Stability of Remote Synchronization in Star Networks of Kuramoto Oscillators
SStability of Remote Synchronizationin Star Networks of Kuramoto Oscillators
Yuzhen Qin, Yu Kawano, Ming Cao
Abstract — Synchrony of neuronal ensembles is believed tofacilitate information exchange among cortical regions in thehuman brain. Recently, it has been observed that distantbrain areas which are not directly connected by neural linksalso experience synchronization. Such synchronization betweenremote regions is sometimes due to the presence of a mediatingregion connecting them, e.g., the thalamus . The underlyingnetwork structure of this phenomenon is star-like and motivatesus to study the remote synchronization of Kuramoto oscillators,modeling neural dynamics, coupled by a directed star network,for which peripheral oscillators get phase synchronized, remain-ing the accommodating central mediator at a different phase.We show that the symmetry of the coupling strengths of theoutgoing links from the central oscillator plays a crucial role inenabling stable remote synchronization. We also consider thecase when there is a phase shift in the model which resultsfrom synaptic and conduction delays. Sufficient conditions onthe coupling strengths are obtained to ensure the stability ofremotely synchronized states. To validate our obtained results,numerical simulations are also performed.
I. INTRODUCTIONSynchronization is a ubiquitous phenomenon which hasbeen observed pervasively in many natural, social and man-made systems [1]–[4]. Remarkable examples include syn-chronized flashing of fireflies, pedestrian footwalk synchronyon London’s Millennium Bridge, and phase synchronizationof coupled Josephson junction circuits [5]–[7]. After it wasfirst proposed in 1975, the Kuramoto model has become oneof the most widely-accepted models in understanding suchsynchronization phenomena in large population of oscillators[8]. It is idealized to allow for detailed mathematical analysis,yet sufficiently capable to capture rich sets of behaviors, andthus has been extended to many variations [9]. As powerfultools for understanding synchronization patterns emerged inhuman brain, the Kuramoto model and its generalizationshave also fascinated researchers in neuroscience. Actually,synchronization across cortical regions in human brain hasbeen believed to be potential mechanism facilitating informa-tion exchange demanded by cognitive tasks [10]. Cohesivelyoscillating neuronal ensembles can communicate effectivelybecause their input and output windows are open at the sametime [11]. Empirically, structural connections of neuronalensembles are believed to play essential roles in renderingsynchronization of cortical regions. However, it has been ob-served that distant cortical regions without direct neural links
The authors are with the Jan C. Willems Center for Systems andControl, Faculty of Science and Engineering, University of Groningen,Groningen, the Netherlands ( { y.z.qin, y.kawano, m.cao } @rug.nl). The workwas supported in part by China Scholarship Council, the European ResearchCouncil (ERC-CoG-771687) and the Netherlands Organization for ScientificResearch (NWO-vidi-14134). also experience functional correlations [12]. This motivatesresearchers to study an interesting behavior dubbed remotesynchronization , which is a situation where oscillators getsynchronized although there is no direct physical links norintermediate sequence of synchronized oscillators connectingthem [13]. Unlike what is pointed out in most findingsthat the coupling strengths in a network are critical forsynchronization of coupled oscillators [14]–[16], a recentarticle reveals that morphological symmetry is crucial forremote synchronization [17]. Some nodes located distantly ina network can mirror their functionality between each other.In other words, theoretically, swapping the positions of thesenodes will not change the functioning of the overall system.A star network is a simple paradigm for such networkswith morphologically symmetric properties. The peripheralnodes have no direct connection, but obviously play similarroles in the whole network. The node at the center acts asa relay or mediator. As an example, the thalamus is sucha relay in neural networks, and it is believed to enableseparated cortical areas to be completely synchronized [18],[19]. This observation of robust correlated behavior takingplace in distant cortical regions through relaying motivatesus to study the stability of remote synchronization in starnetworks in this paper. A star network is simple in structure,but capable of characterizing some basic features of remotesynchronization, and also provides some idea to understandthis phenomenon in more complex networks. Different from[20], we use Kuramoto-Sakaguchi model [9] to describe thedynamics of coupled oscillators, and analytically study thestability of remote synchronization.The contribution of this paper is twofold. First, we con-sider the more challenging setup where the star networkis directed, in contrast to the undirected networks studiedin many existing results such as [21]–[23]. We obtain suf-ficient conditions to facilitate asymptotically stable remotesynchronization between peripheral oscillators when thereis no phase shift. The symmetry of the coupling strengthsof outgoing links from the central oscillator is shown tobe crucial for remote synchronization. In sharp contrast, thecoupling strengths of incoming links to the central oscillatorare not required to be symmetric. It can be intuitivelyparaphrased that the mediator at the central position is ableto render the oscillators around it synchronized by imposinga common input to them, without requiring the feedbackcoming back to be identical. This finding shares somesimilarities with the common-noise-induced synchronizationinvestigated by researchers in physics [24]–[26]. However,different from the phase reduction or averaging methods used a r X i v : . [ n li n . C D ] F e b n these related works, we provide a different proof for thelocal asymptotic stability of the remote synchronization, andmore importantly, we study network-coupled, not isolated,oscillators and derive conditions on the network to enablesynchronization between separated oscillators. Second, wetake a phase shift into consideration. This phase shift isoften used to model synaptic and conduction delays resultingfrom distant connections between remote brain regions [27].Sufficient conditions on the coupling strengths are obtainedto ensure the stability of remote synchronization. We showthat the presence of a phase shift raises the requirement forthe coupling strengths. The rest of this paper is organized asfollows.Section II introduces the model we employ and formulatesthe problem formally. In Section III, we consider the casewhere there is no phase shift. Sufficient conditions are ob-tained to guarantee local stability of remote synchronization.A phase shift is introduced to the model in Section IV.We obtain sufficient conditions under which the remotesynchronization is stable. Section V contains our numericalstudies. Finally, we draw the conclusion in Section VI.II. P ROBLEM F ORMULATION
Synchronization of distant cortical regions having no di-rect links has been observed in human brain. The emergenceof this phenomenon is sometimes due to a mediator orrelay that connecting separated regions, e.g., the thalamus[19]. Motivated by this, we study remote synchronizationby considering n + 1 , n ≥ , oscillators, coupled by astar network, which are labeled by , , . . . , n . Let N = { , . . . , n } be the set of indices of the peripheral oscillators.The central mediator is labeled by . Let S be the unitcircle, and denote S n = S × · · · × S . The dynamics of eachoscillator is described by ˙ θ = ω + n (cid:88) i =1 K i sin( θ i − θ − α ) , ˙ θ i = ω + A i sin( θ − θ i − α ) , i = 1 , , . . . , n, (1)where θ i ∈ S is the phase of the i th oscillator, and ω and ω are the natural frequencies of the central and peripheraloscillators, respectively. Here K i > is the coupling strengthfrom the peripheral node i to the central node (for which werefer to as incoming (with respect to ) coupling strengths),and A i > presents the directed coupling strength from thecentral node and the peripheral node i (for which we referto as outgoing (with respect to ) coupling strengths). It isworth mentioning that incoming and outgoing couplings areallowed to be different, which means that the underlying starnetwork, denoted by G , is directed. The term α is the phaseshift satisfying α ∈ [0 , π/ . In the star network consideredin this paper, remote synchronization is the situation wheresome of the peripheral oscillators are phase synchronized,while the phase of the central mediator connecting themcan be different. We define remote synchronization formallyas follows. Definition 1:
Let θ ( t ) = [ θ ( t ) , . . . , θ n ( t )] (cid:62) ∈ S n +1 be asolution to the system dynamics (1). Let R be a subset of N , whose cardinality satisfies ≤ |R| ≤ n . We say that thesolution θ ( t ) is remotely synchronized with respect to R iffor every pair of indices i, j ∈ R it holds that θ i ( t ) = θ j ( t ) for all t ≥ , but it is not required that θ i ( t ) = θ ( t ) .When R ⊂ N , we say that θ ( t ) is partially remotelysynchronized; in particular, when R = N , we say that θ ( t ) is completely remotely synchronized, for which situationwe refer to as remote synchronization for brevity in whatfollows. A particular case of partially remotely synchronizedsolution is remote cluster synchronization, which is definedas follows. Definition 2:
Let C = {C , . . . , C m } , ≤ m < n be apartition of N . The sets C , . . . , C m are non-overlapping andsatisfy ≤ |C p | < n for all p and ∪ mp =1 C p = N . A partiallyremotely synchronized solution θ ( t ) to the system dynamics(1) is said to be remotely clustered with respect to C if:for any given C p and every pair i, j ∈ C p there holds that θ i ( t ) = θ j ( t ) , ∀ t ≥ ; on the other hand, for any given i ∈ C p , j ∈ C q where p (cid:54) = q , θ i ( t ) (cid:54) = θ j ( t ) .Note that the trivial case when a cluster has only one oscil-lator is allowed. In this paper, we are exclusively interested inthe (partial) remote synchronization when the frequencies ofall the oscillators in the network are synchronized, althoughit has been observed that remote synchronization can occurwithout complete frequency synchronization [28]. In fact, asolution θ ( t ) is said to be frequency synchronized if ˙ θ ( t ) =˙ θ ( t ) = · · · = ˙ θ n ( t ) = ω syn for some ω syn ∈ R . For a given r ∈ S and an angle γ ∈ [0 , π ] , let rot γ ( r ) be the rotationof r counter-clockwise by the angle γ . For θ ∈ S n , we definean equivalence class Rot( θ ) := { [rot γ ( θ ) , . . . , rot γ ( θ n )] (cid:62) : γ ∈ [0 , π ] } . Let θ ∗ be a solution to the equations ω − ω − n (cid:88) j =1 K i sin( θ j − θ − α ) − A i sin( θ − θ i − α ) = 0 , (2)for i = 1 , , . . . , n , which is a solution such that fre-quency synchronization is reached. It is not hard to seethat [rot γ ( θ ∗ ) , . . . , rot γ ( θ ∗ n )] (cid:62) for any γ ∈ [0 , π ] is alsoa solution to the equations. Consequently, the set Rot( θ ∗ ) issaid to be a synchronization manifold for the dynamics (1)[29]. As an extension of the definition of the synchronizationmanifold in [30], we call Rot( θ ∗ ) (partial) remote synchro-nization manifold if there exists a set ( R ⊂ N ) R = N suchthat θ ∗ i = θ ∗ j for any pair i, j ∈ R . In order to study thestability of the (partial) remote synchronization manifold, itsuffices to study the stability of θ ∗ . In the next two sections,we investigate the local stability of remote synchronizationmanifolds. We start with the assumption that there is nophase shift in Section III. The local stability of the remoteand cluster synchronization manifolds is studied. In SectionIV, we consider there is a phase shift α and investigate theinfluence of this phase shift on the stability of the remotesynchronization manifold.III. R EMOTE S YNCHRONIZATION WITHOUT P HASE S HIFT
In this section, we consider the case when there is nophase shift, i.e., α = 0 . We investigate how partial and com-lete remote synchronization in star networks are formed.We show the important roles that the symmetric outgoingcouplings quantified by A i play in enabling synchronizationamong oscillators that are not directly connected.To proceed, define x i = θ − θ i for i = 1 , . . . , n . Thenthe time derivative of x i is given by ˙ x i = ω − ω − n (cid:88) j =1 K i sin( θ − θ j ) − A i sin( θ − θ i ) . (3)Let x = [ x , x , . . . , x n ] (cid:62) ∈ S n , ω = ( ω − ω ) n with n = [1 , . . . , (cid:62) , and sin x = [sin x , . . . , sin x n ] (cid:62) , thenthe dynamics (3) can be represented in a compact form asfollows ˙ x = ω − T sin x := f ( x ) , (4)where f ( x ) = [ f ( x ) , . . . , f n ( x )] (cid:62) and T = A + K K · · · K n K A + K · · · K n ... ... . . . ... K K · · · A n + K n . (5)Let x ∗ be an equilibrium of (4), if it exists, i.e., f ( x ∗ ) = 0 .From the definition of x , we observe that x ∗ corresponds toa (partial) remote synchronization manifold if there existsa set R = N ( R ⊂ N ) such that for any i, j ∈ R , x ∗ i = x ∗ j . In what follows, we show under what conditionson the coupling strengths the equilibrium x ∗ exists and all (apart of) of its elements are identical, which gives rise to thecorresponding (partial) remote synchronization of the model(1). Towards this end, let us first make an assumption. Assumption 1:
We assume that the coupling strengthssatisfy the following inequality A i ≥ ( n − K i , ∀ i ∈ N , (6)and the corresponding matrix T , given by (5), satisfies (cid:107) T − ω (cid:107) ∞ < . (7)Assumption 1 suggests that the strengths of outgoingcouplings are much greater than that of incoming ones. Byobserving that for any i it holds that A i + K i − ( n − K i ≥ ( n − K i + K i − ( n − K i = K i > , we know that the matrix T is column diagonally dominant .By Gershgorin circle theorem [31, Sec. 6.2], one knows allthe eigenvalues of T (cid:62) have positive real parts, which alsomeans that all the eigenvalues of T lie on the right half plane.Thus T is invertible. We are now at a position to present ourmain result in this section. Theorem 1: under Assumption 1, there exists a uniquelocally asymptotically stable equilibrium x ∗ satisfying | x ∗ i | ∈ [0 , π/ for all i ∈ N for the dynamics (3), which is x ∗ = arcsin ( T − ω ) . (8)In addition, if there is a pair of distinct i, j ∈ N such that A i = A j , then x ∗ i = x ∗ j . This x ∗ corresponds to a partial remote synchronization manifold, denoted by Rot( θ ∗ ) , forthe dynamics (1), which implies oscillators i and j areremotely synchronized. Proof:
Due to the page limit, we only provide a sketchof proof. From the hypothesis (7), one knows that there existsa unique solution in [0 , π/ , i.e., x ∗ = arcsin ( T − ω ) .We show the stability of this equilibrium by linearization.Towards this end, we evaluate the Jacobian matrix J ( x ∗ ) = − ∂f∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x ∗ = − T diag (cos x ∗ , . . . , cos x ∗ n ) . (9)By using the facts that T is column diagonally dominant and cos x ∗ i > for all i ∈ N , it is not hard to see that all theeigenvalues of J ( x ∗ ) have negative real parts, which meansthe equilibrium x ∗ i is locally asymptotically stable.Finally, we prove x ∗ i = x ∗ j if A i = A j by showing that sin x ∗ i = sin x ∗ j since | x ∗ i | < π/ for all i . Remark 1:
Theorem 1 shows that oscillators without di-rect connections are able to get phase synchronized justbecause the directed connections from the central mediatortowards them are identical. Authors in [17] show that sym-metries in undirected networks are important for remote syn-chronization. In contrast, we take directions of the couplingsinto account, and show that only the outgoing couplings, A i , matter. This idea is analogous to equitable partition ingraphs investigated in [32] and [33] if the central mediatoris regarded as a cluster and the peripheral ones as another.Different from these two works, we studied the stability ofremote synchronization.What is worth mentioning, by carefully manipulating thesymmetry of the couplings originated from the central node , not only synchronization among distant oscillators canbe facilitated, but also unnecessary synchronization can beeasily prevented. Moreover, interesting patterns of remotesynchronization, such as cluster synchronization, can occur.The following corollary provides some sufficient condi-tions for the existence and stability of remote and clustersynchronization manifold, which follows from Theorem 1straightforwardly. Corollary 1:
Under Assumption 1, there is a locallyasymptotically stable remote synchronization manifold forthe dynamics (1), i.e., in which the solution θ ( t ) is com-pletely remotely synchronized, if A i = A j for every pair i, j ∈ N ; there is a locally asymptotically stable partialremote synchronization manifold for the dynamics (1), inwhich the solution θ ( t ) is remotely clustered with respect to C , if there is a partition of N , denote by C = {C , . . . , C m } ,satisfying |C p | ≥ and ∪ mp =1 C p = N such that: for anygiven C p and every pair i, j ∈ C p it holds that A i = A j ; onthe other hand, for any given i ∈ C p , j ∈ C q where p (cid:54) = q , A i (cid:54) = A j .In next section, we consider the case where there is a phaseshift (or phase lag) term α . The model with the presence of aphase shift is known as the Kuramoto-Sakaguchi model [9].IV. R EMOTE S YNCHRONIZATION WITH P HASE S HIFT
In this section, we consider that there is a phase shift α ∈ (0 , π/ . By introducing a phase shift term, we allowhe oscillators to get synchronized at a frequency that differsfrom the average of their natural frequencies [34]. Thisphenomenon have always been observed in many biologicalsystems such as heart cells and mammalian intestine [35].Moreover, in neural networks the phase shift α is often usedto model delays concerning synaptic connections [27]. Tostudy the remote synchronization of our interest, we simpli-fied the problem by assuming that A i = A and K i = A/n forall i . Note that this simplification ensures that the directionof the network is preserved and the condition (6) is satisfied,which guarantees the property that the outgoing couplingsare much stronger than the incoming ones. Consequently,the dynamics (1) become ˙ θ = ω + An n (cid:88) i =1 sin( θ i − θ − α );˙ θ i = ω + A sin( θ − θ i − α ) , i = 1 , , . . . , n. (10)Conditions on the coupling strength A are subsequentlyobtained to ensure that the dynamics (10) admit a locallyasymptotically stable remote synchronization manifold. Weinvestigate how these conditions depend on the phase shift α .As frequency synchronization is the footstone for the analysisthat follows, let us provide the necessary condition for theexistence of a frequency synchronized solution to (10) andsee how it depends on the phase shift α . Proposition 1:
There is a frequency synchronized solutionto the dynamics (10) only if A ≥
12 cos α | ω − ω | . Letting ˙ θ − ˙ θ i = 0 for all i , the proof of Proposition1 follows immediately. We observe that when α = 0 , thisnecessary condition reduces to A ≥ | ω − ω | / . Obviously,the existence of the phase shift raises the requirement for thecoupling strength A . Next, we show the sufficient conditionson A such that there is a locally asymptotically stable remotesynchronization manifold for (10). Towards this end, let y i =( θ − θ i ) / , y i ∈ S for i = 1 , . . . , n . The time derivativeof y i is ˙ y i = 12 ( ω − ω ) + A n n (cid:88) j =1 sin( θ j − θ − α ) − A sin( θ − θ i − α )= 12 ( ω − ω ) − A n n (cid:88) j =1 sin(2 y j + α ) − A sin(2 y i − α ) := g i ( y ) , i = 1 , , . . . , n. (11)where y = [ y , . . . , y n ] (cid:62) and g ( y ) = [ g ( y ) , . . . , g n ( y )] (cid:62) .Let us provide the main result in this section. Theorem 2:
There is a unique locally asymptotically sta-ble equilibrium y ∗ for the dynamics (11) satisfying | y ∗ i | <π/ for all i , which is y ∗ = 12 arcsin (cid:18) ω − ω A cos α (cid:19) n , (12) if the following conditions are satisfied, respectively:i) when ω > ω , the coupling strength A satisfies A > ω − ω α ; (13)ii) when ω < ω , the coupling strength A satisfies A > ω − ω α . (14)This locally asymptotically stable equilibrium y ∗ for thedynamics (11) corresponds to the locally asymptoticallystable remote synchronization manifold for (10). Proof:
The proof is similar to that of Theorem 1.The idea is to evaluate the Jacobian matrix J ( y ) at theequilibrium y = y ∗ . We prove that all the eigenvalues of J ( y ) have negative real parts by showing that J ( y ) has negativediagonal elements and is diagonally dominant if hypotheses(13) and (14) are satisfied for the cases i) and ii) respectively.This implies the stability of the equilibrium y ∗ . Remark 2:
Theorem 2 provides some sufficient conditionsfor the existence and local stability of the equilibrium ofdynamics (11), or equivalently, for the existence and localstability of remote synchronization manifold of (10). Withthe presence of the phase shift α , the requirement of couplingstrengths is increased. In fact, the larger the phase shift is,the stronger the coupling is required, which can be observedfrom (13) and (14). Interestingly, comparing (14) with (13)we observe that the phase shift has a different impact onthe coupling strength in the two cases when ω > ω and ω < ω . The latter case is more vulnerable to phase shift.V. N UMERICAL E XAMPLES
To validate the results we obtained in Section III andSection IV, we perform some numerical studies in thissection. We consider oscillators coupled by a directedstar network illustrated in Fig. 1. To measure the levels ofsynchronization we introduce the two useful functions asfollows, h ( θ ( t )) = max i,j ∈N | θ i ( t ) − θ j ( t ) | ,h ( θ ( t )) = max i ∈N | θ ( t ) − θ i ( t ) | , If h = 0 , the phase difference between any peripheraloscillator and the central one is zero, which implies com-plete synchronization in the whole network. In particular,if h = 0 , h (cid:54) = 0 , all the phases of peripheral oscillatorsare identical remaining central one different, which yieldsremote synchronization.We first testify the results obtain in Theorem 1. To distin-guish the frequencies, let the frequency of each peripheraloscillator be ω = 0 . π rad/s , and the natural frequencyof the central one be ω = 1 . π rad/s . In order to makecomplete remote synchronization occur, we let A i = 1 . forall i = 1 , , . . . , , and let K = 0 . , K = 0 . , K =0 . , K = 0 . , K = 0 . , K = 0 . . Then the matrix T aa Fig. 1. The star network considered: central node and peripheral ones { , , , , , } .Fig. 2. Trajectories of the maximum absolute values of the phasedifferences when α = 0 : blue represents h = max i,j ∈N | θ i − θ j | andred represents h = max i ∈N | θ − θ i | . becomes T = .
55 0 .
12 0 . .
18 0 . . .
15 1 .
52 0 . .
18 0 . . .
15 0 .
12 1 . .
18 0 . . .
15 0 .
12 0 . .
58 0 . . .
15 0 .
12 0 . .
18 1 . . .
15 0 .
12 0 . .
18 0 . . . It can be verified that T is diagonal dominated and | T − ω | = 0 . < , i.e. conditions in Assump-tion 1 are satisfied. Let the initial phases be θ (0) =[1 . π, . π, . π, . π, . π, . π, . π ] (cid:62) , and then thetrajectories of h ( θ ( t )) and h ( θ ( t )) are presented in Fig.2. It can be observed that h ( θ ( t )) converges to zero,while h ( θ ( t )) converges to a constant, suggesting thatthe peripheral oscillators which are not directly connectedachieve phase synchronization, but the ones that have directconnections (the central one with each peripheral one) donot. Next, we show that cluster synchronization is formed ifthe conditions in Corollary 1 are satisfied. Let the outgoingcoupling strengths be A = A = 2 . , A = A = 2 . , A = A = 4 . , and let the incoming coupling strength be thesame as considered above. One can also check Assumption1 is satisfied since | T − ω | = 0 . < . Let θ (0) =[1 . π, . π, . π, π, . π, . π, π ] (cid:62) , and the phases of theoscillators are plotted on the unit circle S at a sequenceof time instants (see Fig. 3). One can observe that theperipheral oscillators with the same outgoing strength A i get phase synchronized, forming three clusters (in each ofwhich phases are different from the central one’s). Thissuggests that the symmetry of the outgoing couplings of the (a) t = 0 (b) t = 0 . (c) t = 1 (d) t = 2 (e) t = 4 (f) t = 8 Fig. 3. The phases on S at six time instants when α = 0 : black representsthe central oscillator ; blue represents oscillators and ; green represents and ; red represents and .(a) ω < ω, A = 1 . π (b) ω < ω, A = π (c) ω > ω, A = 0 . π (d) ω > ω, A = 0 . π Fig. 4. Trajectories of the maximum absolute values of the phasedifferences when α = π/ : blue represents h = max i,j ∈N | θ i − θ j | and red represents h = max i ∈N | θ − θ i | . peripheral oscillators plays an essential role in facilitatingremote synchronization.Next, we validate the results in Section IV, where there isa phase shift α . Without loss of generality, let α = π/ . First,we consider the case when ω < ω . Let the frequency of eachperipheral oscillator be ω = 0 . π , and the natural frequencyof the central one be ω = 0 . π . From and the condition (14),we calculate the threshold of the coupling strength A , whichis ( ω − ω ) / (2 cos α ) = 1 . π . Let A = 1 . π > . π , andwe plot the absolution value of phase differences h ( θ ( t )) and h ( θ ( t )) in Fig 4(a), from which we observe that remotesynchronization is achieved. On the contrary, if we let A = π ,it can be seen from Fig. 4(b) that remote synchronizationoes not occur. Finally, we consider the case ω > ω by letting ω = 1 . π, ω = 0 . π . The threshold given in(13) becomes ( ω − ω ) / (2 cos α ) = 0 . π . The trajectoriesof h ( t ) and h ( t ) when A = 0 . π and A = 0 . π arepresented in Fig. 4(c) and 4(d), respectively. Shown is Fig.4(c), remote synchronization is achieved. Surprisingly, onecan observe from Fig. 4(d) that the phase differences amongperipheral oscillators approach zero, although the phasedifferences between the peripheral and the central oscillatorsare increasing. This implies remote synchronization can alsotake place without requiring that all the frequencies getsynchronized. We are currently working on the constructionof a mathematical proof for this observed phenomenon.VI. C ONCLUSION
Motivated by synchronization observed in distant corticalregions in human brain, especially neuronal synchrony ofunconnected areas through relaying, we have studied remotesynchronization of Kuramoto oscillators coupled by a starnetworks. We have shown that the symmetry of outgoingconnections from the central oscillator plays a critical role infacilitating phase synchronization between peripheral oscilla-tors. By carefully adjusting the strengths of these couplings,interesting patterns of stable remote synchronization, suchas cluster synchronization, can be achieved. We have alsostudied the case when there is a phase shift. Sufficientconditions have been obtained to ensure the stability ofremote synchronization. Simulations have been performed tovalidate our results. We are highly interested in generalizingour results on remote synchronization to more complexnetworks in the future. R
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