Þ for even n and integerm,including the Lorentzian L12ð!Þ] and ð!; tÞ satisfies cer-tain analyticity conditions, one can obtain (9) by contourintegration. Then rðtÞ can be written is terms of kðtÞ ð!^k; tÞ and one can obtain a closed set of ordinary dif-ferential equations for kðtÞ. In the case of poles withmultiplicity larger than 1, rðtÞ depends also on the partialderivatives with respect to !, skðtÞ  ðsÞð!^k; tÞ.Equations for these new functions skðtÞ can be obtainedby differentiating Eq. (8) with respect to!. For an arbitrarydistribution gð!Þ, we can obtain an approximate evolutionof the system by evaluating integral (9) using a finite,though large, set of values of ! and integrating numeri-cally (8) for each one of these frequencies.In Figs. 2 and 3 we show the order parameters as afunction of the diversity for several frequency distribu-tions, obtained by direct simulation of Eqs. (1) and byapplying the Ott-Antonsen method as described above.For the Gaussian and uniform distributions, but not forthe Lorentzian, the regimen of collective firing (signaledby a nonzero value of the parameter ) is present forintermediate values of the diversity. Beyond the casesshown in the figures, the transition is also present for othersymmetric distributions such as the exponential gð!Þ ¼1ﬃﬃ2p eðﬃﬃ2p=Þj! !j, or the family Lmn ð!Þ for mn  3. EvenL13ð!Þ which has infinite variance (but well-defined firstmoment) presents the reentrant diversity-induced transi-tion (for values of ! close enough to one). Also, if wetruncate the Lorentzian distribution at some finite value of!, i.e., set gð!Þ ¼ 0, if j! !j>C, the system showsthis reentrant transition (we checked for C ¼ 50).Furthermore, Fig. 3 shows that for finite-size systemswith a Lorentzian distribution a maximum in  is indeedpresent, being quite visible up to a few thousand of units. Infact, Lorentzian distributions in systems with a finite num-0 0.5 1 1.5ω00.511.5∆SNICSaddle NodeHopfFIG. 1 (color online). Bifurcation diagram of the model de-fined by Eqs. (1) for a Lorentzian distribution of frequencies andK ¼ 5. There is also a saddle-loop bifurcation line, not shown,that goes from the Takens-Bogdanov point (circle) to the saddle-node separatrix loop point (square) where SNIC line starts [10].Collective firing takes place in the region below the Hopf andright of the SNIC.PRL 105, 084101 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending20 AUGUST 2010084101-2ber of units are effectively truncated, truncation that dis-appears in the limit N ! 1. We conclude that the exis-tence of the transition to common firing is a truly genericphenomenon and the results obtained using a Lorentziandistribution in the infinite system size limit are ratherpathological and misleading.We introduce now an alternative approach to determinethe parameters for which the system shows collective fir-ing. Notice that rðtÞ is time independent in regimes(i) (units at rest) and (iii) (desynchronized state) but notin (ii). Following previous analysis in the limit of infiniteunits [12], we write rðtÞ ¼ ei and, assuming that rðtÞ istime independent, derive the following equations for theglobal amplitude  and phase : sin ¼ !Zj!j>b!ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 b2!2sgð!Þd!  f1ðbÞ; (10)K2 þ  cos ¼Z bbﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃb2 !2pgð!Þd!  f2ðbÞ; (11)with b ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þ K22 þ 2K cosp . It is possible to obtaina closed self-consistent relation for b:b ¼ Kðf21 þ f22Þbf2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃf22 þ ð1 b2Þf21q  hðbÞ: (12)For parameters corresponding to regime (ii) Eq. (12) hasonly one solution corresponding to an unstable fix point.When the branches hþðbÞ and hðbÞ meet two new solu-tions arise. This signals a SNIC bifurcation where a stablefix point and a saddle are created. Imposing hðbÞ ¼hþðbÞ ¼ b, one can determine the line ð !Þ at which thetransition takes place. Noticeably, on the SNIC one obtains ¼ =2 independent of the form of the distribution gð!Þor the values of the parameters. As shown in Fig. 4 theresults of this approach coincide with those of the Ott-Antonsen method.For the Gaussian distribution, the SNIC line that limitsthe collective firing regime starts at ! ¼ 1,  ¼ 0, with anegative slope. Therefore for !< 1, as  increases onefinds first a stable steady state for r [regime (i)], then00.30.60.9ρN=50 LorentzianN=10 000 LorentzianN=10 000 L4,100.10.2ζN=50N=200N=1000N=10 0000 1 2 3 4∆00.10.2ζN=1000N=10 000N=100 000FIG. 3 (color online). Same as Fig. 2 for Lorentzian L12ð!Þ(upper and middle panels) and L14ð!Þ (upper and lower ones)distributions. Note that for the Lorentzian distribution L12ð!Þ theOtt-Antonsen method predicts that  is zero for all values of thediversity  while the numerical simulations for finite system sizeN show a rounded-off transition.00.30.60.9ρ00.20.4ζ Uniform0 1 2 3 4σ00.20.4ζ GaussianFIG. 2 (color online). Order parameters  and  as a functionof the diversity  coming from numerical simulations of the fullsystem of Eqs. (1) for N ¼ 104 units in the cases of a Gaussian(dots) and uniform (crosses) distribution of frequencies. In eachcase, the solid lines are the result of the application of the Ott-Antonsen method using 104 values of ! for the numericalintegration of Eq. (9). K ¼ 5 and ! ¼ 0:97.0.94 0.96 0.98 1ω012345σ1 1.02ω00.511.52∆GaussianK=10K=4LorentzianFIG. 4 (color online). (Partial) Bifurcation diagram forGaussian (left) and Lorentzian (right) distributions. For theGaussian, circles correspond to the Ott-Antonsen method solvednumerically as in Fig. (2), solid lines to the numerical solution ofEqs. (10)–(12) and dashed lines to Eq. (14). For the Lorentziandistribution the lines correspond to the Ott-Antonsen method.PRL 105, 084101 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending20 AUGUST 2010084101-3crosses the SNIC lower boundary entering in regime (ii)and finally crosses the reentrant upper boundary entering inregime (iii) where a stable steady state is present again.Region (ii) moves upwards and broadens increasing thecoupling K. The overall phase diagram still has a Takens-Bogdanov point as in Fig. 1. DecreasingK this point movesdown and to the left. For low values ofK, for exampleK ¼1, the upper boundary of the collective firing region islimited by the Hopf rather than by the reentrant SNIC,nevertheless the diversity-induced transition is still present.This reentrance behavior is generic for all distributionswith well-defined moments.The Lorentzian distribution behaves differently. TheSNIC starts at ! ¼ 1,  ¼ 0 with positive slope.Although apparently this is a small quantitative differenceand the phase diagrams may be topologically equivalent,there are important qualitative consequences and, in par-ticular, no diversity-induced transition to collective firingexists [13].For distributions which decay fast enough we can obtainan analytical approximation for the line ð !Þ at which thetransition to collective firing appears. We assume thatgð!Þ ’ 0 for j!j> b. As b  jK 1j, this approxima-tion turns out to be appropriate for large values ofK. Using ¼ =2 one gets  ¼ ! from Eq. (10). Inserting this in(11) and expanding the integrand to second order in !=bwe obtainð !Þ ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2ð2K2  2Kﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þ K2 !2p 1Þ þ 2q: (13)From Eq. (13) one can obtain d=d !. At the origin of theSNIC line ( ! ¼ 1), the derivative takes a negative valuefor anyK > 0 and tends to1 asK3 forK ! 1. Noticethat this is independent of the form of gð!Þ provided thetails decay fast enough. Higher order corrections do de-pend on the specific form gð!Þ. For the Gaussian the nextorder givesð !Þ2¼ !2ð3þ2K2Þ23þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2½6 !224Kð1þK2 !2Þ3=2þ28ð1þK2 !2Þ2q3: (14)This approximation, plotted as dashed lines in Fig. 4, de-scribes well the location of the transition to firing betweenregimes (i) and (ii) for the case of a Gaussian distribution.In summary, globally coupled active rotators dis-play a diversity-induced transition to a state of collectivefiring. We have shown that this is a genuine transitionthat is found for any disorder distribution with well-defined moments. Curiously enough, the transition isnot present for a Lorentzian distribution, for which thefirst moment integral is only defined as a principal value.We have also found that these nongeneric results given bythe Lorentzian distribution also occur in another well-known excitable system, an ensemble of coupledFitzHugh-Nagumo [14] units, for which the reentrantdiversity-induced transition is present as well for distribu-tions such as Gaussian or uniform, but not for theLorentzian one.While the Lorentzian distribution seems to be suitable tostudy generic properties of the original Kuramoto model, itis not appropriate for excitable systems and beyond thosemay not be adequate for other systems. This is a clearwarning about its use in analytical approximations in-tended to draw conclusions on the generic properties ofcoupled oscillators.We acknowledge financial support by the MICINN(Spain) and FEDER (EU) through Project No. FIS2007-60327. L. F. L. is supported by the JAEPredoc program ofCSIC.[1] A. S. Pikovsky, M.G. Rosenblum, and J. Kurths,Synchronization: A Universal Concept in NonlinearSciences (Cambridge University Press, Cambridge,2001).[2] Y. Kuramoto, Chemical Oscillations, Waves andTurbulence (Springer-Verlag, New York, 1984).[3] J. A. Acebron, L. L. Bonilla, C. J. Perez Vicente, F. Ritort,and R. Spigler, Rev. Mod. Phys. 77, 137 (2005).[4] S. H. Strogatz, Physica (Amsterdam) 143A, 1 (2000).[5] E. Ott and T.M. Antonsen, Chaos 18, 037113 (2008).[6] Y. Kuramoto, in International Symposium onMathematical Problems in Theoretical Physics, editedby H. Arakai, Lecture Notes in Physics Vol. 39(Springer, New York, 1975), p. 420.[7] S. Shinomoto and Y. Kuramoto, Prog. Theor. Phys. 75,1105 (1986).[8] C. J. Tessone, A. Scire, R. Toral, and P. Colet, Phys. Rev. E75, 016203 (2007).[9] E. Ott and T.M. Antonsen, Chaos 19, 023117 (2009).[10] L.M. Childs and H. S. Strogatz, Chaos 18, 043128(2008).[11] T.M. Antonsen, Jr., R. T. Faghig, M. Girvan, E. Ott, andJ. Platig, Chaos 18, 037112 (2008).[12] H. Sakaguchi, Prog. Theor. Phys. 79, 39 (1988).[13] For ! * 1 there is a small region limited by the homo-clinic and the Hopf bifurcations where there is bistabilitybetween a steady state and one with collective motion.However, in the collective motion state, r moves slightlyaround a fixed value rather than performing collectivefiring, and the basin of attraction of this state is verysmall, so this is indeed different from the regime we areconsidering.[14] C. J. Tessone, R. Toral, C. R. Mirasso, and J. D. Gunton, inThe Physics of Complex Systems (New Advances andPerspectives), Proceedings of the International School ofPhysics ‘‘Enrico Fermi,’’ Course CLV, edited by F.Mallamace and H. E. Stanley (IOS Press, Amsterdam,2004), p. 461.PRL 105, 084101 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending20 AUGUST 2010084101-4

Non-universal results induced by diversity distribution in coupled excitable systems

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