Self-induced stochastic resonance in excitable systems

Transcription

1 Self-indced stochastic resonance in excitable systems Cyrill B. Mrato Department of Mathematical Sciences, New Jersey Institte of Technology, Newark, NJ 7 Eric Vanden-Eijnden Corant Institte of Mathematical Sciences, New York Uniersity, New York, NY Weinan E Department of Mathematics and PACM, Princeton Uniersity, Princeton, NJ 85 Abstract The effect of small-amplitde noise on excitable systems with strong time-scale separation is analyzed. It is fond that anishingly small random pertrbations of the fast excitatory ariable may reslt in the onset of a deterministic limit cycle behaior, absent withot noise. The mechanism, termed self-indced stochastic resonance, combines a stochastic resonance-type phenomenon with an intrinsic mechanism of reset, and no periodic drie of the system is reqired. Self-indced stochastic resonance is different from other types of noise-indced coherent behaiors in that it arises away from bifrcation thresholds, in a parameter regime where the zero-noise (deterministic) dynamics does not display a limit cycle nor een its precrsor. The period of the limit cycle created by the noise has a non-triial dependence on the noise amplitde and the time-scale ratio between fast excitatory ariables and slow recoery ariables. It is arged that self-indced stochastic resonance may offer one possible scenario of how noise can robstly control the fnction of biological systems. Key words: Self-indced stochastic resonance, coherence, excitable systems, large deiations, noise-controlled PACS: 5..-a, b, 8..Bj, 8.39.Rt Preprint sbmitted to Elseier Science 9 Jly

2 Introdction Small random pertrbations may hae dramatic effects on dynamical systems and lead to the emergence of new dynamical behaiors which, srprisingly, can be deterministic in sitable limits []. Stochastic resonance is a well-known example [,]. A system drien by a weak periodic forcing will oscillate precisely with the period of the driing force in the presence of anishingly small noise, whereas no sch oscillation arises in the absence of noise. The periodic forcing changes which state is temporarily the most faorable energetically. De to the noise, the system is able to reach this most faorable state by actiated hopping eents. The rate of the hopping depends mainly on the energy barrier to be crossed [3,6], and the barrier height aries in time de to the forcing. As a reslt, in a sitable limit as the period of the forcing goes to infinity and the amplitde of the noise to zero, the hopping takes place precisely when its rate matches the freqency of the driing force. This is the resonance phenomenon whose net effect is a synchronization of the system with the forcing. In this standard mechanism of stochastic resonance, the periodic forcing plays an essential role. Yet in recent years there has been nmerical and experimental eidence that small random pertrbations can also trigger transition to a deterministic periodic behaior een in the absence of periodic drie on the system. The corresponding mechanisms hae been termed atonomos stochastic resonance [5,,] or coherence resonance [3,8]. They both arise in systems close to the threshold of bifrcation toward a periodic behaior. Aboe the bifrcation threshold, a limit cycle is present, bt below threshold the intrinsic oscillation is only a transient featre obsered while the system relaxes toward its eqilibrim state. In sch sitations, the noise can temporarily psh the system aboe the bifrcation threshold and thereby trigger a cycle. If the period of this cycle is mch larger than the time-scale oer which it is triggered by the noise, the ratio between the ariance of the period and its mean is ery small, i.e. the phenomenon displays a high degree of coherence. The resonance phenomenon in these mechanisms refers to the possibility to optimize the degree of coherence by adjsting the amplitde of the noise, which is qite different from its original meaning in standard stochastic resonance. In particlar, both in atonomos stochastic resonance and in coherence resonance the time scale associated with the rate of noise-actiated hopping does not hae to match the intrinsic periodic time-scale of the system (it jst has to be smaller than this time-scale to garantee coherence). As a reslt, noise plays a rather passie role in these mechanisms. In addition, the constraint that the system be close to bifrcation reqires fine-tning of the control parameters. Ths, one may wonder if there is a more robst way by which noise can create deterministic periodic behaior in the absence of periodic driing force.

3 In this paper, we show that there exists indeed a robst mechanism, which we term self-indced stochastic resonance, by which small noise can trigger transition to a deterministic periodic behaior in systems away from bifrcation threshold, i.e. in a parameter regime where the zero-noise (deterministic) dynamics of the system does not display a limit cycle dynamics nor een its precrsor. In addition, we show that the noise amplitde works as a control parameter for the coherent motion, hence the noise plays a trly constrctie role in the deterministic dynamics it initiates. A similar mechanism has been reported recently by Freidlin in []. Here we show its biqity in randomly pertrbed excitable systems. (Self-indced stochastic resonance may for instance explain the reslts obsered in the nmerical experiments with the FitzHgh-Nagmo model performed in [5]). Excitable systems arise in a wide ariety of areas which inclde climate dynamics, chemical reactions, lasers, ion channels, nere cells, neral systems, cardioasclar tisses, etc. and are especially common in biology [7 9]. One can think of them as dynamical systems possessing a rest state, an excited state, and a recoery state. In the absence of pertrbations, an excitable system remains in the rest state. Small pertrbations create only small amplitde linear responses. Larger pertrbations, howeer, case large-amplitde dynamical excrsions dring which the system goes to its excited state, then its recoery state, before retrning to the rest state. Generally, the excited phase is fast whereas the recoery phase is slow becase they arise as a reslt of the competition between positie and negatie feedbacks operating on ery different time-scales. As a reslt, the slow recoery motion in an excitable system can generally be described as a motion on a slow manifold that the system qickly reaches after a large excrsion in the excited state. We shall be interested in excitable systems whose excitatory ariables are pertrbed by small amplitde random pertrbations. In these sitations, we show that noise systematically triggers a new large excrsion while the system is in the slow recoery state. The trigger mechanism inoles a barrier crossing eent by which the system escapes the slow manifold associated with the recoery state and whose rate depends on the position of the system on this manifold. As a reslt, the hopping always arises precisely when its rate matches the recoery time-scale. This is the stochastic resonance part of the mechanism. It is combined with a mechanism of reset proided by the excited state which mitigates the periodic forcing necessary in standard stochastic resonance and instead makes it self-indced. Indeed, after the large excrsion in the excited state triggered by the hopping eent, the system goes back to the state of slow recoery motion and the phenomenon can repeat itself oer and oer again. The reslt is the emergence of a deterministic limit cycle indced by the noise whose amplitde and freqency are controlled by a parameter inoling the amplitde of the noise and the ratio between the fast excitatory and the slow recoery time-scales. 3

4 The remainder of this paper is organized as follows. In section we explain the mechanism of self-indced stochastic resonance in the context of excitable system. We establish in which limit the mechanism arises and express the period of the limit cycle in terms of the noise amplitde and the ratio of time-scales in the system. In section 3 we demonstrate the feasibility of the mechanism on the example of the Brsselator. In this system, a complete analysis is possible which we corroborate by a series of carefl nmerical experiments to illstrate how the coherence of the mechanism can be made arbitrarily high, how the noise can be sed as a control parameter, etc. In section we compare selfindced stochastic resonance with coherence resonance to stress the differences between the two. Finally, some conclding remarks are gien in section 5. The general mechanism of self-indced stochastic resonance A generic excitable system consists of a set of excitatory and recoery ariables, denoted respectiely by and, whose dynamics is goerned by = f(, ) + ε η, = αg(, ). Here f and g are the nonlinearities, α is the ratio of the time-scales, and we hae added some external noise η with amplitde ε pertrbing the excitatory ariables. The noise may hae different physical origins. Here we will assme that η is external white-noise, i.e. a Gassian process with mean zero and coariance η(t)η(t ) = δ(t t ) () Different kinds of noise can be sed and lead to qalitatiely similar reslts. When α, there is a large time-scale separation in the deterministic part of the dynamics goerned by f and g. This is consistent with the excitatory ariables being fast and the recoery ariables being slow. Ths the large excrsions in the excited state arise on the time-scale of order, and their dynamics is goerned by the eqation for in () with frozen (except, perhaps, in some part of the excrsion when trns oer to go back to the recoery state, see below). The recoery state corresponds to a slow motion on the manifold defined by the stable soltion branch of f(, ) = containing the rest state defined by f(, ) = g(, ) = (see Fig. for an illstration). In the absence of the noise, the excited state can only be a transient state, and the trajectory qickly approaches the slow manifold of the recoery state, and then proceeds on the time-scale α along this manifold and into the globally attracting steady state. With the introdction of the noise, the sitation changes. The trajectory may ()

5 leae the recoery state and go back to the excited state by escaping the slow manifold in the direction ia a noise-actiated process. We will assme that for small noise, ε, sch process happens at Kramers rate [3,6] k = ν exp( V ()/ε), (3) where V () is some -dependent energy barrier to be crossed to initiate the escape from the slow manifold and ν is some characteristic freqency. As a reslt, the system will perform an excrsion in the excited state drien mainly by the deterministic part of the dynamics. After this excrsion, the trajectory lands again somewhere on the slow manifold of the recoery motion. The dynamics can then proceed along the slow manifold ntil another escape happens, and so on. This process may lead to a trly deterministic limit cycle in a sitable limit becase of the following two mechanism.. Resonance mechanism. The interplay between the escape eents and the motion along the slow manifold reqires that their time-scales be comparable. Assme that the energy barrier V () in (3) decreases as one approaches the steady state by the dynamical path along the slow manifold (this amonts to assming that the escape eents become easier as the system approaches the steady state). Then by (3), the escape rate is a ery rapidly increasing fnction of when ε is small. Sppose that ε, α in a way that ε log α β () for some finite β. Then escape from the slow manifold will occr with probability one at the point = on this manifold, where satisfies V ( ) = β, (5) proided that this eqation has a soltion on the accessible part (i.e. the stable branch) of the slow manifold. For small bt nonzero α and ε, escape occrs with probability close to one in the icinity of the point = where V ( ) = ε log α. Indeed, before reaches, the slow recoery motion is so mch faster than the escape rate that the system has no time to hop before it reaches. Bt as soon as has passed, the escape rate becomes so mch faster than the recoery motion that the system mst hop. Ths the matching of time-scale implied by (5) is precisely the resonance mechanism in standard stochastic resonance [,].. Reset mechanism. This is inherent to the excitable character of the system. After a large excrsion in the excited state initiated from, the trajectory retrns to the slow manifold at a point which leads again to by the slow recoery motion. Then the process will repeat itself indefinitely in a seqence of recoery and excited states and the dynamics of the system will indeed be a limit cycle. 5

6 .8 o.6... o Fig.. Deterministic flow generated by (6). Here A =.7 and α =.. The thick red and ble cres are the - and -nllclines, respectiely. The slow manifold of the recoery state is the portion of the -nllcline at the left of. The excited state corresponds to the portion of the phase space to the right of the -nllcline. Note that is shown on a logarithmic scale. The mechanism described aboe does not reqire the system to be close to bifrcation, and therefore it is robst against parameter changes as long as (5) defines an accessible point on the slow manifold. In addition, since the dynamics in the recoery state is mch slower than the one in the excited state, the period of the limit cycle will be dominated by the time it takes the system to go from the point it lands on at the slow manifold to where it leaes it again. Bt since depends on the amplitde of the noise ε and the ratio of time-scale α throgh (5), both α and ε can be sed as control parameters for the period of the limit cycle. 3 The Brsselator To demonstrate the feasibility of the self-indced stochastic resonance mechanism discssed in section, we will consider the Brsselator [] = + ( + A) + ε η, = α(a ), where and are scalars and A is a control parameter. This is of the form () for the nonlinearities f = + ( + A) and g = A. The Brsselator is a prototypical excitable system when α and A <. (6) 6

7 This can be seen from its phase portrait shown in Fig. for a particlar choice of the parameters. When A <, the nllclines of (6) intersect on the stable branch of the -nllcline, so the flow is always into the niqe eqilibrim point (, ) = (, A). (7) Note that the slow manifold of the recoery state is the part of the -nllcline located at the left of the -nllcline (see Fig. ). It is also clear from the figre that sfficiently large increases in the ariable away from eqilibrim will reslt in large excrsions into the excited state. For A > A ω, where A ω = + α, (8) the system exhibits a Hopf bifrcation: the steady state (, ) = (, A) becomes nstable and a limit cycle emerges een in the absence of noise. We begin by presenting reslts of the nmerical simlations of (6) with a representatie set of ales of the parameters later in section 3. we perform simlations with different parameters. We se α =., ε =. which are reasonably small ales. We also take A =.7 which is not close to the Hopf bifrcation obsered at A ω, see (8). We integrate (6) sing a forward Eler scheme with adaptie time-step to flly resole the fast excrsion in the excited state. Fig. (a) shows the time series for one realization of the noise. This figre shows a train of large amplitde spikes in the excitatory ariable. It is striking that the spikes are occrring in an almost periodic fashion, with their amplitde and other characteristics being approximately the same at each time. This is corroborated by the phase plane portrait, Fig. (b), which shows a nearly limit cycle behaior in the and ariables. The oerall dynamics confirms the scenario in Sec.. A large excrsion in the excited state is followed by a slow motion in the recoery state by which the system tries to go back to the steady state (, ) = (, A). Bt it neer scceeds as the noise systematically triggers new excrsions in the excited state before the system reaches (, ). These excrsions arise in a predictable fashion when is arond <. To frther qantify the degree of coherence of the phenomenon, we hae analyzed the statistics of the interspike time interals as a fnction of the noise amplitde ε for different ales of the time-scale separation ratio α. For the prpose of this analysis, we define as a spike any excrsion with an amplitde max. Fig. 3 shows the mean interspike interal T and its standard deiation σ T obtained from the nmerical soltion of (6). Also, in the inset we show the ratio σ T /T, which characterizes the signal-to-noise ratio for the interspike distance. For ery small ales of ε the spikes hae the character of a Poisson process since σ T /T as ε with α fixed (see the errorbar and the inset 7

8 5 a) t b) Fig.. Nmerical soltion of (6). The parameters are: A =.7, α =., ε =.. (a) The time series with the nearly periodic spike train. (b) The phase plane plot showing the corresponding (almost) deterministic limit cycle in black. The red and ble lines are the - and -nllclines, respectiely. The solid green line shows the predicted limit cycle. Note that the size of the flctations of the trajectory arond the slow manifold (stable branch of the -nllcline) is accentated by the logarithmic scale sed for the ariable. in Fig. 3). They represent rare incoherent large-amplitde flctations away from the eqilibrim point. Similarly it can be seen that for large noise amplitdes, when the noise is no longer weak, the spike train also looses coherence. Howeer, the data plotted in the inset in Fig. 3 clearly show that for the considered ales of α there exists a broad range of ε where the ratio σ T /T is low, signifying high degree of signal coherence (e.g., σ T /T is less than. for 8

9 α T 8 6 σ / T T α =. α =.5 α =... ε logα. ε log α Fig. 3. Mean interspike distance T and its standard deiation σ T (shown as errorbars) as a fnction of the noise amplitde ε. Consistent with () we measre ε in nits of / log α. Inset: the plot of the ratio of the standard deiation to the mean. In all cases A =.7, and the fll cres correspond to the same ale of α as in Fig.. Note the high degree of coherence that can be achieed when α is small, while at the same time T shows significant dependency on the noise amplitde ε.. ε when α =.). Also, the degree of coherence increases as α decreases. At the same time, the ale of T shows significant dependence on ε, while σ T does not (as seen, e.g., from the errorbars). 3. Analysis Now we present a qantitatie explanation of these obserations, consistent with the general scenario gien in section. 3.. Recoery state and escape Sppose first that the noise is absent in (6), ε =. Then when α the system relaxes qickly to the stable branch of the -nllcline at the left of (, ) where and satisfy = + ( + A). Soling this eqation in gies = + A ( + A), (9) 9

10 which can be inserted in the eqation for in (6) to gie = α(a ) + A + ( + A), () (9) and () specify the slow recoery motion to leading order arising on the O(α ) time-scale when α. When the noise is present bt small in amplitde, ε, (9) and () are not alid all the way p to (, ) on the slow manifold (in fact, this point is neer reached), bt they still goern the slow recoery motion ntil the well-defined point where escape from this slow manifold arises. To see this, let α in (6) assming that and are O(). Then the eqation for redces to = which indicates that is frozen on the O() fast-time scale, and on this time-scale, the dynamics is goerned by the eqation for in (6) in which enters as a fixed parameter. It is conenient to think of this eqation as the motion of a particle in the potential, i.e. write it as where V (, ) is gien by V (, ) = + ε η, () V (, ) = ( + A) () This potential is shown in Fig. for different ales of. It has the shape of a left-slanted S, with a local minimm at the intersection of = const with the stable (left) branch of the -nllcline defining the slow manifold and a local maximm at the intersection of = const with the nstable (right) branch of this nllcline. Therefore, een in the presence of a small noise, ε, the system stays confined near this slow manifold and it can only escape ia a noise-actiated hopping eent whose rate at gien has the form of (3) [6]: where k() = ( + A) π exp( V ()/ε) (3) V () = [( + A) ] 3/ () 3 is the energy barrier to be oercome. It is easy to check that V () is a decreasing fnction of on the entire interal < < (+A), i.e. the rate (3), while ery small when ε, is a ery rapidly increasing fnction of. We can now apply the standard stochastic resonance argment recalled in section. There is a precise ale, =, sch that the rate matches the O(α) inerse time-scale on which the slow recoery motion goerned by (9) and () arises.

11 .5 =..5 V V.5 =.5.5 = Fig.. The potential in (), p to a constant, as a fnction of for arios ales of. The barrier decreases as increases. Ths () is the V () to be inserted into (5), i.e. [( + A) ] 3/ 3 = β (5) fixes as a fnction of β = lim ε,α ε log α, and (9) and () are only alid before reaches. 3.. Excited state and reset After the trajectory escapes the neighborhood of the stable nllcline at =, it contines moing toward increasing ales of corresponding to the excited state. At this point the effect of the noise becomes negligible. With the increase of, the effectie time-scale of decreases [see (6)], so when the system ndergoes a large excrsion in the excited state, the time-scales of these two ariables can no longer be separated. On the other hand, when = O(α ) and = O(), (6) can be simplified by neglecting all the terms except. The reslting system of eqations is = = α, (6) with the asymptotic bondary conditions ( ) =, ( ) =. This is eqialent to = ( α), = α, (7) which can be soled exactly to gie the transition layer for the rising part of the spike. It shows that in the spike rises to max = α on the

12 time-scale of α, while approaches zero asymptotically. This is then followed by a retrn to the recoery state, i.e. the fall of the trajectory onto the -nllcline with fixed = (asymptotically) according to with the asymptotic initial condition () = max. = ( + A), =. (8) Following the excrsion in the excited state, the system starts oer again the slow recoery motion goerned asymptotically by (9) and () with initial condition = Characteristic of the limit cycle Since the system spends most of the time on the slow manifold, asymptotically the period of the limit cycle will be eqal to the time T ( ) it takes to go from = to = by (). This time is explicitly calclated by integrating (): T ( ) = α A + ( + A) + + ( A) log A ( ) A + ( + A) (A ) ( ). (9) A + ( + A) Note also that β in (5) mst satisfy β (β c, ) () with ( A)3 β c = = [( + A) ] 3/ () 3A 3 since the attainable ales of on the -nllcline lie in the interal < < = A. In more colloqial terms, this means that for a fixed ale of α there is a critical amplitde of the noise for the establishment of the limit cycle behaior: ( A)3 ε c =. () 3A log α No limit cycle behaior is possible when β β c. As β approaches β c from aboe, we hae, and T ( ). Therefore, if one fixes the parameters in the deterministic part of the dynamics there is a transition to a limit cycle behaior at a critical ale of the amplitde of the noise.

13 Smmarizing the reslts of this section, we hae shown the feasibility of the mechanism of self-indced stochastic resonance on the specific example of the Brsselator. Small noise pertrbations of the excitatory ariable indce a transition to a deterministic limit cycle whose characteristics are controlled by the noise amplitde and the time-scale ratio according to the ale of β = lim ε,α ε log α. The limit cycle is established away from the bifrcation, i.e. for ales of A which do not need to be close to the critical ale A ω as α, see (8). On the other hand, the limit cycle is obsered only if the noise is aboe the critical ale gien by (). We now proceed to corroborate these predictions ia frther nmerical experiments. 3. Nmerical alidation Since the proposed mechanism operates in the limit α, ε, with the ale of ε log α fixed, we performed frther nmerical stdies of the model at smaller ales of the parameter α to erify its predictions. Or first simlation was performed at the same ales of A and ε log α as in Fig., bt with α =., an order of magnitde smaller. The reslts are shown in Fig. 5. From this figre, one can see a significant improement of coherence (the compted ale of σ T /T.8 here), in agreement with or prediction that the coherence becomes perfect in the limit α. Also, the mean interspike interal T is now within % of the ale gien by (9). Simlations show that decreasing α frther consistently improes the agreement with the theory (we erified this down to α = 6 ). We next inestigate the effect of arying the noise amplitde on the parameters of the limit cycle. The simlations are performed at α =. and A =.7 and are shown in Figs. 6 and 7. Or first prediction is that for ε < ε c, which in the limit α is gien by (), no coherent oscillations will exist. This is what we obsere in the simlations when ε.5, see the first row in Fig. 6. Here, instead of a periodic spike train we obsere a Poisson seqence of spikes. On the other hand, at ε =.3 the behaior rather abrptly changes to almost periodic, consistent with or theory (for this ale of α the predicted ale of ε c =., within reasonable agreement with the obserations). Increasing the noise amplitde, we see that the period and the amplitde of the obsered periodic limit cycle decreases, while the spike train maintains a high degree of coherence, see the two lower panels of Fig. 6. One can also see from the phase portrait in the lower panel of Fig. 6 that the trajectory leaes the -nllcline at a particlar point below the eqilibrim (, ) ia a noise-actiated process, consistent with or mechanism. This is seen more dramatically at larger ales of ε, see Fig. 7. Clearly, the location of the eqilibrim has no effect on the ale of at which the trajectory escapes 3

14 6 a) t b) Fig. 5. Nmerical soltion of the stochastic differential eqation for α =., ε =.67, A =.7 the -nllcline. Remarkably, howeer, the limit cycle maintains its coherence een for ε = (then the noise is no longer weak, bt V ( )/ε is still large, which sffices for or theory to apply). One can also clearly see that noise indeed controls the parameters of the limit cycle. For example, the period and the amplitde of the limit cycle changes seeral-fold in the simlations in Figs 6, 7, while the degree of coherence is essentially nchanged. In other words, the noise amplitde really acts as a control parameter for the oscillatory behaior of the system.

15 α, ε = αt ε =..8 α, α, αt ε = αt. Fig. 6. The effect of arying the noise amplitde: reslts of the simlations at α =., A =.7 Finally, we erify the robstness of or mechanism by showing that it does not reqire tning of the parameters and can be realized far from bifrcation points. To this end, we take α =. and A =.3, which is abot three times smaller than A ω. The reslts of the simlation with ε =.6 are shown in Fig. 8. Note that for these ales of A one needs stronger noise to indce the oscillations (for example, the predicted ale of ε c at these parameters is ε c.). Once again, we see a coherent almost periodic spike train, with a clearly defined ale of which is significantly lower than, consistent with or theory. 5

16 8 7 6 ε =..8 α, αt ε =..8 α, αt ε =..8, α αt. Fig. 7. The effect of arying the noise amplitde (contined): reslts of the simlations at α =., A =.7. (Recall that the size of the flctations arond the stable branch of the -nllcline are accentated by the logarithmic scale we se for.) Comparison with coherence resonance As explained before, self-indced stochastic resonance arises away from the Hopf bifrcation at A = A ω and the limit cycle it indces can be controlled by the parameter β = lim ε,α ε log α (β c, ) (recall that the constraint β > β c garantees that the system hops ot of the slow manifold of the recoery state and back into the excited state before reaching the steady state). Self-indced stochastic resonance persists in the limit as A, which is the limiting ale of A ω as α (see (8)), proided that β > in this limit (in fact, this is tre een when A > ). Howeer, as A, a limit cycle can also be established when β =, bt by a mechanism different from ors and closer to coherence resonance. 6

17 a) α, αt b) Fig. 8. Tning is not reqired for the mechanism: reslts of the simlation at α =., A =.3, ε =.6. Indeed, sppose that A first and then let α, ε in a way that β = lim α,ε ε log α =. In this limit, the steady state is (, ) = (, ), the point located at the top of the -nllcline, which is netrally stable. Starting from =, integrating () shows that the deterministic trajectory reaches the top of this nllcline in finite time T = T ω, where T ω = 3α. (3) Since β =, the noise is too weak to make the trajectory hop from the - nllcline before it reaches (, ). Bt once the trajectory reaches (, ), since there is no energy barrier to oercome to escape this point, a anishingly 7

18 small amont of noise allows the trajectory to take off and perform a large excrsion in the excited state described by (7) and (8), then retrn back to = on the -nllcline and complete the cycle. Since the motion described by (7) and (8) is mch faster than the one described by () which led to (3), the period of the limit cycle is gien by T ω in the limit as α. It is also clear that T ω will be the period of the emerging limit cycle as A +. So, the noise plays a rather passie role in this mechanism of coherence resonance since the limit cycle indced by noise right before bifrcation is essentially the same as the one that will be established right after bifrcation, and its period T ω gien in (3) depends on α only and not on ε. 5 Conclding remarks In smmary, we hae shown that a anishingly small noise in excitable systems with strong time-scale separation can indce a transition to a limit cycle behaior. The mechanism, which we term self-indced stochastic resonance, is robst as it arises away from bifrcations, and it leads to a limit cycle whose period and amplitde can be controlled by the noise. This is different from other mechanisms like coherence resonance which are less robst as they reqire the system to be close to bifrcation and lead to a limit cycle which is essentially the same as the one that emerges after bifrcation and cannot be controlled by the noise. We beliee that self-indced stochastic resonance may hae important implications in seeral areas. Consider for instance copled excitable systems. Or analysis indicates that in sch systems the leel of the noise, both extrinsic and intrinsic, may be sed as an information carrier and be transformed into a (qasi-)deterministic signal. As a prototype, consider a system of all-to-all positiely copled excitable cells. Under the action of the noise of sfficiently small amplitde each cell will occasionally generate a spike. These spikes will hae random phases, so their total inpt on each indiidal cell may aerage to a stationary random signal of low intensity. Now, if the noise leel sddenly increases de to an external distrbance, the cells may switch to the noise-assisted oscillatory mode. This will frther increase the effectie noise amplitde, so that the oscillatory mode may persist een after the distrbance is remoed. In colloqial terms, the system in a dormant state may wake p from the otside rattle. In a similar way, or reslts may be applied to spatially distribted excitable media [9]. In these systems the analog of the noise-actiated eent will be the formation of a radially-symmetric ncles, leading to sbseqent initiation of 8

19 radially-diergent waes [9,,]. In the wake of sch a wae the system will ndergo recoery. It is clear, then, that the system will be most recoered at the position where the wae was initiated. Hence, the new wae will be initiated again at the same spot, with the dynamics repeating periodically. This sggests that the well-known phenomenon of target pattern formation in two-dimensional excitable media [9] might hae an alternatie interpretation in terms of noise-drien periodic wae generation. Acknowledgements C. B. M. is partially spported by NSF ia grant DMS-86. E. V.-E. is partially spported by NSF ia grants DMS-39, DMS-9959 and DMS W. E is partially spported by NSF ia grant DMS-37. References [] M. I. Freidlin, Qasi-deterministic approximation, metastability and stochastic resonance, Physica D 37 () [] M. I. Freidlin, On stable oscillations and eqilibrims indced by small noise, J. Stat. Phys. 3 () [3] M. I. Freidlin, A. D. Wentzell, Random Pertrbations of Dynamical Systems, Springer-Verlag, New York, 98. [] L. Gammaitoni, P. Hänggi, P. Jng, F. Marchesoni, Stochastic resonance, Re. Mod. Phys. 7 (998) [5] H. Gang, T. Ditzinger, C. Z. Ning, H. Haken, Stochastic resonance withot external periodic force, Phys. Re. Lett. 7 (993) [6] C. W. Gardiner, Handbook of stochastic methods for physics, chemistry and the natral sciences, Springer-Verlag, Berlin, 985. [7] J. Keener, J. Sneyd, Mathematical Physiology, Springer-Verlag, New York, 998. [8] B. Lindner, J. Garcia-Ojalo, A. Neiman, L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep. 39 () 3. [9] A. S. Mikhailo, Fondations of Synergetics, Springer-Verlag, Berlin, 99. [] C. B. Mrato, V. V. Osipo, Spike atosolitons and pattern formation scenarios in the two-dimensional Gray-Scott model, Er. Phys. J. B () 3. 9

20 [] G. Nicolis, I. Prigogine, Self-Organization in Non-Eqilibrim Systems, Wiley Interscience, New York, 977. [] V. V. Osipo, C. B. Mrato, Ultrafast traeling spike atosolitons in reactiondiffsion systems, Phys. Re. Lett. 75 (995) [3] A. S. Pikosky, J. Krths, Coherence resonance in a noise-drien excitable system, Phys. Re. Lett. 78 (997) [] T. Wellens, V. Shatokhin, A. Bchleitner, Stochastic resonance, Rep. Prog. Phys. 67 (), 5 5. [5] M. P. Zorzano, L. Vázqez, Emergence of synchronos oscillations in neral networks excited by noise, Physica D 79 (3) 5.