Coherence and anticoherence resonance tuned by noise

Departament de Fı ́sica Aplicada, Universitat Polite `cnica de Catalunya, Avenue Gregorio Maran ̃on 44, E-08034 Barcelona, Spain Departament de Quı ́mica Fı́sica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain Departament d’Estructura i Constituents de la Mate `ria, Facultat de Fı ́sica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain ~Received 7 November 2001; published 24 October 2002 !

Subtle signatures of the ordering role of noise in complex systems constitutes nowadays a new and celebrated paradigm in nonlinear science.Very well documented scenarios are those of stochastic resonance ͓1͔ and, closer to our interest here, coherence resonance either in single units ͓2͔ or arrays ͓3͔.In this later situation, features of coherent noiseinduced emitted pulses can be recognized in autonomous excitable systems that do not exhibit self-sustained oscillations.The whole phenomenon rests on the fact that the time interval between pulses decomposes into an activation and an excursion time which depend differently on the noise amplitude.Restricting to small noise intensities D, i.e., far indeed from the optimal conditions of stochastic resonance, the first contribution largely exceeds the second one, and thus constitutes the emitting rate controlling process mediated by a Kramers-like mechanism ͓2͔.This in turn leads to a nearly Poisson-like distribution of interpulse intervals t p .Consequently, taking its normalized variance defined by Rϭ as an indicator of a noncoherence emition, we have RϽ1, with the limiting Poissonian value Rϭ1 approached from below as D→0.
Apparently thus, there should be nothing specially relevant in this limit for an excitable system subjected to noise.However, and quite surprisingly we will show in this paper that this is not always true.In a very abstract way, the key point to realize is that by referring to a conveniently modified Poisson distribution, one might have a situation opposed to that of coherence resonance, and so termed anticoherence resonance, with RϾ1 and indeed approaching unity from above.This phenomenon is related with the appearance of another temporal scale which enchains pulses into time intervals much shorter than the mean excitation time.In this way the statistics of pulse appearance would be modified, decreasing both the variance and, hopefully more markedly, the averaged value of the interspike intervals distribution.The existence of situations with RϾ1 was conjectured and studied within the context of neuronal dynamics ͓4,5͔, but our emphasis here will be on the fact that the whole phenomenon of coherence and anticoherence is simply controlled by the noise intensity.
We propose that the physical mechanism underlying this phenomenon is that the excitability threshold changes in time during the nonmonotonous but rather oscillatory relaxation to the rest state, as we will see.Although this is a feature displayed by probably quite a number of excitable dynamical schemes when placed closer enough to oscillatory conditions, we have decided to investigate two very different examples of nonlinear models: the standard two variable FitzHugh-Nagumo ͑FHN͒ model and a single model incorporating a delay feedback.In both cases and by appropriately choosing the system parameters, we can separately control the time scales of both its excitatory and relaxational behaviors, favoring, in this way, the effect of fluctuations in tuning both coherence and anticoherence resonance.
Actually the use of feedback to either modify or control nonlinear dynamical responses has been proposed since a long time ago in different chemical ͓6͔ and biochemical ͓7͔ contexts.Much more recently, this question has been revisited after experiments have been conducted introducing global feedback techniques, either delayed ͓8͔ or nondelayed ͓9͔, into different scenarios of spatiotemporal pattern formation.Closer to our scenario, diverse situations of laser dynamics ͓10-12͔ have been explored, searching for the coherent role of the coupling of time delay and noise.From the theoretical point of view this question was already addressed in Refs.͓13,14͔, and emphasizing features of resonant behavior in Refs.͓15,16͔.
First and as a reference model, we consider the FHN model ͓2͔ where u and v are the activator and inhibitor variables.For values of the parameter h slightly larger than one, the system is excitable but very close to the oscillatory regime.
The second model will exhibit more clearly the behavior we want to study.Here we will illustrate the subtle synergy of noise and delay, that will lead to the previously anticipated anticoherence resonance phenomenon, by referring to a single variable bistable mechanism.As a prototype of a simple model with excitable characteristics, we present a one variable bistable system with a negative delay or feedback term.The explicit model equations read where T is the time delay.The reaction term has three possible steady states and two of them are stable.The feedback term is linear and controls the stability of the steady states.This term acts as the inhibitor variable in standard excitable systems with a characteristic time T, acting here as a refractory time.
In both models the noise is prescribed to be Gaussian and white with an intensity D, ͗͑t͒͑tЈ͒͘ϭ2D␦͑tϪtЈ͒.

͑4͒
We have numerically integrated Eqs.͑2͒ and͑4͒ for parameters hϭ1.005 and ϭ0.1, and Eqs.͑3͒ and ͑4͒ for parameters aϭ0.2, bϭ2, cϭ0.2, ␣ϭ0.5, and Tϭ8.These models have been defined to have dimensionless variables and parameters.In the absence of noise, these systems exhibit an excitable behavior in the form of pulses, with oscillating relaxations, as appear in Fig. 1͑a͒ for both models.A pulse can be generated either by a special preparation of the initial condition or by the presence of fluctuations.In the rest situation, the system is in a metastable state and even small fluctuations can allow the system to cross the potential barrier.Thus, in the presence of noise, a series of pulses, at more or less random intervals, can be observed.The statistics of these pulses are the subject of our study as a function of the noise intensity D. Let us analyze two runs of each model, with appropriately chosen and quite different noise intensities.We will start assuming that due to the oscillating decay of the pulse the next one to be generated will more likely take place at the time t 1 ͑slightly larger than the intrinsic refractory time of the pulse͒ when this first maximum appears.Actually this is what one can see in Figs.1͑b͒ and  1͑d͒.What is striking, however, is the completely different long time distribution of interpulse intervals in the extreme situations represented, respectively, by Figs.1͑c͒ and 1͑e͒.In Figs.1͑b͒ and 1͑d͒, we detect a clear signature of coherence resonance.Contrarily in Figs.1͑c͒ and 1͑e͒, we observe again enchained pulses grouping, this time, into small clusters separated by long and random time intervals.
This behavior is quantified in Fig. 2, where we show, for the two models, the numerical results of the coherence indicator R defined above, as a function of the noise intensity.First we observe a minimum of this quantity for some intermediate noise intensity.Such minima are indicative of the accepted signature of coherence resonance, that is, the presence of relatively coherent oscillations induced by a random perturbation of appropriate noise intensity ͓2͔.Second and more importantly in Fig. 2, we observe the presence of a relative maximum, a feature which has not been reported in previous studies of coherence resonance.This maximum appears contrarily for very small values of the intensity of the noise, with the Poisson limit Rϭ1 reached from the above.Although the position of this maximum slightly depends on the particular model, its occurrence is a generic feature of the influence of the noise in both excitable systems, being more pronounced in the model with delay.
Let us see how this maximum is possible from theoretical considerations based on the observation of the numerical data.According to the generic analysis of the simulation data we can conjecture the following functional behavior of the probability distribution of the time intervals between pulses: almost no pulses for times smaller than that corresponding to the first relaxational peak, a very pronounced maximum at this time and an exponential decaying distribution, characteristic of the Kramer's escape, for larger times.
We observe precisely this type of behavior ͑Fig.3͒ from our numerical data accumulated in very large runs, and, accordingly, we propose the following probability distribution for the interpulse time intervals, p͑t p ͒ϭA␦͑ t p Ϫt 1 ͒ϩB͑ t p Ϫt 1 ͒e Ϫt p /t 2 , ͑5͒ t 1 is associated with the characteristic time of the first maximum of the relaxational oscillations, and t 2 is the characteristic time of the barrier crossing mechanism or Kramer's time.A and B are the relative weight parameters of the two functional components of p(t p ). Due to the normalization of p(t p ) only one of them is independent and our choice here is to eliminate B. At this point our system has only three unknown parameters: t 1 , t 2 , and A. An easy way to evaluate them from the simulation data is to look at the cumulative probability distribution, P(t p )ϭ͐ 0 t p p(tЈ)dtЈ.This function is almost zero for small t p with a high slope at t 1 , and a relaxational approach to 1 for large t p .Our numerical data follows quite well this functional dependence and moreover the value of t 1 can be estimated easily at the point in which P(t p ) has a maximum slope.The values obtained are t 1 ϳ24, larger but proportional to T and independent of D for the feedback model and t 1 ϳ5 for the FHN model.As P(t p ) has a simple exponential decay, we can also evaluate the other two parameters t 2 (D) and A(D).
The dependence of the parameters t 2 on D for both models, is seen in Fig. 4. Assuming a Kramer's-like dependence of t 2 , t 2 ϭt 0 e ⌬U/D , ͑6͒ we obtain the parameter values t 0 ϳ8 and ⌬Uϭ0.046 for the feedback model and t 0 ϳ7 and ⌬Uϭ0.00035, for the FHN model.The parameter A is similarly obtained although the explicit dependence on D is not simple but it can readily be tabulated.
Once the parameters of the system have been evaluated, we proceed with the calculation of the statistical moments.The first moment and the standard deviation of this distribution are From Eq. ͑7͒ the quantity R, defined in Eq. ͑1͒, can be written as where kϭt 1 /t 2 Ͻ1, compares these two time scales of the system.
In Fig. 2 we present a comparison between the numerical values of R versus D obtained from simulations and those obtained from our analysis using the parameters, t 1 , t 2 (D) from Eq. ͑6͒, and A(D) from tabulated data.The agreement is remarkable in a wide range around the maximum in spite of the simplicity of the model.According to our numerical estimates and for very small values of the noise intensity, we have that kӶ1 and consequently RϾ1.Although the argument of Ref. ͓2͔, predicting Rр1 when Kramer's barrier crossing mechanism dominates, is correct in most cases, this is not true if a new time scale is involved, t 1 , which is precisely the situation near the oscillatory regime.
Our more physically oriented interpretation of the anticoherence phenomenon is thus the following.When adding additive noise of appropriately small intensity the system responds in what appears to be trains consisting of a few random number of enchained pulses with a deterministic time scale separation t 1 .These clustering episodes appear, however, in a considerably unpredictable way separated by long time intervals t 2 , when recording the whole signal over long times.This is the signature of maximal anticoherence: the mixture of two very different time scales.In this regime the variance ͗(⌬t p ) 2 ͘ decreases with D but ͗t p ͘ decreases much more rapidly, producing the relative enhancement of R.
Two very different dynamical models show the same trends and one can thus conclude that this as a universal and robust phenomena induced by noise and characteristic of excitable systems near its oscillatory regime, where the relaxation to the steady state is not monotonous but oscillatory leading to a dynamical excitability threshold.The anticoherence phenomenon for very small values of the intensity of the noise is also implicit in the figures of Refs.͓11,12͔.
What is also remarkable, although this is indeed not totally unexpected, is that features of coherence resonance are also captured at larger noise intensities.In summary, we observe a crossover from maximal anticoherence to maximal coherence as noise intensity increases.In the context of stochastic resonance, a similar crossover has been found in Ref.
͓17͔.The mechanism opposite to that of standard stochastic resonance is termed resonant trapping.
Finally let us emphasize that to convince ourselves that such combined effects of coherent and anticoherent resonance are not at all spurious effects of using uncorrelated fluctuations ͑white noise͒, we have checked that the behavior of R shown in Fig. 2 also appears for coloured noises ͑results not shown here͒.

FIG. 1
FIG. 1.A single pulse for the FHN model ͓͑a͒, left͔ and for the feedback model ͓͑a͒, right͔.A train of pulses obtained from the FHN model for noise intensities Dϭ0.02 ͑b͒ and Dϭ10 Ϫ4 ͑c͒ and from the feedback model for noise intensities Dϭ0.1 ͑d͒ and Dϭ7•10 Ϫ3 ͑e͒.

FIG. 3 .
FIG. 3. Probability distributions of the interpulse time intervals for the FHN model ͑solid line͒ and for the feedback model ͑dashed line͒.They correspond to noise intensities Dϭ10 Ϫ4 and D ϭ0.015, respectively.

FIG. 4 .
FIG. 4. Parameter t 2 versus noise intensity for the FHN model ͑top͒ and the feedback model ͑bottom͒.Continuous lines are the fits given by Eq. ͑6͒, for the range of D values corresponding to RϾ1.