Periodicity Enhancement of Two-Mode Stochastic Oscillators in a CNN Type Architecture
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Periodicity Enhancement of Two-Mode Stochastic
Oscillators in a CNN Type Architecture
G. Máté, E.Á. Horváth, E. Káptalan, A. Tunyagi and T. Roska
Z. Néda Department of Information Technology
Department of Physics MTA SZTAKI and Pázmány Péter Catholic University
Babeş-Bolyai University Budapest, Hungary
Cluj-Napoca, Romania roska@sztaki.hu
zneda@phys.ubbcluj.ro
Abstract— Two-mode stochastic oscillators coupled through a coupling, both models suggest a phase-transition-like behavior:
simple optimization dynamics exhibit interesting and novel synchronization appears only for coupling strengths that are
collective behavior. Previous studies have shown that for a global bigger than a critical value. For low coupling strengths, the
coupling, apart of synchronization, a puzzling enhancement in system usually will not synchronize. There is however a critical
the periodicity level of the global output can be achieved. Here we coupling value, so that for coupling strengths bigger than this, a
show that a locally coupled system which resembles a CNN-type partial synchronization emerges (Fig. 1). The value of the
architecture of the elementary units, is similarly able to exhibit critical coupling depends on how different the oscillator’s
synchronization for a given parameter range. Moreover, the natural frequencies are. As a general result, it has been shown
resulting enhancement in the periodicity for a 2D lattice topology
that the value of the critical coupling increases linearly with the
of the oscillatory units proves again to be significant. The system
composed by non-perfect oscillators generates thus a perfectly
standard deviation of the oscillator’s natural frequencies. The
periodic signal. An experimental realization of the system gives more different the oscillators are, the stronger the needed
confidence in the numerical simulation results. coupling is to achieve their synchronization.
In the present study we suggest a totally different approach
Monte-Carlo simulation, pulse-coupled oscillators, to synchronization. We consider a novel category of oscillators,
synchronization, CNN architecture named multimode stochastic oscillators [7,8], which are
I. INTRODUCTION capable of synchronization without phase-minimizing forces.
Interaction between the units is through a global optimization
Synchronization is a well-known collective behavior. It process, the system optimizes the output intensity around a
appears in a wide range of physical, biological or sociological desired value. Synchronization appears as an unexpected co-
systems [1-3]. The most intriguing and fascinating product of this optimization. Here, we will show that the
synchronization-type collective behavior appears system is capable of synchronization, even if the optimization
spontaneously (without an external periodic drive) in an process is local. We also prove that a puzzling enhancement in
ensemble of non-identical units. It is believed that spontaneous the periodicity of the system can be achieved if the interaction
synchronization of realistic oscillators having different natural radius is big enough.
frequencies arises only when a strong enough phase-
minimizing coupling exists between the oscillators. The aim of
the present contribution is to show that one can design a CNN-
like architecture where synchronization of oscillators can be
achieved also without phase-minimizing forces.
Synchronization of non-identical units coupled with phase-
minimizing forces can be understood in view of two classical
model-family of synchronization. The first model category, the
Kuramoto-type models [4], assumes rotators coupled through
phase-minimizing forces. For global coupling and some
specific form of the phase-minimizing coupling function, the
system can be exactly solved and conditions for
synchronization can be formulated. The second model
category, the integrate and fire type oscillator model [5,6],
considers more realistic pulse-emitting oscillators. Interaction Figure 1. The characteristic phase-transition in coupled oscillator
between these is pulse-like, i.e. the pulse of one oscillator will systems. Partial synchronization emerges only if the coupling strength
exceeds a critical value.
urge the others to emit pulses. Coupling in this system is
realized through the advances in the oscillator’s phases, which
are induced by the firing of the other elements. For globalII. MULTIMODE STOCHASTIC OSCILLATORS long period mode. Since their total period in this mode is Let us consider in the following the simple case, when the stochastically fluctuating, the system will not synchronize. A oscillators have two possible oscillating modes. Results similar situation will be observable when f* is too big. In such obtained for two modes can be then generalized to multiple cases for each oscillator f(k)
where The number of steps is seemingly increasing with the
considered neighborhood. Unfortunately for S=8, the case
x
1 which would be the most relevant for the CNN type
x ∫0
M = lim x →∞ f (t ) − f (t ) dt , (2) architecture, these results are not convincing, due to the low
enhancement in the periodicity level. Synchronization in this
case is visually not detected while running the applet.
and However, the obtained highly nontrivial synchronization, the
x enhancement in the periodicity level and the non-continuous
1 step-like variation of the synchronization level could all have
x ∫0
f (t ) = lim x →∞ f (t )dt . (3)
important practical application in those CNN-type architectures
where interactions between the cells are extended to more
The general shape of the ∆ (T ) curve as a function of T is neighbors. This could be the case of three dimensional systems
which are nowadays already available.
sketched in Fig. 4. For any f(t) oscillating function we have an
initially increasing tendency at small T values after which for T
= Tm a minimum ( ∆ m ) is reached. One can state that Tm is
the best approximation for the f (t) signals period and the
“periodicity level” of the signal is characterized by
1
p= . (4)
∆m
We can compute this parameter both for one oscillator
working independently (p1) in the long period mode, where the
effect of randomness on the period is smaller, and for the whole
system (p). The ratio p/p1 will characterize the enhancement in
the periodicity. We have investigated this ratio as a function of
f* for a small amount of randomness (t*=0.1) in the system. In
the numerical studies different number of neighbors, S, and
different system sizes (N=LµL) were considered. Simulation
results are summarized in Fig. 5.
Similarly with the globally coupled system, synchronization
emerges in an f* interval. Results are more visible for those
cases, where the interaction radius is bigger. While for S=8
synchronization is not obvious, for the S=12 and S=20
neighborhoods the results are already convincing. In these
cases a periodicity enhancement is observable (p/p1>1). Fig. 5
suggests that the periodicity level of the global signal increases
always with the size of the system. One can also observe a non
continuous step-like variation in the p(f*) curves. As the value
of f* is increased, synchronization disappears in discrete steps.
Figure 4. General shape of the ∆(T) function. Figure 5. Periodicity of the locally coupled system as a function of the f*
threshold. Results for different interaction neighborhoods (quantified by
S) and lattice sizes as indicated in the legend (τ*=0.05 for all curves).IV. EXPERIMENTAL REALIZATION OF THE SYSTEM
A simple CNN-like experimental realization was built and
synchronization within this system was studied. Our
experimental setup is very much alike a previous one [11],
therefore it also contains standalone electronic oscillators
capable of emitting and detecting light pulses. The oscillators
were developed around an 8-bit, RISC-core microcontroller
from ATMEL. The main elements of the oscillator are shown
in Fig. 6: the microcontroller, the photo-resistor and the Light
Emitting Diode (LED).
The circuit diagram of the oscillator is given on Fig. 7. The
internal RC oscillator of the microcontroller was chosen as
time reference. The light intensity in the system is measured by Figure 7. Circuit diagram of the built pulsating oscillators.
a photo-resistor in conjunction with three normal resistors of
10KW, 100KW and 1MW. These resistors enable the use of
several sensitivity ranges. One can also set up a Low Pass Filter
(LPF) on the photo-resistor signal with the help of the two
10nF or 100nF capacitors on the INT0 and INT1 pins. A
reference signal U corresponding to the value of f* in the
model is applied to the oscillators. The voltage on the photo-
resistor (which depends on the selected f(t) light intensity) can
either be measured by a 10 bit resolution analog-digital
converter (ADC) or compared by an in-built hardware
comparator. The output of the oscillator appears as a flash on
the LED. A hardware Pulse Width Modulation (PWM) can be
used to alter this light intensity. All parameters and the
hardware setup can be done by the program inside the
microcontroller. The stochastic nature of the period is
implicitly satisfied due to the analog nature of the internal RC
oscillator. Figure 8. Oscillators placed on the main board in a square lattice
topology.
The oscillators were programmed such that their dynamics
reproduces the collective dynamics of the two-mode system.
During the experiments different number of oscillators were
The parameters of the modes were chosen in agreement with
placed on the circuit board and the U reference voltage (which
the values considered in the simulations. Hence two possible
corresponds to the f* threshold) was automatically varied from
flashing periods were distinguished, one with a longer period
its lowest (0 mV) to its highest value (5000 mV) by a
and another with a shorter one. The oscillators are placed on a
DU=100mV step. After defining the new threshold value a
square lattice topology on circuit board (Fig. 8) which is closed
Dt=10min total measuring time for each reference value is set.
inside a box to isolate the system from external light. In
The driver program running on the PC makes the experiments
principle, each oscillator feels the light pulses emitted in its
automatically for all specified U reference values and will save
neighborhood. This is realized by nailing a mirror on the top
the state of the system with a 10ms time-resolution. From the
inside wall of the box. Through a simple interface the circuit
global f(t) signal of the oscillators, the same p/p1 parameter is
board is connected to a personal computer. This interface
computed as in the case of the numerical simulations.
allows the user to follow in time the state of each oscillator in
part and to fix the relevant physical parameters and alter them On Fig. 9 we illustrate the results obtained for N=9, 16 and
automatically on stream. 22 oscillators placed on the board. The experimental setup
confirms all the major predictions of the numerical simulations.
Visibly there is a clear U reference voltage interval where the
synchronization level strongly increases. As there are more and
more oscillators in the system, the periodicity level of the
system is higher. The experimental setup considered here does
not allow however a proper control of the interaction
neighborhood. The interaction in the experiment is not a proper
local interaction, it is much more a global interaction which
decays with the distance between the units. The step-like
variation which is a characteristic of the local interaction is not
Figure 6. Pulsating electric oscillators which reproduce the dynamics of revealed.
the two-mode stochastic oscillators.are in good agreement with results predicted by numerical
simulations.
ACKNOWLEDGMENT
Support from a Romanian PNCDII/ElBioArch Nr. 12121
research grant is acknowledged. The attendance of E. Á. H. at
the CNNA 2010 conference was supported by the Office of
Naval Research, US. Contribution from R. Sumi and Sz. Boda
is acknowledged.
REFERENCES
[1] S. Strogatz, Sync. The emerging science of spontaneous order. New
York: Hyperion, 2003.
[2] S. Strogatz, “From Kuramoto to Crawford: exploring the onset of
Figure 9. Periodicity level measured in the experimental setup as a synchronization in populations of coupled oscillators,” Physica D, vol.
143, pp. 1–20, 2000.
function of the U reference voltage.
[3] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization. A
universal concept in nonlinear science. Cambridge: Cambridge
University Press, 2002.
V. CONCLUSIONS [4] Y. Kuramoto and I. Nishikava, “Statistical macrodynamics of large
A system composed of two-mode stochastic oscillators was dynamical systems. Case of a phase transition in oscillator
communities,” J. Stat. Phys., vol. 49, pp. 569–605, 1987.
considered in a CNN-type architecture. The oscillators built-in
[5] R. E. Mirollo and S. H. Strogatz, “Synchronization of pulse-coupled
desire is to hold a given output intensity. Each oscillator detects biological oscillators,” SIAM J. Appl. Math., vol. 50, pp. 1645–1662,
the output of the ones in its immediate neighborhood and as a 1990.
function of the detected output intensity will oscillate either in [6] S. Bottani, “Synchronization of integrate and fire oscillators with global
a short period or a long period mode. As a result of this simple coupling,” Phys. Rev. E, vol. 54, pp. 2334–2350, 1996.
optimizing dynamics an unexpected synchronization appears. [7] A. Nikitin, Z. Néda, and T. Vicsek, “Dynamics of two-mode stochastic
Surprisingly, it was found that the periodicity level of the oscillators,” Phys. Rev. Lett., vol. 87, 024101, 2001.
collective output is higher than the periodicity of its standalone [8] Z. Néda, A. Nikitin, and T. Vicsek, “Synchronization of two-mode
elements. This means that the system is appropriate to generate stochastic oscillators: a new model for rhythmic applause and much
signals with enhanced periodicity. The analog nature of the more,” Physica A, vol. 321, pp. 238–247, 2003.
interaction and the interaction topology of the system suggest [9] R. Sumi and Z. Néda, “Synchronization of multi-mode pulse-coupled
stochastic oscillators,” Journal of Optoelectronics and Advanced
that the phenomenon could be of interest from the viewpoint of Materials, vol. 10, pp. 2455–2460, 2008.
practical applications. A simple experimental realization was [10] Z. Néda, Collective dynamics of electronic fireflies.
also given. Electric oscillators capable of emitting and
http://www.phys.ubbcluj.ro/~zneda/sync
detecting light-pulses and with a multimode oscillation
[11] M. Ercsey-Ravasz, Zs. Sarkozi, Z. Neda, A. Tunyagi, and I. Burda,
dynamics were built. Measurements on the experimental setup “Collective behaviour of electronic fireflies,” European Journal of
Physics B, vol. 65, pp. 271–277, 2008.You can also read
