#### 2.1. General Principle

An oscillator neural network is a system of

N coupled oscillators, which may be connected via electric (by resistors and capacitors) [

7,

26], thermal [

17], and optic [

27] couplings, depending on the physical mechanism of oscillator interaction. In the general case, there is a matrix of coupling strengths Δ

_{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).

It is known [

17,

23] that oscillators in a network may undergo the effect of synchronization and, besides synchronization on the first harmonic, synchronous modes of high order may be observed if the signal spectra possess several harmonics. To evaluate synchronization, in this work, we use a family of metrics that consists of two parameters (SHR is the value of high order synchronization and

η is the effectiveness of synchronization). A detailed method of its determination is given in

Section 2.3 and in [

17].

In the general case, high-order synchronization is determined by the ratio SHR =

k_{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).

In addition to SHR, there is also a parameter of synchronization effectiveness

η, that shows what share of oscillations of the whole time signal is synchronized. This parameter is expressed in percentages (see

Section 2.3). If, at any point,

η is less than the threshold value

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

Transition from one synchronous state into another is possible when the oscillator network control parameters are varied. For example, in electric oscillators, the main parameters may be oscillator feed currents I_{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.

The range of control parameters variation, where synchronization does not change its state, is called a synchronization area. There is a whole family of synchronization areas that are called Arnold’s tongues (for the case of two oscillators). A schematic example of synchronization areas for a three-oscillator scheme is shown in

Figure 3a. Here, the control parameters are oscillator feed currents. Each area has its own value of SHR. Besides, each area has its own distribution of the synchronization effectiveness value within

η_{th} <

η < 100%, with a peaked curve.

The number of possible variants of synchronous states (synchronization areas), where the system may exist when the basic control parameters are varied, is denoted as

N_{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.

Vectors **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.

As we have mentioned in the introduction section, the patterns to be stored are usually expressed through a set of vectors. Vector coordinates contain information about the pattern and unambiguously associate it with one of possible variants. For example,

Figure 3c shows storage of the object’s colors in the RGB format (red, green, blue) through the coordinates of vectors

**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.

We suggest the following methods of pattern storage and recognition in a neural network, based on the high-order synchronization effect, and its general scheme is given in

Figure 4.

#### 2.1.1. Vector Storage and ONN Training

The general algorithm for vector storage and ONN training includes the following steps:

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** = (δ

I_{p1}(1), δ

I_{p2}(2), …, δ

I_{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** = (δ

I_{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.

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

The algorithm of test vector **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**.

This method is a more complicated version of the method described in [

22], where the analogy to the frequency-shift keying method of coding is used and, instead of setting the vector through frequencies

**E** = (δ

ω_{1}, δ

ω_{2}, …, δ

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

**E**_{1} = (δ

I_{p1}, δ

I_{p2}, …, δ

I_{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.

Besides, as described in the results section, it is more practical to use vector dimensions **E’** = (δI_{1}, δI_{2}, …, δI_{N−1}) with one less than the number of oscillators (N − 1).

#### 2.2. Model Object

As a model object, we have chosen a neural network composed of three thermally coupled VO

_{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:

where

U_{bv} ~0.82 V is bias voltage of a low-impedance region, and

State is a switch state.

One of three topologies presented in

Figure 1 may be realized, depending on coupling strength magnitudes Δ. At non-zero Δ ≠ 0, the topology is “all-to-all” (

Figure 1c); at Δ

_{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.

Variations of each oscillator are described by the equation of Kirchhoff’s law:

where

U_{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.

Thermal interaction between the

i-th VO

_{2}-oscillator and the neighbor ones ((

i+)—clockwise and (

i−)—counterclockwise of the scheme in

Figure 5b) is realized according to the rule

If the states of oscillators State_{(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).

Oscillograms of oscillations with ~250,000 points and time interval δt = 10 µs were simulated using Equations (1)–(3). After that, the oscillograms were automatically processed, the synchronization order was determined, and cross-sections of oscillator synchronization areas were built.

The switch parameters did not change in numerical simulation of the results, but current intensities I_{p}, coupling strength Δ, and noise amplitude U_{in} varied.

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

To define the synchronization order, we used the family of metrics described above, that consists of two parameters SHR and η.

The problem of finding the high-order synchronization value determined by the ratio of integers SHR =

k_{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].

It should be noted that, at synchronous state, the frequency sets of fundamental (first) harmonics of oscillators (

F_{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]).

Thus, the method of specifying all values of

k and the synchronization order of the system consisting of

N-oscillators comes down to determining the set of pair synchronization fractional values (in regard to the first oscillator) for

N-pairs (

m_{1}/

d_{1},

m_{2}/

d_{2}, …,

m_{N−1}/

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

For example, a set of pair synchronization for oscillator pairs (№1–№2) and (№1–№3) in

Figure 2 looks like (2/1, 4/3), after reduction to a common denominator (4), we get (2/1, 4/3) → (6/3, 4/3), and SHR =

k_{1}:

k_{2}:

k_{3} = 3:6:4.

It should also be noted that the algorithm of pair synchronization definition is based on the search of current oscillation peaks,

I_{sw}, synchronous in time [

17].

The effectiveness of pair synchronization

η is determined as the percentage of the durability of all

N_{SHR} synchronous periods

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

T_{all}:

If there are several synchronization types with different SHR, then the resulting η is associated with the maximum which, in turn, is compared with the threshold value η_{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.