Resonant behavior of a Poisson process driven by a periodic signal

PHYSICAL REVIEW EVOLUME 52, NUMBER 6DECEMBER 1995 Resonant behavior of aPoisson process driven by aperiodic signal J.Javier Brey, J.Casado-Pascual, *and B.Sanchez Fisica Teorica, Universidad de Sevilla, Apartado Correos 1088, E$10-80 Sevilla, Spain (Received 12 April 1995) The statistical properties of the residence times of aperiodically modulated Poisson process (time-dependent shot noise) on aline segment are analyzed. They show the characteristic features of a resonant behavior, which is similar in many aspects to the stochastic resonance taking place in systems confined in monostable or multistable potentials. The dependence of the mean residence time on both the frequency of the periodic stimulus and its amplitude is studied in detail. The behavior of this parameter also displays the efFects associated with the interplay between noise and deterministic modulation. PACS number(s): 05.40.+j, 82.20.Mj I. INTRODUCTION The study of the efFects which appear as aconsequence of the interaction of noise and periodic stimuli has attracted aconsiderable interest over the past decade or so. The best known and understood phenomenon is the so-called stochastic resonance, which consists in the enhancement of the response to asmall periodic signal by noise in anonlinear system [1,2]. In fact, most of the treatments to date deal with nonlinear systems, where the cooperative e8'ects of the coupling between deterministic and random dynamics were expected to be interesting. Nevertheless, it has been shown that similar and partially unexpected cooperative features also appear in linear systems. In particular, Fletcher, Havlin, and Weiss [3] have described the response of aone-dimensional random walker on aline connecting two traps to asinusoidally time-dependent stimulus. They show that the mean residence time as afunction of the &equency of the field goes through aminimum. More recently, Bulsara, Lowen, and Rees [4] have considered aperiodically modulated Wiener process with an absorbing boundary. The behavior they obtain for the response bears agreat resemblance to stochastic resonance, although major diKerences exist in the physical characterization of their model and of bistable systems. It is interesting to notice that the works reported in both papers, Ref. [3] and Ref. [4], were motivated by the modeling of biological processes, namely, pulse field electrophoresis [5] in the former and the response of sensory neurons in the latter [6]. In the present paper we study the response properties to periodic external stimuli of avery simple model. It is aone-dimensional random walk model, like the two mentioned above. In fact, many of the cooperative ef- 'Permanent address: Matematicas Aplicadas II, E.U. Politecnica, Universidad de Sevilla, Virgen de Africa 7, E41011 Sevilla, Spain. fects we will discuss are very similar to the behaviors previously reported [3,4,7], although some significant differences also occur. The theory of erst-passage times in stationary one-dimensional random walks is now well established and exact expressions have been worked out for their statistical properties [8,9]. Nevertheless, the methods developed do not apply in general when an external time-dependent field is present. For this reason, Fletcher et aI,.use numerical and perturbative techniques to study the mean first-passage time in their model. On the other hand, Bulsara et al. consider acontinuous infinite random walk with only one absorbing boundary, and they are able to construct the exact solution of the FokkerPlanck equation for the distribution function of the variable by means of the method of images. Let us also point out for completeness that the mean residence time in aline segment presents as well aresonantlike behavior when the state of the lattice fluctuates [10]. In this case, it has aminimum as afunction of the correlation time characterizing the fluctuations. This phenomenon has been termed resonant activation [ll]. The main difFerence apriori with the resonant effects mentioned above, which are the ones studied here, is the absence of any external deterministic force in the resonant activation. The latter is due to the interaction of two noise contributions. The model we will deal with is asinusoidally modulated Poisson process, in which the walker can move in only one direction. Therefore the residence time in afinite segment can be defined by means of only one absorbing boundary. The simplicity of the model permits us to solve exactly the associated master equation and to study its response properties analytically, without introducing any approximation. This allows abetter understanding of the origin of the resonant efFects which are observed. In particular, we will analyze in detail the relevant role played by the initial phase of the periodic stimulus, a point that has not been addressed up to now. The model can be useful to describe awide variety of phenomena in very diverse areas. For instance, it is possible that the voltage of amembrane controlling some neural firing events [12]monotonically increases in time until reaching 1063-651X/95/52(6)/6071(11)/$06. 00 52 6071 1995 The American Physical Society 6072 J.JAVIER BREY, J.CASADO-PASCUAL, AND B.SANCHEZ 52 athreshold value at which the firing event occurs. If the successive increases of the voltage are assumed to take place at biased random moments, our model can be used to describe its evolution. The presentation is organized as follows. In the next section the model is formulated and exact explicit expressions are given for the first-passage time distribution function (FPTDF) and the mean first-passage time (MFPT). The structure of the FPTDF is analyzed in Sec. III, where it is seen to present many of the characteristic features that have been studied in noisy bistable systems in the presence of deterministic modulation. The MFPT is discussed in Sec. IV. It is shown to exhibit a resonantlike behavior when considered as afunction of the frequency of the external stimulus. The shape of the function strongly depends on the initial phase ofthe stimulus. This parameter also plays an important role in determining the behavior of the MFPT when the amplitude of the external perturbation is varied. In some cases, the MFPT makes transitions &om amonotonic decrease with increasing amplitude of the sinusoidal term to amonotonic increase with this parameter as the &equency of the signal changes. Finally, in the last section, we summarize some of the main conclusions of this work. Also, abrief reference is made to the power spectral density which is obtained when asimple deterministic mechanism of reinjection for the walker is introduced. Its properties bear again agreat resemblance to those found in bistable models. f(t) =~(t)pN(t) and the mean first passage time is OO N dt tf(t) =)dt p„(t). 0n=O (5) Use of Eq. (2) yields 7=)dt I'(N +1— n, I(t)), (6) p-(o) I' %+1— no where I'(n, x) is the incomplete gamma function [13]. Now we particularize for asinusoidally oscillating external field and take c1(t) =p+esin(ddt +Q). The parameter emeasures the amplitude of the periodic stimulus and it is restricted to the interval 0&e&p. Besides, we also consider the specific initial condition p„(0) =b„,e. (8) Then, the FPTDF reads the random process we are considering, the walker never returns to asite after leaving it. Thus the first-passage time distribution function f(t) is simply II. THE MODEL with f(t) =II(t)l" -"' N! (9) The model we will consider is acontinuous-time random walk on aone-dimensional lattice. The walker is restricted to move in agiven direction and it is submitted to atime-dependent external field. The probability p„(t) of finding the random walker at site nat time t obeys the master equation 0(t) =(t) [p —(t) —(t)l E I(t) =pt —— [cos(mt+ P) — cosP] .(10) For given values of the parameters, I(t) is an increasing function of t. Although Eq. (9) provides an exact explicit expression for f(t), and will be used in most of the calculations, it is convenient for future use to consider also the linear stimulus approximation. Expansion of Eq. (9) to first order in egives where the transition rate a(t) is non-negative for all times. This equation can be easily solved using, for instance, the method of the generating function [8] and the general solution reads fl(t) =fo(t) 1+—~sin(~t+4) pE +~[cos(mt+ P) — cosP] ~ ~t where -(t) =): „"„,,[I(t)]"-" -"' n' =— oo (2) fo(t) =,(pt) (12) with t I(t) =dt' n(t') 0 is the FPTDF for the Poisson process in absence of the periodic stimulus. The corresponding linear approximation for the MFPT can be derived by linearizing Eq. (6) or by introducing Eq. (11)into Eq. (5). The result is In what follows we will be concerned with awalker which is initially in the line segment (0,N). Therefore for n) Nthe sum in Eq. (2) extends &om n' =0to n' =N. Our main interest will be in the properties of the arrival times to site N+1. Because of the Poisson nature of cos P—cos N+1arctan —+ N+1 ('+-.:) ' 52 RESONANT BEHAVIOR OF APOISSON PROCESS DRIVEN BY...6073 Here Tp 1s 'tile MFPT foI' the dlstI'lblltloI1 fp('t) 1.e. Tp —— (N+ 1)p III. FIRST-PASSAGE TIME DISTRIBUTION FUNCTION The temporal behavior of the FPTDF depends on the value of the &equency of the perturbation. We have carried out an asymptotic analysis of Eq. (9) in the limit of large ¹The results remain qualitatively valid for Nnot too small. In Appendix Awe give some details of the calculations and here we merely present the main conclusions. At large &equencies, uN /pof the order of unity or larger, the FPTDF is given by f(t) =N— 1/2 1 ir (t) exp —[I(t) — N] 27r 2N exp (t — t)' 20 where N1/2 p+esin((dt~ +P) and tis the time at which the function h(t) =exp —[I(t) — N] 12 2N The dominant region of the distribution is determined by the condition I(t) O(N), which leads to tO(ro). In this region, the FPTDF presents aseries of peaks due to the a(t) factor. For lower frequencies, uNI/2p 1«1, Eq. (14) reduces to the Gaussian form 1 t) —— —2~l +—— (8 2(21) where lis apositive integer. More precisely, Eq. (21) holds in the limit itI — ti«1. ~~N (22) Aproof of this is sketched in Appendix B. In practice, Eq. (21) provides agood approximation for all the peaks which show up in the relevant part of the FPTDF, as is by Figs. 1and 2. In Fig. 1the FPTDF computed &om Eq. (9) is shown for N=50, p=1, s=0.3, P=vr/2, and u=2. Also plotted are Eq. (14) and fo(t). The function f(t) oscillates around fo(t), in agreement with the above discussion. In Fig. 2the values of the parameters are the same with the only difference that now it is u=0.01. The curves presented correspond to Eqs. (9), (15), and (12), respectively. The differences between the exact and the asymptotic expression disappear on the scale of the figures if one takes N=100. The multipeaked structure of the FPTDF follows directly &om the presence of noise and has been extensively studied in the case of modulated noisy bistable systems [2]. Very recently, Bulsara, Lowen, and Rees [4] found the same structure in aperiodically driven continuous random walk with an absorbing boundary. In fact, their model is similar to ours in the sense that the presence of apositive drift guarantees that the walker will reach the boundary for all values of the parameters, contrary to what happens, for instance, in bistable models ofstochastic resonance [2]. The exact location of the peaks of the FPTDF is a quite complicated function of all the system parameters. Nevertheless, the position, t~, of those peaks located in the vicinity of tis given by is maximum, i.e.,it is the solution of the equation I(t )=Nor pldt~ — t[cos((dt~ +f) — cos Q] =ldN) which, in general, has to be solved by means of graphical or numerical methods. Therefore f(t) consists now of a single peak located at t=t.Finally, in the range of very low frequencies, defined by AN@ ((1, the solution of Eq. (18) is N p+ csin/ (19) and Eq. (16) simplifies to p+ssl n$ tm N1/2 (2O) 15 t/To 20 We notice that in this very low &equency limit the FPTDF does not depend on the &equency of the stimulus. When Nis not asymptotically large, the separation of the &equency scales is not well defined, but the three ranges discussed above are clearly identified as illustrated FIG. 1. FPTDF versus normalized tiIne t/Tp, where Ts ——2vr/u. For all curves it is N=50, p=1, &u =2, and @=z./2. For the dashed curve it is s=0, Eq. (12), and for the other two e=0.3. The solid curve has been plotted using the exact expression, Eq. (9), and the point curve using the asymptotic expression, Eq. (14). 6074 J.JAVIER BREY, J.CASADO-PASCUAL, AND B.SANCHEZ where A=%+1— (27) and LL C3 CL C) 6cos B=N —1— +et). (28) 0.04 0.06 0.08 t/To 0.10.12 I——L It happens that the right hand side of Eq. (26) is smaller than efor some values of the parameters, indicating that in those cases the maximum cannot be observed in the parameter region in which the model is defined. Prom the analysis of Eqs. (25) and (26) it can be easily proved that FIG. 2. FPTDF versus time for the same values of the parameters as in Fig. 1, with the only differences that now it is ~=0.01 for all curves and the point curve corresponds to Eq. (15). (29) and this allows the determination of the boundary value t~ of t~ for which the maximum of the peak will be reached upon varying pin the interval e(p(oo. By making p, l=ein Eq. (26) one gets the case, for instance, in Fig. 1. Of course, when t — (2vrk+ 2— P), with kapositive integer, there is apeak at tand it is the highest one of the distribution. If the FPTDF is considered as afunction of the constant part pof the transition rate for an arbitrary given time t, it is found by direct difFerentiation of Eq. (9) that where and /+(/2 +8D)i/2 4642 C=(2N +1)a — 2e cos P (30) (ln fp)' =- cl in f1Nt gP 0! (23) D=e'ld cos P. (ln f„)"—: 8lnf 1 t9p o.' Nt2 I2 (24) These equations indicate that (ln f„)' has two vertical asymptotes at p=pi and p=p2, defined by a.(pi, t) =0 and I(p2, t) =0, respectively. Except at these singularities, (ln f„) is amonotonic decreasing function of p. Furthermore, (ln fz)' tends to tas pgoes to i— nfinity. As aconsequence, (ln f„)' has azero for some critical value p=p, inside the interval max(pi, p2) (p, (oo. For this value, f(t) presents amaximum as pis increased. We have shown above that the location of the peaks inside the relevant region is approximately independent of p. It follows that the heights of the peaks will go through amaximum as functions of p. The critical value pEat which the peak located at t~ presents its maximum will verify the equation So, the maximum can be reached for those peaks such that t~ (t~. In Fig. 3we plot the height of the peaks corresponding to l=3,4, and 5, respectively, for aparticular set of system parameters given in the figure caption. For them it is tM — —102.5, which leads to M=32.62, i.e.,the above discussion predicts that the last peak going I(p,ltl) +tl (per, tlL) [lN I(p,ltl)I — 0. (25) 1A+ (B'+4N)'L' 2ti (26) This is asecond degree equation for p~. One of the solutions is always smaller than eand, therefore, must be eliminated. The other one is FIG. 3. FPTDF peak heights versus pfor N=20, m=2, e=0.2, and P=nj2. From left to right the curves correspond to peak numbers I=5, l=4, and L=3. 52 RESONANT BEHAVIOR OF APOISSON PROCESS DRIVEN BY...6075 through amaximum is the one located at t=t32. This is very close to the exact value obtained numerically, which is t=f33 as seen from the results presented in Fig. 4. Notice that when apeak reaches its maximum height, it does not mean that it is the highest one of the FPTDF for that value of p. The above resonant effect in the peak heights of the FPTDF is well known in noisy bistable dynamical systems, as well as in Wiener processes with drift and an absorbing boundary [4]. Moreover, in the latter, an antiresonant behavior of the FPTDF was pointed out by Bulsara, Lowen, and Rees in Ref. [4]. The height of the maximum of the distribution presents aminimum as a function of noise. Asimilar effect does not take place in our model. For e=0this can be seen by differentiation of Eq. (12). The height of the distribution is amonotonic increasing function of p. The same happens for e&0, as can be shown in the following way. For agiven value of p, we can set the &equency such that aparticular peak l is the highest of the distribution. This amounts to taking the value of w=w~ such that the enveloping of the peaks of f(t) has its maximum at t=ti. The equation of the enveloping, f(t), is obtained by making cos(cut +P) =0 and sin(ut +P) =1in the expression of f(t) Usin. g Eq. (9) one gets gin f(t) N— pt —— 'cosP Bt pt+ — 'cosP =p (33) O C30.20.25 0.30.35 FIG. 4. FPTDF peak heights versus pfor the same values of the parameters as in Fig. 3. From left to right the curves correspond to peak numbers l=35, l=34, and l=33. Therefore f(t) will have amaximum at t=ti if the frequency is pI' 7r ~r =— ~2vrl+. —— P+ —cosP ~. N(2p Changing pleads to FPTDF's which all present their maximum at the lpeak, and the height of this maximum, f„(p), is obtained by substituting the values of t& and wi in Eq. (9). The result is IV. RESONANT BEHAVIOR OF THE MEAN FIRST-PASSAGE TIME In this section we are going to study the behavior of the MFPT was afunction of the several parameters defining the model. We will consider first the dependence on the frequency &u. From Eqs. (6) and (8) one gets ~(0) =~((u =0) =1+— 'sing p(36) and ~(oo) =lim 7(ur) =~o, (37) as expected. It is also obtained that fOw) (N+ 1)(N+ 2)ecosg (8(u) 02(p+ using)s (38) It is clear that if the initial slope of w(u) is negative and w(0) (~(oo), there must be at least aminimum at some frequency u*, which, in general, will depend on N, e, p, and P. On the other hand, ifthe initial slope is positive, it is also possible for 7(w) to present aminimum, but there must be amaximum before it, i.e.,at lower &equencies. For the case of negative initial slope and w(0) )w(oo) no conclusion emerges from Eqs. (36)— (38). Equation (38) shows that the sign of the izntial slope of w(u) is determined by the value of the initial phase Pof the sinusoidal perturbation. The same parameter also determines whether w(0) is larger or smaller than 7(oo). For 0&P(vr/2 the MFPT always presents a minimum. As mentioned in the Introduction, asimilar effect was noticed by Fletcher, Havlin, and Weiss [3] for arandom walker that diffuses on aline connecting two traps, and they referred to u* as aresonant frequency. Their numerical study was restricted to the value P=0 and they did not discuss the relevance of the initial phase of the perturbation. Of course, the structure of Eq. (6) shows that 7as afunction of ucan present not only a first minimum but also aseries of peaks. Their origin can be further easier understood if we consider the linear in eapproximatioii of the MFPT given by Eq. (13). The secondary peaks will be appreciable if the cosine term has time to oscillate before it becomes very small due to the (1+sr /p )~factor, i.e.,if Nis large enough for lixed values of all the other parameters. This is illustrated in Fig. 5, where we have plotted 7/(N +1) for N=10 and N=50. In each of the cases two values of the amplitude, e=0.1and e=0.3, have been considered. In all curves P+~ w— N f„(p) =Ne N! which is independent of the particular peak /chosen. Since the right hand side of Eq. (35) is aznonotonic increasing function ofp, we get neither aresonant behavior nor an antiresonant one. Prom this point of view its behavior is different &om both bistable systems and the Wiener process discussed in Ref. [4]. 6076 J.JAVIER BREY, J.CASADO-PASCUAL, AND B.SANCHEZ 52 fI[&t]II& & [IIIIwhere ais of the order of unity. Use of Eqs. (5), (15), and (18) yields cos((d 'r +Q) — cos f+id TSII1(ld 7+Q) =0(40) and p~* +e7 'sin(u*r* +P) =N, (41) LA CI3 where ~* =w(~'). The balance of Eq. (40) shows that ~* -a'N. (42) K3 o0 IIIIIIIIIIIILiIIIII 0.20.30.40.5 FIG. 5. MFPT divided by N+1versus frequency of the stimulus for e=0.1(solid curves) and e=0.3(point curves). For the two upper curves, showing two minima, it is N=50, while for the two curves presenting only one minimum it is N=10. For all curves p=1, and P=0. The above asymptotic dependencies on Nof ~* and 7* seem to be quite general for the resonant behavior of the MFPT for random walks on aline segment [3,10]. Substitution of Eqs. (39) and (42) into Eqs. (40) and (41) leads to cos(aa' +P) — cos P+aa' sin(aa' +P) =0 and pa' +ea' sin(aa' +P) =1. (44) it is P=0and p=l. For P=m/2, the MFPT has an absolute minimum at u=0, as shown in Fig. 6for three choices of e. The existence of this minimum has atrivial explanation. For P=vr/2 the transition rate cr(t) takes its greatest possible value at t=0and, therefore, the minimum MFPT corresponds to the limit in which o. remains constant in time. We have also investigated the asymptotic interval length dependence of the &equency u* at which the first maximum (or minimum) of the MFPT appears. From the asymptotic expressions for f(t) obtained in Appendix Aand discussed in Sec. III it follows that the first stationary points of f(t) appear for aN Prom this couple of equations one can determine aand a' from the values of p, e, and P. Figures 7 and 8 plot ~* and w*, respectively, as functions of Nfor several values of e. The curves have been obtained by numerical integration of Eq. (6) and for all of them it is P=0 and p=1. The figures clearly confirm the asymptotic behaviors of Eqs. (39) and (42). Moreover, if the slopes for large Kare compared with the solutions of Eqs. (43) and (44), an excellent agreement is obtained. The discrepancies are within the numerical errors. Of course, upon solving Eqs. (43) and (44) one has to choose, for each value of e, the solution leading to the smallest positive values of aand a'. It is also worth mentioning that the equations show that the resonant f'requency u* is a ItI C3 X CO X CO IC3 X Iti & iI»Il~ IC3 X I 1.5X10 2X10 I 2.5X10 0.20.30.4 FIG. 6. MFPT divided by N+1versus frequency of the stimulus for e=0.1(solid curve) and e=0.3(point curve). For the two curves it is N=50, p=1, and P=~. FIG. 7. Resonant frequency u* versus the inverse of the size of the lattice segment for several values of the amplitude e. Prom top to bottom the curves correspond to e=0.05, 0.2, 0.4, 0.6, and 0.9. For all curves it is P=0and p=1. 52 RESONANT BEHAVIOR OF APOISSON PROCESS DRIVEN BY.. . 6077 C3 C3 tr ~ sirs 00 C& 0 G3 C3 0 C3 IIIIII I II I I I I I 400 500 600 700 800 monotonic increasing function of the bias parameter e, while w* decreases with the same parameter. Now let us examine the ~dependence of the MFPT. From Eq. (13) we have FIG. 8. (First) minimum MFPT v* plotted as afunction of the size of the lattice segment for several values of the amplitude e. From top to bottom the curves correspond to e=0.9, 0.6, 0.4, 0.2, and 0.05. For all curves it is P=0and p10 The transition of the MFPT, &om being adecreasing function of the amplitude of aperiodic cosine signal to increase with this parameter as the &equency of the signal varies, has been noticed by Gitterman and Weiss [7] for arandom walk between two trapping points, i.e., the same system as in Ref. [3], but with adifFerence of vr/2 in the initial phase of the signal. In our model, the lowest &equency at which the transition takes place is ~=ptan N&, but transitions &om one regime to the N+1 ~ other occur at all &equencies u=ptan Nz, with ka positive integer. The behavior of w(e) is quite difFerent for P=0. The right hand side of Eq. (45) is in this case negative for all values of the &equency uand no transition appears. This shows the important role that the initial phase of the external periodic stimulus also plays in determining the dependence of the MFPT on the amplitude of the perturbation. Since we have discussed only the initial slope of the function w(e), no information can be obtained about whether it is amonotonic function. We have analyzed numerically Eq. (6), and it turns out that, for those values of Pfor which transitions appear, there is afrequency range near the transition &equency where 7. shows anonmonotonic behavior as eis increased. An example of the described behavior is given in Fig. 9. All the results presented in this work have been restricted to the specific initial condition given in Eq. (8), but they remain qualitatively the same for other choices. In particular, if the uniform initial condition (B~ t 4~') .=o cos N+ 1arctan —+— cos N+1 ("-. :) ' (45) &-(0) =N 1 N+1 ~; (49) Therefore the initial slope of w(e) will depend, among other factors, on the initial phase Pof the stimulus. Consider first the specific value P=7r/2, so that &Br) 0~e) .=o sin N+ 1arctan-p N+1 ("-:)'(46) For fixed Nand p, the right hand side of this equation will be positive or negative depending on the argument of the sine function, that is to say, on the value of the &equency. For uin the intervals (2k +1)m 2(k+ 1)vr ptan (u(ptan N+1 +(47) k=0, 1,2, ...,II:q, with kq the largest integer number equal to or smaller than (1V— 3)/4, i.e.,ki — —[(K— 3)/4], the initial slope uf w(e) is positive, i.e.,wis an increasing function of efor small e. On the other hand, for 0,20.60.8 2k~ (2k+ 1)~ ptan N+1 (u (ptan N+1 (48) k=0, 1,2, ...,k2, with k2 ——[(N —1)/4], 7(e) is adecreasing function for small e. FIG. 9. MFPT versus the amplitude efor cu =0.210, 0.2115, and 0.213 (top to bottom) around the second tranaction. For the three curves it is N=29, p=1, and P=vr/2. 6078 J.JAVIER BREY, J.CASADO-PASCUAL, AND B.SANCHEZ is considered, the main change in the behavior of the MFPT is that all the curves shown become smoother and, for instance, the peaks in Fig. 5for K=50 disappear. Nevertheless, the asymptotic dependencies given by Eqs. (39) and (42) and the existence of transition frequencies as discussed above remain valid. sition, resetting also the periodic stimulus to its initial value [4,14]. In this case, the crossing times are independent of one another and have the same distribution. As a consequence, the global random walk becomes arenewal process. An important property of renewal processes is that the power spectral density (PSD) of the response can be easily obtained from the FPTDF through the relation V. DISCUSSION AND CQNCLUDINC REMARKS S(n) =1bl(nil+1R I+X(n) ),2~) ~1— y(B) (50) The objective in this paper has been to study the statistical properties of the residence time on aline segment for aPoisson random walk with aperiodic timedependent transition rate. The simplicity of the model allows us to solve the master equation governing its evolution and to obtain an exact explicit expression for the probability distribution of the residence times. The structure of the latter shows many of the features which are considered as characteristic ofresonant behavior and, in particular, of the phenomenon of stochastic resonance in bistable systems. Nevertheless, the deep physical differences between the model considered here and bistable models prevent us &om trying to establish any fundamental relationship between both phenomena. First, in our model the boundary is always reached, in contrast with bistable models of stochastic resonance, where the amplitude of the perturbation is too small to produce by itself crossing of the potential barrier. In fact, the amplitude of the modulation does not play an important role in determining the resonant behavior of our system. Secondly, in Poisson processes noise and drift are controlled by the same parameter and, therefore, no competition between them is possible. This also makes arelevant difference with the random walk model studied by Bulsara, Lowen, and Rees [4]. The mean residence time exhibits aresonant behavior too, in the sense that some coherence is induced in the motion of the system, tending to reduce the time it takes the walker to reach the boundary. When the mean time is considered as afunction of the &equency of the stimulus it can go through aminimum. However, this phenomenon is strongly influenced by the initial value of the perturbation. In particular, when the perturbation is maximum at the initial time it is evident that the minimum mean residence time will appear in the limit of zero &equency. Another interesting property which has been analyzed is the existence of atransition in the behavior of the mean residence time regarded as afunction of the amplitude of the stimulus, passing &om being an increasing function to adecreasing function of the same parameter. In this paper we have focused on the statistics of single escape events &om the line segment, without introducing any reinjection mechanism of the walker into it. Such a mechanism is necessary if we want to model firing events taking place when the walker reaches the end of the segment, as might happen, for instance, in sensory neurons [12]. The simplest possibility is to consider adeterministic instantaneous reset of the walker to its starting powhere Adetailed analysis of Eq. (50) for our model will be presented in afuture paper. Here we only comment on some of the relevant conclusions which come &om it. The PSD corresponding to the &equency of the applied periodic stimulus u, i.e.,the function S(A =w) presents aseries of peaks which are approximately located at the &equencies 2kvrp N(52) 0.5 FIG. 10. Power spectral density S(cu) versus the dimensionless variable J) =~" for u=1(solid curve), u=0.75 (point curve), and u=0.5(dashed curve). For all curves it is N=50, P=s'/2, and e=0.2. with kapositive integer. The height of the peaks is a quite fast monotonic decreasing function of k, so that the PSD presents aclear global maximum at ~=~q. In practice, for values of Nnot too large only afew peaks can be resolved on the scale of the highest one, i.e.,of the first. Now we consider the eKect of changing pwhile keeping constant the value of the modulation &equency. As shown in Fig. 10 for three diferent values of ~, the PSD has aresonant behavior. There is again aseries of maxima located near RESONANT BEHAVIOR OF APOISSON PROCESS DRIVEN BY...6079 although their amplitudes decrease rapidly with increasing k, and awell de6ned absolute maximum appears for p=ur N/27r. Besides, the height ofagiven maximum, i.e., for fixed k, is amonotonic increasing function of u. The above behavior of the output signal strength is qualitatively similar to what is observed in stochastic resonance in bistable systems, although, as we have already pointed out, the physical mechanisms responsible for both phenomena are diferent. keeping only the lowest order, yields N1/2 g(s) =@'(s)e 27r (A8) where the Stirling approximation for N! has been used. This expression is avalid approximation to Eq. (A7) for all values of sfor which g(s) is not exponentially small. It is trivially checked that g(s) as given by Eq. (A8) is normalized to 1. To see whether it is possible to simplify further g(s), consider the expansion ACKNOWLEDGMENT +il'(s )@"(s )(s — s)' Partial support &om the Direccion General de Investigacion Cientifica yTecnica (Spain) through Grant No. PB92-0683 is gratefully acknowledged. x(s —s)4 APPENDIX A: ASYMPTOTIC ANALYSIS OF THE FPTDF When this is introduced into Eq. (A8) and the terms of the resulting series are analyzed, it is seen that if In this appendix we are going to study the asymptotic behavior of the FPTDF given by Eq. (9) in the limit of large N. We introduce the dimensionless variable Eq. (A8) can be approximated by (A10) and the dimensionless function (A1) (s) =Nl/2 (e— e~)2 2ue 2 /2vro, (A11) 4(s) =s——[cos(s+ P) — cosP]. p From Eq. (9) we can write the distribution function for the svariable as 2= u)2N [p+ esin(s +ttt)] (A12) (A3) For ein the interval [0,p), C'(s) is amonotonic increasing function of s, while Cy (s) is periodic with period 27r. The function (A4) has amaximum whose location sis given by the equation 4(s )=p cos(s +P) — cos P— ssin P(A13) and Eq. (A5) leads to and sis the solution of Eq. (A5). Upon writing Eq. (All) we have used the fact that, in the limit defined by Eq. (A1D), it is o, 0(,)«2e eod, therefore, we can substitute 4'(s) by ill'(s )in Eq. (A8). In this discussion we have assumed that eis not very close to p. Finally, one more simplification can be carried out if «1. Then. ,Eq. (A5) implies that s«1 and we have Define (A6) ~N p+ csin' (A14) so that the maximum value of h(s) corresponds to iIJ(s) = 1, and rewrite g(s) in the form In this same limit, Eq. (A12) reduces to P~N @y()— N[@(e]— in @(e)] N!~(A7) O8 = 2= ~2N (p+ esinttt)2 (A15) Up to this point everything is exact and no approximation has been introduced. Assume now that Nis very large. Expansion of 4'(s) —ln@(s) around ill(s) =1, Returning to the original variable t, the expressions in Eqs. (14)— (20) are obtained.