1. Stochastic Resonance:New Horizons in Physics & Engineering, Max Planck Institute for the Physics of Complex Systems, Dresden 4-7 October 2004 Andrzej Krawiecki, Andrzej Sukiennicki a, Robert A. Kosiński b, and Thomas Stemler c Controlling spatiotemporal stochastic resonance by time delays a also at Department of Solid State Physics, University of Łódź, Pomorska 149/153, 90-283 Łódź, Poland b also at Central Institute for Labor Protection, Czerniakowska 16, 00-701 Warsaw, Poland c Institute for the Solid State Physics, Technical University Darmstadt, Hochschulstrasse 6, 64289 Darmstadt, Germany

2. Stochastic resonance in spatially extended systems  In spatially extended systems of coupled stochastic units, each exhibiting SR, driven by uncorrelated noises and a common periodic signal, the maximum SNR from a single element can be enhanced for proper coupling, due to noise-assisted spatiotemporal synchronization between the periodic signal and the system (array-enhanced SR) [e.g., J.F. Lindner et al., Phys. Rev. Lett. 75, 3 (1995)].  SR in spatially extended systems can be observed also for signals periodic only in space (without time dependence). In this case, periodic spatial structures are best visible in the system response for optimum noise intensity [e.g., Z. Néda et al., Phys. Rev. E 60, R3463 (1999); J.M.G. Vilar, and J.M. Rubí, Physica A 277, 327 (2000)].  The phenomenon of SR was also observed for spatiotemporal periodic signals, e.g., in the Ising model with thermal noise, driven by a plane wave [L. Schimansky-Geier, and U. Siewert, in Lecture Notes in Physics, ed. L. Schimansky-Geier, T. Pöschel, vol. 484, p. 245 (Springer, Germany, 1997)]. The enhancement of SR due to coupling is also observed, however, the strength of the effect is weaker than in the case of spatially uniform, periodic in time signal.

3. Stochastic resonance with spatiotemporal signal. Example: a chain of coupled threshold elements. A chain of threshold elements is considered with length N, threshold b, coupling constant w, with the spatiotemporal periodicsignal in a form of a plane wave with wave vector k=2  /, frequency  s = 2  /  s, amplitude A<b, and with spatiotemporal Gaussian white uncorrelated noise with intensity D. The output signal y n (i) from element i at a discrete time step n is Measure of stochastic resonance: output SNR from the middle element i=N/2. [A. Krawiecki, A. Sukiennicki, R.A. Kosiński, Phys. Rev. E 62, 7683 (2000)]

4. The SNR vs. D for N =128, T s =128, k =0, A=0.5. Symbols: numerical results for w= - 1.5 (squares), w= -0.1 (triangles), w =1.0 (+), w =1.5 (X). Theoretical results are shown with solid lines. Enhancement of SR for optimum coupling w>0 (here, w=1.0) is observed. Significant enhancement occurs for optimum w>0 if 0  k <  /4 (  8), i.e., when the phase shift  k  1 between neighbouring elements is 0   <  /4. However, the enhancement deteriorates with increasing  and is most effective for k=  0.

5. The SNR vs. D for N =128, T s =128, k =  =  /2 ( =4), A=0.5. Symbols: numerical results for w= - 1.5 (squares), w= -0.1 (triangles), w =1.0 (+), w =1.5 (X). Theoretical results are shown with solid lines. The SNR is practically independent of the coupling w, since for  =  /2 the periodic signals in neighbouring elements are shifted by T s /4, and the probability to have 1 at the output when the periodic signal is maximum is not enhanced by the coupling.

6. The SNR vs. D for N =128, T s =128, k =  =  ( =2), A=0.5. Symbols: numerical results for w= - 1.5 (squares), w= -0.1 (triangles), w =1.0 (+), w =1.5 (X). Theoretical results are shown with solid lines. If  /2 < k   (4 <  2), slight enhancement of SR for any coupling w<0 is observed, particularly for higher noise intensities. Since the periodic signals in neighbouring elements are effectively in anti-phase, negative coupling mostly decreases the probability to have 1 at output when the periodic signal is minimum.

7. Spatiotemporal diagrams and spatiotemporal synchronization w=1.0, k =0, D=0.05 w=1.0, k =  /2, D =0.37 The diagrams correspond to maximum values of the SNR for the middle element. Maximum SNR corresponds to spatiotemporal synchronization with the plane wave

8. Controlling stochastic resonance  The term „controlling stochastic resonance” comprises in general various methods of enhancement of SR by means other than varying the noise strength, e.g., periodic modulation of the barrier height in a bistable potential, [ L. Gammaitoni et al., Phys. Rev. Lett. 82, 4574 (1999).]  Here, time delays are introduced in the coupling terms between neighbouring threshold elements. Optimum choice of the delays, for a given value of the coupling constant w, and for any spatial wavelength of the spatiotemporal periodic signal, leads to the increase of the maximum of the SNR in a single element. By changing the time delays, the height of the maximum SNR can be modified, and thus SR can be „controlled”.

9. And this is how everything will look like in the Ice Age which will eventually begin after the next SR-induced climate jump... courtesy of Dr. P. Jóźwiak, Svalbard 2001

10. The effect of delays in couplings on stochastic resonance with spatiotemporal signal. Example: two coupled threshold elements. A system of two threshold elements is considered, with coupling via delayed output signals (delay times  1,  2 ), driven by periodic signals with frequency  s, amplitude A, shifted in phase by , and white uncorrelated Gaussian noises. Measure of stochastic resonance: the output SNR R (1) from element 1. Due to time delays, the system is equivalent to two elements with no delays in coupling, but with effective phase shifts of the periodic signal  1,  2 [A. Krawiecki, T. Stemler, Phys. Rev. E 68, 061101 (2003)]

11. For the case without delays, the SNR in coupled threshold elements is maximally enhanced if the phase shift  between neighbouring elements is for w>0,  = 0, for w<0,  = . Hence, in the case with delays, the SNR is maximally enhanced if In this way, the maximum possible enhancement of SR due to a given coupling w is achieved for any phase shift  between periodic signals in the two coupled elements. The effect of the phase shift  is cancelled by the optimum choice of the delays, and the „effective” phase shifts     are optimally chosen for a given sign of the coupling constant w. This is an example of controlling stochastic resonance with spatiotemporal signal by time delays in coupling. Note that the optimum delays fulfil the condition  1 +  2 =T s.

12. (a) Contour plots of the maximum SNR R (1) (in dB) vs.  1 and  2 for A =0.1, T s =128, w =0.45,  = , b =0.6, gray scale on the left; ( b ) contour plots of the SNR R (1) (in dB) vs. D and  1 or  2 for A =0.1, T s =128, w =0.45, b =0.6, and ( b)  = ,  1 =T s /2; ( c)  = ,  1 =T s ; (d)  = ,  1 +  2 =T s ; ( e)  =  /2,  1 =3T s /4; ( f)  =  /2,  1 +  2 =T s ; gray scale for ( b –f) on the right.

13. Contour plots of the SNR R (1) (in dB) vs. D and  1 or  2, for A =0.1, T s =128, w= 1.0, b =0.6, and ( a)  /2,  1 +  2 = T s ; ( b) ,  1 = T s /2 (nonoptimu m ), gray scale on the right.

14. Simple theoretical estimation of the SNR For the Gaussian noise the conditional probabilities are The probability to have y n (1) in the simplest approximation can be obtained from the equations

15. Simplifying assumptions: 1. (processes y n (1,2) are cyclostationary); 2. Adiabatic approximation Under these assumptions the above system of equations becomes a closed system of linear equations whose solution is Evaluation of the above probabilities a general case is more difficult since the above system of equations is not closed. The SNR for element 1 is [cf. F. Chapeau-Blondeau, Phys. Rev. E 53, 5469 (1996)]

16. Numerical (symbols) and corresponding theoretical (solid lines) SNR R (1) vs. D for A=0.1, T s =128,  1 +  2 = T s, b=0.6, and (a) w=0.45,  = 0,  1 =0 (squares, optimum delays),  1 = T s /4 (triangles),  1 = T s /2 (dots); (b) w=0.45,  =  /2,  1 =0 (squares),  1 = T s /4 (triangles),  1 =3T s /4 (dots, optimum delays); (c) w=-1.0,  =  /2,  1 =0 (squares),  1 = T s /4 (triangles, optimum delays),  1 = 3T s /4 (dots).

17. Conclusions  SR in coupled threshold elements can be „controlled”, i.e., the maximum of the SNR from a single element can be increased, by introducing proper delays in the coupling, which cancel the effect of the phase shift of the input periodic signal in the two elements. This can be done for any spatial wavelength of the periodic signal.  The above result can be easily extended to the case of a chain of coupled thershold elements and, probably, for other spatially extended systems (e.g., chains of bistable stochastic units).  Time delays in the coupling can naturally appear in many systems, e.g., in biological neural networks, in electric circuits as delays in the transmission lines, etc. The above results show that they can have certain importance for the detection of weak spatiotemporal periodic signals, immersed in noisy background, by means of SR.

18. References A. Krawiecki, A. Sukiennicki, R.A. Kosiński, Int. J. Modern Phys B 14, 837 (2000). A. Krawiecki, A. Sukiennicki, R.A. Kosiński, Phys. Rev. E 62, 7683 (2000). A. Krawiecki, T. Stemler, Phys. Rev. E 68, 061101 (2003). A. Krawiecki, Physica A 333, 505 (2004).

19. Thank you for your attention courtesy of Dr. P. Jóźwiak, Svalbard 2001