We consider an ensemble of coupled nonlinear noisy oscillators demonstrating
in the thermodynamic limit an Ising-type transition. In the ordered phase and
for finite ensembles stochastic flips of the mean field are observed with the
rate depending on the ensemble size. When a small periodic force acts on the
ensemble, the linear response of the system has a maximum at a certain system
size, similar to the stochastic resonance phenomenon. We demonstrate this
effect of system size resonance for different types of noisy oscillators and
for different ensembles -- lattices with nearest neighbors coupling and
globally coupled populations. The Ising model is also shown to demonstrate the
system size resonance.Comment: 4 page

A. A. Zaikin

A. Hamm

A. Longtin

A. Neiman

A. Pikovsky

A. S. Pikovsky

A. Simon

A. Zaikin

B. Hu

C. V. der Broeck

D. A. Dawson

D. F. Russell

H. Gang

J. F. Lindner

J. Javier Brey

J. K. Douglas

J. M. Casado

J. Miller

L. Gammaitoni

M. A. de la Casa

M. E. J. Newmann

M. L. Spano

M. Morillo

P. A. Tass

P. Jung

P. Landa

R. Benzi

R. C. Desai

S. Barbay

U. Siewert

Z. Néda

English

arXiv

Crossref

Physical Review Letters

System Size Resonance in Coupled Noisy Systems and in the Ising Model

We consider an ensemble of coupled nonlinear noisy oscillators demonstrating in the thermodynamic limit an Ising-type transition. In the ordered phase and for finite ensembles stochastic flips of the mean field are observed with the rate depending on the ensemble size. When a small periodic force acts on the ensemble, the linear response of the system has a maximum at a certain system size, similar to the stochastic resonance phenomenon. We demonstrate this effect of system size resonance for different types of noisy oscillators and for different ensembles-lattices with nearest neighbors coupling and globally coupled populations. The Ising model is also shown to demonstrate the system size resonance

Pikovsky, A

Zaikin, A

de la Casa, MA

UCL Discovery

VOLUME 88, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 4 FEBRUARY 2002050601-System Size Resonance in Coupled Noisy Systems and in the Ising ModelA. Pikovsky and A. ZaikinDepartment of Physics, University of Potsdam, Postfach 601553, D-14415 Potsdam, GermanyM. A. de la CasaDepartamento Física Fundamental, Universidad Nacional de Educación a Distancia, 28040 Madrid, Spain(Received 8 March 2001; published 16 January 2002)We consider an ensemble of coupled nonlinear noisy oscillators demonstrating in the thermodynamiclimit an Ising-type transition. In the ordered phase and for finite ensembles stochastic flips of the meanfield are observed with the rate depending on the ensemble size. When a small periodic force acts onthe ensemble, the linear response of the system has a maximum at a certain system size, similar to thestochastic resonance phenomenon. We demonstrate this effect of system size resonance for differenttypes of noisy oscillators and for different ensembles— lattices with nearest neighbors coupling andglobally coupled populations. The Ising model is also shown to demonstrate the system size resonance.DOI: 10.1103/PhysRevLett.88.050601 PACS numbers: 05.40.Ca, 05.45.–a, 05.50.+qStochastic resonance has attracted much interest re-cently [1]. As was demonstrated in [2], a response of anoisy nonlinear system to a periodic forcing can exhibit aresonancelike dependence on the noise intensity. In otherwords, there exists a “resonant” noise intensity at whichthe response to a periodic force is maximally ordered. Sto-chastic resonance has been observed in numerous experi-ments [3]. Noteworthy, the order in a noise-driven systemcan have a maximum at a certain noise level even in theabsence of periodic forcing, this phenomenon being calledcoherence resonance [4].Being first discussed in the context of a simple bistablemodel, stochastic resonance has been also studied in com-plex systems consisting of many elementary bistable cells[5]; moreover, the resonance may be enhanced due to cou-pling [6]. In this paper we discuss another type of reso-nance in such systems, namely, the system size resonance,when the dynamics is maximally ordered at a certain num-ber of interacting subsystems. Contrary to previous reportsof array-enhanced stochastic resonance (cf. also [7]), herewe fix the noise strength, coupling, and other parameters;only the size of the ensemble changes.The basic model to be considered below is the ensembleof noise-driven bistable overdamped oscillators, governedby the Langevin equations,xi  xi 2 x3i 1´NNXj1xj 2 xi 1p2D jit 1 ft .(1)Here jit is a Gaussian white noise with zero mean:ji tjjt0  dijdt 2 t0; ´ is the coupling constant;N is the number of elements in the ensemble, and ft isa periodic force to be specified later. In the absence of pe-riodic force, the model (1) has been extensively studied inthe thermodynamic limitN ! `. It demonstrates an Ising-type phase transition at ´  ´c from the disordered statewith vanishing mean field X  N21Pi xi to the “ferro-magnetic” state with a nonzero mean field X  6X0 [8].1 0031-90070288(5)050601(4)$20.00While in the thermodynamic limit the full description ofthe dynamics is possible, for finite system sizes we havemainly a qualitative picture: In the ordered phase the meanfield X switches between the values 6X0 and its averagevanishes for all couplings. The rate of switchings dependson the system size and tends to zero as N ! `.For us, the main importance is the fact that qualitativelythe behavior of the mean field can be represented as thenoise-induced dynamics in a potential with one minimumin the disordered phase (at X  0) and two symmetric min-ima (at X  6X0) in the ordered phase. Now, applyingthe ideas of the stochastic resonance, one can expect in thebistable case (i.e., in the ordered phase for small enoughnoise or for large enough coupling) a resonantlike behav-ior of the response to a periodic external force when theintensity of the effective noise is changed. Because thisintensity is inverse proportional to N , we obtain the reso-nancelike curve of the response in dependence of the sys-tem size. The main idea behind the system size resonanceis that in finite ensembles of noise-driven or chaotic sys-tems the dynamics of the mean field can be representedas driven by the effective noise whose variance is inverseproportional to the system size [9]. This idea has been ap-plied to the description of a transition to collective behav-ior in [10]. In [11] it was demonstrated that the finite-sizefluctuations can cause a transition that disappears in thethermodynamic limit.Before proceeding to a quantitative analytic descriptionof the phenomenon, we illustrate it with direct numericalsimulations of the model (1), with a forcing term ft A cosVt. Figure 1 shows the linear response function,i.e., the ratio of the spectral component in the mean fieldat frequency V and the amplitude of forcing A, in thelimit A ! 0. For a given frequency V the dependence onthe system size is a bell-shaped curve, with a pronouncedmaximum. The dynamics of the mean field Xt is il-lustrated in Fig. 2, for three different system sizes. Theresonant dynamics (Fig. 2b) demonstrates a typical for© 2002 The American Physical Society 050601-1VOLUME 88, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 4 FEBRUARY 20020.00010.0010.010.1Ω020406080N50010001500linear responseFIG. 1. Linear response of the ensemble (1) (D  0.5, ´  2)in dependence on the frequency and the system size N .stochastic resonance synchrony between the driving peri-odic force and the switchings of the field between the twostable positions.To describe the system size resonance analytically, weuse, following [8], the Gaussian approximation. In thisapproximation, one writes xi  X 1 di and assumesthat di are independent Gaussian random variables withzero mean and the variance M. Assuming furthermorethat N21Pi d2i  M and neglecting the odd momentsN21Pi di, N21Pi d3i , as well as the correlations betweendi and dj, we obtain from (1) the equations for X and M:X  X 2 X3 2 3MX 1s2DNht 1 ft , (2)12M  M 2 3X2M 2 3M2 2 ´M 1 D , (3)0 1000 2000 3000time t−101−101mean field X−101 a)b)c)FIG. 2. The time dependence of the mean field in the ensemble(1) for D  0.5, ´  2, A  0.02, V  p300, and differentsizes of the ensemble: (a) N  80, (b) N  35, and (c) N 15. We also depict the periodic force (its amplitude is not inscale) to demonstrate the synchrony of the switchings with theforcing in (b).050601-2where h is the Gaussian white noise having the sameproperties as jit. In the thermodynamic limit N ! `the noisy term h vanishes. If the forcing term is absent f  0, the equations coincide with those derived in [8].This system of coupled nonlinear equations exhibits apitchfork bifurcation of the equilibrium X  0, M . 0 at´c  3D. This bifurcation is supercritical for D . 23in accordance with the exact solution of (1) given in [8];below we consider only this case. For ´ . ´c the systemis bistable with two symmetric stable fixed points,X20  2 2 ´ 1 S4, M0  2 1 ´ 2 S12 (4)[here S p2 1 ´2 2 24D ], and the unstable pointX  0, M  1 2 ´ 1p1 2 ´2 1 12D 6. Now,with the external noise h and with the periodic forceft, the problem reduces to a standard problem in thetheory of stochastic resonance, i.e., to the problem of theresponse of a noise-driven nonlinear bistable system to anexternal periodic force (because the noise affects only thevariable X, it does not lead to unphysical negative valuesof variance M, since M is strictly positive at M  0).To obtain an analytical formula, we perform further sim-plification of the system (2) and (3). Near the bifurcationpoint, we can use the slaving principle to obtain a standardnoise-driven bistable system:X  aX 2 bX3 1s2DNht 1 ft , (5)where a  1 1 0.5´ 2 1 2 0.5p´ 2 12 1 12D,b  20.5 1 1.5´ 2 1 ´ 2 12 1 12D212. A bet-ter approximation valid also beyond a vicinity of the criti-cal point can be constructed if we use b¯  aX220 insteadof b, where the fixed point X0 is given by (4). Havingwritten the ensemble dynamics as a standard noise-drivendouble-well system (5) (cf. [1,12]), we can use the ana-lytic formula for the linear response R derived in [12]. ItreadsR NX202DaµD2322ps D2122ps ∂2∑1 1p2V22a2exps∏21,(6)where s  aNX202D, and D are the parabolic cylin-der functions. We compare the theoretical linear responsefunction with the numerically obtained one in Fig. 3. Thequalitative correspondence is good; moreover, the max-ima of the curves are rather good reproduced with theformula (6).Above, we concentrated on the properties of the linearresponse. Numerical simulations with the finite forcingamplitude yielded the results similar to that presentedin Figs. 1 and 3. However, for large amplitudes offorcing (e.g., A . 0.1 for V  0.01, D  0.5, ´  2)a saturation was observed: Here the response growsmonotonically with N . This is in full agreement with thecorresponding property of the stochastic resonance indouble-well systems of type (5), where the saturation050601-2VOLUME 88, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 4 FEBRUARY 20020 20 40 60 80N020406080linear response10−2 10−1Ω020406080NmaxFIG. 3. Comparison of the system size dependencies of thelinear response function for frequencies V  0.05 (circles) andV  0.1 (squares) with theory (6). The parameters are D  1and ´ 2 ´c  2.5 (where the exact ´c and the approximate´c  3D are used for the ensemble and the Gaussian approxi-mation, respectively). Inset: Dependence of the system sizeyielding maximal linear response on the driving frequency V[circles: simulations of the ensemble (1), line is obtained bymaximizing the expression (6)].occurs for small noise intensities (cf. Fig. 7 in [1]), dueto the disappearance of multistability for large forcingamplitudes.It is instructive to compare the response of the noise-driven system (1) with the noise-free case D  0. Withoutexternal force, the ensemble relaxes eventually to a steadystate solution with some mean field X; in this state eachoscillator can be in one of the stable steady positions ofthe potential; correspondingly, the oscillators form one ortwo clusters. From the clustering it follows that the linearresponse does not depend on the number of elements inthe ensemble. Our numerical experiments demonstratedalso that the response is system size independent for largeforcing amplitudes as well, where, e.g., the force-inducedcluster mergings occur. Thus, the effect of system sizeresonance essentially relies on the presence of noise, whichbreaks the clustering.Above, we have considered the system of globally cou-pled nonlinear oscillators (1). The same effect of systemsize resonance can be observed in a lattice with nearestneighbors coupling as well. In the thermodynamic limit,the Ising-type phase transition occurs in the lattice (if itsdimension is larger than one). Similar to the globallycoupled ensemble, in finite lattices in the ordered phasethe switchings between the two stable states of the meanfield are observed. With the same argumentation as above,we can conclude that the response of the mean field toa periodic forcing can have a maximum at a certain lat-tice size, while all other parameters (noise intensity, cou-050601-3pling strength, etc.) are kept constant. We illustrate thisin Fig. 4.As the next example we consider the two-dimensionalnearest neighbor Ising model in the presence of a time-dependent external field. The Hamiltonian of the systemreadsH  2JXijsisj 2 A cosVtXisi , (7)where J . 0 and si  61. We are interested in the depen-dence of the response of the mean magnetization mt 1NPi sit on the system size N (for the usual stochasticresonance in the Ising model, i.e., for the dependence ofthe response on the temperature, see [13]). To calculate thelinear response, we used the fluctuation-dissipation theo-rem and obtained this quantity by virtue of the power spec-trum of fluctuations of mt. The latter was found usingthe Metropolis Monte Carlo method on a lattice with he-lical boundary conditions [14]. The results presented inFig. 5 demonstrate the system size resonance of the linearresponse in the two-dimensional Ising model.As the last example of the system size resonance, weconsider a lattice where each individual element doesnot exhibit bistable noisy dynamics, but such a behaviorappears due to interaction and multiplicative noise. Thismodel is described by the set of Langevin equations[15,16]:xi  2xi1 1 x2i 2 1´KXjxj 2 xi1p2D jit 1 1 x2i  1 ft . (8)As has been demonstrated in [15], in some region of cou-plings the ´ system (8) exhibits the Ising-type transition.0 20 40 60 80 100N00.050.10.150.2responseFIG. 4. Filled circles: Response of a two-dimensional latticeof N with nearest neighbors coupling for A  0.02, T  500,D  0.5, and ´  4. Squares: Response of system (8) (a two-dimensional lattice with D  1.25, ´  30, A  0.1, and T 140). Circles: The same as squares, but for a globally coupledlattice with D  1, ´  20, A  0.1, and T  100.050601-3VOLUME 88, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 4 FEBRUARY 20020.00010.0010.01Ω020406080100Nlinear responseFIG. 5. Linear response (in arbitrary units) of the Ising model(7) for the temperature T  2J slightly below the critical tem-perature Tc  2.269J .If an additional additive noise is added to (8), then oneobserves transitions between these states and the so-calleddouble stochastic resonance in the presence of the periodicforcing [17]. As is evident from the considerations above,such transitions occur even in the absence of the additivenoise if the system is finite. Thus, the system size reso-nance should be observed in the lattice (8) as well. Weconfirm this in Fig. 4.Another possible field of application of the system sizeresonance is the neuronal dynamics (see, e.g., [18]). Indi-vidual neurons have been demonstrated to exhibit stochas-tic resonance [3,19]. While in experiments one can easilyadjust noise to achieve the maximal sensitivity to an ex-ternal signal, it may not be obvious how this adjustmenttakes place in nature. The above consideration shows thatchanging the number of elements in a small ensemble ofcoupled bistable elements to the optimum can significantlyimprove the sensitivity (cf. [5]). Moreover, changing itsconnectivity and/or coupling strength, a neuronal systemcan tune itself to signals with different frequencies.In conclusion, we have shown that, in populations ofcoupled noise-driven elements, exhibiting in the thermody-namic limit the Ising-type transition, in the ordered phase(i.e., for relatively small noise and large coupling) the re-sponse to a periodic force achieves maximum at a certainsize of the system. We demonstrated this effect for theIsing model, as well as for lattices and globally coupledensembles of noisy oscillators. We expect the system sizeresonance to occur also in purely deterministic systemsdemonstrating the Ising-type transition, e.g., in the Miller-Huse coupled map lattice [20]. 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Tass, Phase Resetting in Medicine and Biology:Stochastic Modelling and Data Analysis (Springer-Verlag,Berlin, 1999).[19] J. K. Douglas, L. Wilkens, E. Pantazelou, and F. Moss,Nature (London) 365, 337 (1993); D. F. Russell, L. A.Wilkens, and F. Moss, Nature (London) 402, 291 (1999).[20] J. Miller and D. A. Huse, Phys. Rev. E 48, 2528 (1993).050601-4

System size resonance in coupled noisy systems and in the Ising model

System size resonance in coupled noisy systems and in the Ising model

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