We describe the classification method of chimeric synchronization at the beginning of this section, followed by a description of the single-oscillator circuit, ONN structure, formulation of a search problem and network training technique, and technical aspects of thermal coupling organization.

#### 2.1. Chimeric Synchronization Classification

The method of determining the family of metrics of high-order synchronous states (SHR and µ) is described in detail in a previous study [

26]. The method of classification of chimeric synchronization is based on its basis and is not significantly different.

At first, the analog oscillogram of oscillations is represented in the form of the corresponding array, called LE, which stores information on the position of the current pulse leading edges (see

Figure 2).

Then, for two arbitrary oscillators

i and

j, between which it is necessary to determine synchronization, the arrays LE[

i][

t] and LE[

j][

t] are compared (see.

Figure 3). The distance between the two nearest phase-locked pulses is denoted as

T^{z}_{s}, the period of synchronization (where

z is a conditional number of periods

T_{s}). Pulses are considered phase-locked if the distance between them does not exceed 4Δ

t. The value of sub-harmonic ratio between the oscillator

i and the oscillator

j, called SHR

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

where

M_{i} and

M_{j} are numbers of signal periods falling into the synchronization periods

T^{z}_{s} of two oscillators. Equation (1) defines the pattern of high order synchronization of two signals. As the notation “SHR” defines, the formula can determine the ratio of subharmonic numbers in the signal spectra at a common synchronization frequency [

26].

In general, especially when a system behaves erratically, synchronization periods differ and spread in T

^{z}_{s} ≠ T

^{z}^{+1}_{s} and the values of

M_{i} and

M_{j} may change within one oscillogram (see

Figure 4).

Various values of synchronizations SHR

_{i,j} may occur within one oscillogram. To determine the distribution of the SHR

_{i,j}, it is necessary to find the occurrence probabilities

P(

M_{j} :

M_{i}) for each pair (

M_{i} :

M_{j}) that are present in the whole oscillogram. To find the probabilities

P(

M_{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 this by the number of periods in it (

M_{i}), and divide this by the total number of all oscillation periods in the given signal (

N_{j}). Thus, for

P(

M_{j} :

M_{i}) we obtain:

where

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

i.

Therefore, each synchronization value SHR_{i,j} will correspond to the probability of its detection P (M_{j} : M_{i}), expressed as a percentage.

It is convenient to present the probabilities

P(

M_{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 4, the following histogram can be performed:

The histogram in

Figure 5 is calculated by Equation (2), when the pairs occur a 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 [

26]).

In a model experiment, the shapes of the distribution oscillograms P(

M_{j} :

M_{i}) can differ significantly from each other. For example,

Figure 6 presents the main histogram variants occurring during signal processing (oscillator

i corresponds to the reference oscillator with a constant frequency).

The histogram in

Figure 6a corresponds to the case of an absolutely synchronized signal with high order synchronization SHR

_{i,j} = 11:8. The spectrum of oscillator

j has a line character, and the phase diagram corresponds to a single high order synchronization limit cycle. The cases in

Figure 6b,c have a set of different SHR

_{i,j} and correspond to chimeric synchronization states.

A chimeric index, called CH, can be introduced, consisting of the first three values of the synchronization value SHR

_{i,j}:

Accordingly, in

Figure 6, each histogram displays its values: (a) CH

_{i,j} = (11:8), (b) CH

_{i,j} = (3:2 7:5), and (c) CH

_{i,j} = (11:7 24:17 8:5). For

Figure 6a, the values of CH

_{i,j} and SHR

_{i,j} are the same.

For CH, the parameter of synchronization effectiveness

η is defined as the sum of P (

M_{j},

M_{i}) for the first three values in the histogram:

where

k is the sequence number P(

M_{j},

M_{i}) in the histogram.

Therefore, for each histogram (see

Figure 6), we define the effectiveness: (a)

η = 97.8%, (b)

η = 99.7%, and (c)

η = 74.8%.

The new family of metrics (CH

_{i,j},

η) allows sufficient determination of the synchronization states of two oscillators, and classification of chimeric synchronization. Depending on the task, for example, the network training for data coding and pattern recognition, the problem of the presence or absence of synchronization can be solved by formally setting the synchronization effectiveness threshold

η_{th}, so

In the majority of cases, we set

η_{th} = 90%, meaning the signals are synchronized if 90% of their durability has a certain synchronization pattern or a set of patterns of the chimeric synchronization. For the network training, this parameter can be selected within a selected range, and it is one of the important parameters of the network adjustment [

25].

Let us discuss the reasons for the introduction of the concept of CH index chimera synchronization. For signals in

Figure 6b, the synchronization is CH

_{i,j} = (3:2 7:5) and

η = 99.7%, thus, the signals are clearly synchronized (

η > 90%) and have two patterns of synchronization. The spectrum of oscillator

j has a linear character, and the phase diagram of voltages on oscillators has a complex, but not chaotic, attractor, most likely consisting of two limit cycles. If the technique [

26] and only the concept of basic synchronization SHR

_{i,j}, are applied, then the family of metrics would look like SHR

_{i,j} = 3:2 and

η = 54%. As a result, oscillators would be defined as not synchronized, since

η < 90%. However, an accurate calculation of the chimeric synchronization value using the proposed metric allows more accurate and complete characterization of the synchronization state. In addition, such a metric can significantly expand the capabilities of neuromorphic and logical devices that operate on the synchronization effect.

For signals in

Figure 6c, the synchronization is CH

_{i,j} = (11:7 24:17 8:5) and

η = 74.8%, so the signal is weakly synchronized, and at

η_{th} = 90%,, it is not formally synchronized, as in Equation (5). This is confirmed by the type of phase trajectory that fills the entire phase space. In addition, the oscillator

j spectrum is wide, contains many harmonics, and is close to the noise spectrum by its nature.

The main technical problem we faced was the problem of defining the synchronization between the reference oscillator No.0 and the oscillator of the output layer No.5 characterized by the values SHR

_{0,5} and CH

_{0,5}:

Two parameters CH_{0,5} and η are used as the main metrics for evaluation the degree of the two oscillators’ synchronization and are applied in the algorithm of ONN training.

Current oscillograms I_{sw}(t) of oscillators No.0–5 were calculated simultaneously and contained ~250,000 points with time interval Δt = 1 µs. Then, the oscillograms were automatically processed.

#### 2.3. Oscillator Circuit

A model diagram of a single oscillator consists of a current source

I_{p}, a capacitance

C connected in parallel with the VO

_{2} switch, and a noise source

U_{n} (

Figure 7a). The capacitance

C remains constant

C = 10nF, while

I_{p} and

U_{n} vary in the following ranges

I_{p} (435–1220 μA),

U_{n} (0–10 mV). The noise source simulates external or internal circuit noise, for example, switch current noise manifested in fluctuations of switch threshold voltages [

31].

I_{sw} and

U denote the current passing through the VO

_{2} switch and the voltage on it, respectively. The model current-voltage characteristic of the VO

_{2} switch is shown in

Figure 7b. All model switches without coupling have the same I-V characteristic, with stationary natural parameters

U_{th} = 5V,

I_{th} = 550 μA (threshold voltage and current),

U_{h} = 1.5 V,

I_{h} = 1039 μA (holder voltage and current),

U_{cf} = 0.82 V (cutoff voltage). Model curve

I_{sw} =

f(

U) has high-resistance (OFF) and low-resistance (ON) segments with corresponding dynamic resistances

R_{off} = 9100 Ω and

R_{on} = 615 Ω. A more detailed description of the model circuit of a coupled oscillator-based neural network and methods for calculating oscillations in a circuit are given in a previous study [

26]. The system of differential equations for the current in the circuit were numerically calculated with respect to time at regular intervals Δ

t =1 µs.

Tangibly, the operation of the circuit can be enacted as periodic charging and discharging of the capacitor, when the operating point of the circuit is kept in the negative differential resistance (NDR) section, at the expense of the current source I_{p}.

Examples of the calculated oscillogram sections for voltage

U and current

I_{sw} for the reference oscillator are shown in

Figure 8a.

The dependence of the own oscillations frequency

F_{0} on the magnitude of the current

I_{p} is shown in

Figure 8b. Own oscillations occur when the circuit operation point lies in the region of negative differential resistance, in the current range of

I_{th} ≤

I_{p} ≤

I_{h} (550 μA ≤

I_{p} ≤1105 μA), with limit frequencies of 1640 Hz and 10130 Hz. The maximum frequency of 12850 Hz corresponds to a current

I_{p} = 1039 μA. In this way, an increase in the supply current

I_{p} initially leads to an increase in frequency, due to a decrease in the charging time of the capacitor

C to the threshold turn-on voltage

U_{th}, then the frequency decreases due to an increase in the discharge time of the capacitor to the holder voltage

U_{h}.

By setting the supply current below

I_{p} <

I_{th} or above

I_{p} >

I_{h}, the operating point of the circuit is set to the sub-threshold state, where own generation is absent and the switch is always either off or on, respectively (see

Figure 8b). Further, we will call such an oscillator operation mode a sub-threshold oscillator operation mode.

#### 2.4. ONN Structure

The studied ONN consists of a reference oscillator (No.0), whose frequency does not change, the input layer in the form of a one-dimensional matrix, each element of which is represented by one VO

_{2} oscillator (No.1–4), and the output oscillator (No.5) (

Figure 9). Data in the form of binary four-digit numbers is transmitted to a layer of input oscillators in such a way that each oscillator of this layer is associated with two values of supply currents:

I_{p} =

I_{OFF} represents logical 0, or

I_{p} =

I_{ON} represents logical 1. The numbers can be associated with two-dimensional images, as shown in

Figure 9. At the system output, the synchronization order of the output oscillator relative to the reference oscillator is expressed either by the high order synchronization SHR

_{0,5} or by the chimeric synchronization index CH

_{0,5} (see formula 6). Thus, four-digit numbers are converted to the synchronization state of the output oscillator.

The thermal effects of oscillators No.1–5 on each other in the circuit in

Figure 9 are set to be mutually equivalent, that is, Δ

_{i,j} = Δ

_{j,i}, while the reverse effect on the reference oscillator (No.0) from other oscillators are not taken into account: Δ

_{1,i} ≠ 0, Δ

_{i,1} = 0. Thus, the reference oscillator generates the thermal pulses and sets the rhythm of the entire ONN with a constant frequency.

The switch parameters are unchanged in numerical simulation, while current intensities I_{p_i} (I_{ON}, I_{OFF}, I_{p_0,} I_{p_5}), coupling strength constants Δ_{j,i}, noise amplitude U_{n}, and η_{th} vary.

#### 2.5. Task Setting And Technique of ONN Training

One of the tasks that can be set for the network is to perform conversion functions. When 16 variants of input signals are applied, the output oscillator can take different synchronization values, including showing no synchronization if η < η_{th}, as in Equation (5). Denoting the total number of synchronous states at the output as x, x can take a value from 1 to 16. Among the x synchronous states, there can be n unique synchronization values. Then, the output can be codified as “n of x”. In the case where all of the output synchronization values are unique (n = x), this corresponds to the result “x of x”.

We have divided the network responses into 16 variants, corresponding to the condition when a certain number of synchronous states occur and none of them repeat. For example, the task “1 of 1” corresponds to one synchronous state at the output, all other states are not synchronous, and “2 of 2” corresponds to two different states (see explanatory

Table 1).

The most difficult task is the case of “16 of 16”, when each input state corresponds to its unique synchronous state, expressed by one of two indices, SHR_{0,5} or CH_{0,5} (see Equation (6)).

The possible number of representations of the task “

x of

x” is equal to the number of combinations x out of 16 (

${\mathrm{C}}_{16}^{x}$). Therefore, the highest probability to find a solution is for x = 8, and the smallest probability is for x = 16 at

${\mathrm{C}}_{16}^{16}=1$ (see

Figure 10). However, as it will be shown in the results, the type of distribution of the found solutions differs significantly from the distribution in

Figure 10 and is determined by the network parameters. The total number of options for presenting tasks “

x of

x” is

$\sum {\mathrm{C}}_{16}^{x}+1={2}^{16}=65536$, where the absence of synchronization with any input data is taken into account. The total number of network response options is much higher if we take into account answers of the type “

n of

x”.

As ONN with the high order synchronization effect has not been studied before, there are no established methods for network training. One of the ways is to use the Simulated Annealing Algorithm (SAA) [

1] for the network parameter selection: currents (

I_{ON},

I_{OFF},

I_{p_0},

I_{p_5}), coupling strength Δ

_{j,i}, noise amplitude

U_{n}, and synchronization effectiveness threshold

η_{th}. The algorithm’s key point is the random searching of problem solutions at some initial interval of parameters followed by narrowing of these intervals. In the majority of cases considered in the article, we used only a random search in a given range of parameters. In some cases, a better solution could be found by gradient descent near the found solution. The random search algorithm was tested in a previous study [

26] and showed its effectiveness, since a feature of the neural network is the presence of a set of solutions (a set of system parameters) to satisfy the answer for the task “

x of

x”. Therefore, in the results, we present a distribution histogram of the number of solutions, called NS, from the value x.

The values of

I_{ON} and

I_{OFF} currents set the logic levels 1 and 0 by determining the supply current of the corresponding input oscillator. The range of current variation is (435–1220 µA). This range is wider than the range of existence of own oscillations (550–1105 µA) defined by

Figure 8b in

Section 2.1. If we represent a pair of currents (

I_{ON} and

I_{OFF}) in the diagram by the point, as in

Figure 11, then the central area 1 corresponds to the mode where both logic levels make the oscillators oscillate. In other words, when applying either 1 or 0, the oscillator is in the generation zone, and regardless of the input signal, all oscillators oscillate. In area 3, currents

I_{ON} and

I_{OFF} lead the oscillator into sub-threshold mode, while the switch is either ON or OFF.

Area 2 corresponds to the case when one of the current levels I_{ON} or I_{OFF} leads to generation, while the other level sets the oscillator to the sub-threshold mode.

For example, point

**A** in

Figure 11 corresponds to currents

I_{ON} = 894 µA and

I_{OFF} = 958 µA, and both levels of current lead to oscillation. At point

**B** (

I_{ON} = 1153 μA and

I_{OFF} = 1036 μA), a logical 1 (

I_{ON}) leads the oscillator to enter the threshold mode, while the switch is in the on mode, its resistance is

R_{ON}, and a large current passes through it. In turn, a logical 0 (

I_{OFF}) leads to the generation of oscillations. Point

**C** (

I_{ON} = 1163 µA and

I_{OFF} = 459 µA) corresponds to the mode when both 1 and 0 set the oscillator to the sub-threshold mode, while at 1 the switch is on, and at 0 the switch is off.

In the sub-threshold mode, the switch is turned on and heats the neighboring areas. Therefore, the switch affects the threshold voltages of the neighboring areas and plays the role of a constant displacement neuron. In the model, this is inherently taken into account, as the influence of oscillators is affected by the state of the switches (see Equations (A1)–(A3) in the previous study [

26]).

The range of variable currents is chosen in a way that area 1 (pink space) is equal to the sum of areas 2 (blue) and 3 (brown). Therefore, with a random choice of a point in the space of the currents I_{OFF} and I_{ON}, the probabilities of falling into the region of oscillation generation and into the region of the sub-threshold state of the oscillators are equal, which allows us to compare histograms of the solutions NS.

The ONN was set up by brute force, searching the thermal coupling strength Δ_{i,j} and power supply levels (I_{OFF} and I_{ON}) of the input layer oscillators (No.1–4). The random search was performed in the ranges of 0–1 V for Δ_{i,j} (except Δ_{0,j} = 0.2 V) and 435–1220 µA for I_{OFF} and I_{ON}. The values of the supply currents of the reference I_{p_0} and output I_{p_5} oscillators, and the noise level U_{n} were fixed. The number of samples was 10^{5}.

In the current study, the target was to find solutions “

x of

x” defining the operation of the neural network converter, as well as the use of this scheme for filtering images (see

Section 3.6). Other tasks may require different network functions. For example, in the task of driving a vehicle, proximity sensors can input the signals of approaching an obstacle, and output synchronization would determine in which direction the vehicle should turn.

#### 2.6. Method of Oscillators Coupling and Variants of Experimental Implementation

The main model elements of the oscillator network are VO

_{2} switches, which are two-electrode planar structures with a functional layer of vanadium dioxide and two metal contacts. The interaction between the oscillators is carried out by heat flows propagating through the substrate, resulting from the Joule heating when the switches are turned on. This method of coupling was experimentally demonstrated in a previous study [

24].

The interaction of oscillators through thermal coupling assumes a reduction in the threshold switching voltage of each switch

U_{th} by a coupling strength value Δ

_{i,j} when the switches thermally affect each other at the moment of capacitance

C discharge and the release of Joule heat. A detailed mathematical model of thermal coupling of VO

_{2} oscillators is given in a previous study [

26].

An exemplary view of the location of the switches on the substrate is shown in

Figure 12. In the study, we modeled the reference oscillator, which affected all other oscillators, and unidirectional thermal couplings. The question may arise of how to implement unidirectional thermal coupling in a real experiment. The task of the technical implementation of the proposed model objects is the subject of a separate study; nevertheless, possible variants of one-way thermal coupling between two oscillators are presented in

Figure 13. In the diagram (

Figure 13a), the connection is carried out through an additional current resistor included in the circuit and located on the substrate along with switches. Its resistance does not depend on temperature, however, when heated by the current, it transmits a thermal effect from oscillator 0 to oscillator 1. Therefore, the implementation of one-way thermal coupling is possible, in principle. In addition, the amplitude of the thermal interaction signal decays exponentially with the distance between the switches; see detailed descriptions of the physics of thermal coupling in previous studies [

24,

32]. Therefore, the one-way connection can be accomplished by making switches with different radii of thermal exposure

R_{TC}, as in

Figure 13b.

The thermal coupling is chosen due to simplicity of modelling in the case of a fully connected circuit of oscillators; oscillators are galvanically isolated, and this type of coupling has been confirmed experimentally.

In addition to thermal coupling, oscillators can be connected electrically via resistors or capacitors [

33], optically [

34,

35], and via wireless channels [

36], however, the method for determining chimeric synchronization is universal, regardless of the coupling type and the physics of the oscillators.