Coherent Dynamics Enhanced by Uncorrelated Noise
Zachary G. Nicolaou, Michael Sebek, Istvan Z. Kiss, Adilson E. Motter
CCoherent Dynamics Enhanced by Uncorrelated Noise
Zachary G. Nicolaou, Michael Sebek,
2, 3
Istv´an Z. Kiss, and Adilson E. Motter
1, 4, ∗ Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA Department of Chemistry, Saint Louis University, St. Louis, Missouri 63103, USA Network Science Institute, Northeastern University, Boston, Massachusetts, 02115, USA Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA (Received 8 March 2020; revised 17 June 2020; accepted 21 July 2020)Synchronization is a widespread phenomenon observed in physical, biological, and social networks,which persists even under the influence of strong noise. Previous research on oscillators subject tocommon noise has shown that noise can actually facilitate synchronization, as correlations in thedynamics can be inherited from the noise itself. However, in many spatially distributed networks,such as the mammalian circadian system, the noise that different oscillators experience can be ef-fectively uncorrelated. Here, we show that uncorrelated noise can in fact enhance synchronizationwhen the oscillators are coupled. Strikingly, our analysis also shows that uncorrelated noise canbe more effective than common noise in enhancing synchronization. We first establish these resultstheoretically for phase and phase-amplitude oscillators subject to either or both additive and mul-tiplicative noise. We then confirm the predictions through experiments on coupled electrochemicaloscillators. Our findings suggest that uncorrelated noise can promote rather than inhibit coherencein natural systems and that the same effect can be harnessed in engineered systems.DOI: https://doi.org/10.1103/PhysRevLett.125.094101
Phys. Rev. Lett. , 094101 (2020)
Synchronization, the phenomenon in which oscillatorsin a population evolve in step with each other, occursbecause of interactions or common driving forces amongoscillators. Influences from outside an oscillator networkcan often be treated as noise, which is usually expectedto inhibit synchronization. Indeed, small noise can resultin a disproportionately large degree of asynchrony in net-works of nonlocally coupled oscillators [1]. In other con-texts, while network disorder can improve synchronousparallel processing performance [2], noise has also beenfound to limit the permissible time delays in communi-cations for network synchronizability and hinders paral-lel performance [3]. Still, numerous biological systems—such as neural networks [4–6], ecological communities [7],and the cardiac and cardio-respiratory systems [8, 9]—and engineered systems—such as arrays of Josephsonjunctions [10], lasers [11], and nanoelectromechanical de-vices [12]—exhibit robust synchronization even under theinfluence of noise.Previous theoretical and experimental observationshave demonstrated that common noise (in which indi-vidual oscillators experience a shared noise term) can ac-tually induce rather than inhibit synchronization [13, 14].The understanding behind this phenomenon can betraced back to the study of coherence resonance, in whichnoise leads to greater temporal order in systems with ir-regular oscillations [15, 16]; to stochastic resonance [17],which has been used to reduce the threshold to detect tac-tile stimuli in human sensory perception [18]; and to the
Common noiseUncorrelated noise No noise
FIG. 1. Schematics of the main effect. Coupled oscillatorsexperiencing uncorrelated noise exhibit more synchronous dy-namics than those subjected to common noise or no noise.This behavior is distinct from those previously observed inoscillator models that synchronize due to large coupling or inresponse to specific forms of common noise. effects of common driving in synchronizing chaotic or dis-ordered systems [19, 20] as well as the synchronizing ef-fects of periodic driving with a spatially-dependent phase[21]. Synchronization induced by common noise has sincebeen studied in a variety of oscillator networks [22–24].Here, we establish the alternative scenario shown inFig. 1 in which the dynamics are more synchronous inthe presence of uncorrelated noise than in the absence ofnoise or even the presence of common noise. It seems in-tuitive that uncorrelated noise would necessarily inhibitsynchronization since, unlike the common noise case, itdoes not have inherent order. However, uncorrelatednoise is prevalent in many systems, and recent studies a r X i v : . [ n li n . AO ] A ug suggest the potential for uncorrelated noise to have apositive impact on synchronization. For example, cou-pled neuronal networks subject to uncorrelated noise canexhibit enhanced coherence across the networks while re-ducing the coherence within each network [25]. Uncorre-lated noise acting on a pair of oscillators has also beenshown to enhance the phase coherence of one oscillator atthe expense of the other [26]. Furthermore, uncorrelatednoise can promote untwisted phase-locked states overtwisted phase-locked states in small-world networks ofKuramoto oscillators [27] and can stabilize an otherwiseunstable partially synchronized state in a globally cou-pled model of oscillators with biharmonic couplings [28].However, the question of whether uncorrelated noise canenhance synchronization to a greater extent than com-mon noise had so far remained open.We establish our results for several forms of coupledlimit-cycle oscillators governed by d x i dt = f i ( x i ) + KN N X j =1 A ij h ( x i , x j ) + n X k =1 g ik ( X ) ξ ik , (1)where N is the number of oscillators, x i denotes thestate of oscillator i (assumed to be m dimensional), X = ( x , x , · · · , x N ) encodes the full state of the sys-tem, f i describes the evolution of the isolated oscillators, h is the coupling function between two oscillators, K isthe (tunable) coupling constant, and A ij are the entriesof the coupling matrix (assumed to be 1 if nodes i and j are coupled and 0 otherwise). We include n sourcesof noise determined by the state-dependent direction g ik and the random variable ξ ik , with h ξ ik i = h ξ ik ξ jl i = 0for all i, j, k ,and l = k . The ξ ik term represents mul-tiplicative noise in the case that g ik varies with X andrepresents additive noise in the special case that g ik isconstant.We assume that in the absence of coupling ( K = 0)and noise ( ξ ik = 0) the isolated node dynamics approacha limit cycle x i ( t ) → x ci ( t ), with x ci ( t ) = x ci ( t + T i ),where T i is the period of oscillator i . We can alwaysdefine a phase variable θ i ( x i ) for oscillator i that, whenrestricted to the limit cycle, evolves as θ i ( t ) = θ i (0) + ω i t , where ω i = 2 π/T i is the natural frequency. In thepresence of coupling, we consider the oscillators to bemore synchronized when their relative phase differencesare smaller on average. Following Kuramoto [29, 30],we employ the order parameter R ≡ | (1 /N ) P j e i θ j ( t ) | as a measure of synchrony, where i is the imaginary unit.The time-averaged order parameter R is closer to 1 whenoscillators are more synchronized and closer to 0 when theoscillators are less synchronized. In the results below, wesay that noise enhances synchronization if R is largerin the presence of noise than in the absence of noise.We consider two broad forms of noise: common noise,for which ξ ik = ξ jk for all i , j ; and uncorrelated noise,for which ξ ik and ξ jk are independent random variables for all i = j . We are primarily interested in cases inwhich uncorrelated noise enhances synchronization moreso than common noise. Phase-reduced oscillators. —For weakly coupled oscil-lators driven by weak noise, the phase-reduction approx-imation can be applied to reduce the dynamics of Eq. (1)to a Kuramoto-type model with noise, dθ i dt = ω i + KN X j A ij sin ( θ j − θ i ) + g i ( θ i ) η i . (2)The various noise terms in Eq. (1) result in a single ef-fective noise term g i ( θ i ) η i in the phase dynamics if, forinstance, they are all Gaussian variables with autocorre-lations of the same functional form (as shown in Sec. S1of the Supplemental Material [31]). The effective noise η i will be assumed to be Gaussian and white unless oth-erwise noted, with intensity specified by a matrix D ij as h η i ( t ) η j ( t ) i = D ij δ ( t − t ), where δ is the Dirac deltafunction. The function g i ( θ i ), called the phase sensitiv-ity function, arises because the effective noise acts on thephase evolution with varying intensity depending on thephase of the oscillator. In the case of common noise, η i = η j for all i , j and all the elements of the noise inten-sity matrix are identical, with D ij = σ / σ denotingthe noise intensity. In the case of uncorrelated noise, h η i η j i = 0 for i = j and the noise intensity matrix isdiagonal, with D ii = σ / N = 2 phase oscillatorswith g i ( θ i ) = 1, so that the multiplicative noise in Eq. (1)becomes additive in the phase approximation. By mov-ing to a rotating frame, it is possible to take the meannatural frequency equal to zero, so that, without loss ofgenerality, we can take ω = ∆ ω/ ω = − ∆ ω/ K is smaller than ∆ ω , as characterized by their separationangle φ ≡ θ − θ , while their mean angle Θ ≡ ( θ + θ ) / K increases above its critical value K c ≡ ∆ ω , ini-tially with a separation angle φ = − π/
2. In the presenceof Gaussian white noise with constant phase sensitivityand for any value of K , the evolution of the density ofan ensemble of systems ρ ( φ, Θ , t ) can be described by theFokker-Planck equation ∂ρ∂t = ∂∂φ h (∆ ω + K sin φ ) ρ i + σ (cid:20) ∂ ρ∂φ + 14 ∂ ρ∂ Θ (cid:21) , (3)where we have changed variables from θ and θ to φ and Θ. Because Eq. (3) is autonomous with respect toΘ and t , we can find steady solutions which are inde-pendent of the mean phase Θ. Direct integration in thiscase is possible using an integrating factor. After some -π - π π π ϕ ρ K / K c R (c)(b)(a) σ Δω c R UncorrelatedCommon
FIG. 2. Solutions of the Fokker-Planck equation (3) for two phase oscillators subject to Gaussian white noise with constantphase sensitivity g i ( θ ) = 1. (a) Steady ensemble density ρ in Eq. (4) as a function of the phase difference φ for the subcriticalcoupling K/K c = 0 .
95. (b) Time-averaged order parameter R as a function of the normalized coupling constant K/K c . Thearrows in (a) and (b) indicate the change in the solutions as σ increases from zero to 2 √ ∆ ω , where the zero-noise case (thickline) also corresponds to the case of common noise of any intensity. (c) Time-averaged order parameter R as a function ofthe normalized noise intensity σ/ √ ∆ ω at the coupling constant K/K c = 0 .
95, where the lines show the solutions from theFokker-Planck equation for uncorrelated noise (continuous) and common noise (dashed). The circles show the agreement withthe corresponding direct numerical simulations of Eq. (2). simplification, the solution is ρ ( φ )= A Z π dψ (cid:20) π ∆ ω/σ ) − H ( φ − ψ ) (cid:21) × exp (cid:2) ω ( ψ − φ ) /σ − K (cos ψ − cos φ ) /σ (cid:3) , (4)for 0 ≤ φ ≤ π , where A is a normalization constant and H is the Heaviside step function.Figure 2(a) shows how the steady ensemble density ρ varies as the noise intensity varies in the case with asubcritical coupling constant K = 0 . K c . The ensem-ble density, which is peaked near φ = − π/ R = R π cos ( φ/ ρ ( φ ) dφ . Figure2(b) shows how the time-averaged order parameter variesas the noise intensity varies. The sharp, phase-lockingtransition at K = K c is smoothed out as the noise in-tensity increases. For subcritical coupling constants, theorder parameter initially increases with increasing noiseintensity, indicating an enhancement in synchronizationin response to uncorrelated noise that does not occur inthe case of common noise. This is illustrated in Fig. 2(c)for the same K as in Fig. 2(a), but a qualitatively simi-lar effect occurs for all subcritical cases and for couplingconstants just above the critical one. For large couplingconstants, the system is already strongly synchronized inthe absence of noise and thus synchronization is not fur-ther enhanced by noise. Time averaging of trajectoriesfrom direct numerical simulations of Eq. (2) (see Sec. S2of the Supplemental Material [31]) agree extremely wellwith the solutions derived from the Fokker-Planck equa- tion, as illustrated in Fig. 2(c). Phase-amplitude oscillators. —We have shown thatphase oscillators can exhibit enhanced synchronizationunder uncorrelated noise but not under common noise,assuming the noise and coupling terms are weak so thatthe phase-reduction approximation applies. We nextconsider the question of synchronization enhancementin phase-amplitude oscillators experiencing strong noise.As a prototypical example, we consider N = 2 coupledStuart-Landau oscillators each with m = 2 degrees offreedom x i = (cid:0) x (1) i , x (2) i (cid:1) , which are conveniently repre-sented as a complex variable z i ( t ) = x (1) i ( t )+i x (2) i ( t ) andevolve according to dz i dt = F i ( z i ) + K X j =1 ( z i z ∗ j − z j z ∗ i ) z i + G ik ( z , z ) ξ ik , (5)where F i ( z i ) ≡ (1 + i α i ) z i − (1 − i γ i ) | z i | z i describes theintrinsic dynamics, α i and γ i are constants, and ∗ denotescomplex conjugation. The cubic form of the coupling inEq. (5) is selected to result in the Kuramoto-type cou-pling in the phase reduction, which facilitates compar-isons below. In the absence of coupling and noise (when K = 0, ξ ik = 0), the oscillators have a limit-cycle attrac-tor z i ( t ) = r i ( t ) e i θ i ( t ) , where r i ( t ) = 1, θ i ( t ) = θ i (0)+ ω i t ,and ω i = α i + γ i .Figure 3(a) shows the noise forces in the statespace of a Stuart-Landau oscillator for three formsof noise determined by differing G ik . In each case,the tangent of the noise force along the limit cy-cle determines the phase sensitivity function in theweak-noise regime for which the phase reduction wouldhold. As we proceed with our analysis of the strong-noise regime, it is instructive to compare with pre-dictions for phase-reduced oscillators. The phase sen-sitivity function in the phase reduction for Eq. (5)takes the form g ik ( θ i ) = R π dθ j (cid:2) G ik ( e i θ , e i θ ) e − i θ i − G ∗ ik ( e i θ , e i θ ) e i θ i (cid:3) / π , where j = i [31]. - - - - x ( ) x ( ) (a) (b)(c)(d) R R σ R FIG. 3. Impact of noise on phase-amplitude oscillators.(a) State space of a Stuart-Landau oscillator, indicating thevelocity field in the absence of noise and coupling (continuouslines), the limit-cycle attractor (dashed circle), and the noiseforces (arrows). Three forms of noise are represented: Gaus-sian white noise with G i (vertical arrows), Gaussian whitenoise with G i (counterclockwise arrows), and Gamma dis-tributed noise with G i , which acts in the direction of theuncoupled velocity field (continuous lines) when the couplingis small. (b)-(d) Time-averaged order parameter R as a func-tion of the noise intensity σ for correlated (open circles) anduncorrelated (filled circles) noise corresponding to G i (b), G i (c), and G i (d). The oscillator parameters are α = 1and α = γ = γ = 0, and the coupling constant is K = 0 . Taking additive noise with G i ( z , z ) = i resultsin multiplicative noise in the phase reduction witha trigonometric sensitivity function g i ( θ i ) = cos( θ i ),which is expected to induce synchronization under com-mon noise. On the other hand, taking G i ( z , z ) = i z i results in additive noise in the phase reduction, with aconstant sensitivity function g i ( θ i ) = 1. Noise that ismodulated by the noiseless part of the dynamics, with G i ( z , z ) = F i ( z i ) + ( K/ P j (cid:0) z i z ∗ j − z j z ∗ i (cid:1) z i , also re-sults in additive noise in the phase reduction. We thusexpect, based on our results above for phase oscillators,that uncorrelated noise will enhance synchronization butcommon noise will not for G i and G i also in the unre-duced system.Figure 3(b)-(d) assess these predictions through directnumerical simulations. For G i , common Gaussian whitenoise enhances synchronization significantly, as antici-pated above, but uncorrelated noise also enhances syn-chronization to some extent. For G i , uncorrelated Gaus-sian white noise enhances synchronization while commonnoise does not, which once again agrees with the predic-tion above. To assess if these predictions continue to holdunder non-Gaussian noise, we employed noise sampledfrom Gamma distribution for G i , which is dominatedby brief, high-intensity bursts (see Sec. S2 of the Sup-plemental Material [31]). It is interesting to note that,while the enhancement for Gaussian cases are qualita-tively similar to the phase approximation in Fig. 2(c),in the non-Gaussian case, the enhancement continues togrow with increasing noise intensity over the same noise range.In summary, while the synchronization enhancement inEq. (5) depends on specific noise features, uncorrelatednoise continues to enhance synchronization beyond thephase-reduction approximation in cases where commonnoise does not. Electrochemical oscillator experiments. —To testwhether the effect described above can be observed inreal limit-cycle systems, we performed experiments oncoupled electrochemical oscillators. These oscillators,detailed in Supplemental Material Sec. S3 [31] alongwith sample experimental trajectories, are described by m = 2 degrees of freedom, which represent the electrodepotential and the concentration of the electroactivespecies in the vicinity of the electrode [32]. UnlikeStuart-Landau oscillators, the limit cycle in this caseis not circular in the state space, and thus the phaseis a complicated function that we will not attempt todescribe analytically. The experimental system consistsof two such electrochemical oscillators coupled togetherthrough a resistor and the shared fluid environment.For statistical analysis, we create several realizationsof the experimental system, with each realization havingslightly different natural frequencies and being subject tono noise, common noise, and uncorrelated noise. The ex-periments are repeated for three levels of noise intensity,and the time-averaged order parameter is measured foreach experimental run to assess synchronization. Figure4 shows the statistical analysis for these experiments. Wefind that for low noise intensity, there is no statisticallysignificant difference between the cases of no noise, uncor-related noise, and common noise. For intermediate noiseintensities, uncorrelated noise enhances synchronizationsignificantly more so than common noise, confirming theeffect described above. For high noise intensity, commonnoise exhibits a greater synchronization enhancement. (a) (b) (c) N ono i s e U n c o rr e l a t ed C o mm on R N ono i s e U n c o rr e l a t ed C o mm on N ono i s e U n c o rr e l a t ed C o mm on Low noise intensity Intermediate noise intensity High noise intensity
FIG. 4. Electrochemical oscillator experiments with Gaus-sian white noise added to the electrode potentials. (a)–(c) Bar plots of R at three different noise intensities, where D = 0 .
025 V for 14 realizations (a), D = 0 .
05 V for 22 re-alizations (b), and D = 0 .
10 V for 10 realizations (c). Errorbars indicate the estimated errors in the means (i.e., the stan-dard deviation normalized by the square root of the numberof realizations), and the arrows between bar plots go from thesmaller mean value to the larger mean value with percentagesindicating the confidence from a paired t test. We emphasize that, in these experiments, we did notattempt to control the direction of the noise force, giventhat noise can be easily applied only to the electrode po-tential and not to the chemical concentration, nor didwe attempt to determine (or fine-tune) the phase sensi-tivity function, given the complexity of the limit cycle.Nevertheless, we still observe a greater degree of synchro-nization enhancement for the case of uncorrelated noisethan for the case of common noise for intermediate noiseintensity. Thus, these experiments reveal that uncorre-lated noise can outperform common noise in synchroniza-tion enhancement even without careful design.
Discussion. —Our demonstration that uncorrelatednoise can enhance synchronization to a greater degreethan common noise reveals a new mechanism for howcoherent behavior can emerge naturally in spatially-distributed noisy systems. The mechanism that gener-ates this noise-enhanced synchronization can be inter-preted as follows. On the one hand, when coupled oscil-lators are close to phase locking, they often spend timeat relative angles that are far from zero, and their phasesdo not add coherently. On the other hand, uncorre-lated noise allows the oscillators to escape from theselarge phase separations and spend more time with similarphases, even when common noise cannot do so preciselybecause it exerts the same effect on the phases of all os-cillators. Our analysis indicates that this effect occursprominently when the impact of the noise on coupled os-cillators is independent of their phases, which means thatthe coherence is not inherited from a biased filtering ofthe noise.These findings are counterintuitive because the noiseterms acting on different oscillators exhibit permutationsymmetry for common noise but not for uncorrelatednoise; yet, for the systems considered here, the resultingdynamical states are more symmetric in the uncorrelatedcase. Such synchronization enhancement can thus be in-terpreted as a manifestation of asymmetry-induced sym-metry [33], a recently recognized phenomenon in whichsome degree of asymmetry in a system actually increasesthe symmetry in the observed state of that system.In this study, we observed the preferential enhance-ment of coherence by uncorrelated noise over commonnoise in a variety of coupled oscillator systems, includ-ing phase and phase-amplitude oscillators, both theo-retically and experimentally. While we focused here onpairs of oscillators for clarity, we can show that this ef-fect also occurs more generally in larger networks witha frequency gap [34], such as random networks of Janusoscillators [35, 36] (see Sec. S4 of the Supplemental Ma-terial [31]) and multilayer networks relevant to the dis-tributed mammalian neural and circadian systems [37–40] (see Sec. S5 of the Supplemental Material [31]). 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CONTENTS
S1. Effective noise in phase-reduced model 1S2. Direct numerical integration and calculation of R S1 . EFFECTIVE NOISE IN PHASE-REDUCED MODEL The model in Eq. (2) is derived here from the model in Eq. (1) by adapting the phase reductiontechnique in Ref. [30]. To benefit from this technique, we assume that ξ ik and the coupling term areall small and can be accounted for by their leading order contributions. Because these quantitiesare small, we can also assume that the time evolution remains close to the limit cycle. The phase θ i is defined as a function of the state x i such that ∇ x θ i ( x ci ) · f i ( x ci ) = ω i , ensuring that the phaseincreases linearly with the natural frequency on the limit cycle when the oscillators are uncoupled.In the presence of coupling and noise, the evolution of the phase is then given by dθ i dt = ω i + ∇ x θ i · h KN N X j =1 A ij h ( x i , x j ) + n X k =1 g ik ( X ) ξ ik i , (S1)where we used that dθ i /dt = ∇ x θ i · d x i /dt .Taking the leading order approximation, the gradient ∇ x θ i is replaced with the value at thecorresponding point on the limit cycle, which we denote by Z i ( θ i ). Noting that the difference θ i − ω i t is a slow variable, its evolution can be determined by calculating time averages of the fastvariables, which are approximated by averages over a single cycle. This leads to dθ i dt = ω i + KN N X j =1 A ij Γ i ( θ i − θ j ) + n X k =1 g ik ( θ i ) ξ ik , (S2)where the coupling function and phase sensitivity areΓ i ( θ i − θ j ) = Z π Z i ( θ i + φ ) · h (cid:2) x ci ( θ i + φ ) , x cj ( θ j + φ ) (cid:3) dφ π , (S3) g ik ( θ i ) = Z π Z π · · · Z π Z i ( θ i ) · g ik (cid:2) X c ( θ , θ , · · · , θ N ) (cid:3) Y j = i dθ j π , (S4) a r X i v : . [ n li n . AO ] A ug respectively, with X c ( θ , θ , · · · , θ N ) = ( x c ( θ ) , x c ( θ ) , · · · , x cN ( θ N )). It is desirable to define an ef-fective noise η i with phase sensitivity g i ( θ i ) such that the random variables g i ( θ i ) η i and P k g ik ( θ i ) ξ ik are drawn from the same distributions for all θ i . In the general case, the temporal correlations in therandom variable ξ ik cannot be entirely factored from the phase dependencies in g ik ( θ i ). However,if the ξ ik are independent for differing k and are stationary zero-mean Gaussian processes with thesame autocorrelation functions (so that h ξ ik ( t ) i = 0, h ξ ik ( t ) ξ jl ( t ) i = 0 for all i, j, k , and l = k , and h ξ ik ( t ) ξ ik ( t ) i = C i ( t − t ) for all k ), then the sum of Gaussian variables is again Gaussian andthe mean and autocorrelations completely specify the resulting distribution for each θ i . This leadsto an effective phase sensitivity g i ( θ i ) = (cid:2)P k g ik ( θ i ) (cid:3) / and θ i -independent Gaussian η i ( t ) withthe same autocorrelation C i . For the Kuramoto-type coupling function Γ i ( θ i − θ j ) = sin( θ j − θ i ),the reduction dynamics correspond to Eq. (2).In studies of synchronization induced by common noise, the most frequently employed phasesensitivity has been trigonometric, as in g i ( θ i ) = cos( θ i ). A different but important case consideredin the main text is the constant phase sensitivity function, g i ( θ i ) = 1, corresponding to additivenoise in the phase reduction. This choice is natural in systems that do not have a preferred phase,so that the dynamics are invariant under global phase rotations. In the case of n = 1 noise terms, g i ( X c ) = Z i ( θ i ) / k Z i ( θ i ) k results in g i ( θ i ) = 1, which, for example, follows after rescaling thenoise intensity for any rotationally-invariant noise in the Stuart-Landau equation or for g i ( X )proportional to the noiseless part of the dynamics more generally. For common noise with η i ( t ) = η ( t ) for all i , this rotational symmetry permits a change in variables θ i → θ i + R t dt η ( t ), whicheliminates the noise from Eq. (1). It follows that under common noise the order parameter must beidentical to that for the case without noise, and thus common noise cannot affect synchronization inthe case of constant phase sensitivity, whereas, as evidenced by Fig. 2, uncorrelated noise enhancessynchronization. Similar results are expected more generally for noise that results in approximatelyconstant g i ( θ i ). S2 . DIRECT NUMERICAL INTEGRATION AND CALCULATION OF R Direct numerical integration of Eqs. (2) and (5) was carried out in the sense of Itˆo using theGNU Scientific Library. Since computations necessarily use finite time steps, it is not possible tosimulate true white noise, which varies on arbitrarily short time scales. Instead, noise is sampledfrom a Gaussian distribution at a fixed sampling rate of 1 /τ . The variance of the sampled noiseis normalized by 1 /τ in order for the τ → G i , we instead sample 1 + ξ i from a Gamma distribution withmean one and variance σ /τ . In the latter case, noise is dominated by short bursts with largeamplitude, which resembles pulse-like noise relevant to the study of neurons [37]. Figure S1 (a)shows a sample of Gaussian white noise sampled with a rate τ = 10 − , which is sufficiently small toapproximate white noise well, while Fig. S1 (b) shows the Gamma-distributed noise sampled with τ = 10 − .Figure S1 (c) shows example trajectories of the system in Eq. (2) with common noise and uncor-related noise in the case of constant phase sensitivity g i ( θ ) = 1. Since the attractors in the systemare ergodic, the empirical distribution of time the oscillators spend with relative phase separation φ is given by the solution of the Fokker-Planck equation, as shown in Fig. S1 (d). Accordingly,the order parameter R is computed from time averages of cos ( φ/
2) over the trajectories, which,as shown by the dots in Fig. 2(c), agrees extremely well with the exact Fokker-Planck solution.Note than the average value R , on the other hand, would be computed from time averages of thenon-analytic function | cos( φ/ | . Consequently, while the expectation for the subcritical noiseless (a) (b) -π-π / π / π t ϕ -π -π / π / π ϕ P D F t ξ i - - t η i (c) (d) FIG. S1. Direct numerical simulations for Eqs. (2) and (5). (a) Gaussian white noise η i used in simulation ofEq. (2) vs. t , sampled with a rate τ = 10 − . Such noise was also used for ξ i and ξ i in simulations of Eq. (5).(b) Gamma-distributed noise ξ i used in simulations of Eq. (5) vs. t , sampled with a rate τ = 10 − . (c)Phase difference φ = θ − θ vs. time for sample simulations corresponding to Fig. (2) of the main text, with g i ( θ ) = 1 for common noise (blue) and uncorrelated noise (orange). (d) Empirical probability distributionfunction (PDF) for time spent at each phase separation φ derived from the trajectories in (c), with referencelines showing the exact solution ρ of the Fokker-Planck equation in Eq. (4). case ( σ = 0 and K ≤ K c ) with constant phase sensitivity is R ≡ / R varies with K . Thus, R is a more proper order parameter in the case of two phase oscillatorsthan R , since it more clearly delineates the noiseless phase-locking transition. Nevertheless, R and R exhibit the same trends with varying σ , so synchronization is enhanced by uncorrelated noiseaccording to either metric.In summary, the results from direct numerical simulations in Fig. S1 (c)-(d) completely agreewith those from the Fokker-Planck equation presented in Fig. 2 of the main text. The oscillatorsspend more time at small phase separations φ for uncorrelated noise than for common noise, leadingto enhanced synchronization. S3 . ELECTROCHEMICAL OSCILLATOR EXPERIMENTS An electrochemical cell was built with a platinum coated titanium rod counter electrode,Hg/Hg SO /(sat.) K SO reference electrode, 25 working nickel electrodes embedded in epoxy(each 1 mm in diameter), and 3 M H SO held at 10 o C, as shown schematically in Fig. S2 (a). Amulti-channel potentiostat was used to apply constant potential ( V ) to each electrode throughindividual resistors ( R ind ) of 1 kΩ, and the oscillatory current was measured at a rate of 200 Hz. R R c ind
PotentiostatC WR NoiseInterface (a) (b) I ( m A ) I ( m A ) ( s ) I ( m A ) (c)(d) FIG. S2. Electrochemical oscillator experiments. (a) Schematic of the experimental setup, showing theelectrochemical cell with a counter electrode C, a reference electrode R, and two working electrodes Wconnected through R ind = 1 kΩ resistors. The electrodes are coupled through a cross resistance R c , whichgives rise to a coupling strength of K = 1 /R c , and noise is applied through an interface that superimposesan additional potential to the electrode. (b)-(d) Measured currents I vs. time t for the first (blue lines) andsecond (orange lines) oscillators, with no noise (b), uncorrelated noise (c), and common noise (d), where thegray highlights periods for which the magnitude of the phase difference was less than π/ Noise was injected into the system employing a feedback interface to modulate the potential on the i th electrode, V i ( t ) = V + η i ( t ), where η i is Gaussian white noise generated using Matlab. Thenatural frequency of each electrode was determined without noise. Coupling was added through anexternal resistance between two electrodes, which provided multiple pairs of electrodes exhibitingphase slipping behavior near the onset of synchrony. A LabView program was created to switchbetween uncorrelated noise, no noise, and common noise as data was collected.Traces of the experimentally measure current I are shown in Fig. S2 (b)-(d) for realization withno noise, uncorrelated noise, and common noise. Oscillators have natural frequencies around 0 . S2 (b)-(d). Experimental runs were repeated to yield multipledatasets for analysis at weak noise ( D = 0.025 V, 14 pairs), intermediate noise ( D = 0.050 V, 22pairs), and strong noise ( D = 0.10 V, 10 pairs). Since the experiments have limited length of time,the averaging over multiple experimental realizations shown in Fig. 4 is necessary to establish thiseffect statistically. S4 . RANDOM NETWORKS OF JANUS OSCILLATORS While we focused on the case of N = 2 phase oscillators in the main text, uncorrelated noisecan also enhance synchronization when common noise does not in classes of random networks ofoscillators. For concreteness, we illustrate this in a networks of coupled Janus oscillators, which (b)(a) (c) N N N N N N
10 20 30 400.000.050.10 N R c σ R u2 / R c FIG. S3. Effects of noise in Janus oscillator networks. (a) Random networks of Janus oscillators, witharrows from oscillator j to oscillator i indicating coupling from θ j to θ i . (b) Network average of the orderparameter h R c i over 100 random network realizations vs. number of Janus oscillators N for oscillators subjectto common noise, with ∆ ω = 1 and K = β = 1 / R u , relativeto that for common noise, R c , vs. noise intensity σ for random networks of various sizes, with error bars asin (b). have been of recent interest for their potential to produce rich dynamics [35, 36]. Each Janusoscillator is composed of two phase-oscillator components with opposite natural frequencies, andpairs of Janus oscillators interact through a coupling term between phase-oscillator componentsof opposite natural frequencies. Networks of Janus oscillators can also be regarded as multilayernetworks consisting of two layers of phase oscillators, each with a different natural frequency. Weconsider the impact of common and uncorrelated noise acting on each phase-oscillator component,which evolves according to dθ i dt = ∆ ω β sin( θ i − θ i ) + K X j A ij sin( θ j − θ i ) + η i , (S5) dθ i dt = − ∆ ω β sin( θ i − θ i ) + K X j A ji sin( θ j − θ i ) + η i , (S6)where ∆ ω is the frequency gap, β and K are the internal and external coupling constants, respec-tively, A ij are the components of the adjacency matrix, and η ki are Gaussian white noise terms withintensity σ . While here we focus on Eqs. (S5)-(S6) for concreteness, we expect similar conclusionsto hold in wider classes of multilayer and multiplex networks, which have been of recent interestin the study explosive phenomena [34]. In contrast to the globally-coupled Kuramoto model, thereare relationships between the frequency distribution and network structure in these models, whichcan facilitate synchronization enhancement by uncorrelated noise.Figure S3 (a) shows a selection of random regular networks of Janus oscillators, in which eachoscillator has both in- and out-degree equal to 2. The noise is either common (with η i = η j for all i, j ) or uncorrelated (with h η i η j i = h η i η j i = h η i η j i = 0 for all i, j ). Because the degrees of all nodesare identical, all oscillators are close to phase locking with their neighbors for slightly subcritical K ,as it was the case for the systems of N = 2 oscillators considered in the main text. Since commonnoise can be eliminated in a rotating reference frame, the average order parameter in the presenceof common noise R c is equal to the noiseless case for each network realization. The average ofthe order parameter over the network realizations, h R c i , decreases with increasing network size,as shown in Fig. S3 (b). The impact of uncorrelated noise should thus be measured relative tothat of common noise to compare networks of differing sizes, as shown in Fig. S3 (c). Regardlessof network size, uncorrelated noise enhances synchronization for intermediate noise intensity byallowing oscillators to spend less time at large phase separations. Thus, the constructive effect ofuncorrelated noise described in the main text generalize beyond two oscillators to classes of randomnetworks. S5 . LAYERED NETWORKS OF PHASE OSCILLATORS Networks of oscillators in biological systems often have layered structure, and the impact ofuncorrelated noise on such networks is of broad interest. Here, we consider a two-layer version ofthe system in Eq. (2) of the main text. The first layer consists of N − dθ i dt = ω i + K N − N − X j =1 sin ( θ j − θ i ) , i = 1 , · · · , N − , (S7)where K = K [1 + cos ( θ N − θ N − )] / K is a tunable constant. Oscillator heterogeneity is included inthis model by sampling the natural frequencies from a Lorentz distribution centered at zero withwidth γ = 1. The second layer consists of two coupled oscillators that are subject to Gaussianwhite noise, dθ i dt = ω i + K N X j = N − sin ( θ j − θ i ) + g i ( θ i ) η i , i = N − , N, (S8)where K is a constant.The model in Eqs. (S7)-(S8) represents scenarios in which coupling between oscillators dependson time-dependent resources, described here by the variable coupling strength K . For instance,certain properties of neurons that impact their coupling strengths are modulated by the presenceof circadian-regulated hormones such as cortisol [38]. The circadian system relies on the synchro-nization of multiple interacting clocks located both in the brain [39] and in tissues outside thebrain, including the liver and kidneys [40]. These clocks can experience common and uncorre-lated noise owing to the differential impact of the various environmental signals they receive, whichinclude light, available nutrients, sleep schedule, and tissue-dependent conditions. In this exam-ple, θ N − and θ N in Eq. (S8) would represent the phases of circadian clocks whereas θ , . . . , θ N − in Eq. (S7) correspond to the phases of neurons whose coupling strength is modulated by thecircadian-regulated hormones.We analyze the response of the time-averaged order parameter R to common noise, uncorrelatednoise, and no noise, as shown in Fig. S4 (a) for a constant function g i ( θ i ) = 1. For all couplingconstants considered, the order parameter is larger in the case of uncorrelated noise than in the i t i ππ/ π/ -π/ -π/ θ - Θ π- (a) (b) (c) U n c o rr e l a t ed C o mm on N ono i s e K ′′ R FIG. S4. Synchronization enhanced by uncorrelated noise in a two-layer network. (a) Time-averaged orderparameter R as a function of the coupling parameter K for N − K = 0 .
95, the naturalfrequencies are ± .
5, and the noise intensity is σ = 1 .