NCSP'22
RISP International Workshop on Nonlinear circuits,Commuinications and Signal Processing 2022
February 28 - March 1

Synchronization and Clustering of Chaotic Circuit Networks with Hebbian Rule
Yoko Uwate† , Thomas Ott‡ and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp‡Zurich University of Applied Sciences
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
Email: thomas.ott@zhaw.chAbstract
In this study, we investigate clustering phenomena in a network composed of coupled chaotic circuits. In this networkmodel, the coupling strength is reflected by the distance information when the chaotic circuits are placed in a twodimensional space. For more advanced clustering, we propose a new method that applies Hebbian rule to synchronization. By applying Hebbian rule, it is confirmed that clusters
that cannot be extracted by standard chaotic circuit networks
can be extracted.
1. Introduction
Today, every time we open a newspaper, the page contains
Artificial Intelligence:AI characters, and new technologies
that apply deep learning are being developed everywhere in
the industry. One of the basic laws of learning in the brain is
the Hebbian rule. This rule has proposed by Hebb in 1949 [1].
This is a fundamental rule of learning and long-term memory based on the hypothesis that synapses become more efficient at transmitting electrical stimuli each time a neuron
fires, and conversely, less efficient if they do not fire for a longtime. The mechanism as to the neurobiology of the Hebbianrule has also been clarified. Coupling strengths are enhancedby the electrical signals and strengthened the part of morecommunications. The neurons have output part called axon.
Some axons are covered with oligodendrocyte on the Myelin
sheaths. Oligodendrocyte is white material which is rich inlipid. Covered axons can communicate quicker than noncovered ones. This phenomenon is called Myelination [2].
For example, we practice to achievement something when weusually try to do it. This action from challenge to achievement is myelination. Therefore, there is a big relation between learning and myelination. The study about myelination has possibility to make more performance for learningby adapting application.
In recent years, it has become necessary to handle increas-

ingly large amounts of information in our daily lives. Toenable structuring and analysis of such data, it is useful topartition each data set into clusters. The aim of a clustering
algorithm is to find data clusters that consist of similar elements. Clustering algorithms have widespread applications
in a variety of fields, including data mining, image processing and biological data analysis [3]-[5]. Various different
clustering algorithms are thus available, along with many different applications. Previously, many clustering studies havebeen performed using discrete time models, such as coupled
map lattices (CMLs) and self-organizing maps (SOMs) [6][8]. However, few studies of clustering have been performed
using a continuous time model. Therefore, this work focuses
on research into clustering phenomena using real electronic
circuits in a continuous time model.
Coupled chaotic circuits can be realized using electronic
circuits and various interesting phenomena can be observedin these circuits. In recent years, many studies have reported
on application of the clustering and synchronization phenomena that can be observed in coupled chaotic circuits to naturalsciences. The reason for this interest is that the characteristicsof the chaos phenomena observed in coupled chaotic circuits
also exist in real life, in phenomena such as human behavior,emotions and heartbeats. At the same time, synchronizationand clustering phenomena have been studied associated withthe chaos phenomena. Coupled chaotic circuits thus have thepotential to be applied to a variety of different fields. We
believe that we can apply the synchronization phenomena of
coupled chaotic circuits to social networks in real life if wecan clear up the chaos phenomena. Therefore, our study considers a new approach to investigation of the synchronizationand clustering phenomena that occur in coupled chaotic circuits.
In our previous studies [12]-[15], we have investigatedclustering in two-dimensional networks of complex chaoticcircuits, where the coupling strength reflects the network
distance information. We showed that circuits that are arranged close to each other can achieve phase synchroniza-373

The normalized circuit equations can be described bytion, whereas coupled circuits located far away from eachother cannot be synchronized. We also observed that these
dx

networks of coupled chaotic circuits can be split into dif= αx + z
ferent synchronized groups, thus revealing a clustering phe dτdynomenon.
(3)= z − f (y)dτHowever, the layout of the clustering we were targeting dz = −x − βywas easy, and we needed to consider a more practical anddτdifficult problem. To achieve more complex and advanced
clustering, we propose a clustering method based on a new where f (y) is described as follows:
chaotic coupled circuit network applying Hebbian rule. For amore complex clustering example, we consider a circuit lay1
1
δout with overlapping clusters. By using computer simultions,y+− y−.
(4)
f (y) =2
δδ
we confirm that chaotic circuit network with the Hebbia rule
is more effective than the starndard chaotic circit network.
The following equations represent the circuit equations
when all the chaotic circuits are coupled globally with eachother (all-to-all coupling).
2. Chaotic Circuits ModeldxiIn this section, we explain the chaotic circuit model. Fig= αxi + zi

dτ


ure 1 shows the chaotic circuit model that has been investigated in the literature [9]-[11]. This circuit consists of three dyi = zi + f (y)memory elements, one linear negative resistance element, anddτ
(5)
one nonlinear resistance element consisting of two diodes.
N
X
dzi
The negative resistance is realized using the linear region of a= −xi − βyi −
γij (zi − zj )dτnegative impedance converter made from an operational am
j=1
plifier.
(i, j = 1, 2, · · ·, N )In the computer simulations, we set the parameters to be
α = 0.460, β = 3.0 and δ = 470. The characteristic function
f (y) can be described as a three-segment piecewise-linearfunction. In this study, the value of γij reflects the distance
between the circuits in an inverse manner, as described using
the following equation:(a) Chaotic circuit.

γij =(b) Chaotic attractor.

Figure 1: Chaotic circuit model.

g(dij )2.

(6)

Here, dij denotes the Euclidean distance between the i−thcircuit and the j − th circuit, while g is a scaling parameterthat determines the coupling strengths.

The approximate I − V characteristics of the nonlinear
resistance element are indicated by the following equation, 3. Overlapped Clustering Arrangement and Hebbian
where the parameter rd is the slope of the nonlinear resis- Learningtance.
The previous circuit layout was a case where clusteringrdVVwas easy. This time, we consider a more difficult case: a cirvd (i2 ) =i2 +− i2 −.
(1)cuit layout with overlapping clusters. The number of clusters
2rdrd
is set to three, and the chaotic circuits are randomly placed
By changing the variables, such thataccording to a normal distribution from the center of eachr√cluster. The circuit arrangement with overlapped clusters isC
L1 C
shown in Fig. 2.
i1 =V x; i2 =V y; v = V z;
L1
L2
The purpose of clustering for this circuit layout is not tor√pdivide it into three clusters, but to extract as many nodes asCL1L1 C(2) possible from the central part of the cluster.
r= α;= β; rd= δ; t = L1 CτL1L2L2374

0.8

0.8

0.7

0.7

transient state H1 H2 H3 H4"locationA.c" u 2:3"locationB.c" u 2:3"locationC.c" u 2:3

τts

Clster-3
0.6

0.6

0.5

0.5

H10τh τh τh τh ... τhτ (iteration)yy

Clster-1
0.4

0.3

Clster-2

0.2

0.4

Applying Hebbian Learning
voltage differences: fabs(xi-xj) < 0.3synchronization ratio > 50%

0.3

0.2

0.1

0.1

000.1

0.2

0.3

0.4
x0.5

0.6

0.7

0.8(a) Overlapped clusters.

000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

synchronization
coupling: γ +=0.001

x(b) Circuit arrangement.

un-synchronization
coupling: γ -=0.001
Parameters:
τts=100,000
τg=10,000
Hebbian Learning (H)=10Figure 2: Circuit arrangement with overlapped clusters(N =100, the number of clusters: 3, red points: 30 (cluster1), green points: 30 (cluster-2), blue points: 40 (cluster-3)).

yIn this study, we propose a new clustering method that apFigure 3: Process of Hebbian Learning.
plies Hebbian rule as well as the determination of clusters
by synchronization of chaotic circuit networks. The Hebbian rule states that synapses, which connect neurons be- tering with Hebbian rule, it was successful in extracting morecome more efficient when neurons fire repeatedly, and less nodes from the center of the cluster.
efficient when they do not fire for long periods of time. We
0.7
apply this Hebbian rule to chaotic circuit synchronization. In
other words, the coupling between the synchronized chaotic
0.6
circuits is made stronger, and the coupling between the unsynchronized chaotic circuits is made weaker. The Hebbian
0.5
rule is applied to the chaotic circuits network as following
steps.
0.4[step-1] At the initial state, all nodes are fully connected withcoupling strengths depending on distance.
0.3[step-2] After a transient phase, we apply two rules for a0.2
sequence of generations. Each generation has length τh =10, 000. The conceptual diagram of the computer simulation
0.1
0.1
0.20.30.4
0.5
0.6
0.7
is shown in Fig. 3.
x• (check synchronization:) In order to check whether
two nodes are alike, we calculate the synchronization
ratio for every pair of oscillators. If the synchronization ratio is larger than 50%, the corresponding couplingstrength becomes stronger with ∆γ = 0.01.

Figure 4: Clustering result with Hebbian rule.

The extraction rates of the clusters are summarized in theTabs. I and II. By using the Hebbian rule, over half of theIn order to analyze the synchronization ratio, we define
nodes in the two clusters have been successfully detected.
a synchronization state as
While in the case of the clustering without Hebbian rule, thenumber of detected nodes in the every cluster was less than|xk − xn | < 0.3 (k ∈ Sn )half. Therefore, it can be said that the clustering method forchaotic circuit networks using the Hebbian rule is effective.
[step-3] Step-2 is repeated until 10 iterations are reached(H = 10).
[step-4] At the final state (G = 10), we check the synchroTable 1: Clustering result with Hebbian rule.
nization ratio for every pair of oscillators.
nn%
4. Simulation Results
Figures 4 and 5 show the clustering results of chaotic circuit network with/without Hebbian rule. In the case of clus-375

cluster-1
cluster-2
cluster-3

cen

30
30
40

22
19
19

73.3
63.3
47.5

[4] N. Dhanachandra, K. Manglem & Y. J. Chanu, “ImageSegmentation Using K -means Clustering Algorithm
and Subtractive Clustering Algorithm,” Procedia Computer Science, Vol. 54, pp. 764-771, 2015.

0.7

0.6

y0.5

[5] G. Manogaran, V. Vijayakumar, R. Varatharajan, P. M.
Kumar & R. Sundarasekar, “Machine Learning BasedBig Data Processing Framework for Cancer Diagnosis Using Hidden Markov Model and GM Clustering,”
Wireless Personal Communications, Vol. 102, Issue 3,
pp. 20992116, 2018.

0.4

0.3

0.2

0.1
0.1

0.2

0.3

0.4
x

0.5

0.6

[6] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of Chaotic Elements,” Physical D, vol. 41, pp. 137-172, 1990.

0.7

Figure 5: Clustering result without Hebbian rule.

[7] T. Ott, M. Christen & R. Stoop, “An Unbiased Clustering Algorithm Based on Self-organization Processes in
Spiking Neural Networks,” Proc. of NDES’06, pp. 143146, 2006.

Table 2: Clustering result without Hebbian rule.
nc nen%cluster-1 30
7 23.3cluster-2 30
4 13.3cluster-3 40
9 22.5[8] L. Angelini, F. D. Carlo, C. Marangi, M. Pellicoro& S. Stramaglia, “Clustering Data by Inhomogeneous
Chaotic Map Lattice,” Phys. Rev. Lett., 85, pp. 554-557,
2000.
[9] Y. Nishio, N. Inaba, S. Mori & T. Saito, “Rigorous Analyses of Windows in a Symmetric Circuit,” IEEE Transactions on Circuits and Systems, vol. 37, no. 4, pp. 473487, Apr. 1990.

5. Conclusion

In this study, we propose a clustering method based on anew chaotic coupled circuit network applying Hebbian rule inorder to achieve more complex and advanced clustering, Fora more complex clustering example, we considered a circuit [10] C. Bonatto & J. A. C. Gallas, “Periodicity Hub andNested Spirals in the Phase Diagram of a Simple Resislayout with overlapping clusters. By using computer simultitive Circuit,” Phys. Rev. Lett., 101, 054101, Aug. 2008.
ons, we confirmed that chaotic circuit network with the Hebbian rule is more effective than the starndard chaotic circit [11] A. Koseska, E. Volkov, & J. Kurths, “Real-World Exisnetwork. By using the synchronization state between nodes,
tence and Origins of the Spiral Organization of Shrimpwe were able to confirm that it can be applied to more adShaped Domains,” Phys. Rev. Lett., 105, 074102, Aug.
vanced clustering.
2010.
In future work, we would like to study clustering phenomena in large-scale networks. Furthermore, we hope to apply [12] Y. Takamaru, H. Kataoka, Y. Uwate & Y. Nishio, “Clustering Phenomena in Complex Networks of Chaotic Cirthe clustering methods developed here to data mining, imagecuits,” Proc. ISCAS’12, Vol. 22, pp. 320-334, 2012.
processing and other applications in real life situations.
References[1] D. O. Hebb, “Organization of Behavior,” 1949.

[13] Y. Takamaru, Y. Uwate, T. Ott & Y. Nishio, “Dependence of Clustering Patterns on Density of Chaotic Circuits in Networks,” RISP Journal of Signal Processing,vol. 17, no. 4, pp. 103-106, Jul. 2013.

[14] K. Ago, Y. Uwate & Y. Nishio, “Investigation of Clus[2] K. A. Nave & H. B. Werner, “Myelination of the nertering in Complex Networks Using Coupled Chaotic
vous system: mechanisms and functions,” Annu Rev
Circuits,” Proc. of IFAT’15, pp. 101-102, 2015.
Cell Dev Biol, DOI: 10.1146/annurev-cellbio-100913013101, 2014.
[15] K. Oi, Y. Uwate & Y. Nishio, “Synchronization andClustering in Coupled Parametrically Excited Oscilla[3] Daniel T. Larose & Chantal D. Larose “Discoveringtors with Small Mismatch,” Proc. of ISCAS’15, pp. 910Knowledge in Data: An Introduction to Data Mining,”
913, 2015.
John Wiley and Sons, 2014.

376