Realization of Associative Memory by Oscillators Changing the Coupling Strength
Kohei Yamamoto, Naoto Yonemoto, Yoko Uwate, and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: {yamamoto, yonex34, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
In this paper, we adapt the theory of the Associatron to thecoupling strength of Kuramoto model and realize associative
memory in Kuramoto model. We consider that this realization
enables associative memory in the coupling system of the oscillators as well. Therefore, we solve the circuit equations of
the coupling system of the van der Pol oscillators and realize
associative memory by using oscillation and synchronization.
In this study, we propose that nine van der Pol oscillators areconnected by resistors and simulate 3 × 3 image processing.

Figure 1: Coupling system of oscillators

We use Kuramoto model, which is a well-known mathematicalmodel of synchronization. Since there are nine os1. Introduction
cillators to be used this time, N=9. We devise the coupling
There are various synchronization phenomena in nature, strength K in this equation based on the Associatron[6].
such as the emission of ﬁreﬂies and the movement of pendu9lums. Similarly, synchronization phenomena have been ob∂θiK∑=
ω+sin(θj − θi )served in the coupling system of oscillators. Interestingly,
∂t
9 j=1
(1)
it has been proven that pattern recognition and associative
memory can be realized by using this oscillator synchronizai = 1, 2, ..., 9tion phenomena[1-2]. It is also known that pattern recogFor Eq.(1), we redeﬁnenition and associative memory can be realized by using this{technology. Pattern recognition and associative memory are
E0 × sexcellent human abilities, but there is still a big difference in
K=E×s
their abilities compared to computers. For a long time, many
researchers have been actively developing and studying CNN
and other technologies to solve this problem. In recent years, where s is the coupling strength and E is the coupling strength
the synchronization of oscillator coupling systems begin to matrix determined based on each pattern; E0 is the storage
attract attention, and research on pattern learning, recall, and matrix for the input pattern and E is the storage matrix forrecognition using various oscillators begin to be conducted the stored pattern. Since we have nine oscillators, we use a[3-4]. In this study, we ﬁrst use Kuramoto model to realize 3 × 3 pattern. First, each square of the pattern corresponds0
associative memory[5]. Next, we verify whether the same re- to the poscillator W1∼W9 in Fig.1 from the upper left, and fsults can be obtained with a coupling system of van der Pol and f are created with the black area of the image as 1 andIn this case, p is the number of patterns
oscillators.As a result, associative memory was achieved in the white area as -1.
pto
bestored,andfiscreated for the number of patterns.
both cases. However, it was found that the coupling strength
had to be reduced when using van der Pol oscillators.
{0000f 0 = fW
1 , fW 2 , fW 3 , ..., fW 9pppp
f p = fW 1 , fW 2 , fW 3 , ..., fW
92. Associative memory using Kuramoto model
We conduct an experiment with nine oscillators W1∼W9
mutually coupled, which is shown in Fig.1. We observe thesynchronization of the coupling system of these oscillators.

125This f 0 and f p are used to deﬁne E0 and E.

E0 = f 0 f 0T
 0 0fW 1 fW 100
 fW
2 fW 1

=
..
.

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


00. . . fW
1 fW 900

. . . fW
2 fW 9 ..
..
.
.

00fW
9 fW 100fW
9 fW 2...

E=p∑k=100fW
9 fW 9Based on these condition, E0 and E are calculated. The
initial value of θ is given randomly between 0 and 0.1[rad],and ω = π[rad/s], s = 0.4. The calculation is performed
until t = 0 ∼ 60[s], and at t = 25[s], the storage matrix ischanged from E0 to E for the memory recall. The oscilla(2) tion waveform obtained from the calculated phase is shownin Fig.4, and the graph showing the phase difference of each
oscillator with oscillator W1 is shown in Fig.5.

f k f kT
0kfWk fW 1kkp
∑  fW 2 fW 1=..
.

kkfW
1 fW 2
kkfW 2 fW2..
.

...
...
..
.

kkfW
1 fW 9(3)kkfW
2 fW 9 
..
.

kkfW
9 fW 1

kkfW
9 fW 2...

kkfW
9 fW 9k=1

Now we are ready for associative memory. First, we use
the memory matrix E0 of the input pattern to synchronize it.
After that, we change to the storage matrix E of the storage
pattern and synchronize it again. The memory recall startfrom the point of the change. From the calculated phase, wederive the phase difference from the reference oscillator of
W1. When the phase difference is 0, 2π..., it is said to besynchronized with W1. On the other hand, when the phasedifference is π, 3π, ..., it can be said to be asynchronous. The
phase difference is converted into a pattern by determining
whether it is synchronized with W1 or not.

Figure 4: The oscillation waveforms (Kuramoto Model).

3. Simulation of associative memory using Ku-ramoto model
Figure 5: The phase difference (Kuramoto Model).
The two patterns to be stored in the system are shown in
Fig.2, and the input pattern is shown in Fig.3.

Figure 6 shows the pattern we created based on the phasedifference at t = 60[s]. It can be seen that associative memory is achieved.

Figure 2: Stored patterns.
Figure 6: The recall pattern (Kuramoto Model).

4. Associative memory by a coupling system ofvan der Pol oscillators
Figure 3: Input pattern.

In the previous section, we were able to achieve associative memory by devising the coupling strength of Kuramoto126model. Next, we perform the same veriﬁcation on the cou- 5. Simulation of associative memory using vanpling system of van der Pol oscillators as shown in Fig.7. der Pol oscillator
Nine oscillators are connected by resistors, so the circuit
The image to be stored and input are exactly the same as inequations are Eq.(4). iG is the equation of the nonlinear elethe experiment in section 3, so E・E0 , and s are the same valment, which is Eq.(5).
ues. The initial values of x and y are given randomly between
0 and 0.1, ε = 0.1, and s = 0.01. At τ = 100, the storage matrix is changed from E0 to E for the memory recall. From theabove, calculations are performed for τ = 0 ∼ 200, and thegraph of xn is shown in Fig.8. And Fig.9 shows the result ofcalculating the difference between the oscillation waveformof oscillator W1 and x1 .

Figure 7: Circuit model of Van der Pol Oscillator
9∑
1
C dvn = i − i −(vn − vk )
LG
dt
R
k=1

 dinL dt = vnn = 1, 2, ..., 9iG = −g1 v + g3 v 3

(4)

(5)
Figure 8: The oscillation waveforms (van der Pol oscillator).

The parameters for normalization of Eq.(4) are follows:√g1
v=x,g3√L
ϵ = g1,C√
g1 Cy,g3 L√1 Lγ=.
R C
i=t=√LCτ,The normalized circuit equations are described as Eq.(6):9∑ dxn = ϵx (1 − x2 ) − y +K(vn − vk )
nnndτ
k=1
 dy
nn = 1, 2, ..., 9.
dτ = xn
In this case, K is{
K=Figure 9: The synchronization state (van der Pol oscillator).
(6)
The pattern created based on the synchronization state atτ = 200 is shown in Fig.10. It can be seen that associative
memory can be realized even with a coupling system of vander Pol oscillators.

E0 × sE×s

as in the previous section. The calculated x may have different amplitudes depending on each oscillator. Therefore,
instead of considering the phase difference as in the associative memory of Kuramoto model, we consider the difference
between the oscillation waveform x1 of oscillator W1. When
it is synchronous, the difference be zero, and when it is asynchronous, the amplitude have doubled from the original oscillation waveform. In this way, determine whether it is synchronized with W1, and convert it into a pattern.

127

Figure 10: The recall pattern (van der Pol oscillator).

In this section, we experimented with s = 0.01, because recall was not successful with s = 0.4, the same condition as insection 3. Under this condition, we conﬁrmed that the oscillator W1 cyclically repeats synchronously and asynchronously
from the moment it is changed to the storage matrix E. An
example of this is shown in Fig.11. After switching the storage matrix to E at τ = 30, the oscillation waveforms become
asynchronous around τ = 80, 180 and synchronous aroundτ = 130, 230.

References[1] Paolo Maffezzoni, Bichoy Bahr, Zheng Zhang, andLuca Daniel, Oscillator Array Models for AssociativeMemory and Pattern Recognition , IEEE Transactions
on Circuits and Systems I: Regular Papers (Volume: 62,Issue: 6, June 2015) pp.1591-1598.
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Oscillatory neurocomputers with dynamic connectivity , Physical Review Letters (Volume: 82, Issue: 14,April 1999) pp. 2983-2986.
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Neural Network for Pattern Recognition Applications ,2018 IEEE 61st International Midwest Symposium onCircuits and Systems (August 2018) pp. 238-241.
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Figure 11: The oscillation waveforms(Van der Pol oscillator,W1,W9)

In addition to W9, this phenomenon was also observedin four other oscillators W2, W3, W4, and W7. Therefore,
we created a pattern based on the synchronization states ofτ = 80, 130, 180, 230, . . . which are periodically synchronized and asynchronized. As a result, the two patterns shownin Fig.2(a) and 2(b) were repeatedly recalled, conﬁrming thatassociative memory cannot be realized at s = 0.4. These results indicate that associative memory can be achieved with avan der Pol oscillator, but the coupling strength needs to bereduced to achieve this.
6. Conclusion
In this study, we have realized associative memory for a3 × 3 pattern by changing the coupling strength. Using Kuramoto model to realize associative memory, we consider thatassociative memory can be realized in any coupling systemof oscillators where the coupling strength can be changed.
Therefore, as an example, we realized associative memory
with a van der Pol oscillator. However, it was also conﬁrmedthat in the coupled system of the van der Pol oscillator, when
the coupling strength was too large, it was not recalled well.
However, this study was not able to investigate the extent towhich the coupling strength affects the associative memory.
Therefore, the task is to verify the effects of changes in thecoupling strength and storage matrix on associative memory.
It is also necessary to verify the results with analog circuits.

128[5] Yoshiki Kuramoto, Self-entrainment of a population of
coupled non-linear oscillators , International Symposium on Mathematical Problems in Theoretical Physics(January 1975) pp. 420-422.
[6] Kaoru Nakano, Associatron- A Model of AssociativeMemory , IEEE Transactions on Systems, Man, andCybernetics (Volume: SMC-2, Issue: 3, July 1972) pp.
380-388.