IEEE Workshop on Nonlinear Circuit Networks
December 8-9, 2023

Coupled Oscillators with Sparse Couplingfor Realization of Associative Memory
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
There has been research on synchronization 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 thisphenomenon [1]. It is being studied with the goal of bringing
associative memory capabilities closer to those of humans.
Network sparsity has also been studied in CNNs to remove
unnecessary computation and memory [2]. In previous studies,
associative memory was achieved by using an associatron in
the coupling system of oscillators. Furthermore, the relationship between the coupling strength between oscillators andassociative memory was investigated [3]. When implementing
the oscillator model in an actual circuit, there are many wires,
which makes implementation difficult.
To reduce wiring due to loose coupling between oscillators,
in this study, sparse coupling is used in the coupling systemof oscillators to disconnect the coupling. The couplings tobe 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.1II. S YSTEM M ODEL OF A SSOCIATIVE MEMORY USING THE
COUPLING OSCILLATORS

Figure 1 (a) shows the circuit diagram of the van derPol (VDP) oscillator. A model of a mutually coupled system
consisting of 6 × 10 VDPs is shown in Fig. 1 (b). Alloscillators are coupled to each other via resistors, and in thisstudy, they are partially disconnected.
The normalized circuit equations are given as follows:
60
∑

 dxn

= εxn (1 − x2n ) − yn + αK
(xn − xk ),
dτ
(1)
k=1

 dyn

= xn

dτ
1 The extended version of this study is being reviewed for ISCAS’24[4].

(a)

(b)

System model ((a) The circuit diagram of the VDP,(b) A model of a globally coupled system consisting.).
Fig. 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.
To realize associative memory, first synchronous simulationof the oscillator with the storage matrix for the input pattern isperformed, and then the simulation is switched to simulation
with it for the stored patterns. The output pattern is determinedby the phase difference of the reference oscillator W1.
III. S IMULATION R ESULTS
The input pattern is shown in Fig. 3 (a) and The storedpatterns are shown in Fig. 3 (b), (c) and (d).

Fig. 2. 6 × 10 patterns ((a) is input pattern, (b), (c) and (d)are stored patterns.).

In the case of patterns in Fig. 2, the input pattern Fig. 2(a) similar to the stored pattern Fig. 2 (c). If Fig. 2 (c) is arecall pattern, associative memory is considered to be realized.
Equation (1) is calculated using Runge-Kutta method with thestep size h = 0.1 for τ = 0 ∼ 2000. The initial values of xand y are random between 0 and 0.1, the parameters are ε =0.1, s = 0.005. The storage matrix was changed from E0 to Eat τ = 100 for recalling a stored pattern. Count is the numberof times the reference oscillator passed through the Poincar
section.

- 24 -A. Coupling Disconnection of Noisy Part of Oscillators

Recall patterns ((a) 29 × 29 disconnect coupling
between oscillators, (b) 30 × 30 disconnect coupling betweenoscillators.).
each of the simulation, the number of disconnections in thesimulations and the rate of coupling disconnections in themodel are shown in Table I.
Fig. 6.

The number of disconnections in simulation
8×38 8×40 8×51 8×52 29×29 30×30

CountCount

(a)

(b)

(a)

(b)

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

Fig. 4.

It can be seen from Fig. 2 (c) and Fig. 4 (b) that the colorof W13 has changed from the stored pattern.
B. Coupling Disconnection of Randomly Selected Oscillators
The coupling between some randomly selected oscillators
is disconnected from an equal number of other oscillators inthis model. The phase difference of disconnected couplings of29 randomly selected oscillators for each are shown in Fig. 5(a), and 30 randomly selected oscillators are shown in Fig. 5(b). The results of the recall patterns at count = 300 are shownin Fig. 6.
Phase Differences[° ]Phase Differences[° ](b)TABLE IT HE RATE OF COUPLING DISCONNECTION

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

The rate ofdisconnectedcouplings[%]
Realizationofassociative memory

17.2

18.0

23.0

23.5

47.5

50.8

⃝

×

⃝×

⃝×

The same number of disconnected oscillators can realize
associative memory even though the rate of coupling disconnection is higher. Therefore, it can be observed that the larger
the rate of coupling disconnection, the larger the recall patternis broken.
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. 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. When the oscillator and the coupling of the noisy partwere disconnected, the recall pattern was not broken and thecoupling disconnection rate was small, however when the 30
× 30 couplings were disconnected, the recall pattern was
broken significantly. In summary, it was possible to confirm
that the rate of coupling disconnection varied depending on
the combination of the 60 oscillators.
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.
R EFERENCESCountCount

(a)

(b)

Phase difference ((a) 29 × 29 disconnect coupling
between oscillators, (b) 30 × 30 disconnect coupling between
oscillators.).
The maximum value at which associative memory could
be realized was 28 when the same number of oscillator
connections were disconnected.
Finally, Add simulations using 8 randomly selected oscillators, summarize whether associative memory is realized inFig. 5.

(a)Phase Differences[° ]Phase Differences[° ]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). The results of the recall patterns atcount = 300 are shown in Fig. 4.

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Symposium on Circuits and Systems (ISCAS), 2024.(being submitted)- 25 -