Correlations, fluctuations, and stability of a finite-size network of coupled oscillatorsMichael A. Buice and Carson C. ChowLaboratory of Biological Modeling, NIDDK, NIH, Bethesda, Maryland 20892, USA共Received 17 April 2007; published 13 September 2007兲The incoherent state of the Kuramoto model of coupled oscillators exhibits marginal modes in mean fieldtheory. We demonstrate that corrections due to finite size effects render these modes stable in the subcriticalcase, i.e., when the population is not synchronous. This demonstration is facilitated by the construction of anonequilibrium statistical field theoretic formulation of a generic model of coupled oscillators. This theory isconsistent with previous results. In the all-to-all case, the fluctuations in this theory are due completely to finitesize corrections, which can be calculated in an expansion in 1 /N, where N is the number of oscillators. TheN→⬁ limit of this theory is what is traditionally called mean field theory for the Kuramoto model.DOI: 10.1103/PhysRevE.76.031118 PACS number共s兲: 05.70.Ln, 05.45.Xt, 05.10.Gg, 11.10.⫺zI. INTRODUCTIONSystems of coupled oscillators have been used to describethe dynamics of an extraordinary range of phenomena 关1兴,including networks of neurons 关2,3兴, synchronization ofblinking fireflies 关4,5兴, chorusing of chirping crickets 关6兴,neutrino flavor oscillations 关7兴, arrays of lasers 关8兴, andcoupled Josephson junctions 关9兴. A common model ofcoupled oscillators is the Kuramoto model 关10兴, which de-scribes the evolution of N coupled oscillators. A generalizedform is given by␪˙i=␻i+KN兺jf共␪j−␪i兲, 共1兲where i labels the oscillators,␪iis the phase of oscillator i,f共␪兲 is the phase dependent coupling, and the intrinsic driv-ing frequencies␻iare distributed according to some distri-bution g共␻兲. In the original Kuramoto model, f共␪兲=sin共␪兲.Here, we consider f to be any smooth odd function. Thesystem can be characterized by the complex order parameterZ共t兲 =兺jei␪j共t兲⬅ r共t兲ei⌿共t兲, 共2兲where the magnitude r gives a measure of synchrony in thesystem.In the limit of an infinite oscillator system, Kuramotoshowed that there is a bifurcation or continuous phase tran-sition as the coupling K is increased beyond some criticalvalue Kc关10兴. Below the critical point the steady state solu-tion has r =0 共the “incoherent” state兲. Beyond the criticalpoint, a new steady state solution with r⬎ 0 emerges. Stro-gatz and Mirollo analyzed the linear stability of the incoher-ent state of this system using a Fokker-Planck formalism关11兴. In the absence of external noise, the system displaysmarginal modes associated with the driving frequencies ofthe oscillators. However, numerical simulations of the Kura-moto model for a large but finite number of oscillators showthat the oscillators quickly settle into the incoherent statebelow the critical point. The paradox of why the marginallystable incoherent state seemed to be an attractor in simula-tions was partially resolved by Strogatz, Mirollo, and Mat-thews 关12兴 who demonstrated 共within the context of the N→ ⬁ limit兲 that there was a dephasing effect akin to Landaudamping in plasma physics which brought r to zero with atime constant that is inversely proportional to the width ofthe frequency distribution. Recently, Strogatz and Mirollohave shown that the fully locked state r =1 is stable 关13兴 butthe partially locked state is again marginally stable 关14兴. Al-though dephasing can explain how the order parameter cango to zero, the question of whether the incoherent state istruly stable for a finite number of oscillators remains un-known. Even with dephasing, in the infinite oscillator limitthe system still has an infinite memory of the initial state sothere may be classes of initial conditions for which the orderparameter or the density exhibits oscillations.The applicability of the results for the infinite size Kura-moto model to a finite size network of oscillators is largelyunknown 关34兴. The intractability of the finite size case sug-gests a statistical approach to understanding the dynamics.Accordingly, the infinite oscillator theories should be thelimits of some averaging process for a finite system. Whilethe behavior of a finite system is expected to converge to the“infinite” oscillator behavior, for a finite number of oscilla-tors the dynamics of the system will exhibit fluctuations. Forexample, Daido 关15,16兴 considered his analytical treatmentsof the Kuramoto model using time averages and he was ableto compute an analytical estimate of the variance. In contrast,we will pursue ensemble averages over oscillator phases anddriving frequencies. As the Kuramoto dynamics are deter-ministic, this is equivalent to an average over initial phasesand driving frequencies. Furthermore, the averaging processimparts a distinction between the order parameter Z and itsmagnitude r. Namely, do we consider 具Z典 or 具r典= 具兩Z兩典 to bethe order parameter? This is important as the two are notequal. In keeping with the density as the proper degree offreedom for the system 共as in the infinite oscillator theoriesmentioned above兲, we assert that 具Z典 is the natural orderparameter, as it is obtained via a linear transformation ap-plied to the density.Recently, Hildebrand et al. 关17兴 produced a kinetic theoryinspired by plasma physics to describe the fluctuationswithin the system. They produced a Bogoliubov-Born-Green-Kirkwood-Yvon 共BBGKY兲 moment hierarchy andtruncation at second order in the hierarchy yielded analyticalresults for the two point correlation function from which thefluctuations in the order parameter could be computed. AtPHYSICAL REVIEW E 76, 031118 共2007兲1539-3755/2007/76共3兲/031118共25兲 ©2007 The American Physical Society031118-1this order, the system still manifested marginal modes. Goingbeyond second order was impractical within the kinetictheory formalism. Thus, it remained an open question as towhether going to higher order would show that finite sizefluctuations could stabilize the marginal modes.Here, we introduce a statistical field theory approach tocalculate the moments of the distribution function governingthe Kuramoto model. The formalism is equivalent to the Doi-Peliti path integral method used to derive statistical fieldtheories for Markov processes, even though our