# A New Method of the Pattern Storage and Recognition in Oscillatory Neural Networks Based on Resistive Switches

^{*}

## Abstract

**:**

_{2}-switches is used to demonstrate a novel method of pattern storage and recognition in an impulse oscillator neural network (ONN), based on the high-order synchronization effect. The method allows storage of many patterns, and their number depends on the number of synchronous states N

_{s}. The modeling demonstrates attainment of N

_{s}of several orders both for a three-oscillator scheme N

_{s}~ 650 and for a two-oscillator scheme N

_{s}~ 260. A number of regularities are obtained, in particular, an optimal strength of oscillator coupling is revealed when N

_{s}has a maximum. Algorithms of vector storage, network training, and test vector recognition are suggested, where the parameter of synchronization effectiveness is used as a degree of match. It is shown that, to reduce the ambiguity of recognition, the number coordinated in each vector should be at least one unit less than the number of oscillators. The demonstrated results are of a general character, and they may be applied in ONNs with various mechanisms and oscillator coupling topology.

## 1. Introduction

_{2}-switches [7,8], 1T-TaS

_{2}charge density wave devices [9,10,11], thyristors [12], tunneling diodes [13], resistive memory elements [14], spin-torque nano-oscillators [15]. Such ONNs appear to be interesting because of hardware solution simplicity, as well as compactness and energy efficiency of the developed micro- and nanoelectronic self-oscillators. VO

_{2}-based oscillators, as the elements of ONNs, have been chosen because they ensure rapid electric switching (~10 ns) [16], manufacturability with high degree of nanoscaling [17] and, above all, because of the pronounced effect of thermal coupling that simplifies ONN assembly and circuit engineering of galvanically isolated oscillators. Consequently, VO

_{2}-oscillators started being used as the prototypes of neuro-oscillators for cognitive technology [8,16,18].

_{2}-oscillators [17]. In many studies [22,24,25], patterns to be stored are expressed through a set of vectors. Vector coordinates contain information about the pattern and unambiguously associate it with one of possible variants. For instance, the vector of the object’s color in RGB coordinates (white color—RGB (255,255,255)) may be used as a 3-dimension vector. There are some methods of vector storage based on oscillator elements’ synchronization in ONNs, and one of them is presented in paper [20]. To store

**E**vector, a phase-shift keying method of a test vector

**T**is specified by weight matrix setting; at the second stage, the weights are sharply changed to the initial values (corresponding to the stored vectors), and the system arrives at one of the stable combinations of phase shift

**E**. However, this phase method has the following drawbacks: N

^{2}couplings with tunable weights and a two-stage procedure of pattern recognition.

**E**is stored through oscillator frequency shifts against the central frequency of oscillator array F

^{0}synchronization (on the first harmonic) per the values corresponding to the vector coordinates

**E**= (δ

_{ω}

_{1}, δ

_{ω}

_{2}, …, δ

_{ω}

_{N}). Recognition of test vector

**T**occurs at the reverse shift of frequencies and, in the case when the vectors coincide

**T**≈

**E**, the synchronization, indicating the fact of pattern recognition, takes place. This method allows usage of an oscillator star configuration and only N couplings, however, the disadvantage of this method is that just one vector is stored.

## 2. Materials and Methods

#### 2.1. General Principle

_{i}

_{,j}(weights), where i, j are the numbers of interacting oscillators, and Δ

_{i}

_{,j}denotes the value of the i-th oscillator effect on the j-th one. Oscillator networks may form various topologies: fully connected—all-to-all; and not fully connected—bus, star, and ring. Figure 1a–c show examples of two and three oscillators connections using topologies “star”, “all-to-all”, and an example of N oscillator connections using a mixed topology (Figure 1d).

_{1}:k

_{2}:k

_{3}:..:k

_{N}, where k

_{N}is a harmonic order of N-th oscillator at the common frequency of the network synchronization F

_{s}, (SHR—subharmonic ratio). As an example, Figure 2 shows spectra of three electric oscillators that have synchronization of the order SHR = k

_{1}:k

_{2}:k

_{3}= 3:6:4. The following rule should be noted: if all paired oscillators have different synchronization frequencies, there is always a common synchronization frequency F

_{s}for the whole system (all pairs), and the network synchronous state will also be determined by the ratio SHR = k

_{1}:k

_{2}:k

_{3}:..:k

_{N}at frequency F

_{s}(see Section 2.3).

_{th}, then SHR is absent, and the signal is considered conventionally non-synchronized.

_{p}, their variation causes changes of the basic oscillation frequency F

^{0}. Nevertheless, in some cases, transition between states may be achieved by variation of coupling forces or noise intensity.

_{th}< η < 100%, with a peaked curve.

_{s}. The value of N

_{s}depends on many parameters: the oscillator number N, the range of control parameters and their number, network topology, strength of coupling between oscillators, noise level in the system and on the threshold value of synchronization effectiveness η

_{th}. We will cover the issue in detail later, nevertheless, we have shown in [28], that for a two-oscillator network, N

_{s}has a maximum at certain values of coupling strengths between oscillators, and decreases when the system noise amplitude increases. When the coupling strength grows considerably, the value N

_{s}decreases because of the nearby synchronization areas’ integration.

**E**

_{1},

**E**

_{2}, ...,

**E**

_{NS}, that connect the origin of coordinates with the points of synchronization effectiveness maximum η, can be associated with the synchronization areas. Thus, the system stores N

_{s}of vectors, and the dimensionality of the stored vectors M is determined by the number of chosen control parameters. The coordinates determine the shift of oscillators’ control parameters, for example, currents

**E**= (δI

_{p1}(1), δI

_{p2}(2), …, δI

_{pN}(M)), against the origin of coordinates.

**E,**whose values give the information about the intensity of red and green colors

**E**= (red, green), and parameter blue is fixed as blue = 100. This example, in Figure 3c, shows the intensities of RGB components on the axes that can be linearly transformed into the values of the oscillator currents and vice versa.

#### 2.1.1. Vector Storage and ONN Training

- For storage, arbitrary vectors
**E**_{1},**E**_{2}, …,**E**_{i}, ...,**E**_{Ns}should be specified. If necessary, control parameters should be transformed into the corresponding coordinate system (for example, a color one, see Figure 3b,c). In general, vectors have dimensionality M and appear as a set of a network parameters that affects the system SHR. For example, they can be either currents, as shown in Figure 3**E**= (δI_{p1}(1), δI_{p2}(2), …, δI_{pN}(M)), or they can be coupling strengths between some definite oscillators**E**= (δΔ_{i}_{,j}(1), δΔ_{i,j}(2), …, δΔ_{i}_{,j}(M)), or mixed parameters**E**= (δI_{p1}(1), δΔ_{i}_{,j}(2), …, δΔ_{i}_{,j}(M)) (see Figure 4). - Then, the network should be trained by the adjustment of the ONN parameters that are not used for the vectors’ determination (coupling strengths, currents of other oscillators in the network, noise level, and synchronization effectiveness threshold η
_{th}). The adjustment is performed until the synchronization areas coincide with the vectors’ ends at the point of maximum value of synchronization effectiveness η (similar network training was used in the work [15]). The adjustment can be performed in two steps.- First, by using random search until the vectors enter the synchronization area.
- Then, one of gradient methods [29] may be applied to search the maximum η. As a result, each stored vector corresponds to its unique value of SHR and maximum of η(
**E**).

- If the training does not provide a positive result, one more oscillator should be included into the system and coupled with all oscillators already present, thus increasing the number of varied parameters and the number of possible synchronous states N
_{s}. Then, the training should be repeated (see step 2).

#### 2.1.2. Vectors Recognition

**T**recognition includes the following steps:

- Set the test vector
**T**to the system input through applying shifts to the control parameters (see Figure 4). The vector’s coordinates may be either shifts of currents, or coupling strengths or their combination, as it has been indicated above. - If one of the conditions is met (
**T**≈**E**_{1}or**T**≈**E**_{2}or … or**T**≈**E**_{NS}), i.e., coordinates values of**T**are equal to one of the stored patterns, a transition to the synchronous state will occur and, actually, the act of the corresponding pattern recognition will take place. Which patterns have been exactly recognized can be determined by the value of SHR. The existence of the synchronization areas ensures the vector recognition even at its coordinates’ insignificant displacement from the stored pattern. - The degree of match d
_{m}between the objects may be such magnitude as the difference between the synchronization effectiveness of the stored and the test vectors d_{m}= η(**E**) − η(**T**). If the magnitude of η(**E**) is not known, then to compare the degree of match, the formula d_{m}= 100% − η(**T**) can be used. The less d_{m}is, the closer vector**T**is to vector**E**.

**E**= (δω

_{1}, δω

_{2}, …, δω

_{N}), in our method, the vector is set through the control parameters

**E**

_{1}= (δI

_{p1}, δI

_{p2}, …, δI

_{pN}), that has the same meaning. The principle difference is that here, a high-order synchronization effect is used, thus allowing storage of a multitude of patterns in the ONN.

**E’**= (δI

_{1}, δI

_{2}, …, δI

_{N−1}) with one less than the number of oscillators (N − 1).

#### 2.2. Model Object

_{2}-oscillators, where each oscillator has the scheme of a relaxation oscillator. Our choice is conditioned by the fact that we have done some research in thermal coupling [17,30] and its modeling, however, the coupling may be an electric one (capacitive or resistive [7]). It is known that an electric switching effect is observed in VO

_{2}film-based structures, that is conditioned by a phase metal–insulator transition (MIT) at the moment when the temperature reaches T

_{t}~ 340 K, because of Joule heating by the passing current I

_{sw}[16]. This gives high-impedance (OFF) and low-impedance (ON) branches on I–V characteristics with threshold voltages (OFF→ON) U

_{th}~ 5 V and holding voltages (ON→OFF) U

_{h}~ 1.5 V (see Figure 5a). Both branches of I–V characteristics are reasonably well approximated by f

_{sw}curve, consisting of two linearized regions with dynamic resistance R

_{off}~ 9.1 kΩ and R

_{on}~ 615 Ω:

_{bv}~0.82 V is bias voltage of a low-impedance region, and State is a switch state.

_{2,3}= Δ

_{3,2}= 0, the topology is “star” (Figure 1b); and at Δ

_{2,3}= Δ

_{3,2}= Δ

_{1,3}= Δ

_{3,1}= 0, the scheme turns into a two-oscillator one (Figure 1a). The control parameters here are source currents I

_{p1}, I

_{p2}, I

_{p3}, and their variation leads to alteration of the fundamental oscillation frequency F

^{0}of oscillators.

_{i}(t) is the output voltage taken from the capacitor (C = 100 nF), I

_{sw(i)}(t) = f

_{sw}(U

_{i}(t) − U

_{in}) is the current passing through a switch, determined by I–V characteristics (1), I

_{p(i)}is the i-th oscillator supply current, respectively, U

_{in}is the amplitude of switch internal noise, and i is the oscillator’s number.

_{2}-oscillator and the neighbor ones ((i+)—clockwise and (i−)—counterclockwise of the scheme in Figure 5b) is realized according to the rule

_{(i+)}and State

_{(i−)}are on the OFF branch of I–V characteristics, then the threshold voltage of the i-th VO

_{2}-oscillator does not change: U

_{th(i)}= U

_{th}. Rule (3) is the same for all oscillators (with regard to cyclic permutation).

_{p}, coupling strength Δ, and noise amplitude U

_{in}varied.

#### 2.3. Method of Calculating a Family of Metrics

_{1}:k

_{2}:k

_{3}:..:k

_{N}(see Section 2.1) may be solved in several ways. For example, by direct analysis of all oscillation spectra, or by searching the synchronization order of each pair of oscillators based on the method which we suggested in [17].

_{1}

^{0}, F

_{2}

^{0}, F

_{3}

^{0}, …, F

_{N}

^{0}) must be commensurable. This is evident because at the synchronous state, there is a common synchronization frequency F

_{s}, and the equality (F

_{s}= F

_{1}

^{0}·k

_{1}= F

_{2}

^{0}·k

_{2}= … = F

_{N}

^{0}·k

_{N}) is fulfilled. If we divide F

_{1}

^{0}into all frequencies in the set (F

_{1}

^{0}, F

_{2}

^{0}, F

_{3}

^{0}, …, F

_{N}

^{0}), then we will get (1, F

_{1}

^{0}/F

_{2}

^{0}, F

_{1}

^{0}/F

_{3}

^{0}, …, F

_{1}

^{0}/F

_{N}

^{0}) = (1, k

_{2}/k

_{1}, k

_{3}/k

_{1}, …, k

_{N}/k

_{1}), that is, a new set of rational numbers determining pair synchronization of all oscillators in regard to the first oscillator (see [17]).

_{1}/d

_{1}, m

_{2}/d

_{2}, …, m

_{N−1}/d

_{N−1}), and to its reduction to a common denominator:

_{1}:k

_{2}:k

_{3}= 3:6:4.

_{sw}, synchronous in time [17].

_{SHR}synchronous periods T

_{s}with the definite SHR, to the whole durability of the processed oscillogram T

_{all}:

_{th}(in our case 90%). Oscillations are considered synchronized when η exceeds the threshold η ≥ η

_{th}. If the system consists of more than two oscillators, then the total effectiveness η is calculated as the mean value of all oscillator pairs. It should be noted that the proposed methods of SHR and η identification may be used in oscillator systems with noise. It has been noted that the noise increase leads mainly to the decrease of η, while SHR does not normally change.

## 3. Results

_{p1}, I

_{p2}, and noise and coupling strength values are U

_{in}= 40 mV and Δ = 0.2 V. It can be seen that there is a whole family of synchronization areas that are called Arnold tongues [23]. The number of possible variants of synchronous states, N

_{s}, in which the system may exist while the control parameters are varied, is N

_{s}= 9. The dimension of the stored vectors in this case is 2, and the coordinates determine current shifts

**E**

_{1}= (δI

_{p1}, δI

_{p2}), with respect to the origin of coordinates.

**E’**

_{1}= (δI

_{1}, δI

_{2}, …, δI

_{N−1}) of a dimension one less than the number of oscillators (N − 1), in this case,

**E’**

_{1}= (δI

_{1}). In practice, this means that we fix the current for one oscillator, and vary the currents for the others (see Figure 6a, dashed line I

_{p2}= const). Thus, we eliminate the ambiguity of synchronization definition by one of the vector coordinates, and the areas of possible synchronization are narrowed.

_{m}= η(

**E’**) − η(

**T**) may serve as the parameter of the degree of match between the test and stored vectors.

_{p1}= 950 µA, and parameters U

_{in}= 40 mV and Δ = 0.2 V, that are similar to a two-oscillator scheme.

**E’**of dimension 2. In this case, with all other things being equal, N

_{s}depends on the topology, and is N

_{s}= 16 for a “star” connection and N

_{s}= 14 for an “all-to-all” connection. The area shape also depends on the topology.

_{s}, increases. Yet, this is evident as the number of freedom degree increases at determining the synchronization value SHR = k

_{1}:k

_{2}:k

_{3}:..:k

_{N}. Nevertheless, as we show below, at certain parameters, there are some exceptions from the general rule.

#### 3.1. Vector Storage and ONN Training

**E’**

_{1}= (28, 149),

**E’**

_{2}= (28, 28), and

**E’**

_{3}= (150, 31). It should be noted that the number of coordinates in each vector is one unit less than the number of oscillators, and is equal two. As it has been explained above, this is necessary to narrow the area of possible synchronization and to reduce the recognition ambiguity. Liner transformation of coordinates, from the current parameters into color parameters, should be thought over initially. In our case, we used the following formulas: red $\iff $ (I

_{p2}[µA] − 550 [µA])/2 and green $\iff $ (I

_{p3}[µA] − 550 [µA])/2. After the working area has been transformed, we set three vectors, as shown in Figure 7b.

_{p1}and coupling strength Δ

_{i,j}) in such a way that the synchronization areas are obtained on the vectors at the point of maximum η. A fine adjustment for the maximum η can be done by using the gradient search. As a result, we have found that at Δ

_{i,j}= Δ = 0.2 V, and I

_{p1}= 850 µA, the system complies with the assigned task.

#### 3.2. Vectors Recognition

**T**≈

**E’**

_{1}is supplied, the neural network is transformed into a synchronous state with the synchronization order 1:2:1. At

**T**≈

**E’**

_{2}, we get SHR = 1:2:2. At

**T**≈

**E’**

_{3}, we get SHR = 1:1:2. In each case, the degree of match d

_{m}= η(

**E’**) − η(

**T**) is calculated, that determines the degree of match of vector

**T**with the stored vectors. For example, at

**T**= (31,145), the system transfers to SHR = 1:2:1 and the vector is recognized as the vector

**E’**

_{1}= (28,149), with the degree of match d

_{m}= η(

**E’**) − η(

_{1}**T**) = 2% (see Figure 7d).

**T**= (31,145).

**E’**

_{1}= (28,149).

_{m}= η(

**E’**) − η(

_{1}**T**) = 2%.

_{s}on Δ at three different configurations of a neural network at the constant noise level U

_{in}= 20 mV. The existence of the main maximum N

_{s}is evident at some optimal value Δ

_{opt}, in this case, this value is roughly the same as Δ

_{opt}~0.1 V for all configurations, and does not depend on the oscillator number. The existence of the curve maximum N

_{s}(Δ) reduplicates our results obtained in [28] for a two-oscillator scheme. Inalterability of Δ

_{opt}for a different number of oscillators N, with all other parameters being equal, might be explained by the fact that, with the increase of the number of freedom degrees for synchronization order, SHR = k

_{1}:k

_{2}:k

_{3}:..:k

_{N}, the value of N

_{s}has the tendency to grow.

_{s}to decrease when the coupling strength Δ grows above Δ

_{opt}and, at its large values, the system tends to the lowest possible N

_{s}= 1 with synchronization value 1:1:1. This is related to the fact that, with the increase of Δ, the surface of certain synchronization areas increase. Neighboring areas merge; in this case, the synchronization order of the resulting area predominantly consists of lower harmonic numbers. As the dimension of the control parameters is limited, such growth of synchronization area surfaces irrevocably results in a decrease of their number and value of N

_{s}. The insertions in Figure 8 show the evolution of synchronization areas and demonstrate the effect of their merging with Δ growth.

_{opt}. In turn, this is related to the fact that, in the presence of noise in the neural network, the increase of Δ may result in the development of new synchronization areas at the control parameters values that previously corresponded to the non-synchronous state of the system. Therefore, in a general case, the curve N

_{s}(Δ) may have a complicated shape with several maximums, as we can see in Figure 8.

_{opt}. The latter is due to synchronization effect degradation at Δ→0.

_{s}in the system; this does not contradict the rule suggested above. For example, N

_{s_max}= 17 is for two oscillators, for three-oscillator schemes, and N

_{s_max}= 28 and N

_{s_max}= 45 are for the “all-to-all” and “star” schemes, respectively. At the same time, at certain values of coupling strength (for example, at Δ = 1.1 V), the value of N

_{s}for a two-oscillator scheme may be even higher. Also, the regularity that the “star” topology has higher N

_{s}than the topology of “all-to-all” is observed. All of these things mean that the increase of the coupling number may contribute to the effect of system desynchronization and decrease of N

_{s}, as it seems that oscillators prevent each other from synchronization.

_{s}vs noise level in the system U

_{in}at the same coupling strength Δ = 0.2 V, for three configurations of an oscillator neural network. The general trend for the decrease of N

_{s}at the noise amplitude increase is due to the decrease of the surface of synchronous areas which eventually disappear (see the insertions in Figure 9).

_{s}for a three-oscillator configuration “star” is higher than that for a two-oscillator one. The shapes of curves N

_{s}(U

_{in}) are similar, which indicated that the physics of noise effect on the network is similar, and does not depend on the number of oscillators.

_{s}(Δ) and N

_{s}(U

_{in}), it may be seen that the number of synchronization areas N

_{s_max}in our models may reach N

_{s_max}~450, at an optimal coupling strength value Δ = Δ

_{opt}, and at lowered noise U

_{in}= 10 μV, it increases to N

_{s_max}~650.

## 4. Conclusions

_{2}-switches.

_{s}, where each state of the system is characterized by synchronization order SHR = k

_{1}:k

_{2}:k

_{3}:..:k

_{N}.

_{s}increases with the increase of the number of interacting oscillators. The modeling demonstrates achievement of N

_{s}of several orders: N

_{s}~650 for a three-oscillator scheme and N

_{s}~260 for a two-oscillator scheme.

_{s}decrease with the increase of coupling strength and switches’ inner noise amplitude, is also shown.

_{s}, of a neural network using the minimum number of neural oscillators. In addition, the proposed concept of pulse synchronization definition, through calculation of a family of metrics, opens a natural way for gradient method application to an oscillator network training (optimization).

_{2}thermally coupled relaxation oscillators), the demonstrated method of pattern storage and recognition is sufficiently general, and the fundamental character of the obtained regularities may be the subject of further research of ONNs, of various mechanisms and oscillator-coupling topology.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Examples of two (

**a**) and three (

**b**) oscillators connection into a neural network using topologies “star” (

**b**), “all-to-all” (

**c**), and N-oscillators using mixed topology (

**d**), where Δ

_{i}

_{,j}indicates the value of the i-th oscillator effect on the j-th one.

**Figure 2.**Example of oscillation spectra of three electric oscillators at synchronization order subharmonic ratio (SHR) = k

_{1}:k

_{2}:k

_{3}= 3:6:4, where I

_{SW}is the current amplitude of a signal in an oscillator, F

^{0}is first harmonic, k is the harmonic number at the synchronization frequency F

_{s}.

**Figure 3.**(

**a**) Schematic representation of the synchronization areas for a three-oscillator scheme; (

**b**) Examples of the vector and object’s RGB color association, that illustrate the algorithm of the network training and recognition (

**c**).

**Figure 4.**Schematic representation of pattern recognition principle by using oscillator neural network (ONN), where M is the dimensionality of the test vector

**T**, N is the number of oscillators, N

_{s}is the maximal number of the stored vectors

**E**.

**Figure 5.**Experimental and model I–V characteristics of VO

_{2}-switch (

**a**); a model scheme of a neural network based on three oscillators circuits with VO

_{2}-switches interacting via thermal coupling (

**b**).

**Figure 6.**Example of synchronization areas for a two-oscillator scheme (

**a**). The arrows show sampled vectors

**E**and

**E’**, in regard to the origin of coordinates. Distribution of η for a two-oscillator scheme (

**b**). Cross-section η at I

_{2}= 900 µA (

**c**).

**Figure 7.**(

**a**) Synchronization areas with SHR for a three-oscillator scheme “star” at I

_{p1}= 950 µA; (

**b**) Distribution of synchronization effectiveness η with the set vectors

**E’**at I

_{p1}= 950 µA; (

**c**) Distribution of synchronization effectiveness η with the set vectors

**E’**at I

_{p1}= 850 µA, on an enlarged scale with vector

**T**(

**d**); Levels of coupling Δ = 0.2 V and noise U

_{in}= 40 mV.

**Figure 8.**Dependence N

_{s}on coupling strength value between oscillators Δ at various configurations of oscillator neural network and constant noise level U

_{in}= 20 mV. The insertions show the evolution of synchronization areas and demonstrate the effect of their merging with Δ growth.

**Figure 9.**Dependence N

_{s}on noise level U

_{in}at various configurations of an oscillator neural network at the same coupling strength Δ = 0.2 V.

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## Share and Cite

**MDPI and ACS Style**

Velichko, A.; Belyaev, M.; Putrolaynen, V.; Boriskov, P. A New Method of the Pattern Storage and Recognition in Oscillatory Neural Networks Based on Resistive Switches. *Electronics* **2018**, *7*, 266.
https://doi.org/10.3390/electronics7100266

**AMA Style**

Velichko A, Belyaev M, Putrolaynen V, Boriskov P. A New Method of the Pattern Storage and Recognition in Oscillatory Neural Networks Based on Resistive Switches. *Electronics*. 2018; 7(10):266.
https://doi.org/10.3390/electronics7100266

**Chicago/Turabian Style**

Velichko, Andrei, Maksim Belyaev, Vadim Putrolaynen, and Petr Boriskov. 2018. "A New Method of the Pattern Storage and Recognition in Oscillatory Neural Networks Based on Resistive Switches" *Electronics* 7, no. 10: 266.
https://doi.org/10.3390/electronics7100266