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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)

1045-9227/04$20.00 © 2004 IEEE

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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)