2024 IEEE International Symposium on Circuits and Systems (ISCAS) | 979-8-3503-3099-1/24/$31.00 ©2024 IEEE | DOI: 10.1109/ISCAS58744.2024.10558213

Associative Memory Function
Using Coupled Oscillators with Sparse Coupling
Kento Fukuta, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: {fukuta1215, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we applied sparse coupling to thecoupling system of van der Pol oscillators and investigated
the realization of associative memory. The phase difference ofoscillator waveforms and the difference of output patterns bydisconnecting the coupling between oscillators was clarified. As aresult, it was confirmed that the realization of associative memory
is possible using sparse coupling, however the maximum rate ofcoupling disconnection for which associative memory is possible
depends on the combination of oscillators in the coupling system.

I. I NTRODUCTION
Synchronization phenomena are abundant in nature, andthere has been extensive research on this phenomena observed
from the coupling system of oscillators. It has been proven that
coupling system of oscillators is able to realize pattern recognition and associative memory by using the synchronization
phenomenon [1],[2]. This is known as the Oscillatory Neural
Network (ONN) where the oscillators mimic the neurons of thebrain [3],[4]. And it is being studied with the goal of bringing
pattern recognition and associative memory capabilities closer
to those of humans.
In previous studies, associative memory was achieved byusing an associatron in the coupling system of oscillators [5].
This is because the application of the asociatron enabled thedefinition of the coupling strength between oscillators and thecalling of a storage matrix. Furthermore, the relationship between the coupling strength between oscillators and associative
memory was investigated [6].
In this study, sparse coupling is used in the coupling systemof oscillators to disconnect the coupling. The couplings to
be disconnected are divided into three cases, and the phasedifference and recall patterns are compared by the computer
simulations. We also investigate the relationship between therate of disconnection and associative memory.

(a)(b)

System model ((a) The circuit diagram of the VDP,(b) A model of a globally coupled system consisting.).
Fig. 1.

The normalized circuit equations are given as follows:
60
∑
dxn

2

=εx(1−x)−y+αK(xn − xk ),nnndτ
k=1
dyn

= xn
dτ

(1)

Where α is the connectivity matrix, it represents the state ofcoupling between oscillators, and it is possible to disconnect
the coupling between the specified oscillators.
Here, the method is explained based on the proposed method
in previous study. For Eq. (1) K is defined as{K=E0 × s
E ×s

II. S YSTEM M ODEL OF A SSOCIATIVE MEMORY USING THE
COUPLING OSCILLATORS

Figure 1 (a) shows the circuit diagram of the van der Pol(VDP) oscillator used in this study. A model of a mutuallycoupled system consisting of 6 × 10 VDPs is shown in Fig.
1 (b). All oscillators are coupled to each other via resistors,
and in this study, they are partially disconnected.

s is the coupling strength and E0 and E are based on theasociatron. Additionally E0 is the storage matrix for the input
pattern and E is the storage matrix for the stored patterns.
f 0 and f p are image data and there are the same number ofoscillators W1 to W60 in the model. They are ”-1” when W1∼ W60 in Fig. 1 (b) represents black and ”1” when W1 ∼W60 represents white.

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00
f = (fW1pp
f = (fW
10fW
2pfW20fW
3pfW
3······The storage matrix was changed from E0 to E at τ =100 forrecalling a stored pattern. Count is the number of times thereference oscillator passed through the Poincar section.

0fW60 )
pfW60 )

E0 and E are defined using the transpose matrix f T as
follows:
E0 = f


0Tf

000fW
1 fW 100
 fW
2 fW 1

= ..
.

00fW
1 fW 200fW 2 fW 2..
.

······..
.

00fW
1 fW 60
00fW 2 fW 60..
.

00fW60 fW 100fW60 fW 2···00fW60 fW 60

E =p∑Oscillators in the noisy part of the model are those whosepixels have the wrong color when compared to the input
pattern in Fig. 2 (a), and the correct stored pattern in Fig. 2 (c).
Figure 3 shows the phase difference when the oscillator in thenoisy part of Fig. 2 (a) is disconnected from the 38 randomly
selected oscillators in Fig. 3 (a), and the 40 randomly selectedoscillators in Fig. 3 (b).

f kT f kk=1kkfW1 fW 1kkp ∑ff
 W2 W1=..
.

kkfW
1 fW 2kkfWf2 W2..
.

······..
.

kkfW
1 fW 60
kkfWf
2 W 60..
.

kkfW60 fW 1kkfW60 fW 2···

kkfW60 fW 60

k=1(2)

A. Coupling Disconnection of Noisy Part of Oscillators

Phase Differences[° ]{ (3)To realize associative memory using these storage matrices
E0 and E, first synchronous simulation of the oscillator with Eis performed, and then the simulation is switched to simulation
with E0 . The output pattern is determined by the phasedifference of the reference oscillator W1, and the color of
the pixels is determined by the separation between 0 and 180
degrees.

Count

Phase Differences[° ](a)III. S IMULATION R ESULTS
The input pattern is shown in Fig. 2 (a) and The stored
patterns are shown in Fig. 2 (b), (c) and (d).

Count(b)Phase difference ((a) 8 × 38 disconnect coupling
between oscillators, (b) 8 × 40 disconnect coupling between
oscillators.).
Fig. 3.

The results of the recall patterns at count = 300 are shownin Fig. 4.
6 × 10 patterns ((a) is input pattern, (b), (c) and (d)are stored patterns.).
Fig. 2.

The values of f 0 and f p are determined from Fig. 2, and E0
and E are created by Eps. (2) and (3). In the case of patternsin Fig. 2, the input pattern Fig. 2 (a) similar to the storedpattern Fig. 2 (c). If Fig. 2 (c) is a recall pattern, associative
memory is considered to be realized. Equation (1) is calculatedusing Runge-Kutta method with the step size h = 0.1 for τ =0 ∼ 2000. The initial values of x and y are random between
0 and 0.1, the parameters are ε = 0.1, s = 0.005 in Eq. (1).

The correct recall pattern was obtained in Fig. 4 (a).
However, it can be seen from Fig. 2 (c) and Fig. 4 (b) that thecolor of W13 has changed from the stored pattern. Therefore,
the associative memory function was impaired when more than
38 oscillators were disconnected.
B. Coupling Disconnection of Randomly Selected Oscillators
Here, the coupling between some randomly selected oscillators is disconnected from an equal number of other oscillatorsin this model. The phase difference of disconnected couplingsof 29 randomly selected oscillators for each are shown in Fig.

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(a)

(b)

Recall patterns ((a) 8 × 38 disconnect coupling
between oscillators, (b) 8 × 40 disconnect coupling between
oscillators.).

Fig. 4.

Phase Differences[° ]5 (a), and 30 randomly selected oscillators for each are shownin Fig. 5 (b).

Count

Phase Differences[° ](a)(b)

Recall patterns ((a) 29 × 29 disconnect coupling
between oscillators, (b) 30 × 30 disconnect coupling betweenoscillators.).
Fig. 6.

Next, to compare with subsection A, which uses 8 oscillators in the noisy part, we use the same number of randomly
selected oscillators. These oscillators are disconnect the coupling of 8 and 51 or 52 randomly selected oscillators. The
phase difference of disconnecting the coupling of 51 oscillators
is shown in Fig. 7 (a) and the coupling of 52 oscillators isshown in Fig. 7 (b).

Phase Differences[° ](a)CountCount(b)
Phase difference ((a) 29 × 29 disconnect coupling
between oscillators, (b) 30 × 30 disconnect coupling between
oscillators.).

Phase Differences[° ](a)Fig. 5.

The results of the recall patterns at count = 300 are shownin Fig. 6.

Count(b)Fig. 7. Phase difference ((a) 8 × 51 disconnect coupling between oscillators,(b) 8 × 52 disconnect coupling between oscillators.).

It was impossible to realize associative memory when 30
× 30 couplings were disconnected in Fig. 6 (b). Therefore,
the correct recall pattern could be obtained by disconnecting
28 × 28 connections in Fig. 7 (a), and the maximum value atwhich associative memory could be realized was 28 when thesame number of oscillator connections were disconnected.

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TABLE I
T HE RATE OF COUPLING DISCONNECTION
8 × 38The rate ofdisconnected
couplings[%]
Realizationofassociative memory

The number of disconnections in simulation
8 × 40 8 × 51
8 × 52 29 × 29
30 × 30

17.2

18.0

23.0

23.5

47.5

50.8

⃝

×

⃝×

⃝×

The results of the recall patterns at count = 300 are shownin Fig. 8.

(a)

(b)

Recall patterns ((a) 8 × 51 disconnect coupling
between oscillators, (b) 8 × 52 disconnect coupling betweenoscillators.).

Fig. 8.

The correct recall pattern was obtained by disconnecting 8× 51 connections in Fig. 8 (a). Associative memory was notrealized when 8 × 52 couplings were disconnected.
Finally, Summarize whether associative memory is realizedin each of the simulation, the number of disconnections inthe simulations and the rate of coupling disconnections in themodel are shown in Table I.
When disconnecting the same number of oscillator connections, the model had a 50% connection disconnection rate,
and associative memory was not realized. The same numberof disconnected oscillators can realize associative memory
even though the rate of coupling disconnection is higher.
Compared to disconnecting the couplings in the noisy parts ofFig.1 (a), disconnecting the couplings of 8 randomly selectedoscillators resulted in a higher rate of coupling disconnection,
and associative memory was realized. When the coupling of
the noisy parts was disconnected, the recall pattern was notbroken as abruptly as the 30 × 30 or the 8 randomly selectedoscillators. Therefore, it can be observed that the larger therate of coupling disconnection, the larger the recall pattern isbroken.

As a result, it was possible to realize associative memory
using sparse coupling, but associative memory was not realized
when the rate of coupling disconnection was high. In addition,
there were differences in the recall pattern breaking. When theoscillator and the coupling of the noisy part were disconnected,
the recall pattern was not broken and the coupling disconnection rate was small, however when the 30 × 30 couplings
were disconnected, the recall pattern was broken significantly.
In addition, although the oscillators in the noisy part andthe 8 oscillators selected at random had the same number ofoscillators, the realization of associative memory was realizedeven if many oscillators and couplings were disconnected. Insummary, it was possible to confirm that the rate of coupling
disconnection varied depending on the combination of the 60oscillators.
For the future work, we would like to investigate thecouplings that can be preferentially disconnected when disconnecting couplings. In addition, a circuit simulation is to be
performed to confirm the difference from this study.
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IV. CONCLUSIONSIn this study, we proposed to use sparse coupling for thecoupling system of van der Pol oscillators to disconnect thecoupling. The realization of associative memory was investigated based on the phase difference of oscillation waveforms
and recall patterns.

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