# A Model of an Oscillatory Neural Network with Multilevel Neurons for Pattern Recognition and Computing

^{*}

## Abstract

**:**

## 1. Introduction

^{3}) of oscillators, where not only global synchronization but the synchronization by middle field [13], mode of quasi-synchronization [14] and even chimeras synchronization [15] are feasible.

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

_{2}) oscillators [19]. In relaxation oscillators with VO

_{2}-based film elements, electric self-oscillations are activated by the effect of electric switching governed by metal—insulator transition (MIT) [20,21,22,23]. The processing speed of VO

_{2}devices switching which amounts to ~10 ns [24] and manufacturing technology that allows switching elements to be created with high levels of nano-scalability make VO

_{2}-switch based oscillators the perfect objects for research on neuro-oscillators to solve cognitive technology problems [25,26,27,28]. Relaxation oscillators with high order synchronization effects can be realized by using electric coupling [25,26,27,28] and by using not only VO

_{2}-switches, but any other switching elements such as thyristors [29], tunneling diodes [30], resistive memory cells [31], and spin torque oscillator [16,32].

_{2}-oscillators and present a general concept of visual pattern recognition based on high order synchronization effects. We used the multiplicity of synchronous states to extract object classes by using a single output oscillator (compared to, for example, an array of oscillators at the output [10,11]). The concept of a multi-level neuron allows for using a smaller number of output neurons (oscillators) to implement the more complex cognitive functions of the neural network. We developed a set of special metrics [19,33], such as the high-order synchronization value and the synchronization effectiveness value. Compared to the neural network presented in [33], this network is able not only to memorize and classify patterns, but also to perform logical operations, computer calculations and emulate other functions of artificial intelligence systems.

## 2. Materials and Methods

#### 2.1. Oscillator Circuit and Method of ONN Organization

_{2}-switch. We described the process of fabrication and electric I–V characteristics of an electric VO

_{2}-switch in detail in [19]. I-V characteristics are well approximated by a piecewise linear function (see Appendix A.1) that has two conduction states (low-resistance and high-resistance) and a region of negative differential resistance (NDR). These switches may be used in the circuit of a relaxation generator with power supply I

_{p}holding the operation point in the NDR region of the I-V characteristic and with capacity C, connected to the switch in parallel. In addition, a source of noise U

_{n}models the circuit’s interior or exterior noises, such as current noise of a switch [34]. The oscillator’s output signal is current I

_{sw}(t) flowing through the VO

_{2}-switch, which is used to calculate the synchronization level of two oscillators (see Section 2.3). The current signal directly determines the effect of thermal coupling inside the network.

_{2}oscillators was convincingly demonstrated by us in the experiment [19], and it is based on the mutual thermal effect of switches due to their Joule heating and the dependence of the threshold voltage U

_{th}on temperature. In the pre-threshold mode, when the VO

_{2}temperature of the switch channel is close to the MIT temperature T

_{th}~ 340 K [23], the external thermal influence can “push” it to switch, which is equivalent to lowering the threshold voltage by the value of s, the thermal interaction force. The switching causes an even higher temperature rise in the switch, which leads to the appearance of a temperature pulse in the surrounding space, which propagates as a temperature wave. In the oscillator circuit, the thermal effect of the switches on each other occur in the mode of repetitive pulses initiated by self-oscillations of oscillator currents, which ultimately leads to their synchronization [19]. The value of s can be controlled experimentally, for example, by varying the distances between the switches or by varying the parameters of the external circuit [19].

_{2}switches is shown in Figure 1. A detailed presentation of the mathematical model of thermally coupled relaxation oscillators is given in Appendix A.1.

_{TC}in [19], therefore, in this model we limited ourselves to interaction with the nearest oscillators. Consideration of more complex cases can be done in future publications.

_{th}= 5 V, U

_{h}= 1.5 V, U

_{bv}= 0.82 V, R

_{off}= 9.1 kΩ, R

_{on}= 615 Ω). In this circuit, capacity is a constant parameter C = 100 nF, and its value significantly determines the frequency range of oscillator operation [37] and its natural frequency F

_{0}. In our case, the frequency range was 165 Hz ≤ F

_{0}≤ 1266 Hz at the range of feeding currents 550 µA ≤ I

_{p}≤ 1061 µA.

#### 2.2. ONN Structure

_{i,j}, that is set by the following matrix:

_{i,j}= s

_{j,i}= s

_{m}, where i, j are the numbers of neighboring oscillators). Neighboring oscillators are only connected by horizontal and vertical lines. So, the central oscillator No.5 has four couplings in the matrix, the corner oscillators have two couplings, and the oscillators in the center of the edges have three couplings (with the central oscillator and two corner ones) and they all unidirectionally affect oscillator No.10 (output neuron) in the output layer (s

_{i,10}= s

_{o}and s

_{10,i}= 0, where i = 1 … 9). Importantly, there is the reference oscillator No.0 in the circuit and the synchronization order of all other oscillators is measured against this oscillator. Oscillator No.0 (Figure 2) unidirectionally affects all other oscillators (s

_{0,i}= s

_{r}, and s

_{i,0}= 0, where i = 1 … 10).

_{i}are the coordinates of the input vector X = (x

_{1}, …, x

_{9}), that correspond to white (x

_{i}= 0) and blue (x

_{i}= 1) colors of pattern squares.

_{0,10}between the reference oscillator No.0 and the oscillator in the output layer No.10 serves as the control value. The values of the feeding currents for these oscillators are fixed and may differ from currents in the matrix, therefore they are indicated as I

_{0}and I

_{10}(see Figure 3).

#### 2.3. Method of Synchronization Order Definition

_{i}and k

_{j}are harmonics order of oscillators at the common frequency of their synchronization F

_{s}.

_{i}:k

_{j}, when the following relation is observed:

_{i}

^{0}, F

_{j}

^{0}are frequencies of main harmonics.

_{i,j}= 2:7

_{s}and subharmonics numbers at this frequency k

_{i}= 2 and k

_{j}= 7. Usage of signal spectra for estimating the magnitude of SHR

_{i,j}is not effective because the signals might not have a strictly periodic sequence and when noise is added to the system, the spectral lines broaden. Below, we will describe the mathematical procedure that estimates the value of SHR

_{i,j}without the use of spectral characteristics but using a current signal and array LE. Array LE stores information on the position of the current pulse leading edges (see Figure 5).

^{z}

_{s}—the period of synchronization (where z is a conditional number of periods T

_{s}).

_{i,j}may be estimated using a phase-locking method:

_{i}and M

_{j}are the number of signal periods falling into the synchronization periods T

^{z}

_{s}of two oscillators. Formula (5) can be easily obtained from Formula (3) taking into account Formula (4) and the following ratio (T

_{s}= M

_{i}·T

_{i}

^{0}= M

_{j}·T

_{j}

^{0}, F

_{i}

^{0}= 1/T

_{i}

^{0}, F

_{j}

^{0}= 1/T

_{j}

^{0}).

^{z}

_{s}≠ T

^{z}

^{+1}

_{s}and the values of M

_{i}and M

_{j}may change within one oscillogram (see Figure 7).

_{i,j}may occur within one oscillogram. To determine which SHR

_{i,j}value prevails, it is necessary to find the occurrence probabilities P(M

_{j}:M

_{i}) for each pair (M

_{i}:M

_{j}) that is present in the whole oscillogram and to select the pair with the maximum value of P = P

_{max}(M

_{j}:M

_{i}). Then, the final value of SHR

_{i,j}will be written as:

_{j}:M

_{i}), we can count how many times NP(M

_{j}:M

_{i}) the given pair appeared within the whole oscillogram of the oscillator i, multiply by the number of periods in it (M

_{i}) and divide by the total number of all oscillations periods in the given signal (N

_{j}). Thus, for P(M

_{j}:M

_{i}) we obtain:

_{i}is the total number of periods in the oscillogram of oscillator i.

_{j}:M

_{i}) as a histogram where the values are positioned in the descending order of the magnitude P. For example, for the oscillogram section in Figure 7 the following histogram can be built. The histogram in Figure 8 is calculated by Formula (7), when the pairs occur the following number of times, NP(2:7) = 2, NP(2:9) = 1, NP(2:5) = 1, and the total number of periods is N

_{i}= 28 (in real calculations, N

_{i}was in the range of 1000–3000 for greater accuracy, see Appendix A.2).

_{i,j}, the parameter of synchronization effectiveness η is defined as the maximum probability P

_{max}(M

_{j}:M

_{i}):

_{i,j}= 2:7 with effectiveness of η = 50%.

_{i,j}, η) allows sufficient determination of the synchronization state of two oscillators and calculation of the distance between the states, i.e., the difference between the synchronization degree. This property allows the use of metrics in an oscillator neural network training, pattern recognition systems and artificial intelligence [8,9,10,11].

_{i,j}can be solved by formally setting the synchronization effectiveness threshold η

_{th}, so

_{th}= 90%, meaning the signals are synchronized if 90% of their durability have a certain synchronization pattern. For the network training, this parameter can be chosen within a selected range, and it is one of the important parameters of the network adjustment [33].

_{0,10}:

_{0,10}can be expressed in several ways: as a ratio, a simple fraction or a real number, for example, SHR

_{0,10}= 10:3 = 10/3 = 3.33. Later, we will use this property to present the results more vividly.

_{0,10}has the properties of an output neuron while the reference neuron may be considered as a master generator to which we calculate synchronization of other network neurons.

_{sw}(t) of oscillators No.0–10 were calculated simultaneously and contained ~250,000 points with the time interval Δt = 10µs (see Appendix A.1). Then, the oscillograms were automatically processed.

_{p_i}(I

_{ON}, I

_{OFF}, I

_{0,}I

_{10}), coupling strength constants s (s

_{r}, s

_{m}, s

_{o}), noise amplitude U

_{n}and η

_{th}varied.

#### 2.4. Pattern Classifier Implementation and Problem Definition

_{n}= (x

_{1}, …, x

_{9}) where each cell may take the value x

_{i}= 0 (white color) or x

_{i}= 1 (blue color), and n is the number of the vector equal to the decimal value of the coordinates presented as a binary sequence. For example, Figure 2 and Figure 3 show an input pattern that corresponds to the vector X

_{489}= (1,1,1,1,0,1,0,0,1). The total number of patterns (vectors) n in the input layer matrix is 2

^{9}= 512 (X

_{0}… X

_{511}). Presuming that the pattern processing layer together with the output layer has certain symmetry, a set of 512 vectors X

_{n}may be divided into 102 classes C

_{j}, where j is the number of classes (j =1 … 102) (see Figure 9). The complete list of classes and their elements is described in Supplementary Materials (Data1.txt).

_{j}consists of a lot of figures (vectors) that have the same number of blue (white) cells and rotation-reflection axial symmetry of the 4th order (symmetry at rotation by 90°).

_{4}in Figure 10).

_{0,10}. Distribution into classes allows us to find figures that at any current values (I

_{ON}, I

_{OFF}, I

_{0}, I

_{10}) and coupling strength constants (s

_{r}, s

_{o}, s

_{m}), noise amplitude U

_{n}and η

_{th}, will have the same values of synchronization effectiveness η and SHR

_{0,10}within one class of figures as a result of neural network symmetry. The initial distribution of all 512 figures into classes allows us to recognize not one specific figure, but a class (out of 102 possible ones) into which this figure is classified. For example, for class C

_{5}the equality SHR

_{0,10}(X

_{5}) = SHR

_{0,10}(X

_{260}) = SHR

_{0,10}(X

_{320}) = SHR

_{0,10}(X

_{65}) will be observed.

- Synchronization of oscillators No.0 and No.10 with the corresponding value of SHR
_{0,10}and η > η_{th}, exists only for one specific class C_{j}with number j = m out of 102 classes:$${\mathrm{SHR}}_{0,10}{(\mathrm{C}}_{\mathrm{j}})=\{\begin{array}{l}{k}_{0}:{k}_{10}\text{}\mathrm{and}\text{}\eta \ge {\eta}_{\mathrm{th}}\text{}\mathrm{if}\text{}j=m\\ \mathrm{no}\text{}\mathrm{syncronysation}\text{}\mathrm{and}\text{}\eta {\eta}_{\mathrm{th}}\text{}\mathrm{if}\text{}j\ne m\text{}\end{array}$$Here we have to show the solutions of this problem with various values of m. - There is a set of classes
**C**= {C_{Z1}, C_{Z2}… C_{ZP}}, where Z_{1}, Z_{2}… Z_{P}are arbitrary non-repeating indices, where the number is P < 102. When inputting this set into the oscillator system, it comes to the synchronization states corresponding to the set**SHR**= {SHR^{(1)}_{0,10}, SHR^{(2)}_{0,10}… SHR^{(P)}_{0,10}}. The set**SHR**does not have the same elements, i.e., each class of figures from set**C**corresponds to a unique synchronization order SHR_{0,10}. By analogy with (4) the problem may be expressed as:$${\mathrm{SHR}}_{0,10}{(\mathrm{C}}_{\mathrm{j}})=\{\begin{array}{l}{\mathrm{SHR}}_{0,10}^{(1)}\text{}\mathrm{and}\text{}\eta \ge {\eta}_{\mathrm{th}}{\text{}\mathrm{if}\text{}\mathrm{C}}_{\mathrm{j}}\in \mathbf{C}\text{}\mathrm{and}\text{}j=\text{}{Z}_{1},\\ {\mathrm{SHR}}_{0,10}^{(2)}\text{}\mathrm{and}\text{}\eta \ge {\eta}_{\mathrm{th}}{\text{}\mathrm{if}\text{}\mathrm{C}}_{\mathrm{j}}\in \mathbf{C}\text{}\mathrm{and}\text{}j=\text{}{Z}_{2},\\ .............................................................................\\ {\mathrm{SHR}}_{0,10}^{\left(\mathrm{P}\right)}\text{}\mathrm{and}\text{}\eta \ge {\eta}_{\mathrm{th}}{\text{}\mathrm{if}\text{}\mathrm{C}}_{\mathrm{j}}\in \mathbf{C}\text{}\mathrm{and}\text{}j=\text{}{Z}_{\mathrm{P}},\\ \mathrm{no}\text{}\mathrm{syncronysation}\text{}\mathrm{and}\text{}\eta {\eta}_{\mathrm{th}}{\text{}\mathrm{if}\text{}\mathrm{C}}_{\mathrm{j}}\notin \mathbf{C}\end{array}$$

**C**= {C

_{1}, C

_{4}, C

_{5}}, in this case the corresponding set is

**SHR**= {16:15, 13:10, 14:13}. Therefore, unambiguous recognition of figures belonging to three different classes takes place. Synchronization is not realized for all other classes and η < η

_{th}.

**C**variants, this problem can be reduced to the search of possible solutions with P > 1 and to the determination of the maximum value of P.

- III.
- The third variant of the problem corresponds to a fully trained network when it solves problem II for all possible input classes C
_{j}, when P = 102.

_{0,10}. Therefore, the set of input classes

**C**= {C

_{1}, C

_{2}… C

_{102}} transfers into a set of synchronous states

**SHR**= {SHR

^{(1)}

_{0,10}, SHR

^{(2)}

_{0,10}… SHR

^{(102)}

_{0,10}}, where all the elements have non-duplicate values.

_{0,10}allows input pattern classification into P classes within set

**C**. This is the most striking difference between the described neural network and variants presented in the literature. This increases the net data throughput of a single neuron and enables us to create multilevel output cascades of neural networks with high functionality.

#### 2.5. Technique for ONN Training

_{ON}, I

_{OFF}, I

_{0,}I

_{10}), coupling strength (s

_{r}, s

_{o}, s

_{m}), noise amplitude U

_{n}and the synchronization effectiveness threshold η

_{th}. The algorithm’s main point is the random searching of the problem II solution with the maximum value of P at some initial interval of parameters, followed by the narrowing of these intervals.

_{p}≤ 1061 µA. Therefore, the currents (I

_{ON}, I

_{OFF}, I

_{0,}I

_{10}) varied in this range. We determined the variation steps as δI

_{p_i}= 1 µA. So, there were 512 current steps.

_{Σ}should not reduce the threshold voltage of the I–V characteristic of a switch U

_{th}below the holding voltage U

_{h}, must be met:

_{th}= 5 V, U

_{h}= 1.5 V) and coupling configurations (see Figure 2), the limits of the coupling strength variations are subject to the following conditions: (s

_{r}+ s

_{o}·9) < 3.5 V and (s

_{r}+ s

_{m}·4) < 3.5 V. This condition is met with the following ranges that we have chosen: s

_{r}= 0 ÷ 0.2 V, s

_{m}= 0 ÷ 0.5 V, s

_{o}= 0 ÷ 0.3 V. The variation step was chosen as 0.1% of the range magnitude.

_{n}≤ 900 µV with the number of steps equal to 12 was chosen for the noise amplitude.

_{th}, the variation range was 10 % ≤ η

_{th}< 100%, with the number of steps equal to 25 and minimal spacing δη

_{th}= 0.1%. Parameter η

_{th}does not belong to the network parameters but rather to the parameters of the algorithm of synchronization order SHR

_{0,10}calculation. The value η

_{th}strongly affects the results of synchronization and the results of pattern recognition because it is a conditionally chosen characteristic. Identifying its optimal value for the recognition problems is an important step in network tuning and training.

_{0,10}at each stage. A full, direct search of all parameters’ variants would take a lot of time and computational resources. Therefore, we fixed the values U

_{n}and η

_{th}, and varied currents (I

_{ON}, I

_{OFF}, I

_{0}, I

_{10}) and couplings strength (s

_{r}, s

_{o}, s

_{m}). The algorithm for searching for the solution of problem II with the maximum value P included the following steps:

_{n}and η

_{th}

- Step 1:
- Random searching of parameters (I
_{ON}, I_{OFF}, I_{0}, I_{10}, s_{r}, s_{o}, s_{m}) in the maximal range of their variations and finding the values meeting the maximum value P. The number of searching attempts is 1000. - Step 2:
- Narrowing of the parameters ranges by 5 times with their symmetric distribution in relation to the results of the previous step. The number of searching attempts is 1000.
- Step 3:
- Narrowing of the parameters ranges by 25 times with their symmetric distribution in relation to the results of the previous step. The number of searching attempts is 1000.

_{n}is kept fixed because this is the parameter that is not often controlled in the experiment. It is determined by external and internal circuit noises, and we considered its effect individually in more detail. We used U

_{n}= 80 µV, which is close to the experimentally observed value [19].

_{th}is fixed because it is the main parameter in the synchronization algorithm, and we considered its effect individually in more detail. For the pattern recognition experiments, we used a workstation (Intel Xeon quad core processor, Albuquerque, NM, USA, 4 × 2 GHz, 8GB RAM) running 64-bit Windows Server 2008. CPU time for a single run on one core for the direct search procedure with 1000 search attempts took ~5 h.

## 3. Results

_{n}= 80 µV, η

_{th}= 90%) and varied the currents (I

_{ON}, I

_{OFF}, I

_{0}, I

_{10}) and coupling strength (s

_{r}, s

_{o}, s

_{m}). The distribution of the solutions after the first, second and third steps of training is shown in the diagram (Figure 12), the values of P are on the x-axis, and the corresponding number of solutions N

_{P}are on the y-axis. The largest number of solutions corresponds to P = 0 when there is no synchronization at any input class C

_{j}. After steps 2–3. the number of solutions with a low value of P decreases and the solutions with a higher value of P appear. Step 3 gives more solutions in the range P = 6–10 in comparison with step 2, although the maximum value P = 14 is similar in both cases.

_{0,10}, and the probability of such an event at the first step of training is ~55% (the calculation was based on the data in Figure 12), which results in the ambiguous recognition of figures and their classes. Another frequent case of wrong training is the absence of oscillatory synchronization for any input class (P = 0), which has a probability of ~30% in the first step (see the values in Figure 12).

#### 3.1. Solution of Problem I

_{m}. For example, Figure 13b shows a neural network that recognizes class C

_{1}with the corresponding synchronization order SHR

_{0,10}= 20:29, and there is no synchronization for all other classes. The probability of any solution with P = 1 during the first step is ~10% (here the total number of attempts is 1000 and the value of N

_{p}~ 100, see Figure 12), and this is the maximum probability of the solution in comparison with other P > 1. Figure 14 shows the distribution of solutions at P = 1 for various values of m (4) after 60 repetitions of the first step of training at the maximum range of all parameter variations. The maximum probability (~4%) can be seen for solutions with sets C

_{1}and C

_{102}when all cells of the input pattern are either empty or colored, see Figure 9. Solutions for other m are much rarer, with the probability being two orders lower (~0.03%). Parameters corresponding to the same solution can differ significantly. For example, the system can recognize class C

_{102}at the input, while at the output SHR

_{0,10}would be different for different parameters, however, the problem is still considered solved. The histogram shows that the network can be trained to solve problem I with a certain, predetermined value of m.

#### 3.2. Solution of Problem II

**C**should correspond to the unique synchronization order from set

**SHR**while there is no output oscillator synchronization for other classes. For example, at the first step of training, N

_{p}= 26 solutions were found for P = 2, two of them are shown in Figure 13c,d, and Figure 13e demonstrates the variant of set

**C**and

**SHR**for P = 4.

^{2}

_{102}= 5151 solution combinations (where C

^{2}

_{102}is the number of combinations 2 out of 102). For P = 14, C

^{14}

_{102}is a 17-digit number. Therefore, we indicate the maximum value of P, and do not indicate which solution was found.

#### 3.3. Solution of Problem III

#### 3.4. Study of the Noise Effect on the Training Results

_{p}on the noise amplitude value in network U

_{n}(η

_{th}= 90%). The values were taken from the first step of training.

_{n}= 400 µV, the number of solutions N

_{P}and the value P sharply decreases. Most of the solution variants are distributed in the range 1 ≤ P ≤ 2. The maximum value for N

_{p}corresponds to P = 1 at noise level U

_{n}= 400 µV. In this case, the integral value ∑N

_{P}for all values P also has a maximum for U

_{n}= 400 µV. Therefore, “stochastic resonance” is present when a certain level of noise induces maximum number of solutions for problems I–II. This may be caused by two differently directed tendencies of N

_{p}reduction: the occurrence of “extra” synchronizations with decreasing U

_{n}and suppression of the number of synchronizations with increasing U

_{n}.

#### 3.5. Examination of the Synchronization Threshold on the Training Result

_{p}on the value of threshold synchronization effectiveness (U

_{n}= 80 µV).

_{p}with a decrease in η

_{th}below 40% can be seen, and the growth of η

_{th}up to η

_{th}= 99% on average does not change the values of P and N

_{p}. This seems to be caused by the fact that synchronization occurring during problems I and II solution has a high value of effectiveness η > 99%. In this case, reduction of η

_{th}just adds “extra” synchronous states, which do not meet the problem conditions.

_{th}= 30% is interesting, and we observed some resonance for values of P. Maximum P declines as with reduction of η

_{th}as with growth of η

_{th}, caused by reduction of in the solution number N

_{p}.

#### 3.6. Study of the Dynamics of the Neural Network

_{o}= 0 V. The result was a circuit equivalent to two coupled oscillators, when oscillator No.0 affects oscillator No.10 with a force of s

_{r}= 0.3 V. By varying the oscillator currents (I

_{0}, I

_{10}), we obtained the standard distribution pattern of SHR

_{0,10}in the form of Arnold tongue, where the regions of equal synchronization are extended sections in the form of rays (see Figure 17a) with diagonal symmetry. A special feature is a smooth (gradient) increase in SHR

_{0,10}from the lower right distribution corner (I

_{0}= 1061 µA, I

_{10}= 500 µA) to the upper left corner (I

_{0}= 500 µA, I

_{10}= 1061 µA). Similar patterns of synchronization distribution in the system of two oscillators were observed by us earlier, for example, in [33]. After adding the effect of the processing layer, when s

_{o}= 0.1 V, we get the type of synchronization distribution shown in Figure 17b. The synchronization areas are island type. The layer of input oscillators divides Arnold’s tongue into local areas while the tendency of the gradient increase in synchronization is preserved.

_{0}and I

_{10}, when s

_{o}= 0.29 V, and vary the currents I

_{ON}and I

_{OFF}, the pattern of the synchronization distribution changes its appearance (see Figure 18a). We observed a chaotic scatter of the synchronization magnitude over the field of parameters, with preservation of the diagonal symmetry and a wide scatter of values within 1–200 µA. As I

_{ON}and I

_{OFF}values change the frequency of many oscillators in the processing layer at once, it is difficult to predict the tendency of the synchronization distribution. Nevertheless, the range of SHR

_{0,10}variations is much smaller than when I

_{0}and I

_{10}are varied, which is obviously due to the constancy of the main frequency between the oscillators whose synchronization is measured. With a decrease in the value of s

_{o}to s

_{o}= 0.1 V, the form of the distribution has a similar appearance, but the range of SHR

_{0,10}change is reduced.

_{o}= 0 V, s

_{r}= 0.13 V, I

_{0}= 650 µA, I

_{10}= 950 µA, there is only one unidirectional effect of oscillator No.0 on No.10, and the base synchronization has the value SHR

^{b}

_{0,10}= 23:12 = 1.917. In the second case, we added random variations of coupling strength (s

_{o}, s

_{m}) and currents (I

_{ON}, I

_{OFF}) at the same values of s

_{r}= 0.13, I

_{0}= 650 µA, I

_{10}= 950 µA. As a result, after the first step of training we obtained a set of solutions (

**C**and

**SHR**) for various P > 0 and positioned all elements of SHR

_{0,10}on the plot (see Figure 19). It can be seen that all elements of the sets

**SHR**are distributed within some interval above the boundary SHR

^{b}

_{0,10}. The dispersion is δSHR

_{0,10}~0.6.

_{0}= 950 µA and I

_{10}= 650 µA). The base synchronization had the value of SHR

^{b}

_{0,10}= 8:15 = 0.533, and when other parameters were varied, the elements of

**SHR**diverge upward with δSHR

_{0,10}~ 0.4. At high and similar values of current I

_{0}= 1058 µA and I

_{10}= 1058 µA, the base synchronization is SHR

^{b}

_{0,10}= 1:1 = 1, and δSHR

_{0,10}~ 0.3.

_{0,10}value upward from the base synchronization value of oscillators No.0 and No.10. The effect of oscillators positioned in the processing layer only increases the frequency of oscillator No.10, but cannot decrease it because of the nature of thermal coupling, which can initiate switching but cannot suppress it.

## 4. Discussion

_{n}could be selected more carefully on the basis of the regularities shown in Figure 15, where the maximum of the solution number N

_{P}at a certain noise level is presented. This effect is similar to stochastic resonance phenomena in spike networks [40,41,42], and requires additional research and analysis.

^{b}

_{0,10}of oscillators No.0 and No.10 and showed that the effect of oscillators’ processing layer leads to SHR

_{0,10}≥ SHR

^{b}

_{0}. The selection of currents for the reference oscillators No.0 and No.10 determines the variation range of δSHR

_{0,10}, and most likely, the larger this range is, the higher the probability of finding the solutions with high P. If we set the current values I

_{0}, I

_{10}, for example, at the boundary of the maximum range I

_{0}= 1058 µA, I

_{10}= 1058 µA, then it will be more difficult for the matrix of oscillators in the processing layer to change the dynamics of the output oscillator that operates in a high frequency mode and under the high frequency effect of the reference oscillator. As a result, we observed narrowing of δSHR

_{0,10}to ~0.3 (see Figure 19). Figure 18 demonstrates the areas of parameters where SHR

_{0,10}practically does not change and where it changes very often, therefore, this may explain the productivity of the annealing training algorithm. In Step 1, we found areas with a large number of solutions, and then in Step 2 and 3, by thorough scanning of these areas, we found solutions with the largest P (see Figure 12).

_{th}, which is used in the synchronization magnitude determination method, is of great importance. As the results and regularities arising in Figure 16 show, too high (η

_{th}≥ 99.8) and too low (η

_{th}≤ 20) parameters significantly reduce the number of solutions N

_{P}. Nevertheless, we suppose that the usage of η

_{th}< 50% is not completely justified, as the parameter η determines the share of a synchronous signal in the total oscillogram [19]. Moreover, the technique of SHR determination may affect the result, and future research aimed at its improvement may lead to positive shifts in this field. The addition of alternative methods of pattern transfer into a network that vary, for example, current I

_{10}or couplings magnitudes may significantly expand the range of variations of SHR

_{0,10}, hence, the maximal attainable value P.

_{2}–based device, therefore, oscillator No.0 affected all other oscillators. This can be easily implemented by arranging VO

_{2}-switches on one substrate. Nevertheless, the development of an actual device requires specific setting of coupling strengths that depend on many parameters [19] and this is beyond the scope of the current work.

## 5. Conclusions

_{0,10}allows its usage to classify the input pattern into P classes with the set

**C**. The training implementation of this network for solving cognitive tasks is an interesting possibility, as the network consists only of oscillators and does not use other computational modules.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Model Circuit of a Coupled Oscillators-Based Neural Network

_{i,j}, which is set by matrix S (1).

_{p_i}, capacitance C connected in parallel with a VO

_{2}switch and noise source U

_{n_i}. The capacity magnitude C was constant C = 1 00 nF, but I

_{p_i}and U

_{n_i}varied within the following ranges: I

_{p}(550 µA ÷ 1061 µA), U

_{n}(20 µV ÷ 900 µV). The current through the VO

_{2}switch and its voltage are defined as I

_{sw_i}and U

_{i}, respectively. Voltage-current characteristics of the VO

_{2}switch are shown in Figure A2, where the experimental and model curves are presented.

**Figure A1.**Circuit of a single oscillator of a VO

_{2}-based structure. I is the number of an oscillator; I

_{p_i}is a source of current, C is a capacitance, U

_{n_i}is a source of noise, I

_{sw_i}is the current through the VO

_{2}switch, U

_{i}is the voltage at the switch.

**Figure A2.**Typical experimental I–V characteristics of a separate switch (Experiment curve) and its model curve (Model curve).

_{th}= 5 V (threshold voltage), U

_{h}= 1.5 V (holder voltage), U

_{cf}= 0.82 V (cutoff voltage).

_{sw_i}= f(U

_{i}), which has high-resistance (OFF) and low-resistance (ON) segments with corresponding dynamic resistances R

_{off_i}= 9100 Ω and R

_{on_i}= 615 Ω, can be presented by the following formula:

_{i}denotes the VO

_{2}switch state 1 (OFF—high resistance), 0 (ON—low resistance).

_{th_i}and U

_{h_i}are threshold turn-on and holder voltages of switches (see Figure A3):

_{th}by the value s due to thermal effect of other switches. The value s is the thermal coupling strength. The physics of thermal interaction is caused by the generation of a heat wave when the switch is turned on, which propagates and acts (heats) on the surrounding switching structures. In [19], we showed that it is possible to introduce an interaction radius R

_{TC}beyond which the induced temperature is less than 0.2 K. Therefore, in the model, we assume that each switch only interacts with surrounding switches that are within the radius of R

_{TC}. The heating of structures leads to a decrease in the threshold voltage [19], this change characterizes the value of s, which we call the coupling force. The higher the s, the stronger the effect of one switch on the other.

_{th_i}in regard to the switch state (flag

_{i}) with which they interact.

_{th i}can be presented by the following formula:

_{i}= 0), and U

_{th}is the natural turn-on voltage without coupling.

_{j,i}on i-switch are summed up from all the other j-th switches at the turn-on state.

_{c_i}is the capacitor voltage, I

_{sw_i}is the current through the switch, U

_{n_i}is the noise in the switch, and f(U) is the I–V characteristic function (A1).

^{−5}s using implicit Euler method and discrete noise U

_{n}(t) was generated according to the algorithm U

_{n}(t) = U

_{n0}·randn(t), where U

_{n0}—noise amplitude and randn(t)—normal random numbers generated with zero mean and dispersion equal to 1, realized through the algorithm of uniformly distributed random value transformation [43].

_{p_i}, noise amplitude U

_{n0}, and coupling strengths (s

_{r}, s

_{m}, s

_{o}) we could calculate oscillograms of current I

_{sw_i}(t), voltage at the capacity U

_{c_i}(t) and voltage at switches:

**Figure A3.**Examples of calculated oscillograms sections (1000 points), voltage U

_{0}and current I

_{sw_0}, for the oscillator with i = 0 at I

_{p_0}= 1200 µA. Here we do not show other circuit parameters because other oscillators do not affect this one.

**Figure A4.**Spectrum of a current signal for the oscillogram shown in Figure A3.

_{0}and current I

_{sw_0}are shown in Figure A3. The spectrum of the current signal has a large number of harmonics, see Figure A4. The main frequency F

^{0}= 1269 Hz determines the period of current oscillations T

^{0}~ 788 µs (because T

^{0}= 1/F

^{0}).

#### Appendix A.2. Dependence of SHR_{0,10} and η on the Number of Pulses in the Oscillogram

_{0,10}and η according to the method in Section 2.3, with the desired accuracy, is the determination of the number of pulses used for the calculation. In fact, it is necessary to set the minimum duration of the processed oscillogram. This is also important for determining the minimum calculation time for a family of metrics. For example, in a real experiment to calculate SHR

_{0,10}and η, it is required to record the oscillogram, and then make the calculation. In a model experiment, this time is determined by the oscillogram simulation time.

_{0}= 1017 µA, I

_{10}= 891 µA, I

_{ON}= 725 µA, I

_{OFF}= 1035 µA s

_{r}= 0.1036 V, s

_{m}= 0.207 V, s

_{o}= 0.29298 V, η

_{th}=90%, U

_{n}= 80 µV, the input image is X

_{3}). With an increase in the number of points, the number of pulses in the oscillograms of oscillators No.0 and No.10 grows (see Figure A5b.)

**Figure A5.**The dependence of η (

**a**) and the number of pulses (

**b**) on the number of points on the oscillogram. The dashed line corresponds to 250,000 oscillogram points.

_{0.10}= 38:37 with the number of points more than 60,000 (when η ≥ η

_{th}at η

_{th}= 90%); with a smaller number of points, the synchronization is not detected (η < η

_{th}).

^{−5}s).

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**Figure 1.**The oscillator circuit and example of oscillators’ interaction via thermal coupling of VO

_{2}switches. I

_{sw}(t)—current signal in a VO

_{2}switch, that leads to its Joule heating resulting in thermal impulses that spread along the substrate; U

_{n}—source of noise, s—thermal coupling strength.

**Figure 2.**ONN organization circuit for pattern recognition as a matrix of 3 × 3 elements. Digits indicate the sequence numbers of oscillators.

**Figure 3.**Principle of a pattern transfer from the input layer to the oscillator matrix via setting of their currents (I

_{OFF}is for white squares, I

_{ON}is for blue squares). Separate colors are used for oscillators No.0 and No.10 and for their currents I

_{0}and I

_{10}, respectively.

**Figure 4.**Qualitative spectra of two coupled synchronized oscillators with parameter SHR

_{i,j}= 2:7.

**Figure 5.**Oscillogram of oscillator current No. 10 and the corresponding array of positions of the leading edges of the current pulse LE [10] [Δt]. Δt—calculation time interval (Appendix A.1).

**Figure 7.**Arrays LE[i] and LE[j] for two oscillators with a non-constant period of synchronization T

^{z}

_{s}.

**Figure 8.**Histogram of probabilities distribution P(M

_{j}, M

_{i}) calculated by using Formula (11) for signals LE, shown in Figure 7.

**Figure 12.**Diagram of the solution number N

_{P}distribution against the value of P for three subsequent steps of the network training algorithm. The total number of attempts at each step is 1000. The maximum value P = 14 was obtained at the following parameters: (I

_{ON}= 722 µA, I

_{OFF}= 1034 µA, I

_{0}= 1020 µA, I

_{10}= 887 µA, s

_{r}=0.10176 V, s

_{o}= 0.29202 V, s

_{m}= 0.202 V, U

_{n}= 80 µV, η

_{th}= 90%).

**Figure 13.**Training results at various values of current I

_{ON}: examples of incorrect solutions of training (

**a**) and solutions of problem I (

**b**); examples of correct solutions of problem II (

**c**–

**e**). The complete list of the model parameters is presented in Supplementary Materials (Data2.txt).

**Figure 14.**Distribution of the solutions number for problem I (P = 1) according to the values m (4), calculated for 60 repetitions of the first step of training (U

_{n}= 80 µV, η

_{th}=90%). Insets show the classes of input images C

_{m}for m = 1, m = 5, m = 102.

**Figure 15.**Dependence of solution number N

_{P}for various values of P on noise level U

_{n}(η

_{th}= 90%).

**Figure 16.**Dependence of solution number for various values of P on the threshold synchronization effectiveness η

_{th}(U

_{n}= 80 µV).

**Figure 17.**The synchronization distribution SHR

_{0,10}in the regions of currents I

_{0}and I

_{10}with (

**a**) s

_{o}= 0 V and (

**b**) s

_{o}= 0.1 V. For all cases, I

_{ON}= 725 µA, I

_{OFF}= 1036 µA, s

_{r}= 0.3 V, s

_{m}= 0.207 V, η

_{th}=90%, U

_{n}= 80 µV, and the class C

_{94}is the input.

**Figure 18.**The synchronization distribution SHR

_{0,10}in the regions of currents I

_{ON}and I

_{OFF}with (

**a**) s

_{o}= 0.29V and (

**b**) s

_{o}= 0.1 V. For all cases, I

_{0}= 1017 µA, I

_{10}= 891 µA, s

_{r}= 0.1036 V, s

_{m}= 0.207 V, η

_{th}= 90%, U

_{n}= 80 µV, and the class C

_{94}is on the input.

**Figure 19.**Distribution of synchronization values SHR

_{0,10}at random variations of coupling strengths (s

_{o,}s

_{m}) and currents (I

_{ON}, I

_{OFF}). Fixed parameters are shown in the plot. The base sync value (SHR

^{b}

_{0,10}) is shown by a solid line. SHR

^{b}

_{0,10}is the sync value at s

_{o}= 0.

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Velichko, A.; Belyaev, M.; Boriskov, P.
A Model of an Oscillatory Neural Network with Multilevel Neurons for Pattern Recognition and Computing. *Electronics* **2019**, *8*, 75.
https://doi.org/10.3390/electronics8010075

**AMA Style**

Velichko A, Belyaev M, Boriskov P.
A Model of an Oscillatory Neural Network with Multilevel Neurons for Pattern Recognition and Computing. *Electronics*. 2019; 8(1):75.
https://doi.org/10.3390/electronics8010075

**Chicago/Turabian Style**

Velichko, Andrei, Maksim Belyaev, and Petr Boriskov.
2019. "A Model of an Oscillatory Neural Network with Multilevel Neurons for Pattern Recognition and Computing" *Electronics* 8, no. 1: 75.
https://doi.org/10.3390/electronics8010075