1018IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 5, SEPTEMBER 2004
Grouping Synchronization in a Pulse-Coupled
Network of Chaotic Spiking Oscillators
Hidehiro Nakano, Student Member, IEEE, and Toshimichi Saito, Senior Member, IEEE
of simplechaotic spiking oscillators(CSOs). If a unit oscillator and
its neighbor(s) have (almost) the same parameter values, they ex-
hibit in-phase synchronization of chaos. As the parameter values
differ, they exhibit asynchronous phenomena. Based on such be-
havior, some synchronous groups appear partially in the network.
Typical phenomena are verified in the laboratory via a simple test
circuit. These phenomena can be evaluated numerically by using
an effective mapping procedure. We then apply the proposed net-
segmented into some sub-regions.
Index Terms—Chaos, grouping, integrate-and-fire, pulse-cou-
pled network (PCN), spiking neuron, synchronization.
–. The PCNs exhibit various synchronous and asyn-
chronous phenomena ,  and are applicable to associative
memories , , image processing , , etc. The PCNs
can be realized by simple electric circuits . In the published
literature, IFNs with periodic behavior have been the main
focus. On the other hand, we have presented chaotic spiking os-
cillators (CSOs) that can output a chaotic spike-train –.
The CSO can be regarded as a higher order IFN that can exhibit
chaos and rich bifurcation phenomena; the CSO has richer
dynamics than usual IFNs. Connecting plural CSOs using
each spike-train, a chaotic pulse-coupled network (CPCN) can
be constructed. The pulse-coupling method of the CPCN is
based on and is simpler than that of PCN in , , and .
The CPCN can exhibit various chaos synchronous phenomena
that may be applicable to image processing . Our CSO is
concerned with a novel resonate-and-fire neuron (RFN) model
. Study of CSOs having rich dynamics may contribute
to the study of basic nonlinear phenomena and flexible engi-
neering applications including image processing , pattern
recognition , and pulse-based communications , .
This paper studies synchronous phenomena in a CPCN con-
sisting of simple CSOs. In Section II, as a preparation, we in-
troduce the basic dynamics of the single CSO presented in .
ULSE-COUPLED networks (PCNs) of integrate-and-fire
neurons (IFNs) are a type of artificial neural network
Manuscript received June 2, 2003; revised December 12, 2003. This work
was supported by JSPS.KAKENHI under Grant 13650427.
H. Nakano is with the Department of Computer Science and Media
Engineering, Musashi Institute of Technology, Tokyo, Japan (e-mail:
T. Saito is with the Department of Electronics, Electrical, and Computer En-
gineering, Hosei University, Tokyo, Japan (e-mail: firstname.lastname@example.org).
Digital Object Identifier 10.1109/TNN.2004.832807
The CSO can be implemented easily, and typical chaotic be-
havior is verified in the laboratory. In Section III, we present
the CPCN having a local connection structure. Each CSO is
connected with the neighbor CSOs. First, we consider a CPCN
consisting of two CSOs. If both the CSOs have (almost) the
phenomena. Next, we consider a ladder CPCN consisting of
four CSOs. If a CSO and its neighbor CSO(s) have (almost) the
phenomena. Based on such behavior, some synchronous groups
appear partially in the CPCN. Typical phenomena are verified
in the laboratory with a simple test circuit. In Section IV, we
introduce a normal form equation for the CPCN in order to ex-
tract essential parameters. By defining a coincident spike rate
between the CSOs, synchronous phenomena can be evaluated
numerically. In order to more efficiently calculate this rate, we
introduce a mapping procedure. This map can be described pre-
the CPCN to image segmentation. For the input, we prepare a
lattice CPCN where the parameter of each CSO corresponds
to each pixel value of the input. By the grouping synchronous
phenomena, the input image data can be segmented into some
sub-regions. We show typical simulation results.
This paper provides basic experimental and analysis results
for a PCN of simple chaotic oscillators. These results contribute
to the study of basic nonlinear phenomena. Our CPCN has
a simple local connection structure and can exhibit various
grouping synchronous patterns depending on the network
parameters. This means that the CPCN has rich functionality
and may be developed into flexible applications such as image
Fig. 1 shows a CSO. The CSO will be a unit element of the
current sources (VCCSs) construct a linear circuit . The cir-
We assume that (1) has unstable complex characteristic roots
1045-9227/04$20.00 © 2004 IEEE
NAKANO AND SAITO: GROUPING SYNCHRONIZATION IN A PCN OF CSOs1019
Fig. 2.Implementation example of a CSO.
In this case, the capacitor voltages can vibrate below the firing
. If the capacitor voltage
the pulse-generator (PG) outputs a single firing spike
that closes the firing switch
. Then, is reset to the base
Repeating in this manner, the CSO generates a firing spike-train
In this paper, for simplicity, we assume that all the circuit ele-
ments are ideal and define the switching dynamics as an ideal
Fig. 2 shows an implementation example of the CSO. The
linear VCCSs are realized by operational transconductance am-
plifiers (OTAs, NJM13600). The conductance value , which
controls the self-running angular frequency
by means of
, and . We have confirmed an approxi-
, can be adjusted
. The PG and the firing switch
1This definition is a routine in the electrical engineering.
??? ??, ?
attractor. (b) Time-domain waveform.
Chaotic attractor in a unit CSO (? ? ??? ??, ?
? ??? ?, ? ? ???? ?, ?
? ?? ??, ??? ?
? ??? ?). (a) Phase space
represent the CSOs and the couplings, respectively.
CPCN having a local connection structure. The circles and solid lines
are realized by one comparator (LM339), two analog switches
(4066), one capacitor, and one resistor. Let the time constant
be sufficiently small and let
reaches the firing threshold voltage
comparator closes the switch
in a short time. The voltage
andis reset to the base voltage
Then the comparator opens the switch
dynamics can satisfy (3) and (4), approximately. In order to vi-
sualize the very narrow firing spikes, we have used a monstable
multivibrator (MM, 4538) which does not affect the dynamics
of the CSO. Fig. 3 shows a typical chaotic attractor in the labo-
ratory measurements.  has presented corresponding chaotic
attractor in numerical simulations and shown theoretical evi-
dence for the chaos generation.
at. If the
and the voltagecharges up
closes the firing
in a short time.
circuit dynamics is described by
CSO is connected with the neighbor CSOs. Fig. 5 illustrates
the connection method. If the capacitor voltage
denotes the index of the CSO. Each
1020IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 5, SEPTEMBER 2004
Switching rule of the CPCN. The ?th and ?th CSOs are connected to
firing threshold voltage
At the same time, the firing spike
th CSO (the neighbor of the th CSO) instantaneously. At this
moment, if the capacitor voltage
tory threshold voltage
, the th CSO outputs the firing spike
, and is reset toinstantaneously. If
, the th CSO does not output the firing spike and
fected. In the case where the th CSO has other neighbor(s), the
By this instantaneous propagation, some of the CSOs output
the firing spikes simultaneously. Repeating in this manner, each
CSO generates a firing spike-train
, the th CSO outputs the firing spike
is reset to the base voltageinstantaneously.
is input to the
is higher than the refrac-
is less than
Equations (5)–(7) give the dynamics of the CPCN.
Fig. 6 shows an implementation example of the PG of the
th CSO, where. Let the time constant
sufficiently small, thus, firing spikes can be propagated between
the CSOs with a sufficiently short time delay. Let
. The switch is closed when
than the refractory threshold voltage
closed when the firing spike
are closed simultaneously, thevoltage
andis reset to the base voltage
time, the voltage
is input to the th CSO with a short
reaches the firing
. The switch
is applied from the th
in a short time. At the same
2This circuit can deal with the other neighbors by connecting multiple
switches in parallel, as shown in Fig. 6.
Fig. 6. Implementation example of a PG for a CPCN.
Fig. 7 shows typical experimental results for a CPCN of two
have (almost) the same parameter values, they exhibit in-phase
synchronization of chaos, as shown in Fig. 7(a). Since both the
CSOs have (almost) the same trajectories, they always output
coincident firing spikes. As the conductancevalue of the second
CSO varies, the synchronous phenomena are broken down to
asynchronous phenomena, as shown in Fig. 7(b). Since both the
CSOs have different trajectories, they output both coincident
and incoincident firing spikes.
Fig. 8 shows typical experimental results for a ladder CPCN
of four CSOs:
rameter values, they exhibit in-phase synchronization of chaos,
as shown in Fig. 8(a). Note that all the CSOs have (almost) the
same trajectories independently of the number of the neighbors,
and always output coincident spikes.
As the conductance values of the third and fourth CSOs vary,
the synchronous phenomena change into grouping synchronous
phenomena, as shown in Fig. 8(b). The first and second CSOs
form another in-phase synchronous group. The two groups do
not synchronize to each other. Fig. 9 shows the time-domain
waveforms. In the figure, at
, and both andare lower than
andjump tosimultaneously, and both
are higher than
. Then, all the states jump to
neously. However, because of parameter differences, trajecto-
ries of the third and fourth CSOs part from those of the first
and second CSOs. This means that CSOs in the same group
can have (almost) the same trajectories after synchronization
is achieved. However, in the rare case where states of either
group are close to
in the firing moment of the other group,
the synchronization may be broken down because of the influ-
ences such as noise and/or parameter mismatches. In the figure,
,reaches,is lower than
. Then,moves continuously and
. These switching dynamics can satisfy (6) and
. If both the CSOs
. Then, both
and the other states
NAKANO AND SAITO: GROUPING SYNCHRONIZATION IN A PCN OF CSOs 1021
??? ?). (a) In-phase synchronization of chaos ???? ? ??? ? ??? ???.(b)
Asynchronous phenomena (???
? ??? ??, ???
Typical experimental results for a CPCN of two CSOs (? ? ??? ??,
? ?? ??, ?
? ??? ?, ? ? ???? ?, ?
? ??? ?, ?
? ??? ??).(c) Network
a likewise manner, the synchronization of the third and fourth
CSOs may be broken down. However, CSOs having (almost)
the same parameters can repeat the synchronous firings and the
synchronization can rapidly recover. Therefore, we observe the
grouping synchronous phenomena in the laboratory, as shown
in Fig. 8(b).
IV. NUMERICAL ANALYSIS
In this section, we investigate synchronous phenomena asso-
ciated with the proposed CPCN. Using dimensionless variables
??? ??, ?
??? ??, ??? ? ??? ? ??? ??). (c) Network topology.
Typical experimental results for a ladder CPCN of four CSOs (? ?
? ?? ??, ?
? ??? ?, ? ? ???? ?, ?
? ??? ?). (a) In-phase synchronization of chaos ????
? ??? ???. (b) Grouping synchronization (???
? ??? ?, ?
Equations (5)–(7) are transformed into (9)–(11), respectively
Fig. 10 shows the normalized trajectories. Except for the self-
running angular frequency
, the CPCN has three parameters:
1022IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 5, SEPTEMBER 2004
Fig. 9.Trajectories for a ladder CPCN of four CSOs.
the damping , the base , and the refractory threshold . For
simplicity we fix these parameters
For, (9) has the following exact piecewise solution.
Using this solution, trajectories can be calculated precisely
. Fig. 11 shows typical synchronous and asynchronous
phenomena in numerical simulations. The numerical data in
Fig. 11(a) and (b) correspond to the laboratory data in Fig. 7(a)
and (b), respectively. Also, in the numerical simulations, we
have verified grouping synchronous phenomena corresponding
to the laboratory data in Fig. 8.
be the th time when either of the th or th CSO
outputs a firing spike. Let
and are nonnegative integers. We then define the following
coincident spike rate between the th and th CSOs:
,) denotes an initial state vector at.
be the number of spikes of
Calculating this rate, we can numerically evaluate the syn-
chronous phenomena. The case
implies that the th and
Fig. 10.Normalized trajectories for a CPCN of two CSOs.
th CSOs always output coincident spikes. The case
implies that the th and th CSOs output no coincident spike.
In order to more efficiently calculate this rate, we introduce a
return map. First, we define the following objects:
Let us consider trajectories starting from
Fig. 10). As the trajectories start from
, either trajectory
This map is given analytically. The firing time
is given by
is a positive-minimum root of
NAKANO AND SAITO: GROUPING SYNCHRONIZATION IN A PCN OF CSOs 1023
(b) Asynchronous phenomenon (? ? ?, ? ? ????).
Typical numerical results for a CPCN of two CSOs (? ? ????, ? ? ????, ? ? ???). (a) In-phase synchronization of chaos ??
? ? ???, ? ? ?, ? ? ?? , ? ? ?? ). A and B correspond to Fig. 7(a) and
CoincidentspikerateforaCPCNoftwoCSOs(? ? ????,? ? ????,
The root can be solved by the Newton–Raphson method. The
are given by
Since the return map (15) is given analytically, the coincident
spike rate (13) can be calculated rapidly.
Fig. 12 shows the coincident spike rate for the CPCN of two
spike rate for the ladder CPCN of four CSOs:
. Asapproaches 1,
COINCIDENT SPIKE RATE FOR A LADDER CPCN OF four CSOS (? ? ????,
? ? ????, ? ? ???, ? ? ?? , ? ? ?? ). THE PARAMETER VALUES OF (a)
AND (b) CORRESPOND TO THOSE OF FIG. 8(a) AND (b), RESPECTIVELY
,,and, where for
simplicity of analysis we set the parameters as follows:
We have set random initial states for each CSO for the calcula-
tion of the coincident spike rate. In the calculation of (13), the
have converged reasonably in about
. As the parameter
shown in Fig. 13. The rate
is similar to Fig. 12. On the other
stant value 100 [%] independently of the parameter value .
In order to investigate effects of parameter perturbation, we
consider the basic case
changes, asvaries, the rate
Fig. 14 shows the coincident spike rate for parameter perturba-
tion. If the perturbation
is not 0, the rate
denotes parameter perturbation of the fourth CSO.
is not 100 [%].
1024IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 5, SEPTEMBER 2004
? ? ????, ? ? ?? , ? ? ?? ). C and D correspond to Fig. 8(a) and (b),
Coincident spike rate for a ladder CPCN of four CSOs (? ? ????,
? ? ?? , ? ? ?? ). D corresponds to Fig. 8(b). (a) ? ? ???. (b) ? ? ???.
Effects of a parameter perturbation (? ? ????,? ? ????,? ? ????,
, we can evaluate that the second and third CSOs are in dif-
ferent groups, and the third and fourth CSOs are in the same
is satisfied for a sufficiently small value
V. IMAGE SEGMENTATION
In this section, we apply a CPCN to image segmentation. Let
the input image data be binary and 10
Fig. 15(a). For the input, we prepare a lattice CPCN consisting
10 CSOs. Each CSO is allocated on each pixel and con-
nected with the nearest neighbors, as shown in Fig. 15(b). The
We have set a random initial state for each CSO. If a CSO and
its neighbor CSO(s) have the same parameter values, they ex-
10 pixels, as shown in
(a) Input image data. (b) Allocation of each CSO. The number denotes the CSO
index. (c) Synchronous groups appearing for the input. (d) Output spike-trains
of each group.
Image segmentation using a CPCN (? ? ????, ? ? ????,? ? ???).
havior, five synchronous groups appear partially in the CPCN,
as shown in Fig. 15(c). That is, the binary image input is seg-
mented into five sub-regions. In the simulation, transient time
to the grouping synchronization approximates
normalized time). Fig. 15(d) shows the output spike-trains of
each group. These synchronous phenomena can be evaluated
by the coincident spike rate, as shown in Table II. In the table,
we have chosen two representative CSOs from a group. Note
that any coincident spike rates between CSOs in the same group
are a constant value 100 [%]. Each CSO in groups 2 to 5 has
the same parameter values. However, CSOs in different groups
do not synchronize to each other. Also, we have verified such
grouping synchronous phenomena for a gray scale image input
NAKANO AND SAITO: GROUPING SYNCHRONIZATION IN A PCN OF CSOs1025
COINCIDENT SPIKE RATE FOR THE SIMULATION IN FIG. 15 (? ? ????, ? ? ????, ? ? ???, ? ? ?? , ? ? ?? )
Image segmentation using a PCN of basic periodic oscillators ?? ?
Such grouping is difficult for basic periodic oscillator net-
works. Fig. 16 shows a typical simulation result using a PCN
of basic integrate-and-fire oscillators described by
The input image data and the allocation of each oscillator are
the same as those of Fig. 15(a) and (b), respectively. The pa-
of each oscillator corresponds to each pixel
value. We have set a random initial state for each oscillator. In
this simulation, group 1 and the other groups exhibit a kind of
multiphase synchronization to each other. Groups 2 to 5 exhibit
in-phase synchronization through group 1. These groups are not
separated. If basic periodic oscillator networks are applied to
the image segmentation, globally inhibitory couplings are nec-
essary , . However, our CPCN cannot exhibit multiphase
synchronization because of the chaotic behavior of each CSO.
Even if all the CSOs start from the almost same initial state,
CSOs in different groups cannot have the same trajectories be-
cause of sensitivity to the initial states in chaos. Therefore, our
CPCN can realize the image segmentation without global cou-
plings. The couplings of the CPCN can be simpler than those
of existing basic periodic oscillator networks for the image seg-
We have studied synchronous phenomena in a CPCN. The
CPCN exhibits grouping synchronous phenomena character-
ized by partial in-phase synchronization of chaos. Constructing
a simple test circuit, we have verified typical phenomena in
the laboratory. We have evaluated these phenomena numeri-
cally by using a mapping procedure. Based on the grouping
synchronous phenomena, we have applied the CPCN to image
segmentation. In future work, we are considering the following.
• For simplicity, we have fixed some parameters. We should
analyze the synchronous phenomena and related bifurca-
tion phenomena for a wider parameter region.
• We have confirmed grouping synchronous phenomena
in the CPCN both experimentally and numerically. We
should analyze characteristic of transient time to synchro-
nization in more detailed.
• The grouping synchronous phenomena are broken down
because of the influences such as noise and/or parameter
mismatches. We should analyze the phenomena including
these influences in more detailed. Also, we should imple-
nomena and their stability.
• We have obtained fundamental results for image segmen-
tation usingthe CPCN. We should apply theCPCN to var-
ious natural image data and evaluate its performance.
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Hidehiro Nakano (S’02) received the B.E., M.E.,
and Ph.D. degrees in electrical engineering, all from
Hosei University, Tokyo, Japan, in 1999, 2001, and
He is currently a Research Assistanr with the
Department of Computer Science and Media Engi-
neering, Musashi Institute of Technology, Tokyo,
Japan. His research interests include chaotic circuits
and neural networks.
Dr. Nakano is a Student Member of the INNS and
Toshimichi Saito (M’88–SM’00) received the B.E.,
M.E, and Ph.D. degrees in electrical engineering, all
from Keio University, Yokohama, Japan, in 1980,
1982, and 1985, respectively.
He is currently a Professor with the Department of
Electronics, Electrical, and Computer Engineering,
Hosei University, Tokyo, Japan. His current research
interests include analysis and synthesis of nonlinear
circuits, chaos and bifurcation, artificial neural
networks, power electronics, and digital communi-
Dr. Saito is a Member of the INNS and IEICE.