Entrainment by a pacemaker, representing an element with a higher frequency,
is numerically investigated for several classes of random networks which
consist of identical phase oscillators. We find that the entrainment frequency
window of a network decreases exponentially with its depth, defined as the mean
forward distance of the elements from the pacemaker. Effectively, only shallow
networks can thus exhibit frequency-locking to the pacemaker. The exponential
dependence is also derived analytically as an approximation for large random
asymmetric networks.Comment: 4 pages, 3 figures, revtex 4, submitted to Phys. Rev. Let

A. N. Zaikin

A. T. Winfree

Alexander S. Mikhailov

E. E. Abrahamson

Hiroshi Kori

M. Stich

S. C. Manrubia

S. J. Kuhlman

S. N. Dorogovtsev

Y. Kuramoto

English

arXiv

Entrainment by a pacemaker, representing an element with a higher frequency, is numerically investigated for several classes of random networks which consist of identical phase oscillators. We find that the entrainment frequency window of a network decreases exponentially with its depth, defined as the mean forward distance of the elements from the pacemaker. Effectively, only shallow networks can thus exhibit frequency locking to the pacemaker. The exponential dependence is also derived analytically as an approximation for large random asymmetric networks

Kori, H.

Mikhailov, A.

MPG.PuRe

arXiv:cond-mat/0402519v3  [cond-mat.dis-nn]  6 Oct 2004Entrainment of randomly coupled oscillator networks by a pacemakerHiroshi Kori1, 2, ∗ and Alexander S. Mikhailov21Department of Physics, Graduate School of Sciences, Kyoto University, Kyoto 606-8502, Japan2Abteilung Physikalische Chemie, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany(Dated: 20.FEB.2004)Entrainment by a pacemaker, representing an element with a higher frequency, is numericallyinvestigated for several classes of random networks which consist of identical phase oscillators. Wefind that the entrainment frequency window of a network decreases exponentially with its depth,defined as the mean forward distance of the elements from the pacemaker. Effectively, only shallownetworks can thus exhibit frequency-locking to the pacemaker. The exponential dependence is alsoderived analytically as an approximation for large random asymmetric networks.PACS numbers: 05.45.Xt, 89.75.Fb, 87.18.SnPacemakers are wave sources in distributed oscillatorysystems typically associated with a local group of ele-ments having a higher oscillation frequency. Target pat-terns, generated by pacemakers, were the first complexwave patterns observed in the Belousov-Zhabotinsky sys-tem [1]. Pacemakers play an important role in func-tioning of the heart [2] and in the collective behaviorof Dictyostelium discoideum [3]. They are also observedin large-scale ecosystems [4]. In addition to pacemak-ers produced by local heterogeneities in the medium [5],self-organized pacemakers in uniform birhythmic mediahave been theoretically studied [6]. While the majority ofrelated investigations have so far been performed for sys-tems with local diffusive coupling between the elements,pacemakers can also operate in oscillator networks withcomplex connection topologies. The circadian rhythmin mammals is a daily variation of 24 hours that reg-ulates basic physiological processes in such animals [7].It is produced by a complex network of neurons form-ing the so-called suprachiasmatic nucleus (SCN) [8]. Asrecently shown, this oscillator network undergoes spon-taneous synchronization in absence of any environmen-tal input, but its intrinsic synchronization period is thensignificantly longer than 24 hours [9]. Therefore, the ac-tual shorter rhythm results from the environmental en-trainment and must be externally imposed. The entrain-ment is mediated by direct photic inputs from eyes intothe SCN, which undergo periodic daily variation. How-ever, it is known that only a distinct subset of neuronsin this network is directly influenced by photic inputs[10]. Hence, functioning of this particular neural systemis crucially dependent on the ability of the entire complexnetwork to become entrained by an external pacemaker.Analogous behavior can also be expected, for example, inheterogeneous arrays of globally coupled electrochemicaloscillators where synchronization and entrainment havebeen experimentally demonstrated [11].To understand operation of pacemakers in networkswith complex connection topologies, action of a pace-maker in a random oscillator network should first be in-vestigated. In this Letter, networks of identical phaseoscillators with random connections are considered. Apacemaker is introduced as a special element whose os-cillations have a higher frequency and are not influencedby the rest of the system. Depending on the pacemakerfrequency and the strength of coupling, the pacemakercan entrain the entire network, so that the frequenciesof all its elements become equal to that of the pace-maker. We find that the entrainment window decreasesexponentially with the depth of a network, defined asthe mean forward distance of its elements from a pace-maker, and thus only shallow networks can effectively beentrained. This result is confirmed in numerical simu-lations for several different classes of random networks,including small-world graphs. It is further analyticallyderived as an approximation for random networks withasymmetric connections.We consider a system of N+1 phase oscillators, one ofthem being a pacemaker. The model is given by a set ofevolution equations [12] for the oscillator phases φi andthe pacemaker phase φ0,φ˙i = ω −κpNN∑j=1Aij sin(φi − φj)− µBi sin(φi − φ0),φ˙0 = ω +∆ω. (1)The topology of network connections is determined bythe adjacency matrix A whose elements Aij are either 1or 0. The element with i = 0 is special and representsa pacemaker. Its frequency is increased by ∆ω with re-spect to the frequency ω of all other oscillators [19]. Thepacemaker is acting on a randomly chosen subset of N1elements, specified by Bi taking values 1 or 0. The totalnumber of connections to the pacemaker, N1 =∑iBi,is fixed. The coupling between elements inside the net-work is characterized by strength κ. The strength ofcoupling from the pacemaker to the network elements isdetermined by the parameter µ. In absence of a pace-maker, such networks undergo autonomous phase syn-chronization at the natural frequency ω. Without lossof generality, we put ω = 0. Moreover, we rescale timeas t′ = t ∆ω and introduce rescaled coupling strengths2FIG. 1: (color online) Phases of elements in the entrainedstate; N = 100, p = 0.05, N1 = 3, κ = 100 and µ = 1000.κ′ = κ/∆ω, µ′ = µ/∆ω. After such rescaling, the modeltakes the form of Eqs. (1) with ∆ω = 1 and ω = 0 (wedrop primes in the notations for the rescaled couplings).In terms of the original model (1), increasing the rescaledcoupling between the elements is equivalent either to anincrease of coupling κ or to a decrease of the relativepacemaker frequency ∆ω.The presence of a pacemaker imposes hierarchical or-ganization. For any node i, its distance h with respectto the pacemaker is given by the length of the minimumforward path separating this node from the pacemaker.All N1 elements in the group directly connected to thepacemaker have distances h = 1, the next elements whichare connected to the elements from this group have dis-tances h = 2, etc. Thus, the whole network is dividedinto a set of shells [13], each characterized by a certainforward distance h from the pacemaker. The set of num-bers Nh is an important property of a network. Thedepth L of a given network, which is the mean distancefrom the pacemaker to the entire network, is introducedas L = (1/N)∑h hNh. It should be noticed that suchordering of network nodes is based solely on the forwardconnections down the hierarchy and does not depend onthe distribution of reverse (upward) connections in thesystem.First, we investigated standard random asymmetricnetworks, where independently for all connections Aij =1 with probability p and Aij = 0 otherwise. Only sparserandom networks with relatively low mean connectivity pand a small number N1 of elements directly connected tothe pacemaker were considered. Numerical simulationswere performed for the networks of size N = 100 startingwith random initial conditions for the phases of all oscilla-tors. For each oscillator, its effective long-time frequencyωi was computed as ωi = T−1 [φi(t0 + T )− φi(t0)] withsufficiently large T and t0. The simulations show thatthe response of a network to the introduction of a pace-maker depends on the strength κ of coupling between theoscillators. When this coupling is sufficiently large (andcoupling µ to the pacemaker is also sufficiently strong asassumed below), the pacemaker entrains the whole net-work (i.e., ωi = 1 for all elements i). The frozen relativephases ψi ≡ φi − φ0 are displayed in Fig. 1. Here, theelements are sorted according to their hierarchical shells.Despite random variations, there is a clear correlation be-tween phases of oscillators and their positions in the hi-erarchy. Generally, the phase decreases for deeper shells,and the phase difference between the neighboring shellsrapidly becomes smaller as deeper shells are considered.As the coupling strength κ is decreased, the entrainmentbreaks down at a certain threshold value κcr. Our simula-tions show that synchronization between the first and thesecond shells was almost always the first to break down,and the frequencies of the second and deeper shells re-mained equal in most cases for the considered randomnetworks.Figure 2 displays in the logarithmic scale the thresh-olds κcr for a large set of networks with different depthsand different numbers of elements in the first shell. Eachgroup with a certain N1 is displayed by using its ownsymbol. Every such group generates a cluster of datapoints. Correlation between the entrainment thresholdand the network depth is apparent. The distributionsinside each cluster and the accumulation of the clustersyield the dependence κcr(L) of the entrainment thresholdon the network depth. Note that the statistical varia-tion of the data becomes larger for deeper networks withlarger L and for smaller pN . Similar dependence wasfound for the networks with different mean connectiv-ity p (see inset). Remarkably, the observed dependencescould be well numerically approximated by the exponen-tial dependenceκcr ∝ (1 + pN)L. (2)As the second class, asymmetric small-world networks[14] were considered. To generate them, we first con-structed a one-dimensional lattice of N elements whereeach element had incoming connections from up to itskth neighbor (the degree was thus 2k). Then a randomlychosen link in the lattice was eliminated and a distantconnection between two independently randomly chosenelements was introduced. This construction was repeatedqN times, with the parameter q specifying the random-ness of a network. When q was small, the network wasclose to a lattice and, in this case, we have seen that sta-ble wave solutions with different winding numbers werepossible, depending on initial conditions (cf. [12, 15]). Toavoid this, we chose almost synchronized states as initialconditions. The entrainment thresholds for such small-world networks are displayed in Fig. 3 and again show aclear correlation between κcr and L. The dependence onthe depth is approximately linear in lattices (q = 0), but3FIG. 2: (color online) Dependence of the entrainment thresh-old on the depth L for an ensemble of random networks withN = 100 and p = 0.1. In the inset, respective data for net-works with p = 0.06 and 0.2 is plotted. Solid lines are theexponential functions cp(1 + pN)L with appropriate fittingparameters cp.it becomes strongly nonlinear even when small random-ness is introduced. For q = 0.1, the dependence is alreadyapproximately exponential, though the dispersion of datais strong. As randomness q is increased, the dependenceapproaches that of the standard random networks withpN = 2k.We have also investigated asymmetric scale-free ran-dom networks [16], asymmetric regular random networks(where every element has exactly the same number ofeither incoming or outgoing connections), and symmet-ric standard random networks. For all of them, ap-proximately exponential dependences of the entrainmentthreshold on the network depth were observed in a largeparameter region.The exponential dependence (2) can be approximatelyderived for asymmetric random networks with large Nand pN . In the large-size limit, random graphs havelocally a tree-like structure [13]. The global tree approx-imation has previously been used for determining statis-tical properties of random networks [16]. We apply herethe same approximation and assume that the graph offorward connections extending from the pacemaker noderepresents a tree, so that any oscillator has only one in-coming connection from the previous shell. Then theshell populations Nh are given by Nh = N1(pN)h−1 forh = 2, . . . , H, where H is the total number of shells de-termined by∑Hh=1Nh = N . Because pN is large, wehave Nh ≪ NH ≃ N for h < H , and thus L ≃ H . Next,we estimate the numbers mhk of incoming connectionsleading from all elements in the kth shell to an oscilla-tor in the hth shell. By definition of hierarchical shells,mhk = 0 if k < h−1. In the tree approximation,mhk = 1FIG. 3: (color online) Dependence of the entrainment thresh-old on the depth L for an ensemble of small-world networkswith N = 100, k = 3 and different randomness q. The solidline is the same exponential function as that fitted to the datawith pN = 6(= 2k) in Fig. 2. The dotted line is linear fittingfor the regular lattice (q = 0).for k = h− 1. Because most of the population is concen-trated in the last shell, reverse connections from othershells can be neglected. On the average, the number ofreverse connections from the shell H to an oscillator inthe shell h is mhH = pNH . Moreover, the relative sta-tistical deviation from this average is of order (pN)−1/2,and is thus negligible. Therefore, in this approximationall oscillators inside a particular shell have effectively thesame number of connections from other shells, and a statewith phase synchronization inside each shell is possible.In this state, all oscillators inside a shell have the samephase, i.e. φi = θh for all oscillators i in a shell h. Un-der entrainment, the phases of such a state can be foundanalytically as a solution of algebraic equations−κpNH∑k=h−1mhk − sin(θh − θk) = 1 for h = 2, . . . , n,− µ sin(θ1 − θ0)−κpNH∑k=2m1k sin(θ1 − θk) = 1, (3)where θ0 ≡ φ0. For large pN , we can linearize sin(θh−θk)for h, k ≥ 2 in the solution of Eqs. (3) [it can be shownthat θ2 − θH is of order O(1/pN)]. Furthermore, usingthat NH ≃ N ≫ Nh for h < H and L ≃ H , we obtainfor h ≥ 2 thatsin(θh−1 − θh) =pNκ(1 + pN)L−h. (4)Equations (4) determine the phases of oscillators in theconsidered synchronized state. Note that the explicitvalue of the phase θ1 in this state is not needed below.4The entrainment breakdown can, in principle, oc-cur through destabilization of the synchronized state.Though the analytical proof of its stability is not yetavailable, our numerical simulations show that the syn-chronous entrained state with 0 < θh−1 − θh < pi/2 isalways stable when it exists. Thus, the breakdown ofentrainment in the considered system takes place in asaddle-node bifurcation, through the disappearance of so-lutions of Eqs. (3). This occurs when | sin(θh−1−θh)| = 1for certain h. For large enough µ, we always have0 < sin(θ0 − θ1) < 1 (a sufficient condition is µ > 1+ κ).Among the other terms, the term sin(θ1 − θ2) is alwaysthe largest one. Therefore, the solution disappears andbreakdown occurs when sin(θ1 − θ2) = 1. SubstitutingEq. (4) into this equation and solving it with respect toκ, we finally derive the dependence (2). Thus, we seethat the entrainment breakdown occurs through the lossof frequency locking between the first shell and the restof the network. The analytical estimate for the criticalcoupling strength, obtained using the tree approxima-tion, agrees well with the numerical data, even for thenetworks which are not very large.So far we have used the coupling strength which wasrescaled as κ → κ/∆ω. Therefore, if the non-scaledcoupling strength is fixed, Eq. (2) determines the max-imum ∆ωc at which the entrainment is still possible,∆ωc ∝ κ(1 + pN)−L. The entrainment by a pacemakercan take place only if its frequency lies inside the interval(ω, ω +∆ωc).Thus, the entrainment window decreases exponentiallywith the depth of a network. This is the principal re-sult of our study, which holds not only for standard ran-dom networks, where the above analytical estimate isavailable, but also for small-world graphs and other nu-merically investigated random topologies. In practice, itimplies that only shallow random networks with smalldepths are susceptible to frequency entrainment.Our results remain valid when, instead of a pacemaker,external periodic forcing acts on a subset of elements. Wehave checked that the reported strong dependence on thenetwork depth remains valid for systems with larger net-work sizes, heterogeneity in frequencies of individual os-cillators, and several other coupling functions. The studywas performed for coupled phase oscillators which serveas an approximation for various real oscillator systems,including neural networks (see, e.g., [12, 17, 18]). Itsconclusions should be applicable for a broad class of os-cillator networks with random architectures.Financial support by the Japan Society for Promotionof Science (JSPS) is gratefully acknowledged.∗ E-mail address: kori@fhi-berlin.mpg.de[1] A. N. Zaikin and A. M. Zhabotinsky, Nature 255, 535(1970).[2] A. T. Winfree, The Geometry of Biological Time(Springer, New York, 1980).[3] K. J. Lee, E. C. Cox, and R. E. Goldstein, Phys. Rev.Lett. 76, 1174 (1996).[4] B. Blasius, A. Huppert and L. Stone, Nature 399, 6734(1999).[5] J. J. Tyson and P. C. Fife, J. Chem. Phys. 73, 2224(1980); M. Stich, A. S. Mikhailov, Z. Phys. Chem. 216,521 (2002).[6] see, e.g., M. Stich, M. Ipsen, and A. S. Mikhailov, Phys.Rev. Lett. 86, 4406 (2001); M. Stich, M. Ipsen, andA. S. Mikhailov, Physica D 171, 19 (2002).[7] R. Y. Moore, Ann. Rev. Med. 48, 253 (1997).[8] E. E. Abrahamson and R. Y. Moore, Brain Reserch 916,172 (2001).[9] S. Yamaguchi et.al., Science 302, 1408 (2003).[10] S. J. Kuhlman et.al., J. Neurosci. 23, 1441 (2003).[11] I. Z. Kiss, Y. M. Zhai, and J. L. Hudson, Science 296,1676 (2002).[12] Y. Kuramoto, Chemical Oscillations, Waves, and Turbu-lence (Springer, New York, 1984).[13] S. N. Dorogovtsev, J. F. F. Mendes, A. N. Samukhin,Nucl. Phys. B 653, 307 (2003); S. N. Dorogovtsev,J. F. F. Mendes, Evolution of Networks: From Biolog-ical Nets to the Internet and WWW (Oxford UniversityPress, Oxford, 2003).[14] D. J. Watts and S. H. Strogatz, Nature (London) 393,440 (1998).[15] S. C. Manrubia, A. S. Mikhailov, D. H. Zanette, Emer-gence of Dynamical Order: Synchronization Phenomenain Complex Systems (World Scientific, Singapore 2004).[16] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys.Rev. E 64, 026118 (2001).[17] Y. Kuramoto, Physica D 50, 15 (1991).[18] H. Kori and Y. Kuramoto, Phys. Rev. E 63, 046214(2001); H. Kori, Phys. Rev. E 68, 021919 (2003).[19] Note that the system (1) is invariant under transforma-tion ω → −ω,∆ω → −∆ω,φ → −φ, and therefore thesame entrainment behavior takes place when ∆ω < 0.

Entrainment of randomly coupled oscillator networks by a pacemaker

Kori, H

Mikhailov, AS

Kyoto University Research Information Repository

Title Entrainment of randomly coupled oscillator networks by apacemakerAuthor(s)Kori, H; Mikhailov, ASCitationPHYSICAL REVIEW LETTERS (2004), 93(25)Issue Date2004-12-17URL http://hdl.handle.net/2433/49877RightCopyright 2004 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityEntrainment of Randomly Coupled Oscillator Networks by a PacemakerHiroshi Kori1,2,* and Alexander S. Mikhailov21Department of Physics, Graduate School of Sciences, Kyoto University, Kyoto 606-8502, Japan2Abteilung Physikalische Chemie, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany(Received 20 February 2004; published 15 December 2004)Entrainment by a pacemaker, representing an element with a higher frequency, is numericallyinvestigated for several classes of random networks which consist of identical phase oscillators. Wefind that the entrainment frequency window of a network decreases exponentially with its depth, definedas the mean forward distance of the elements from the pacemaker. Effectively, only shallow networks canthus exhibit frequency locking to the pacemaker. The exponential dependence is also derived analyticallyas an approximation for large random asymmetric networks.DOI: 10.1103/PhysRevLett.93.254101 PACS numbers: 05.45.Xt, 87.18.Sn, 89.75.FbPacemakers are wave sources in distributed oscillatorysystems typically associated with a local group of elementshaving a higher oscillation frequency. Target patterns,generated by pacemakers, were the first complex wavepatterns observed in the Belousov-Zhabotinsky system[1]. Pacemakers play an important role in the functioningof the heart [2] and in the collective behavior ofDictyostelium discoideum [3]. They are also observed inlarge-scale ecosystems [4]. In addition to pacemakers pro-duced by local heterogeneities in the medium [5], self-organized pacemakers in uniform birhythmic media havebeen theoretically studied [6]. While the majority of re-lated investigations have so far been performed for systemswith local diffusive coupling between the elements, pace-makers can also operate in oscillator networks with com-plex connection topologies. The circadian rhythm in mam-mals is a daily variation of 24 h that regulates basic physi-ological processes in such animals [7]. It is produced by acomplex network of neurons forming the so-called supra-chiasmatic nucleus (SCN) [8]. As recently shown, this os-cillator network undergoes spontaneous synchronization inthe absence of any environmental input, but its intrinsicsynchronization period is then significantly longer than24 h [9]. Therefore, the actual shorter rhythm resultsfrom the environmental entrainment and must be exter-nally imposed. The entrainment is mediated by directphotic inputs from eyes into the SCN, which undergoperiodic daily variation. However, it is known that only adistinct subset of neurons in this network is directly influ-enced by photic inputs [10]. Hence, functioning of thisparticular neural system is crucially dependent on theability of the entire complex network to become entrainedby an external pacemaker. Analogous behavior can alsobe expected, for example, in heterogeneous arrays of glob-ally coupled electrochemical oscillators where synchro-nization and entrainment have been experimentally dem-onstrated [11].To understand the operation of pacemakers in networkswith complex connection topologies, the action of a pace-maker in a random oscillator network should first be in-vestigated. In this Letter, networks of identical phaseoscillators with random connections are considered. Apacemaker is introduced as a special element whose oscil-lations have a higher frequency and are not influenced bythe rest of the system. Depending on the pacemaker fre-quency and the strength of coupling, the pacemaker canentrain the entire network, so that the frequencies of all itselements become equal to that of the pacemaker. We findthat the entrainment window decreases exponentially withthe depth of a network, defined as the mean forwarddistance of its elements from a pacemaker, and thus onlyshallow networks can effectively be entrained. This resultis confirmed in numerical simulations for several differentclasses of random networks, including small-world graphs.It is further analytically derived as an approximation forrandom networks with asymmetric connections.We consider a system of N  1 phase oscillators, one ofthem being a pacemaker. The model is given by a set ofevolution equations [12] for the oscillator phases i andthe pacemaker phase 0,_i  ! pNXNj1Aij sini j 	Bi sini 0;_0  ! !: (1)The topology of network connections is determined by theadjacency matrix A whose elements Aij are either 1 or 0.The element with i  0 is special and represents a pace-maker. Its frequency is increased by !with respect to thefrequency ! of all other oscillators [13]. The pacemaker isacting on a randomly chosen subset of N1 elements, speci-fied by Bi taking values 1 or 0. The total number ofconnections to the pacemaker, N1 PiBi, is fixed. Thecoupling between elements inside the network is charac-terized by strength . The strength of coupling from thepacemaker to the network elements is determined by theparameter 	. In absence of a pacemaker, such networksundergo autonomous phase synchronization at the naturalfrequency !. Without loss of generality, we put !  0.Moreover, we rescale time as t0  t! and introducePRL 93, 254101 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending17 DECEMBER 20040031-9007=04=93(25)=254101(4)$22.50 254101-1  2004 The American Physical Societyrescaled coupling strengths 0  =! and 	0  	=!.After such rescaling, the model takes the form of Eq. (1)with !  1 and !  0 (we drop primes in the notationsfor the rescaled couplings). In terms of the original model(1), increasing the rescaled coupling between the elementsis equivalent either to an increase of coupling  or to adecrease of the relative pacemaker frequency !.The presence of a pacemaker imposes hierarchical or-ganization. For any node i, its distance hwith respect to thepacemaker is given by the length of the minimum forwardpath separating this node from the pacemaker. All N1elements in the group directly connected to the pacemakerhave distances h  1, the next elements that are connectedto the elements from this group have distances h  2, etc.Thus, the whole network is divided into a set of shells [14],each characterized by a certain forward distance h from thepacemaker. The set of numbersNh is an important propertyof a network. The depth L of a given network, which is themean distance from the pacemaker to the entire network, isintroduced as L  1=NPhhNh. It should be noticed thatsuch an ordering of network nodes is based solely on theforward connections descending down the hierarchy anddoes not depend on the distribution of reverse (upward)connections in the system.First, we investigated standard random asymmetric net-works, where independently for all connections Aij  1with probability p and Aij  0 otherwise. Only sparserandom networks with relatively low mean connectivityp and a small number N1 of elements directly connected tothe pacemaker were considered. Numerical simulationswere performed for the networks of size N  100 startingwith random initial conditions for the phases of all oscil-lators. For each oscillator, its effective long-time frequency!i was computed as !i  T1it0  T it0 withsufficiently large T and t0. The simulations show that theresponse of a network to the introduction of a pacemakerdepends on the strength  of coupling between the oscil-lators. When this coupling is sufficiently large (and cou-pling 	 to the pacemaker is also sufficiently strong asassumed below), the pacemaker entrains the whole net-work (i.e., !i  1 for all elements i). The frozen relativephases  i 	 i 0 are displayed in Fig. 1. Here, theelements are sorted according to their hierarchical shells.Despite random variations, there is a clear correlationbetween phases of oscillators and their positions in thehierarchy. Generally, the phase decreases for deeper shells,and the phase difference between the neighboring shellsrapidly becomes smaller as deeper shells are considered.As the coupling strength  is decreased, the entrainmentbreaks down at a certain threshold value cr. Our simula-tions show that synchronization between the first and thesecond shells was almost always the first to break down,and the frequencies of the second and deeper shells re-mained equal in most cases for the considered randomnetworks.Figure 2 displays in the logarithmic scale the thresholdscr for a large set of networks with different depths anddifferent numbers of elements in the first shell. Each groupwith a certain N1 is displayed by using its own symbol.Every such group generates a cluster of data points.Correlation between the entrainment threshold and thenetwork depth is apparent. The distributions inside eachcluster and the accumulation of the clusters yield thedependence crL of the entrainment threshold on thenetwork depth. Note that the statistical variation of thedata becomes larger for deeper networks with larger Land for smaller pN. Similar dependence was found forthe networks with different mean connectivity p (see in-set). Remarkably, the observed dependences could be wellapproximated numerically by the exponential dependencecr / 1 pNL: (2)As the second class, asymmetric small-world networks[15] were considered. To generate them, we first con-structed a one-dimensional lattice of N elements, whereeach element had incoming connections from up to its kthneighbor (the degree was thus 2k). Then, a randomlychosen link in the lattice was eliminated and a distant−1−0.75−0.5−0.2500 25 50 75 100ψii (elements)Pacemakershell 1shell 2shell 3shell 4shell 5FIG. 1 (color online). Phases of elements in the entrained state;N  100, p  0:05, N1  3,   100, and 	  1000.101002 3 4pN= 20pN= 6101002 2.5 3 3.5 4κcrLN1=123pN=1020FIG. 2 (color online). Dependence of the entrainment thresh-old on the depth L for an ensemble of random networks withN  100 and p  0:1. In the inset, respective data for networkswith p  0:06 and 0:2 are plotted. Solid lines are the exponentialfunctions cp1 pNL with appropriate fitting parameters cp.PRL 93, 254101 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending17 DECEMBER 2004254101-2connection between two independently randomly chosenelements was introduced. This construction was repeatedqN times, with the parameter q specifying the randomnessof a network. After that, N1 nodes of the network wererandomly chosen and connected to one additional noderepresenting the pacemaker. When q was small, the net-work was close to a lattice and, in this case, we have seenthat stable wave solutions with different winding numberswere possible, depending on initial conditions (cf. [12,16]).To avoid this, we chose almost synchronized states asinitial conditions. The entrainment thresholds for suchsmall-world networks are displayed in Fig. 3 and againshow a clear correlation between cr and L. The depen-dence on the depth is approximately linear in lattices (q 0), but it becomes strongly nonlinear even when smallrandomness is introduced. For q  0:1, the dependenceis already approximately exponential, though the disper-sion of data is strong. As randomness q is increased, thedependence approaches that of the standard random net-works with pN  2k.We have also investigated asymmetric scale-free randomnetworks [17], asymmetric regular random networks(where every element has exactly the same number ofeither incoming or outgoing connections), and symmetricstandard random networks. For all of them, approximatelyexponential dependences of the entrainment threshold onthe network depth were observed in a large parameterregion.The exponential dependence (2) can be approximatelyderived for asymmetric random networks with large N andpN. In the large-size limit, random graphs have locally atreelike structure [14]. The global tree approximation haspreviously been used for determining statistical propertiesof random networks [17]. We apply here the same approxi-mation and assume that the graph of forward connectionsextending from the pacemaker node represents a tree, sothat any oscillator has only one incoming connection fromthe previous shell. Then the shell populations Nh are givenbyNh  N1pNh1 for h  2; . . . ; H, whereH is the totalnumber of shells determined byPHh1Nh  N. BecausepN is large, we have Nh  NH ’ N for h < H, and thusL ’ H. Next, we estimate the numbers mhk of incomingconnections leading from all elements in the kth shell to anoscillator in the hth shell. By definition of hierarchicalshells, mhk  0 if k < h 1. In the tree approximation,mhk  1 for k  h 1. Because most of the population isconcentrated in the last shell, reverse connections fromother shells can be neglected. On average, the number ofreverse connections from the shell H to an oscillator in theshell h is mhH  pNH. Moreover, the relative statisticaldeviation from this average is of order pN1=2 and is thusnegligible. Therefore, in this approximation, all oscillatorsinside a particular shell have effectively the same numberof connections from other shells, and a state with phasesynchronization inside each shell is possible. In this state,all oscillators inside a shell have the same phase, i.e., i h for all oscillators i in a shell h. Under entrainment, thephases of such a state can be found analytically as asolution of algebraic equations pNXHkh1mhk sinh  k  1 for h  2; . . . ; n;	 sin1  0  pNXHk2m1k sin1  k  1; (3)where 0 	 0. For large pN, we can linearize sinh k for h; k  2 in the solution of Eqs. (3) [it can be shownthat 2  H is of order O1=pN]. Furthermore, usingNH ’ N  Nh for h <H and L ’ H, for h  2 we obtainsinh1  h  pN 1 pNLh: (4)Eq. (4) determines the phases of oscillators in the consid-ered synchronized state. Note that the explicit value of thephase 1 in this state is not needed below.The entrainment breakdown can, in principle, occurthrough destabilization of the synchronized state. Thoughthe analytical proof of its stability is not yet available,our numerical simulations show that the synchronous en-trained state with 0< h1  h < =2 is always stablewhen it exists. Thus, the breakdown of entrainment in theconsidered system takes place in a saddle-node bifurca-tion, through the disappearance of solutions of Eqs. (3).This occurs when j sinh1  hj  1 for certain h. Forlarge enough 	, we always have 0< sin0  1< 1 (asufficient condition is	> 1 ). Among the other terms,the term sin1  2 is always the largest one. Therefore,the solution disappears and breakdown occurs whensin1  2  1. Substituting Eq. (4) into this equationand solving it with respect to , we finally derive thedependence (2). Thus, we see that the entrainment break-down occurs through the loss of frequency locking betweenFIG. 3 (color online). Dependence of the entrainment thresh-old on the depth L for an ensemble of small-world networks withN  100, k  3, various N1, 1  N1  20, and different ran-domness q. The solid line is the same exponential function asthat fitted to the data with pN  6 2k in Fig. 2. The dottedline is linear fitting for the regular lattice (q  0).PRL 93, 254101 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending17 DECEMBER 2004254101-3the first shell and the rest of the network. As seen in Fig. 2,the analytical dependence for the critical coupling strength,obtained using the tree approximation, agrees well with thenumerical dependence, even for the networks which arenot very large.So far we have used the coupling strength which wasrescaled as ! =!. Therefore, if the nonscaled cou-pling strength is fixed, Eq. (2) determines the maximum!c at which the entrainment is still possible, !c /1 pNL. The entrainment by a pacemaker can takeplace only if its frequency lies inside the interval!;!!c.Thus, the entrainment window decreases exponentiallywith the depth of a network. This is the principal result ofour study, which holds not only for standard random net-works, where the above analytical estimate is available, butalso for small-world graphs and other numerically inves-tigated random topologies. In practice, it implies that onlyshallow random networks with small depths are susceptibleto frequency entrainment.Our results remain valid when, instead of a pacemaker,external periodic forcing acts on a subset of elements. Wehave checked that the reported strong dependence on thenetwork depth remains valid for systems with larger net-work sizes, heterogeneity in frequencies of individual os-cillators, and several other coupling functions. The studywas performed for coupled phase oscillators which serve asan approximation for various real oscillator systems, in-cluding neural networks (see, e.g., [12,18,19]). Its conclu-sions should be applicable for a broad class of oscillatornetworks with random architectures.Financial support by the Japan Society for Promotion ofScience (JSPS) is gratefully acknowledged.*Electronic address: kori@fhi-berlin.mpg.de[1] A. N. Zaikin and A. M. Zhabotinsky, Nature (London)225, 535 (1970).[2] A. T. Winfree, The Geometry of Biological Time (Springer,New York, 1980).[3] K. J. Lee, E. C. Cox, and R. E. Goldstein, Phys. Rev. Lett.76, 1174 (1996).[4] B. Blasius, A. Huppert, and L. Stone, Nature (London)399, 354 (1999).[5] J. J. Tyson and P. C. Fife, J. Chem. Phys. 73, 2224 (1980);M. Stich and A. S. Mikhailov, Z. Phys. Chem. (Frankfortam Main) 216, 521 (2002).[6] See, e.g., M. Stich, M. Ipsen, and A. S. Mikhailov, Phys.Rev. Lett. 86, 4406 (2001); Physica (Amsterdam) 171D,19 (2002).[7] R. Y. Moore, Ann. Rev. Med. 48, 253 (1997).[8] E. E. Abrahamson and R. Y. Moore, Brain Res. 916, 172(2001).[9] S. Yamaguchi et al., Science 302, 1408 (2003).[10] S. J. Kuhlman et al., J. Neurosci. 23, 1441 (2003).[11] I. Z. Kiss, Y. M. Zhai, and J. L. Hudson, Science 296, 1676(2002).[12] Y. Kuramoto, Chemical Oscillations, Waves, andTurbulence (Springer, New York, 1984).[13] Note that the system (1) is invariant under transfor-mation !! !, !! !, ! , and there-fore the same entrainment behavior takes place when!< 0.[14] S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin,Nucl. Phys. B653, 307 (2003); S. N. Dorogovtsev andJ. F. F. Mendes, Evolution of Networks: From BiologicalNets to the Internet and WWW (Oxford University Press,Oxford, 2003).[15] D. J. Watts and S. H. Strogatz, Nature (London) 393, 440(1998).[16] S. C. Manrubia, A. S. Mikhailov, and D. H. Zanette,Emergence of Dynamical Order: SynchronizationPhenomena in Complex Systems (World Scientific,Singapore, 2004).[17] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys.Rev. E 64, 026118 (2001).[18] Y. Kuramoto, Physica (Amsterdam) 50D, 15 (1991).[19] H. Kori and Y. Kuramoto, Phys. Rev. E 63, 046214 (2001);H. Kori, Phys. Rev. E 68, 021919 (2003).PRL 93, 254101 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending17 DECEMBER 2004254101-4

Physical Review Letters

Kori, Hiroshi

Mikhailov, Alexander S.

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Entrainment by a pacemaker, representing an element with a higher frequency, is numerically investigated for several classes of random networks which consist of identical phase oscillators. We find that the entrainment frequency window of a network decreases exponentially with its depth, defined as the mean forward distance of the elements from the pacemaker. Effectively, only shallow networks can thus exhibit frequency locking to the pacemaker. The exponential dependence is also derived analytically as an approximation for large random asymmetric networks.
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Entrainment of Randomly Coupled Oscillator Networks by a Pacemaker

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Entrainment of randomly coupled oscillator networks by a pacemaker

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