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Andrzej Krawiecki, Andrzej Sukiennicki a, Robert
A. Kosinski b, and Thomas Stemler c Controlling
spatiotemporal stochastic resonance by time
delays
a also at Department of Solid State Physics,
University of Lódz, Pomorska 149/153, 90-283
Lódz, 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
Stochastic ResonanceNew Horizons in Physics
Engineering, Max Planck Institute for the
Physics of Complex Systems, Dresden 4-7 October
2004
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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.
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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 k2p/l, frequency ws
2p/Ts, amplitude Altb, and with spatiotemporal
Gaussian white uncorrelated noise with intensity
D.
The output signal yn(i) from element i at a
discrete time step n is
Measure of stochastic resonance output SNR from
the middle element iN/2.
A. Krawiecki, A. Sukiennicki, R.A. Kosinski,
Phys. Rev. E 62, 7683 (2000)
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The SNR vs. D for N128, Ts 128, k0, A0.5.
Symbols numerical results for w -1.5
(squares), w -0.1 (triangles), w1.0 (),
w1.5 (X). Theoretical results are shown with
solid lines.
Enhancement of SR for optimum coupling wgt0 (here,
w1.0) is observed. Significant enhancement
occurs for optimum wgt0 if 0? k lt p/4 (l ? 8),
i.e., when the phase shift Df k?1 between
neighbouring elements is 0? Df lt p/4. However,
the enhancement deteriorates with increasing Df
and is most effective for k Df 0.
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The SNR vs. D for N128, Ts 128, k Df p/2
(l4), A0.5. Symbols numerical results for
w -1.5 (squares), w -0.1 (triangles), w1.0
(), w1.5 (X). Theoretical results are shown
with solid lines.
The SNR is practically independent of the
coupling w, since for Df p/2 the periodic
signals in neighbouring elements are shifted by
Ts/4, and the probability to have 1 at the
output when the periodic signal is maximum is not
enhanced by the coupling.
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The SNR vs. D for N128, Ts 128, k Df p
(l2), A0.5. Symbols numerical results for
w -1.5 (squares), w -0.1 (triangles), w1.0
(), w1.5 (X). Theoretical results are shown
with solid lines.
If p/2 lt k ? p (4 lt l ? 2), slight enhancement of
SR for any coupling wlt0 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.
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Spatiotemporal diagrams and spatiotemporal
synchronization
The diagrams correspond to maximum values of the
SNR for the middle element.
w1.0, k 0, D0.05
Maximum SNR corresponds to spatiotemporal
synchronization with the plane wave
w1.0, k p/2, D 0.37
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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.
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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 t1,t2), driven by periodic signals with
frequency ws, amplitude A, shifted in phase by
Df, 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 Df1, Df2
A. Krawiecki, T. Stemler, Phys. Rev. E 68,
061101 (2003)
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• For the case without delays, the SNR in coupled
  threshold elements is maximally
• enhanced if the phase shift Df between
  neighbouring elements is
• for wgt0, Df 0,
• for wlt0, Df p.

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 Df between periodic signals in the
two coupled elements. The effect of the phase
shift Df is cancelled by the optimum choice of
the delays, and the effective phase shifts Df1,
Df2 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
t1t2Ts.
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(a) Contour plots of the maximum SNR R(1) (in dB)
vs. t1 and t2 for A0.1, Ts128, w0.45, Df p,
b0.6, gray scale on the left (b) contour plots
of the SNR R(1) (in dB) vs. D and t1 or t2 for
A0.1, Ts128, w0.45, b0.6, and (b) Df p,
t1Ts/2 (c) Df 0,t1Ts (d) Df 0, t1t2Ts
(e) Df p/2, t13Ts /4 (f) Df p/2, t1t2Ts
gray scale for (bf) on the right.
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Contour plots of the SNR R(1) (in dB) vs. D and
t1 or t2 , for A0.1, Ts128, w1.0, b0.6, and
(a) Df p/2, t1 t2 Ts (b) Df p, t1 Ts /2
(nonoptimum), gray scale on the right.
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Simple theoretical estimation of the SNR
The probability to have yn(1) in the simplest
approximation can be obtained from the equations
For the Gaussian noise the conditional
probabilities are
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Simplifying assumptions
(processes yn(1,2) are cyclostationary)
1.
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)
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Numerical (symbols) and corresponding theoretical
(solid lines) SNR R(1) vs. D for A0.1, Ts128,
t1 t2 Ts, b0.6, and (a) w0.45, Df 0, t10
(squares, optimum delays), t1 Ts/4 (triangles),
t1 Ts/2 (dots) (b) w0.45, Df p/2, t10
(squares), t1 Ts/4 (triangles), t13Ts/4 (dots,
optimum delays) (c) w-1.0, Df p/2, t10
(squares), t1 Ts/4 (triangles, optimum delays),
t1 3Ts/4 (dots).
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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.
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References

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

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