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: email@example.com).
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
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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.