Very large stochastic resonance gains in finite sets of interacting identical subsystems driven by subthreshold rectangular pulses

Very large stochastic resonance gains in finite sets of interacting identical subsystems driven by subthreshold rectangular pulses David Cubero, Jesús Casado-Pascual, José Gómez-Ordóñez, José Manuel Casado, and Manuel Morillo Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, Sevilla 41080, Spain 共Received 4 October 2006; revised manuscript received 26 March 2007; published 7 June 2007兲 We study the phenomenon of nonlinear stochastic resonance 共SR兲in a complex noisy system formed by a finite number of interacting subunits driven by rectangular pulsed time periodic forces. We find that very large SR gains are obtained for subthreshold driving forces with frequencies much larger than the values observed in simpler one-dimensional systems. These effects are explained using simple considerations. DOI: 10.1103/PhysRevE.75.062102 PACS number共s兲: 05.40.⫺a, 05.45.Xt The phenomenon of stochastic resonance 共SR兲seems to be important in a wide variety of contexts in physics, chemistry, and the life sciences 关1兴. A lot of work has been devoted to the study of SR both in simple 关2兴and complex systems 关3兴, such as in ion channel assemblies 关4兴or globally coupled networks of noisy neural elements 关5兴, to name a few examples. In this work, we consider a complex system formed by a finite number of Ncoupled noisy bistable subsystems, the attention to finite sets being inspired by the fact that certain processes in neuroscience seem to involve a rather small number of subsystems 关6兴. The signal-to-noise ratio 共SNR兲and the SR gain are two common quantifiers used to characterize the SR response of noisy systems driven by time-periodic forces. We will study SR effects on a collective variable of finite sets of interacting subunits driven by periodic rectangular pulses. Our results show a tremendous enhancement of SR effects in this collective variable with respect to those observed in single unit systems. Let us consider a set of Ninteracting subsystems, each one of them characterized by a single degree of freedom xi 共i=1,...,N兲, whose dynamics is governed by the Langevin equations 关7–9兴 x ˙i=xi−xi 3+ ␪ N兺 j=1 N 共xj−xi兲+ ␰ i共t兲+F共t兲,共1兲 where ␰ i共t兲are Gaussian white noises with zero average and 具 ␰ i共t兲 ␰ j共s兲典=2D ␦ ij ␦ 共t−s兲, ␪ is the parameter defining the strength of the interaction between subsystems, and F共t兲is an external driving force of period T. In this work, we will restrict ourselves to forces of the type 关10兴 F共t兲= 冦 A;0艋t⬍tc, 0; tc艋t⬍T/2, −A;T/2 艋t⬍T/2 + tc, 0; T+tc艋t⬍T. 冧 共2兲 The parameter r=2tc/T, usually called duty cycle, measures the fraction of a period during which this driving force has a nonvanishing value. In addition, we will only consider subthreshold amplitudes A, so that the driving force 共2兲cannot induce sustained oscillations between dynamical attractors in the absence of noise. The model described by Eq. 共1兲共without the external periodic driving兲was used years ago by Kometani and Shimizu 关7兴as an empirical model to describe muscle contraction. Later on, Desai and Zwanzig 关8兴gave a more detailed statistical mechanical description in the asymptotic N→⬁limit and used it to model order-disorder transitions. The addition of an external driving can, in principle, be used to describe the phenomenology of a forced contracting muscle. We focus on the collective variable S共t兲defined as S共t兲=1 N兺 i=1 N xi共t兲,共3兲 which has previously been used 关9兴in the global analysis of coupled bistable systems. It might be considered as the total output process of a parallel array of Nidentical interacting subunits, subject to independent noise sources ␰ i共t兲and the same external forcing F共t兲. Its signal-to-noise ratio 共Rout兲is defined in the usual way as Rout =Qu Ql ,共4兲 with Qu=2 T 冕 0 T d ␶ Ccoh共 ␶ 兲cos共⍀ ␶ 兲,共5兲 Ql=2 ␲ 冕 0 ⬁ d ␶ Cincoh共 ␶ 兲cos共⍀ ␶ 兲,共6兲 where ⍀=2 ␲ /T,Ccoh共 ␶ 兲=1 T兰0 Tdt具S共t+ ␶ 兲典⬁具S共t兲典⬁and Cincoh共 ␶ 兲=C共 ␶ 兲−Ccoh共 ␶ 兲with C共 ␶ 兲=1 T兰0 Tdt具S共t+ ␶ 兲S共t兲典⬁. For a set of Ncoupled linear oscillators driven by the external driving force F共t兲and subject to the noise terms ␰ i共t兲 as in Eq. 共1兲, the SNR of the corresponding collective process Rout 共L兲coincides with that of the random process formed by the arithmetic mean of the individual noise terms ␰ i共t兲 plus the deterministic driving force F共t兲, namely, F共t兲+ ␰ 共t兲 with ␰ 共t兲=N−1兺i=1 N ␰ i共t兲. The process ␰ 共t兲is a Gaussian white noise of effective strength D/N. Then, it is easy to prove that PHYSICAL REVIEW E 75, 062102 共2007兲 1539-3755/2007/75共6兲/062102共4兲©2007 The American Physical Society062102-1 Rout 共L兲=2A2N关1 − cos共 ␲ r兲兴 ␲ D.共7兲 Thus, for our nonlinear case, it seems convenient to analyze the SR gain Gdefined as 关9兴 G=Rout Rout 共L兲,共8兲 which compares the SNR of a non linear system with that of a linear system subject to the same stochastic and deterministic forces. We have carried out extensive simulations of the Langevin equations 共1兲with the external driving 共2兲. In all cases reported here the coupling strength is fixed to ␪ =0.5 and the subthreshold driving amplitude to A=0.3. There is nothing special about this particular ␪ value. The qualitative results would be the same for any other value of ␪ ⫽0. In Fig. 1we show the collective signal-to-noise ratio and SR gain as a function of the noise strength Dfor a set of N=10 identical subsystems, a driving fundamental frequency ⍀=0.01 and several values of the duty cycle r. It can be seen that, while the SNR curves are nearly identical for r艋0.4, the SR gain increases drastically, reaching a very large value for r=0.1. This dramatic increase is easily understood by taking into account the observed almost constant behavior of Rout and the fact that Rout 共L兲, Eq. 共7兲, decreases monotonically with r. However, for sufficiently small r,Rout must decrease faster than Rout 共L兲, because the interval tcbecomes smaller than the time it takes for the system to react to a constant force of amplitude A, and thus, the driving produces almost no effect in the system. This behavior is shown in the inset of Fig. 1, where it can be seen that the SR gain decreases for r 艋0.08. Also, as seen in Fig. 1, when r=1 共rectangular driving signal兲,Greaches a peak at a noise value D⬇0.2. Even though Rout 共L兲is as large as it can be, the huge increase of Rout is enough to overcome the increase in Rout 共L兲, yielding a substantial value for the SR gain. The very large gain values in Fig. 1for pulses with short duty cycles are observed only for a small range of noise strengths around D⬇0.08. In Fig. 2we present the behavior of Qland Quwith D. Notice that around D⬇0.08 there is a strong reduction of two orders of magnitude in the level of fluctuations of the collective variable as measured by Ql. Therefore, the large SR effects quantified by the SNR and the SR gains are essentially due to the very large reduction of the fluctuation spectrum of the output signal at the fundamental driving frequency for a range of noise values. We have also analyzed the SNR and the SR gain for different values of ⍀. Our results for Rout and Gas a function of Dare depicted in Fig. 3for a set of N=10 interacting identical subunits driven by rectangular pulses with duty cycle r=0.1 and several values of the fundamental driving fre0 0.1 0.2 0.3 0.4 D 0 20 40 60 80 G 0 0.1 0.2 0.3 0.4 0 20 40 60 80 R out r= 0.1 r=0.2 r=0.3 r= 0.4 r=1 0 0.2 0.4 0.6 0.8 1 r 0 50 100 150 FIG. 1. Dependence of the signal-to-noise ratio Rout and the SR gain Gfor a set of N=10 identical subsystems and an external driving of frequency ⍀=0.01 for several values of the duty cycle r: 0.1 共open circles兲, 0.2 共crosses兲, 0.3 共triangles兲, 0.4 共squares兲, and 1 共stars兲. Inset shows the SR gain as a function of rfor a fixed noise strength D=0.08. The lines have been drawn as a guide to the eye. 0 0.1 0.2 0.3 0.4 0.5 0.6 D 0.01 0.1 1 10 100 Ql 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 Qu r= 0.1 r=0.2 r=0.3 r= 0.4 r=1 FIG. 2. Dependence of the denominator and the numerator of the collective signal-to-noise ratio for the same parameter values as in Fig. 1. The lines have been drawn as a guide to the eye. 00.05 0.1 0.15 0. 2 D 0 5 10 15 G 00.05 0.1 0.15 0. 2 0 2 4 R out Ω = 0.015 Ω = 0.02 Ω = 0.03 Ω = 0.04 FIG. 3. Dependence of the signal-to-noise ratio and the SR gain for a set of N=10 identical subsystems and duty cycle r=0.1 for several values of the driving frequency ⍀: 0.015 共open circles兲, 0.02 共crosses兲, 0.03 共triangles兲, and 0.04 共squares兲. Solid lines have been drawn as a guide to the eye. The horizontal dashed line marks the unity for the SR gain. BRIEF REPORTS PHYSICAL REVIEW E 75, 062102 共2007兲 062102-2 quency ⍀: 0.015 共open circles兲, 0.02 共crosses兲, 0.03 共triangles兲, and 0.04 共squares兲. As we increase the driving frequency ⍀, the SR gain is gradually reduced. For a sufficiently large driving frequency 共pulses of very short duration兲, gains larger than unity are not observed for any value of the noise strength. As mentioned above, the large SNR values observed in Fig. 1are related to the existence of a sharp minimum of Ql at a certain noise value. As observed in Fig. 2, this value is D⬇0.08 for a driving signal with short duty cycle, amplitude A=0.3 and fundamental frequency ⍀=0.01. For other amplitude and frequency, the location of the minimum will change, although the mechanisms leading to the existence of this minimum will be qualitatively the same. To understand the Qldependence on D, it is important to analyze in detail the dynamics imposed by the external driving force in Eq. 共2兲. For the Dand ␪ values of interest to the present discussion, simulations show that when the driver is absent, the collective variable S共t兲performs a noise induced random movement between the two symmetric attractors of the dynamics, which we will regard as located at ±S0. During a time interval of duration tc, the external force F共t兲=Afavors the positive attractor. Thus, if S共t兲was at the negative attractor at the beginning of the interval, the external forcing will drive it to the positive one in a random time that we will denote by ⌼1. Certainly, different realizations of the noise will yield different values of ⌼1. Running simulations with many independent trajectories, we have computed the probability Prob共⌼1⬍ ␶ 兲=f共 ␶ 兲that the variable S共t兲has jumped before a time ␶ to the attractor favored by a constant driving of amplitude A. In Fig. 4共a兲we present f共 ␶ 兲for the set of parameters relevant to the discussion: A=0.3, D=0.08, and N=10 共solid line兲. It can be seen that for the case r=0.1 and ⍀=0.01 共thus, tc⬇31.4兲, a transition between the attractors for ␶ =tcis performed with probability almost unity. Then, during the rest of the half-period, an interval of duration T/2−tc, the external force is zero and the system is free to jump between the attractors due solely to noise. Let us now denote by ⌼2the random time it takes to jump from one attractor to the opposite one when F共t兲=0. Figure 4共b兲shows the probability Prob共⌼2⬍ ␶ 兲=g共 ␶ 兲that this jump has taken place before a time ␶ vs ␶ . Since for r=0.1 and ⍀=0.01 we have T/2−tc⬇282.7, it can be checked in Fig. 4共b兲that almost no transitions take place during this time interval under these conditions. We can carry out the same analysis for the symmetric situation during the second half-period of the driving force. Consequently, S共t兲performs a neat trajectory between its attractors with transitions induced systematically every half-period by the external driving when it has a nonvanishing value. As a result, we would expect Cincoh共0兲, itself an average of the second cumulant of S共t兲over a period, to be of the order of the effective noise D/N, and Cincoh共 ␶ 兲 short-lived. This is, in fact, what is observed in Fig. 5 共see inset兲. The curves in Fig. 4共a兲indicate that the probability of transitions between the attractors before tcout of the most unstable attractor is smaller for D=0.04 than for D=0.08. This is to be expected as the probability of transitions out of an attractor goes to zero when D→0 with subthreshold driving amplitudes. As a result, Cincoh共0兲and the decay time of Cincoh共 ␶ 兲increase as Dgets smaller than D=0.08. Consequently, noticing its definition in Eq. 共6兲,Qlhas to increase as Dis reduced from D=0.08. Figure 2shows that for noise values larger than D ⬇0.08 the Qldependence on Dfor inputs with short duty cycles is rather different from the one observed for a rectangular input 共r=1兲. This fact indicates that for short duty cycles and these larger noise values, the time intervals during which F共t兲=0 are crucial for the system response. By comparing the g共 ␶ 兲curves for D=0.08 and D=0.16 in Fig. 4共b兲, we see that the probability of transitions between attractors during the time interval T/2−tcis negligible, whereas there 0 100 200 300 400 50 0 τ 0 0.2 0.4 0.6 0.8 1 g(τ) D=0.08, N=1 D=0.08, N=10 D=0.16, N=10 010203040 τ 0 0.2 0.4 0.6 0.8 1 f(τ) D=0.08, N=1 D=0.08, N=10 D=0.04, N=10 (a) (b) FIG. 4. 共a兲Probability that the collective variable S共t兲has jumped before a time ␶ to the attractor favored by an external constant driving of amplitude A=0.3 starting from the opposite attractor. The solid line corresponds to a system with D=0.08 and N =10, the dotted line to D=0.08 and N=1, and the dashed line to D=0.04 and N=10. 共b兲Probability that S共t兲jumps before a time ␶ to the opposite attractor in the absence of driving. The solid line corresponds to a system with D=0.08 and N=10 共being almost zero for the range of times shown兲, the dotted line to D=0.08 and N =1, and the dashed line to D=0.16 and N=10. 0 100 200 300 400 τ 0 0.05 0.1 0.15 0.2 0.25 0.3 Cincoh(τ) Ω=0.01 Ω=0.03 0 1020304050 0 0.005 0.01 0.015 FIG. 5. Temporal behavior of the incoherent part of the correlation function of the collective variable for a system of N=10 identical subsystems with D=0.08 and r=0.1. The solid lines correspond to ⍀=0.01 and the dashed line to ⍀=0.03. The inset is a magnification of the case ⍀=0.01 displayed for the sake of clarity. BRIEF REPORTS PHYSICAL REVIEW E 75, 062102 共2007兲 062102-3 is a non-negligible probability for D=0.16. Therefore, the bimodal character of the probability distribution for S共t兲is enhanced as Dincreases from D⬇0.08 to D⬇0.16 and one can then conclude that Cincoh共0兲and the decay time of Cincoh共 ␶ 兲increase as Dincreases. Consequently, for short duty cycles, Qlalso increases as Dincreases within the interval D⬇0.08 to 0.16. On the other hand, for r=1 and this same range of noise values, the situation is different because the driving signal never vanishes. Then, transitions out of the most unstable attractor happen more frequently as noise increases 关f共 ␶ 兲saturates at 1 earlier兴, and this leads to a decrease of both Cincoh共0兲and the decay time of Cincoh共 ␶ 兲as D increases, with the corresponding decrease in Ql. Similarly, raising the driving fundamental frequency has a drastic effect on the behavior of Cincoh共 ␶ 兲with respect to its optimal behavior discussed above for D=0.08 and r=0.1, as depicted in Fig. 5for two frequency values: ⍀=0.01 and 0.03. The picture that emerges for ⍀=0.03, differs considerably from that observed for ⍀=0.01. For the higher frequency case, with tc⬇10.4, the value of the solid line at ␶ =10.4 in Fig. 4共a兲indicates that, for a considerable number of trajectories, the driving force is not able to induce a transition during the time interval tc. Consequently, the second cumulant of S共t兲becomes of the order of the distance between the attractors. On the other hand, for the lower frequency, transitions occur for the corresponding ␶ =tc⬇31.4 with almost total certainty. This leads to a Cincoh共0兲for ⍀ =0.03 much larger than for ⍀=0.01. Additionally, since still almost no transitions occur at the intervals when the driving is absent, as shown in Fig. 4共b兲, nor obviously when the driving favors the initial attractor, any unsuccessful transition is carried on until the next period, and we would expect a long-lived Cincoh共 ␶ 兲. Namely, noise induced correlations persist during a few driving periods. These observations justify the full behavior observed in Fig. 5. In conclusion, we have analyzed the enhancement of SR effects in a collective variable characterizing the response of finite sets of interacting noisy subsystems. The cooperative effect of noise and nonlinearity in finite sets of interacting, driven, bistable subunits reflects in a substantial decrease in the noise level of the collective output S共t兲with respect to that observed in the output of a single independent unit. Very large values of the SR gain can be achieved for trains of short rectangular pulses of the type defined by Eq. 共2兲. Even though the results presented here have been obtained with external pulses with very brisk changes of their amplitudes at certain instants of time, similar results can also be observed when the pulse amplitude changes continuously, as long as there are time intervals within a period where the amplitude changes very slowly, followed by short time intervals where the amplitude changes very drastically. This research was supported by the Dirección General de Enseñanza Superior of Spain 共Grant No. FIS2005-02884兲, the Junta de Andalucia, and the Juan de la Cierva program of the Ministerio de Ciencia y Tecnología 共D.C.兲. 关1兴L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 共1998兲. 关2兴P. Hänggi, ChemPhysChem 3, 285 共2002兲. 关3兴P. Jung and G. Mayer-Kress, Phys. Rev. Lett. 74, 2130 共1995兲; J. F. Lindner, B. K. Meadows, W. L. Ditto, M. E. Inchiosa, and A. R. Bulsara, ibid. 75,3共1995兲; A. Pikovsky, A. Zaikin, and M. A. de la Casa, ibid. 88, 050601 共2002兲. 关4兴G. 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