Grouping synchronization in a pulse-coupled network of chaotic spiking oscillators.

Abstract

This paper studies a pulse-coupled network consisting of simple chaotic spiking oscillators (CSOs). If a unit oscillator and its neighbor(s) have (almost) the same parameter values, they exhibit in-phase synchronization of chaos. As the parameter values differ, they exhibit asynchronous phenomena. Based on such behavior, 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 network to image segmentation. Using a lattice pulse-coupled network via grouping synchronous phenomena, the input image data can be segmented into some sub-regions.

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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
Abstract—Thispaperstudiesapulse-couplednetworkconsisting
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-
worktoimagesegmentation.Usingalatticepulse-couplednetwork
viagroupingsynchronousphenomena,theinputimagedatacanbe
segmented into some sub-regions.
Index Terms—Chaos, grouping, integrate-and-fire, pulse-cou-
pled network (PCN), spiking neuron, synchronization.
I. INTRODUCTION
P
[1]–[8]. The PCNs exhibit various synchronous and asyn-
chronous phenomena [2], [3] and are applicable to associative
memories [4], [6], image processing [7], [8], etc. The PCNs
can be realized by simple electric circuits [9]. 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 [10]–[13].
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 [4], [7], and [8].
The CPCN can exhibit various chaos synchronous phenomena
that may be applicable to image processing [12]. Our CSO is
concerned with a novel resonate-and-fire neuron (RFN) model
[14]. Study of CSOs having rich dynamics may contribute
to the study of basic nonlinear phenomena and flexible engi-
neering applications including image processing [15], pattern
recognition [16], and pulse-based communications [17], [18].
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 [11].
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:
nakano@ic.cs.musashi-tech.ac.jp).
T. Saito is with the Department of Electronics, Electrical, and Computer En-
gineering, Hosei University, Tokyo, Japan (e-mail: tsaito@k.hosei.ac.jp).
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
sameparametervalues,theyexhibitin-phasesynchronizationof
chaos.Astheparametervaluesdiffer,theyexhibitasynchronous
phenomena. Next, we consider a ladder CPCN consisting of
four CSOs. If a CSO and its neighbor CSO(s) have (almost) the
sameparametervalues,theyexhibitin-phasesynchronizationof
chaos.Astheparametervaluesdiffer,theyexhibitasynchronous
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-
ciselybyusingexactpiecewisesolutions.InSectionV,weapply
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
processing systems.
II. CSO
Fig. 1 shows a CSO. The CSO will be a unit element of the
CPCN.Thetwocapacitorsandthetwolinearvoltage-controlled
current sources (VCCSs) construct a linear circuit [11]. The cir-
cuitdynamicsisdescribedbythefollowingequationifthefiring
switch
is open.
for(1)
We assume that (1) has unstable complex characteristic roots
, where
(2)
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NAKANO AND SAITO: GROUPING SYNCHRONIZATION IN A PCN OF CSOs1019
Fig. 1.CSO.
Fig. 2. Implementation example of a CSO.
In this case, the capacitor voltages can vibrate below the firing
threshold voltage
. If the capacitor voltage
the pulse-generator (PG) outputs a single firing spike
that closes the firing switch
voltage
instantaneously while
instantaneously
reaches,
. Then,is reset to the base
cannot change
if (3)
Repeating in this manner, the CSO generates a firing spike-train
if
for
(4)
where
In this paper, for simplicity, we assume that all the circuit ele-
ments are ideal and define the switching dynamics as an ideal
model.1
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-
mate relation
and arehigh-andlow-voltagelevels,respectively.
, can be adjusted
. The PG and the firing switch
1This definition is a routine in the electrical engineering.
Fig. 3.
??? ??, ?
attractor. (b) Time-domain waveform.
Chaotic attractor in a unit CSO (? ? ??? ??, ?
? ??? ?, ? ? ???? ?, ?
? ?? ??, ??? ?
? ??
? ??? ?). (a) Phase space
Fig. 4.
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
capacitor voltage
reaches the firing threshold voltage
comparator closes the switch
to
in a short time. The voltage
switch
andis reset to the base voltage
Then the comparator opens the switch
withthesufficientlysmalltimeconstant
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. [11] has presented corresponding chaotic
attractor in numerical simulations and shown theoretical evi-
dence for the chaos generation.
at. If the
, the
and the voltagecharges up
closes the firing
in a short time.
returns to
.Theseswitching
and
III. CPCN
Fig.4showsaCPCNhavingalocalconnectionstructure.The
circuit dynamics is described by
if(5)
if(6)
where
CSO is connected with the neighbor CSOs. Fig. 5 illustrates
the connection method. If the capacitor voltage
denotes the index of the CSO. Each
reaches the

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1020IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 5, SEPTEMBER 2004
Fig. 5.
each other.
Switching rule of the CPCN. The ?th and ?th CSOs are connected to
firing threshold voltage
, and
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
, andis 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
firingspike
isinputtotheneighbor(s)instantaneously.
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 voltage instantaneously.
is input to the
is higher than the refrac-
is less than
is unaf-
if
or
otherwise
and
(7)
where
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 switchis closed when
thresholdvoltage
.Theswitch
than the refractory threshold voltage
closed when the firing spike
CSO.2Iftheswitchisclosed,orifboththeswitches
are closed simultaneously, thevoltage
inashorttime.Thevoltage
andis reset to the base voltage
time, the voltage
is input to the th CSO with a short
timedelay.Then
returnsto
denotestheindexoftheneighbor(s)ofthe thCSO.
be
at
reaches the firing
isclosedwhen
. The switch
is applied from the th
ishigher
is
and
charges upto
closesthefiringswitch
in a short time. At the same
withthesufficientlysmalltime
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.
constant
(7), approximately.
Fig. 7 shows typical experimental results for a CPCN of two
CSOs:
,and
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:
,
and.IfalltheCSOshave(almost)thesamepa-
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
formanin-phasesynchronousgroup.ThethirdandfourthCSOs
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
than
, and bothandare lower than
andjump tosimultaneously, and both
continuously. At
,reaches
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,
at
,reaches,is lower than
than
. Then,moves continuously and
synchronizationofthefirstandsecondCSOsisbrokendown.In
. These switching dynamics can satisfy (6) and
. If both the CSOs
,,
,reaches,is higher
. Then, both
and
and the other states
move
simulta-
, and
jumps to
is higher
; the

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NAKANO AND SAITO: GROUPING SYNCHRONIZATION IN A PCN OF CSOs1021
Fig. 7.
?
??? ?). (a) In-phase synchronization of chaos ???? ? ??? ? ??? ???.(b)
Asynchronous phenomena (???
? ??? ??, ???
topology.
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
and parameters
(8)
Fig. 8.
??? ??, ?
??
???
??? ??, ??? ? ??? ? ??? ??). (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
if(9)
if(10)
if
orand
otherwise.
(11)
Fig. 10 shows the normalized trajectories. Except for the self-
running angular frequency
, the CPCN has three parameters:

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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.
if
(12)
where (
Using this solution, trajectories can be calculated precisely
[10]. 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.
Let
be theth time when either of the th or th CSO
outputs a firing spike. Let
for
ofspikesof
for
andare 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
,andlet
,where
bethenumber
,and
(13)
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:
(14)
Let us consider trajectories starting from
Fig. 10). As the trajectories start from
oftheCSOsmustreachthethresholdatsomefinitetime
andthetrajectoriesreturnto
map
at(see
, either trajectory
,
.Wecanthendefineareturn
(15)
This map is given analytically. The firing time
is given by
(16)
where
is a positive-minimum root of
(17)