2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

Study on Precision Recall by Using van der Pol Oscillatorswith Third-Power and Fifth-Power for Associative Memory
Yukinojo Kotani, Kento Fukuta, Yoko Uwate and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
E-mail: {kotani, fukuta1215, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract

2. COUPLING CIRCUIT MODEL

Figure 1 shows the coupling circuit model of van der PolIt is a great goal for many researchers to give computers asoscillators
via resistors. The current-voltage characteristics
sociative memory functions. This study compares associativeofnonlinear
resistor in van der Pol oscillator is also shown inmemory functions between the coupling circuit model withFig.
2.
van der Pol oscillators with third-power and the model withfifth-power. As a result, synchronization state of the modelwith fifth-power was much stabler. Besides, the model with
fifth-power outperforms the model with third-power in termsof recall response time and recall rate.
1. INTRODUCTION
Synchronization phenomenon is that two or more objects
do the same motions at the same time. It is observed in nature,such as luminescence of firefly and cry of frogs. Besides, synFigure 1: Coupling circuit model.
chronization phenomenon is applied in development studies
iG
iG
by various researchers. In the field of medical science, heartbeat is one of the synchronization phenomena, myocardialcells synchronized each other. Belousov-Zhabotinsky (BZ)
reaction is popular for chemical researchers as chemical osvvcillation. Similarly, synchronization phenomenon is observedon the coupling circuit via resistors [1].
Associative memory is to recall more closely related thingsfrom a part of memory. It is particular to human ability, andthen superior to computer. On the other hand, some peo(a) Third-power.
(b) Fifth-power.
ple cannot recognize the names and emotions when they see
human face without age relationship. This illness is called
Figure 2: Current-voltage characteristics.
prosopagnosia. To assist the patients, many researchers have The circuit equations of the model are Eq. (1) and the equastudied associative memory and pattern recognition by using tions of current-voltage characteristics of nonlinear resistors
coupling circuits [2], [3].
is Eq. (2).
In previous study, associative memory was realized by us60∑
ing the coupling circuit model of van der Pol oscillators with
1dv

 C n = −iGn − iLn +(vn − vk )third-power [4]. Besides, we confirmed that associative memdt
R
(1)
k=1
ory was also realized by using van der Pol oscillator with
 L diLn = v(n = 1, 2, · · ·, 60).
nfifth-power [5], and then investigated the properties of vandt
der Pol oscillators with fifth-power [6].
{In this study, we discussed average of recall response time−g1 v + g3 v 3(Third-power)
and recall rate by analyzing phase differences time evolutioniG =(2)35
g1 v − g3 v + g5 v (Fifth-power).
on both models.
61

2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

The parameters for normalization of Eq. (1) with thirdpower are follows:
√√
g1
g1 C
v=x, i =y, t = LCτ
g3
g3 L
√√
L1 L
ε = g1, K=.
CR C

Storage matrixes E0 and E are defined as Eq. (7) and Eq. (8)using f 0 and f p .
E0 = f 0 × f 0T .
p
1∑ p
E=f × f pT .
p

√

(8)

k=1

The normalized circuit equations with third-power are described as Eq. (3).
60∑ dxn = εx (1 − x2 ) − y +K(xn − xk )
nnndτ
k=1
 dyn= xn (n = 1, 2, · · ·, 60).
dτ(7)

(3)

On the other hand, Eq. (1) with fifth-power are normalized
by using these parameters:

Based on the above, recall process is described below.
First, parameter of normalized circuit equations is set, andthen calculate the solution by using Runge-Kutta method.
Hence, input pattern is recognized on the coupling circuit
model by using Eq. (7). Second, each oscillator on the model
begin to synchronize or asynchronized with the reference oscillator W1 by using Eq. (8). Finally, pixels output white andblack by using Eq. (6). White is output when each oscillator
is synchronized with the reference oscillator W1, and blackwhen each oscillator is asynchronized.
Phase difference time evolution is calculated by usingPoincaré map. Figure 3 shows the solution space for normalized circuit equations. Besides, zn and θn are defined asEq. (9).

√√g
g1 C 41v= 4x, i = 4y, t = LCτ
4
gg5 L
√5
√
L1 L
g3
ε = g1, K=, δ= √.
4
CR C
g1 2 g5 2√２

The normalized circuit equations with fifth-power are described as Eq. (4).
dxn
= −εxn (1 − δx2n + x4n ) − yndτ60∑+K(xn − xk )
(4)
k=1
dyn

= xn dτ(n = 1, 2, · · ·, 60).

Figure 3: Solution space.
zn = (xn , yn )θn = arctan(yn − y1) (n = 1, 2, · · ·, 60).
xn − x1(9)Poincaré section is the positive half of the x-axis in Fig. 3, so
x1 > 0 and y1 = 0. Absolute value of phase difference |θn | ismore less than or equal to 0° and greater than or equal to 90°,
Here, K is redefined as Eq. (5) based on associatron [4, 5, nth-oscillator outputs white. |θn | is more less than 90° andgreater than or equal to 180° , nth-oscillator outputs black.
6].
In this study, three stored patterns are used in Fig. 4.
{E0 × sK=(5)E × s.

Parameter s is the coupling strength. E0 and E are the storage
matrixes for input pattern and stored patterns respectively.
Each pixel corresponds to van der Pol oscillators in Fig. 1.
The oscillator W1 in the top left is reference. f 0 and f p are
created by setting white pixel to +1 and black pixel to -1. p
is the number of stored patterns and f p is created for eachstored pattern.
{ 0000f = (fW
1 fW 2 · · · fW 60 )
(6)
pp
p
p
f = (fW 1 fW 2 · · · fW60 ).

Figure 4: Stored patterns.

622023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

3. RESULTS
In these simulations, parameters of the model are set toTable 1: Average of recall response time.
ε = 0.1, s = 0.01, τ = 625 and δ = 1.
Hamming distance
101520Characteristics
Third Fifth Third Fifth Third FifthFirst, we investigated recall response time by using 10 kinds
Ave. recall response time
26.4
15.5
33.3
16.1
25.3
14.5
of input patterns whose hamming distance for stored patternis 10. Recall response time is defined as first count that themodel recall stored pattern. Figure 5 shows recall process ofIn Table 1, average of recall response time of the modelboth models when input broken 0 whose hamming distance with fifth-power is faster than the model with third-power
is 10. In addition, synchronization state of both models are without input patterns.
shown in Fig. 6. Count is the number of times that z1 passthrough poincaré section.
Next, we investigated recall rate by using 10 kinds of input patterns whose hamming distance for stored pattern is 20.
Figure 7 shows recall results of both models when input broken 0 whose hamming distance is 20, and the synchronization
state is shown in Fig. 8.

(a) Third-power.

(a) Third-power.

(b) Fifth-power.

Figure 7: Recall results.

(b) Fifth-power.

Figure 5: Recall process.

(a) Third-power.

(b) Fifth-power.

Figure 8: Synchronization state.

(a) Third-power.

(b) Fifth-power.

Figure 6: Synchronization state.
As a result, both models recalled stored pattern. Besides,
at count 16, the model with fifth-power had already recalled
stored pattern but the model with third-power had not yet.
So, we calculated average of recall response time by using
10 input patterns whose hamming distance is 10, 15 and 20.
Table1 shows average of recall response time.

As you can see, the model with fifth-power recalled stored
pattern although the model with third-power recalled broken
pattern. Besides, phase differences of the model with thirdpower split into away from 0° or 180°. On the other hand,phase differences of the model with fifth-power split into
0° or 180° clearly. It has same tendency when input broken
1 whose hamming distance is 20. However, both models recalled stored pattern every times when input broken 2 whosehamming distance is 20. Therefore, we reinvestigated thatrecall rate when input broken 2 whose hamming distance is25. Recall results of both models when input broken 2 whosehamming distance is 25 are shown in Fig. 9.
632023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

4. CONCLUSION(a) Third-power.

(b) Fifth-power.

Figure 9: Recall results.

We investigated recall response time and recall rate byusing the coupling circuit model with third-power and themodel with fifth-power. First, the model with fifth-power recalled stored patterns faster than the model with third-power.
Second, synchronization state of the model with third-power
has a tendency to split into 0°or 180°when input broken pattern. Otherwise, the one of the model with fifth-power isstable even when input more broken pattern. Consequently,
recall rate of the model with fifth-power is higher than themodel with third-power. In the future, we would like to examine the solution flow in detail, and realize associative memoryon the model with fifth-power by using unstable limit cycle.

As a result, the model with third-power recall 1 and themodel with fifth-power recall 2. In this simulation, each hamming distance of input pattern for 1 and 2 is 30 and 25 respectively. Hence, the model with fifth-power recalled correct
pattern but the model with third-power recalled failed pattern.
Table 2 shows recall rate. Recall rate is defined as an index
of recall times from count 0 to 80 divided by the number oftrials.

Hamming distance
Characteristics
Recall rate 0
Recall rate 1
Recall rate 2References[1] Seiichiro Moro, Yoshifumi Nishio and Shinsaku Mori,“Synchronization Phenomena in Oscillators Coupled byOne Resistor”, IEICE Transactions on Fundamentals,
vol. E78-A, no. 3, pp. 244-253, February 1995.
[2] Paolo Maffezzoni, Bichoy Bahr, Zheng Zhang andLuca Daniel, “Oscillator Array Models for AssociativeMemory and Pattern Recognition”, IEEE Transactions
on Circuits and Systems-I: Regular Pepars, vol. 62, issue. 6, pp. 1591-1598, June 2015.

Table 2: Recall rate.
2025
Third Fifth Third Fifth
0.7
1
none none
0.4
1
none none
1
1
0.2
0.9

[3] Paolo Maffezzoni, Bichoy Bahr, Zheng Zhang andLuca Daniel, “Analysis and Design of Boolean Associative Memories Made of Resonant Oscillator Arrays”,
IEEE Transactions on Circuits and Systems-I: RegularPepars, vol. 63, no. 11, pp. 1964-1973, November 2016.

In all cases, recall rate of the model with fifth-power ishigher than the model with third-power. In other words, themodel with fifth-power recall correct pattern even when input
more broken pattern.
Here, we discussed the cause of the difference of the synchronization state between the model with third- power andthe model with fifth-power. The difference of the synchronization state is caused by numerical results of Eqs. (3) and(5) arising from the current-voltage characteristics of nonlinear resistor Eq. (2). We confirmed that the model with thirdpower and the model with fifth-power oscillated while converging on a stable limit cycle, when examined the synchronization state. In addition, we used same values of parameters
for all experiments except for δ. The difference between themodel with third-power and fifth-power is only presence orabsence of δ. Therefore, we consider that the cause of the difference of the synchronization state between the model withthird- power and the model with fifth-power is numerical results of Eqs. (3) and (5) resulting δ.

[4] Kohei Yamamoto, Naoto Yonemto, Yoko Uwate andYoshifumi Nishio, “Realization of Associative Memory
by Oscillators Changing the Coupling Strength”, RISPInternational Workshop on Nonlinear Circuits Communicatiobs and Signal Processing 2021, pp.125-128,
March 2021.
[5] Yukinojo Kotani, Kento Fukuta, Yoko Uwate and Yoshifumi Nishio, “Comparison of Associative Memory
Function by Using Synchronization of van der Pol Oscillators with Third-Power and Fifth-power Nonlinear
Characteristics”, Shikoku-section Joint Convention ofthe Institutes of Electrical and Related Engineers, p.10,
September 2022.
[6] Yukinojo Kotani, Kento Fukuta, Yoko Uwate andYoshifumi Nishio, “Research on Synchronization Phenomenon in Coupling Circuit Model of van der Pol Oscillators for Associative Memory”, IEEE Workshop onNonlinear Circuit Networks, pp.1-2, December 2022.

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