2023 International Symposium on Nonlinear Theory and Its Applications
NOLTA2023, September 26-29, 2023, Catania and Online

Analysis of Hierarchical Hetero Associative Memory
Using van der Pol Oscillator Synchronization
Kazuki Yasufuku, Yoko Uwate and Yoshifumi Nishio†Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {yasufuku, uwate, nishio}@ ee.tokushima-u.ac.jp
Abstract– In this study, we proposed a new method of hetero-associative memory using the synchronization phenomenon of hierarchically connected van der Pol oscillators. Asa result, we confirmed that the model can be configured
with any number of oscillators for input and output. We also
confirmed that this model has noise correction capability as
well as an auto-associative memory model. In this study, wemeasured the noise correction capability of the proposedmodel and further examined its storage capacity.
1. Introduction
In nature, various synchronous phenomena exist, such as
the luminescence of fireflies and the movement of pendulums. Similarly, synchronization phenomena based onphase differences have been observed in coupled oscillatorsystems [1]. Interestingly, it has been proven that patternrecognition and associative memory can be realized by utilizing the synchronization phenomena of oscillators [2]-[4].
This method differs from the currently widely used Neumann-type computers, which perform retrieval of stored information by inputting information similar to the stored information. Since the elements perform operations in parallel at this time, the robustness of the elements and savingsin the number of operations are expected.
Therefore, our previous work proposed to realize patternrecognition and associative memory of images by adding anew memory matrix based on association to the couplingstrength between van der Pol oscillators [5]-[6]. There are
two patterns of associative memory depending on the pairof input and output patterns to be stored. One is when theinput and output patterns are the same, this associative
memory is called auto-associative memory. The other iswhen the input and output patterns are different, and thisassociative memory is called hetero-associative memory.
Hetero-associative memory is realized by forming amemory matrix with different pattern pairs of input and output patterns. In our previous work, we proposed a heteroassociative memory model based on the synchronization
phenomenon of van der Pol oscillators [7]. However, thismodel uses the same oscillator for input and output. As aresult, there is a restriction that the same number of oscillators must be used for input and output. Therefore, we
ORCID iDs Kazuki Yasufuku: 0009-0007-14194677, Yoko Uwate: 0000-0002-2992-8852, Yoshifumi Nishio: 0000-0002-0247-0001
apropose a new hierarchical mutual recall associative
memory model consisting of an input layer and an output
layer using van der Pol oscillators.
Furthermore, this model can correct noise as well as theauto-associative memory model. Therefore, we numerically evaluate the recall patterns when noise is inserted into
the input patterns in Section IV.
We also evaluate the recall rate when the number of patterns to be memorized is increased, and investigate the relationship between the recall rate and the time to complete
recall. This is discussed in Section V.
2. Hierarchical Hetero Associative Memory by van der
Pol OscillatorsIn this section, we explain the proposed model.
First of all, Fig. 1 shows the previous study's hetero-associative memory model. Each cell consists of van der Pol
oscillators, and all oscillators are coupled to each other via
resistors. Proper setting of the coupling strength based on
the input pattern and the pattern to be stored enables patternrecognition and recall. In this model, we confirmed that different patterns can be recalled at the input and output. However, it has the limitation that the same number of oscillators must be used for input and output because the sameoscillator coupling system is used for input and output.

Figure 1: Associative memory model with mutually coupled oscillators.
Therefore, we propose a hetero-associative memory
model with hierarchically connected van der Pol oscillators,
as shown in Fig. 2 In this model, each cell consists of a vander Pol oscillator. In the input layer, each oscillator is mutually coupled as in previous studies. Furthermore, thismodel has an output layer, where each oscillator in the input
layer is coupled to all oscillators in the output layer. Thishierarchical structure allows the use of different numbersof patterns for inputs and outputs.

This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International.

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Next, the input and output methods of the pattern are described.

Figure 2: Hierarchical structure model consisting of inputand output layers.
The normalized circuit equations are described as follows.
𝑁𝑑𝑥𝑛= 𝜀𝑥𝑛 (1 − 𝑥𝑛 2 ) − 𝑦𝑛 − ∑ 𝑲(𝑥𝑛 − 𝑥𝑘 )𝑑𝜏(1)𝑘=1𝑑𝑦𝑛= 𝑥𝑛{𝑑𝜏N is the number of oscillators to be connected to. Also, K
is defined as

𝑬 ×𝑠
𝑲={ 𝟎𝑬×𝑠(2)s is the coupling strength, and 𝐸0 and E are the accumulation matrices determined based on the input and output
sides of the storage pattern. Pattern creation was performed
with black as -1 and white as 1. Let 𝑥𝑖 denote the input pattern, 𝑎𝑖 the input side of the storage pattern, and 𝑏𝑖 the output side of the storage pattern.

𝒙𝒊 = (𝒙𝟏 , 𝒙𝟐 , 𝒙𝟑 , ⋯ , 𝒙𝑵 )(3)

𝒂𝒊 = (𝒂𝟏 , 𝒂𝟐 , 𝒂𝟑 , ⋯ , 𝒂𝑵 )(4)

𝒃𝒊 = (𝒃𝟏 , 𝒃𝟐 , 𝒃𝟑 , ⋯ , 𝒃𝑵 )(5)

Taking Figure 3 as an example, 𝑎𝑖 corresponds to the inputs, “dog” and “flower”, and 𝑏𝑖 corresponds to the outputs, “D” and “F”. Using these, a memory matrix can becreated as shown in Eq. (7). Also, 𝑥𝑖 represents the actualinputs. For example, in Section 4, noise is included in theinputs, and in this case, the noise pattern corresponds to𝑥𝑖 . If the number of pairs of patterns to be stored is P, thestorage matrix is expressed as follows.

𝑬𝟎 = 𝒙𝒊 𝒙𝒊 𝑻𝒙𝟏 𝒙𝟏
𝒙𝟐 𝒙𝟏=( ⋮𝒙𝑵 𝒙𝟏

𝒙𝟏 𝒙𝟐
𝒙𝟐 𝒙𝟐
⋮𝒙𝑵 𝒙𝟐

⋯
⋯⋱
⋯

𝒙𝟏 𝒙𝑵
𝒙𝟐 𝒙𝑵)(6)

𝒂𝒌𝟏 𝒃𝒌𝑵𝒂𝒌𝟐 𝒃𝒌𝑵(𝟕)⋮

Step 1: As shown in Fig. 2, the input layer is mutuallycoupled with each oscillator. From Eq. (2), the memory matrix 𝐸0 of K is used to recognize the input pattern.
Step 2: Connect the input layer and the output layer. At
this time, the oscillators of all input layers are interconnected with the oscillators of all output layers, and fromEq. (2), the output pattern is recalled by using the memory
matrix of E for K.
Step 3: Check the phase of the voltage of each oscillator
in the output layer. At this time, the difference between thephase of the voltage of each oscillator and that of the first
oscillator is checked to determine whether they are synchronous or asynchronous. The output pattern is white if
the oscillators are synchronized, and black if they are not.
Note that the selection of the reference oscillator may resultin the output of an inverted pattern, depending on the input
pattern.
In addition, a threshold value is used to determine whether
or not the synchronization is achieved. In this simulation,
the threshold value was set to 1.0.
3. Simulation Results
We trained a 24×24 pattern of a dog and a flower in theinput layer and a 14×12 pattern of the initials "D" and "F"in the output layer. Figure 3 shows the patterns of thelearned pairs.

Figure 3: The patterns of the learned pairs (dog and flower).
In this study, simulations were performed using theRunge-Kutta method to analyze the oscillator behavior.
Also, the step size is fixed with h=0.1. The variables in Eq.
(1) are ε=0.1, s=0.002, τ≦40 in step1, 40<τ in step2.
Figure 4 shows the phase difference between the voltageof the reference first oscillator and the other oscillators inthe output layer when the dog pattern is input.

𝒙𝑵 𝒙𝑵

𝑷

𝑬 = ∑ 𝒂𝒌𝒊 𝒃𝒌𝒊

𝑻

𝒌=𝟏
𝑷

=∑
𝒌=𝟏

𝒂𝒌𝟏 𝒃𝒌𝟏
𝒂𝒌𝟐 𝒃𝒌𝟏
⋮

𝒂𝒌 𝒃𝒌
( 𝑵 𝟏

𝒂𝒌𝟏 𝒃𝒌𝟐
𝒂𝒌𝟐 𝒃𝒌𝟐⋮
𝒌 𝒌𝒂𝑵 𝒃𝟐⋯⋯⋱⋯⋮

𝒂𝒌𝑵 𝒃𝒌𝑵)

Figure 4: The phase difference in voltage.

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Figure 5 shows the "D" recall process when inputting thedog pattern. It can be seen that "D" recalls the pattern normally.

Figure 5: "D" recall process.

The similarity between the pattern recalled when eachnoise pattern was input and the normal "D" pattern was calculated as the recall rate and shown in Fig. 8. It can be seenthat the recall rate worsens as the noise increases. Finally,in Fig. 7(e), the recall was 4.5. At this point, we confirmedthat the recall pattern was "F" because the input was close
to the inverted pattern of the flower pattern. This is because,by definition, if "F" is recalled instead of "D," the recall rate
is not 0, but 4.5, the similarity between "F" and "D."Figure 6 shows the recall process of "F" when the flower
pattern is inputted without changing the learned memory
matrix. It can be confirmed that "F" is successfully recalled as in the case of the dog pattern.

Figure 6: "F" recall process.
Figures 5 and 6 confirm that different numbers of inputs
and recalls are possible by using input and output layers.
4. Numerical Evaluation of Noise Insertion
To numerically evaluate the characteristics of the output
patterns when noise is inserted into the input patterns, wedefined the degree of similarity ρ between the patterns.
If the two patterns at this time are Ai and Bi, the similarity ρ is expressed as follows.

Figure 8: The recall rate for noise pattern input.
5. Evaluation of Recall Rate Change with Number ofLearning Patterns
The third pairs of star and “S” patterns were learned tosimulate recall. Figure 9 shows the patterns of the learnedpairs.

Figure 9: The patterns of the learned 3 pairs.

𝑁

𝜌=1∑ 𝐴𝑖 𝐵𝑖𝑁(𝟖)𝑖=1When two patterns are similar, the similarity ρ has a large
value, and when they are the same, ρ=1.
Figure 7 shows the input noise patterns. 5 types of patterns
with increasing noise in sequence. The percentage of noise
inserted was quantified by the similarity between the noise
pattern and the exact dog pattern. The noise was created byinverting the black-and-white pattern. The locations of thepattern to be inverted were selected where the pattern was
not close to the floral pattern. As a result, when a largeamount of noise was inserted, the pattern was inverted fromthe flower pattern.

Figure 10 shows the phase difference between the voltageof the reference first oscillator and the other oscillators in
the output layer when the dog pattern is input to the learning model in the three-pair pattern.

Figure 10: The phase difference between the voltage.

Figure 7: input patterns with noise. (a) ρ=0.90 (b) ρ=0.70(c) ρ=0.50 (d) ρ=0.30 (e) ρ=0.11

Figure 11 shows the "D" recall process when inputting thedog pattern. It can be seen that "D" recalls the pattern normally.

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6. Conclusions

Figure 11: "D" recall process.
Figures 12 and 13 show the recall process of "F" and "S"when the flower pattern and flower pattern are input without changing the learned memory matrix. As in the case of
the dog pattern, it can be seen that the "F" and "S" are successfully recalled.

In this study, we analyzed a newly proposed hierarchicalhetero-associative memory model. We confirmed that thehierarchical model can handle different numbers of data byusing different numbers of oscillators for input and output.
In addition, since this model can correct noise, we evaluated the recall rate for noise patterns.
Furthermore, we investigated the effect of increasing thenumber of training patterns on recall. Increasing the number of patterns to be learned increased the time to recall,and increasing the number of patterns prevented complete
recall. In future studies, we intend to investigate the relationship between the number of oscillators and memory capacity.

Figure 12: "F" recall process.
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Figure 13: "S" recall process.
Next, we conducted a numerical analysis of recall patterns
when the number of pairs of patterns to be learned was increased. In addition to the second set of dog and flower patterns proposed in section III, the third set of star patterns,
the fourth set of apple patterns, and five sets of heart patterns were used to simulate an increased number of learning
pattern pairs. The newly added apple and heart patterns areshown in Fig. 14.

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Figure 14: Additional learned patterns.
Figure 15 shows the respective recall rates for each of thedog patterns when the number of patterns to be learned wasincreased when the dog patterns were input. It can be seenthat when the number of patterns to be learned was increased, the time to complete recall increased. Also, when
the number of pairs was increased to five, complete recall
became more difficult.

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Associative Memory using Synchronization of van der Pol
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2022.

Figure 15: The respective recall when the number of patterns to be learned were increased.

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