Synchronization Patterns in Nonlinear Circuit Networks
Altered by Learning Rules
Yoko Uwate 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
Abstract

Therefore, in this study, we investigate the behavior ofcoupled chaotic circuit networks by employing phase-based
synchronization detection. Take one chaotic circuit as thereference circuit and calculate the phase difference using aPoincare section. First, we examine the synchronization rateof the entire network, and furthermore, we investigate how
the distribution of weights changes by utilizing learning rules.

This study investigates the synchronization phenomena when
the weights of coupled chaotic circuit networks are learned.
In contrast to previous approaches using voltage differencesof capacitors for synchronization detection, we employ phasedifferences in this study. This enables more accurate weightlearning based on precise synchronization detection. We explore how synchronization phenomena and weights changewith and without weight learning.
2. Chaotic Circuits Model
1. Introduction

Figure 1 shows the chaotic circuit model that has been investigated in the literature [8]-[10]. This circuit consists ofthree memory elements, one linear negative resistance element, and one nonlinear resistance element consisting of twodiodes. The negative resistance is realized using the linear
region of a negative impedance converter made from an operational ampliﬁer.

Synchronization phenomena in nonlinear circuit networks
exhibit intriguing phenomena such as N -phase synchronization, oscillation quenching, and independent oscillation.
These phenomena hold promise for applications in modelingand controlling high-dimensional nonlinear systems. In particular, chaos theory is applied in biomedical engineering forthe analysis of physiological signals such as heart rate variability and brain waves. This contributes to the understanding of the complex dynamics of biological systems and plays
a role in the development of medical diagnostics and treatment methods. The fundamentals of chaotic synchronization
have been studied with regard to the geometry and stabilityof synchronization [1], [2]. Examples of applications using
chaos synchronization include secure communication and image encryption [3], [4].
(a) Chaotic circuit.
(b) Chaotic attractor.
As part of our previous research, we have investigated clustering using synchronization in coupled chaotic circuit netFigure 1: Chaotic circuit model.
works [5]-[7]. Placing chaotic circuits on a two-dimensional
plane and reﬂecting the distance between chaotic circuits inthe magnitude of coupling allows for efﬁcient clustering. AdThe approximate I − V characteristics of the nonlinear
ditionally, for more advanced clustering, we have conﬁrmed
resistanceelement are indicated by the following equation,
the effectiveness of altering coupling weights using learningwheretheparameter rd is the slope of the nonlinear resisrules. In Refs. [6] , [7], the method for synchronization detance.
tection involves using the voltage difference between coupledchaotic circuits. However, in synchronization detection forchaotic coupling systems, using phase rather than voltage difVrd
V
vd (i2 ) =i2 +− i2 −.
(1)ference enables more accurate synchronization measurement.
2rdrd250

21.5

10.5

Y

By changing the variables, such thatr
√
C
L1 C
i1 =
V x; i2 =V y; v = V z;
L1L2r
√
pC
L1 C
L1r= α;= β; rd= δ; t = L1 Cτ
L1L2L2(2)-1

The normalzied equations represent the circuit equations
when all the chaotic circuits are coupled globally with eachother (all-to-all coupling).
dxi= αxi + zi

dτ

dyi



= zi + f (y)
dτ
(3)

N
 dziX= −xi − βyi −γij (zi − zj )
dτ

j=1
(i, j = 1, 2, · · ·, N )where f (y) is described as follows:11δ
y+− y−f (y) =.
2δδ

-1.5-2-2g(dij )2.

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.
3. Circuit Arrangement and Learning Process
Figure 2 shows an example of the arrangement of 100
chaos circuits. This is generated by a Gaussian distribution
with a parameter σ = 0.7. In the simulation, investigations
are conducted for ﬁve different networks.
Next, we explain about learning process of coupled chaoticcircuits networks. A ﬂowchart of these steps is shown inFig. 3.
We apply this Hebbian rule to chaotic circuit synchronization. In other words, the coupling between the synchronizedchaotic circuits is made stronger, and the coupling between
the un-synchronized chaotic circuits is made weaker. TheHebbian rule is applied to the chaotic circuits network as following steps.

-1

-0.5

0X

0.5

1

1.5

2

[step-1] At the initial state, all nodes are fully connected withcoupling strengths depending on distance.
[step-2] After a transient phase, we apply two rules for a sequence of generations. Each generation has length G = 10.
• (Determination of synchronization:) In order to determine whether two nodes are alike, we calculate the synchronization ratio for every pair of oscillators. If the synchronization ratio is larger than 80%, the corresponding
coupling strength becomes stronger with ∆γ = 1e−6 .

(4)

(5)

-1.5

Figure 2: One example of circuit arrangement in 2dimensional space (N = 100, σ=0.7).

In the computer simulations, we set the parameters to beα = 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:

γij =0-0.5

In order to analyze the synchronization ratio, we deﬁnea synchronization state by calculating phase differenceas|θk − θn | < 20◦ (k ∈ Sn )[step-3] Step-2 is repeated until iterations are reached (G =2000).
[step-4] At the ﬁnal state (G = 2000), we check the synchronization ratio for every pair of oscillators.
Here, two learning methods are considered. One method
involves increasing the weights for synchronized connections
and decreasing the weights for unsynchronized connections(Learning − PM). The other method involves increasing
the weights only for synchronized connections without decreasing the weights (Learning − P). The differences between the two learning methods is also investigated.
4. Simulation Results
First, we investigate the average synchronization rate of theentire circuit and the number of connections with synchronization exceeding 80 percent. The average results for ﬁve
different networks are summarized in Table 1. Focusing on
the average synchronization rate, it is observed that without
weight learning, the rate is 67.7 %. On the other hand, bothmethods with learning show an improvement in synchronization rates. Furthermore, when comparing Learning-PM andLearning-P, it is evident that Learning-P achieves a higher

251peak is at 80 %, indicating the absence of edges with low
synchronization rates.
Figure 5 shows the distribution of weights value. In this
proposed circuit system, the weight value is determined by
the distance between two chaotic circuits. We investigate theeffect of learning by checking the change of distribution ofweights. In the case of no-learning, most of weights has smallvalues. By using learning process, the peak of distribution isshifted to around 0.002. The difference between LearningPM and Leraning-P is that there are not may small weights
for Learning-P.

Start

G < 2000Runge-Kutta

Applying
Learning1) Determination of synchronization
phase differences: θ < 20 degree
synchronization ratio > 80%

3500

35003000

3000

2500

2500
Number of edges

YesNo

Number of edges

G%10==02000

1500

10002) Updating the weights
synchronization edge: γ +=1e-6

1500

1000

500

500

00020406080

100

02040

Synchronization rate [%]un-synchronization edge: γ -=1e-66080

100

0.00050.0006

Synchronization rate [%](a) No-learning.

(b) Learning-PM.

35003) Updating the generation
G++

3000

2500
Number of edges

End2000

2000

1500

1000

Figure 3: Flowchart of learning process.

500

00204060

80

100

Synchronization rate [%]synchronization rate. Similarly, the number of highly synchronized connections is more than three times greater withweight learning compared to no learning. The results indicate that learning weights enhances the synchronization rate.
The higher synchronization rate observed with Learning-P isattributed to its lack of an effect reducing the size of connections with low synchronization rates.

(c) Learning-P.

Table 1: Synchronization rate and high-synchronized edges
Average of
Number ofsync. rate [%] high-sync. edges
No-learning
67.70
1301.4
Learning-PM
75.13
3122.6
Learning-P
83.91
3862.6

2500

2500

2000

2000

Number of edges

Number of edges

Figure 4: Distribution of synchronized edges.

1500

1000

1500

1000

500

500

0000.0001

0.00020.00030.00040.00050.000600.0001

0.0002

Weight value

0.00030.0004Weight value(a) No-learning.

(b) Learning-PM.

2500

Next, we investigated the synchronization rate distributionof each edge within the circuit network. The simulation results are presented in Fig. 4, where the horizontal axis represents the synchronization rate, and the vertical axis represents
the number of edges. In the case of no learning, it can be observed that there is a peak at 70 % synchronization rate. ForLearning-PM, the peak has increased to 80 % and it is notablethat the synchronization rates around 10 are higher comparedto the case with no learning. In the case of Learning-P, the252

Number of edges

2000

1500

1000

500

00

0.00010.00020.00030.00040.00050.0006Weight value(c) Learning-P.
Figure 5: Distribution of weight value.

As future works, we would like to propose more effective
Next, we consider the timing of learning process. The simulation results of synchronization rate and number of high- learning method by using neuro science phenomena such assynchronized edges depending on timing of learning process STDP coupling.
are shown in Figs. 6 and 7, respectively. This is the average
result of ﬁve different networks. From these ﬁgures, we canReferences
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Learning_P

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80[3] M. F. Hassan, M. Hammuda, “A New Approachfor Constrained Chaos Synchronization with Application to Secure Data Communication,” Journalof the Franklin Institute, vol. 356, pp. 6697-6723,
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706050510204050

Timing of learning process

Figure 6: Synchronization rate depending on timing of learning process.

5000

Number of synchronizaed edges

4500

without Learning
Learning_PM
Learning_P

4000[5] Y. Uwate, Y. Takamaru, T. Ott and Y. Nishio, “Clustering Using Chaotic Circuit Networks with Weighted
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3500
3000
2500
2000

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1500
1000
500
0[4] S. Moon, J-J. Baik and J. M. Seo, “Chaos Synchronization in Generalized Lorenz Systems and an
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5

1020

40

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50

Timing of learning process

Figure 7: High-synchronized edges depending on timing oflearning process.

5. Conclusion[8] 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.
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In this study, we have investigated the characteristics ofthe proposed chaotic circuits networks with learning by focusing on synchronization rate and weight value. By using [10] A. Koseska, E. Volkov, & J. Kurths, “Real-World Existhe computer simulations, we have conﬁrmed that in the casetence and Origins of the Spiral Organization of Shrimpof synchronization rate, the peak of distribution becomes two
Shaped Domains,” Phys. Rev. Lett., 105, 074102, Aug.
peaks by applying Learning-PM. In the case of weight value,2010.
the distribution form is changed from power distribution tonormal distribution.

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