A Method for Evaluating Chimeric Synchronization of Coupled Oscillators and Its Application for Creating a Neural Network Information Converter

: This paper presents a new method for evaluating the synchronization of quasi-periodic oscillations of two oscillators, termed “chimeric synchronization”. The family of metrics is proposed to create a neural network information converter based on a network of pulsed oscillators. In addition to transforming input information from digital to analogue, the converter can perform information processing after training the network by selecting control parameters. In the proposed neural network scheme, the data arrives at the input layer in the form of current levels of the oscillators and is converted into a set of non-repeating states of the chimeric synchronization of the output oscillator. By modelling a thermally coupled VO 2 -oscillator circuit, the network setup is demonstrated through the selection of coupling strength, power supply levels, and the synchronization e ﬃ ciency parameter. The distribution of solutions depending on the operating mode of the oscillators, sub-threshold mode, or generation mode are revealed. Technological approaches for the implementation of a neural network information converter are proposed, and examples of its application for image ﬁltering are demonstrated. The proposed method helps to signiﬁcantly expand the capabilities of neuromorphic and logical devices based on synchronization e ﬀ ects.


Introduction
Artificial neural networks are actively used in image and speech recognition applications [1,2], as well as in computer calculations [3] and data coding [4].The functional importance of synchronization in information processing has stimulated the development of neural networks models with oscillatory dynamics and neuromorphic algorithms based on synchronization effect [].
In 2002, Kuramoto and Battogtokh reported that arrays of non-locally related oscillators could spontaneously divide into synchronized and desynchronized subpopulations [5].This amazing discovery challenged the previous believe that the connected identical oscillators are either synchronized or will work incoherently, chaotically.Since the network had a hybrid nature uniting both coherent and non-coherent parts, it was proposed to call such states chimera, because of their resemblance to mythological Greek animals, assembled from incomparable parts [6].Recent studies have demonstrated that chimera states are not limited to phase oscillators, but can be found in a wide variety of different systems and are observed in space-time dynamics.It is worth to mention the studies on chimeric states in networks of, Kuramoto phase oscillators [7], leaky integrate-and-fire [8] FitzHugh-Nagumo [9] and Hindmarsh-Rose [10] models.Nevertheless, the classification of chimera states and its accurate estimation for application in technical devices remains an important issue.
A class of oscillatory neural networks (ONN) can be identified, where the basic elements are relaxation oscillators that generate sequences of pulses (spikes), and ONN can encode information at pulses repetition rate.Such ONN are interesting due to simplicity of hardware implementations, as developed micro-and nano-electronic autogenerators ensure networks compactness and energy efficiency.In addition, a pulsed-type ONN, where the periodic oscillation spectrum has a multi-frequency character, have a special mode of harmonics synchronization or, in other words, a high order synchronization effect [11][12][13].This effect has been demonstrated experimentally using the example of thermally coupled VO2 oscillators [11].In relaxation generators with elements based on vanadium dioxide film, oscillations are initiated by the electric switching effect caused by the metal-insulator phase transition [14].
Oscillators based on VO2 structures are chosen as ONN elements due to the high speed of electrical switching (10 ns) [15], high degree of nanoscale in manufacturing [16], and, most importantly, the presence of a significant thermal coupling effect that simplifies the ONN layout and circuitry of galvanically isolated oscillators [11].The thermal coupling effect, having a local nature, makes the network to resemble a cellular neural network (CNN) [3,17] and opens 3D integration possibilities.For these reasons, VO2-oscillators are actively used in prototyping of neuro-oscillators for the tasks of cognitive technologies.
The high-order synchronization of two oscillators can be characterized by a family of metrics: the ratio of subharmonics (SHR) and synchronization effectiveness  [13].These metrics are used to create a neuromorphic device for pattern recognition.Synchronization is measured relative to the reference oscillator, which has a constant pulse generation frequency; this makes it possible to compare the synchronization of all oscillators in the network with each other.Although the system dynamics during synchronization can be described only by SHR and  value, there are a number of examples where oscillograms contain sections with different synchronization, which corresponds to the chimera state of the system.Chimera synchronization states appear quite often in the model, but they are not involved in the operation of the device and belong to an asynchronous type of oscillations.For example, Figure 1 shows the current pulses of two oscillators with numbers 0 and 1, with alternating synchronization pattern defined by two parameters SHR 1 0,1 = 4:3 and SHR 2 0,1 = 9:7.The dotted lines mark the synchronization moments (phase locked) of the individual pulses.Such an alternation of the synchronization pattern is a stable state in time and is characterized by a periodic jump of the phase of synchronous pulses.
Several authors have dealt with the classification of chimera states [18], including a direction with evaluating Lyapunov spectra [19,20], analyzing the instantaneous  distribution of the amplitudes of the ensemble elements [21,22] and complex order parameters [23].
The terms of the amplitude and phase chimeras in an ensemble of chaotic oscillators are proposed in [19,21].The term phase chimera is very close to the phenomenon we observe.It describes the mechanism of the appearance of chimera states with instability when the ensemble of oscillators goes into the mode of random switching between in-phase and antiphase oscillations, and the existence of at least two attractors.In our case, we do not register anti-phase oscillations, and the observed effect can be called high order synchronization chimeras, with random or periodical alternation of a high order synchronization pattern.
In this paper, we describe the method for classification of chimera states and use it to analyze the dynamics of six thermally coupled VO2 oscillators as applied to the implementation of a neural network converter.

Materials and Methods
We describe the classification method of chimera states 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.

Chimera states classification
The method of determining the family of metrics of high-order synchronous states (SHR and µ) is described in detail in [13].The method of classification of chimera states 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 LE, that 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 two nearest phase-locked pulses is denoted as T z s -the period of synchronization (where z is a conditional number of periods Ts).Pulses are considered phase-locked if the distance between them does not exceed 4t.Therefore, SHRi,j value may be estimated using a phase-locking method: where Mi and Mj are numbers of signal periods falling into the synchronization periods T z s of two oscillators.
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 Mi and Mj may change within one oscillogram (see Figure 4).Various values of synchronizations SHRi,j may occur within one oscillogram.To determine the distribution of the SHRi,j, it is necessary to find the occurrence probabilities P(Mj : Mi) for each pair (Mi : Mj) that are present in the whole oscillogram.To find the probabilities P(Mj : Mi), we can count how many times NP(Mj : Mi) the given pair appeared within the whole oscillogram of the oscillator i, multiply by the number of periods in it (Mi) and divide by the total number of all oscillations periods in the given signal (Nj).Thus, for P(Mj : Mi) we obtain: where Ni is the total number of periods in the oscillogram of oscillator i.Therefore, each synchronization value SHRi,j will correspond to the probability of its detection P (Mj: Mi) expressed as a percentage.
It is convenient to present the probabilities P(Mj : Mi) 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 done: The histogram in Figure 5 is calculated by formula (2), when the pairs occur number of times NP(2:7)=2, NP(2:9)=1, NP(2:5)=1, and the total number of periods is Ni=28 (in real calculations, Ni was in the range of 1000-3000 for greater accuracy [13]).
In a model experiment, the shape of the distribution oscillogram P(Mj: Mi) 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 SHRi,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 Figures 6b,c have a set of different SHRi,j and correspond to chimera synchronization states.
For CH the parameter of synchronization effectiveness  is defined as the sum of P (Mj, Mi) for the first three values on the histogram: where k is the sequence number P(Mj, Mi) on the histogram.

Frequency, Hz
Amplitude, mA In the majority of cases, we set th= 90%, meaning the signals are synchronized, if 90% of their durability have a certain synchronization pattern or a set of patterns of the chimera states.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 [24].Let us discuss the reasons for the introduction of the concept of CH index chimera synchronization.For signals in Figure 6b, synchronization is CHi,j = (3:2 7:5) and  = 99.7%,thus, the signals are clearly synchronized (> 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 [13] and only the concept of basic synchronization SHRi,j, are applied, then the family of metrics would look like SHRi,j = 3:2 and  = 54%.As a result, oscillators would be defined as not synchronized, since  <90%.However, an accurate calculation of the chimera synchronization value, using 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, synchronization is CHi,j = (11:7 24:17 8:5) and  = 74.8%,so the signal is weakly synchronized, and at th = 90%, , it is not formally synchronized (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 SHR0,5 and CH0,5: Two parameters CH0,5 and η are used as the main metrics for evaluation the degree of two oscillators' synchronization and are applied in the algorithm of ONN training.
Current oscillograms Isw(t) of oscillators No.0-5 were calculated simultaneously and contained ~250 000 points with time interval t=1µs.Then the oscillograms were automatically processed.

Method of chimera states color mapping
To represent the value of chimera synchronization CHi,j, we chose the RGB color display method, since the value of CHi,j contains three components, see formula 6.Each color component is represented by the following algorithm: where SHRmax is the maximum value of SHR on the graph.For example, CH (1:3 3:1 3:2) with SHRmax = 3 is converted to RGB (0, 200, 150).

Oscillator Circuit
A model diagram of a single oscillator consists of a current source Ip, a capacitance C connected in parallel with the VO2 switch, and a noise source Un (Figure 7).The capacitance C remains constant C = 10nF, while Ip and Un vary in the following ranges Ip (435 µA ÷ 1220 µA), Un (0 mV ÷ 10 mV).The noise source simulates external or internal circuit noise, for example, switch current noise manifested in fluctuations of switch threshold voltages [25].Isw and U denote the current passing through the VO2 switch and the voltage on it, respectively.The model current-voltage characteristic of the VO2 switch is shown in Figure 7b.All model switches without coupling have the same I-V characteristic, with stationary natural parameters Uth=5V (threshold voltage), Uh=1.5 V (holder voltage), Ucf=0.82V (cutoff voltage).Model curve Isw=f(U) has high-resistance (OFF) and low-resistance (ON) segments with corresponding dynamic resistances Roff=9100  and Ron=615 .A more detailed description of the model circuit of a coupled oscillators-based neural network and methods for calculating oscillations in a circuit is given in [13].The system of differential equations for the current in the circuit were numerically calculated with respect to time at regular intervals 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 Ip.    8a.
The dependence of the own oscillations frequency F0 on the magnitude of the current Ip is shown in Figure 8b.Own oscillations exist in the current range of 550 µA ≤Ip≤1105 µA, with limit frequencies of 1640 Hz and 10130 Hz.The maximum frequency of 12850 Hz corresponds to a current Ip = 1039 µA.In this way, an increase in the supply current Ip 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 Uth, then the frequency decreases due to an increase in the discharge time of the capacitor to the holder voltage Uh.
By setting the supply current below Ip <550 µA or above Ip> 1105 µA, 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 prethreshold oscillator operation mode.

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 VO2 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: Ip = IOFF -logical 0 or Ip = ION -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 SHR0,5 or by the chimera index CH0,5 (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, Δi,j = Δj,i, while the reverse effect on the reference oscillator (No.0) from other oscillators are not taken into account: Δ1,i ≠ 0, Δi,1 = 0. Thus, the reference oscillator generates the thermal pulses and sets the rhythm of the entire ONN with a constant frequency.

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 <th, see formula (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".The case, when all the output synchronization values are unique (n=x), 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 occure and none of them repeats.For example, the task "1 of 1" corresponds to one synchronous state at the output, all other states are not synchronous, "2 of 2" corresponds to two different states, see explanatory Table 1.
The most difficult task is the case "16 of 16", when each input state corresponds to its unique synchronous state expressed by one of two indices SHR0,5, or CH0,5 (see formula 6).
Table 1.Example of network responses for the "x of x" task.Empty cells correspond to the absence of synchronization by criterion (5).The responses are taken at high order SHR0,5 and chimera CH0,5 synchronization.

High order synchronization
Chimera synchronization "1 of 1" "2 of 2" "16 of 16" " Figure 10 Histogram of the number of "x of x" task representations.
The possible number of representations of the task "x of x" is equal to the number of combinations x out of 16 (C x 16).Therefore, the highest probability to find a solution is for x = 8, and the smallest probability is for x = 16 at 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 ∑ C x 16 + 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 simulated annealing algorithm [1] for the network parameters selection: currents (ION, IOFF, Ip_0, Ip_5), couplings strength Δj,i, noise amplitude Un and synchronization effectiveness threshold ηth.The algorithm's key point is the random searching of problem solution at some initial interval of parameters followed by these intervals narrowing.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 [13] 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 solutions NS from the value x.
The values of ION and IOFF currents set the logic levels 1 and 0 by determining the supply current of the corresponding input oscillator.The range of currents variation is (435 µA ÷ 1220 µA).This range is wider than the range of existence of own oscillations (550 µA ÷ 1105 µA) defined by figure 8b in paragraph 2.1.If we represent a pair of currents (ION and IOFF) in the diagram by the point, see Figure 11, then the central area 1 corresponds to the mode, when both logic levels make the oscillators to 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 the areas 3, currents ION and IOFF lead oscillator into subthreshold mode, while the switch is either ON or OFF.
Area 2 corresponds to the case when one of the current levels ION or IOFF leads to generation, while the other level sets the oscillator to the prethreshold mode.For example, point A in Figure 11 corresponds to currents ION = 894 µA and IOFF = 958 µA, both levels of current lead to oscillator oscillation.At point B (ION = 1153 µA and IOFF = 1036 µA), a logical 1 (ION) leads the oscillator to enter the threshold mode, while the switch is in the on mode, its resistance is RON and a large current passes through it.In turn, a logical 0 (IOFF) leads to the generation of oscillations.Point C (ION = 1163 µA and IOFF = 459 µA) corresponds to the mode when both 1 and 0 set the oscillator to the prethreshold mode, while at 1 the switch is on, and at 0 the switch is off.
In the prethreshold 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 formulas A1-A3 in [13]).
The range of variable currents is chosen in a way that the area 1 (pink space) is equal to the sum of the areas 2 (blue) and 3 (brown).Therefore, with a random choice of a point in the space of the currents IOFF and ION, the probabilities of falling into the region of oscillation generation and into the region of the subthreshold state of the oscillators are equal, this allows us to compare histograms of the solutions NS.
The ONN was set up by brute force searching the thermal coupling strength Δi,j and power supply levels (IOFF and ION) of the input layer oscillators (No.1-4).The random search was performed in the ranges 0-1 V for Δi,j (except Δ0,j = 0.2 V) and 435 µA ÷ 1220 µA for IOFF and ION.The values of the supply currents of the reference Ip_0 and output Ip_5 oscillators, and the noise level Un 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 which direction the vehicle should turn.

Method of oscillators coupling and variants of experimental implementation
The main model elements of the oscillator network are VO2 switches, which are twoelectrode 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 [11].The interaction of oscillators through thermal coupling assumes a reduction in the threshold switching voltage of each switch Uth by an coupling strength value Δi,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 VO2 oscillators is given in [13].
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, 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, heated by the current, it transmits a thermal effect from oscillator 0 to oscillator 1. Thereby, 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 description of the physics of thermal coupling in [11,26].Therefore, the one-way connection can be accomplished by making switches with different radii of thermal exposure RTC, see 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 [27], optically [28,29] and wireless channel [30], however, the method for determining chimera synchronization is universal, regardless of the coupling type and the physics of oscillators.

Investigation of the chimera states of two coupled oscillators
Let us consider how the distribution of synchronous states of two oscillators changes in the space of supply currents with the parameters th= 90%, Un = 1 mV, Δ0,5=0.2V, Δ5,0=0 V (Figure 14a).Oscillators are connected by one-way coupling, oscillator No.0 affects oscillator No.5, and oscillators No.1-4 in the diagram in Figure 14a are disabled.Figure 14b shows the synchronization distribution in the form of Arnold tongues, when the chimera states are not taken into account (the colors correspond to the procedure described in section 2.2).Arnold tongues have a linear extended form, however, they are transformed at high currents and shape regions with alternating synchronization (indicated by a dotted line).Behavior is explained by the nonlinear dependence of the frequency on the supply current, as shown in Figure 8b.In addition, a well-known phenomenon is observed when synchronization occurs predominantly in areas where the frequency of the oscillator No.0 is greater than the frequency of the oscillator No.5.This is clearly seen for synchronization SHR0,5 = 1:1 which occurs predominantly above the diagonal, see Figure 14b.The unidirectional action of the oscillator 0 is more effective in this case, as it leads and initiates oscillations in the adjacent circuit.
Application of the technique described in section 2.1 makes areas with chimera states visible, indicated by gray color in Figure 14c.For example, the chimera state CH0,5 = (21:20 20:19 1:1) occurs at the currents Ip_0 = 948 µA, Ip_5 =1004 µA, and the synchronization efficiency = 98%.In general, theareas with a detected synchronization with the efficiency ≥th are inreased, due to the appearance of areas with chimera synchronization.Now the areas, previously diagnosed as nonsynchronous operating modes of oscillators, have a welldefined classification with chimera synchronization, and can be used in creation of logical and neuromorphic devices.

The study of the neural network information converter, without accounting for chimera synchronization
The ONN under investigation has the structure described in section 2.5.The output oscillator's state is determined by the conventional synchronization index SHR0,5.The training was carried out according to the method described in section 2.6 with the number of samples 10 5 .
The results of the distribution of the number of NS solutions for the "x of x" problem after training are shown in Figure 15a.Four curves are presented, the curve ("whole area", black marker) corresponds to the entire space of the ION-IOFF currents, and the remaining curves correspond to different areas of the ION-IOFF space (see Figure 11).The calculation was performed at a noise level of Un=0 mV, th =30% and Ip_0=1039 µA, Ip_5=750 µA.It can be seen that the solution to the "16 of 16" problem is achieved only in regions 2 and 3 (see the section marked by the dotted line NS 2,3 (16)~100), when one of the oscillators is in the prethreshold mode.No solution NS 1 (16)=0 is found in area 1.The maximum number of solutions belongs to the "13 of 13" task, corresponding to area 2 (NS 2 (13)~400).For area 1, the maximum number of solutions falls within the task "9 of 9" (NS 1 (9)~50).In area 2, there are more solutions than in all other areas, which means the circuit has the greatest potential, when one of the currents (ION or IOFF) sets the oscillator into the threshold state, and the other current creates the generation mode of oscillator.
The type of distribution depends on a number of parameters, like Un, th, Ip_0, Ip_5.When adding noise to the system Un = 1 mV (Figure 15b), the number of solutions to the "16 of 16" problem has significantly decreased (NS 2,3 (16)~10), but the number of solutions with a low x

NS
x has significantly increased.Therefore, noise can increase the number of solutions, which is associated with the decrease of solutions in one range and an increase in another.Apparently, there is an optimal noise value, when the maximum number of solutions is observed (since with an unlimited increase in noise, the number of solutions will obviously decrease), and this effect is similar to the stochastic resonance effect observed in [13].

The study of the neural network information converter, accounting for chimera synchronization
The distribution results of the number of NS solutions to the "x of x" problem after training are presented in Figure 16a.Inclusion of chimeric synchronization leads to an increase in the NS maximum by more than 10 times, and the number of solutions to the "16 of 16" problem increases by 30 times.The greatest number of solutions, as in the previous case, falls on area 2. Another interesting result is the appearance of solutions in area 1, where all generators are active at any input data.
Hence, accounting for chimeric synchronization significantly increases the probability of finding solutions.Four curves are presented, the curve (black marker) corresponds to the entire space of the ION-IOFF currents, and the other curves correspond to different areas of the ION-IOFF space (see Figure 11).The distribution of solutions to the "16 of 16" problem in ION-IOFF space is presented in Figure 17a.The distribution is symmetrical to the diagonal and concentrated to the area 2. No solutions exist, when both currents are greater than 1105 mA or less than 550 mA.In this case, both the ION and IOFF currents lead to either switching on or switching off The synchronization efficiency ηth and the noise level Un affect the form of the distribution NS. Figure 16b presents the NS distribution for ηth = 90%.The distribution maximum is shifted to the left, and the maximum of the total number of solutions is in the "12 of 12" area.At the same time, the maximum value decreased by ~ 2 times, and the number of solutions to the "16 of 16" task fell by 10 times.Such a decrease happens due to the cut-off of solutions by the criterion η ≥ ηth.In Figure 17b (blue dots), the dependencies of the synchronization efficiency values on the input data are presented, which are elements of a single solution.So, at ηth = 30% (Un = 0mV) all points satisfy the condition η ≥ ηth, and the array of elements CH0,5 satisfies the "16 of 16" problem.At the same time, at ηth = 90% only six points satisfy η ≥ ηth condition and constitute solution for the "6 of 6" task.Therefore, with the increase of ηth, the number of solutions to "x of x" problems with lower x grows and shifts the NS distribution to the left, while NS decreases for the "16 of 16" task.
Figure 16c-d represents the NS distributions for different noise levels Un.
An increase in noise leads to a decrease in the number of solutions to the "16 of 16" problem, with the distribution shifted to the left.In contrast to the influence of ηth, an increase in noise leads to a significant increase in the number of solutions to "x of x" problems with lower x; and NS experiences the most significant increase for the "1 of 1" task.The noise has the least impact on solutions in area 3, as the input layer oscillators here are constantly in the prethreshold stable state regardless of the noise voltage dynamics, and the noise affects only the reference oscillator No.0 and output oscillator No.5.With increasing noise (Figure 17b), the synchronization efficiency of almost all elements of the solution decreases, as reflected in the observed patterns of NS distribution.

Scheme of converting with a minimum number of couplings
A special case of a fully connected circuit is the circuit shown in Figure 18a, which contains only unidirectional links with the output oscillator, and the input oscillators do not interact.
The scheme is simple and contains only five couplings, however, its functionality allows solving the "x of x" tasks.The distribution of the number of NS solutions to the "x of x" problem after training is shown in Figure 18b.Solutions exist in all areas, however, the number of solutions is less by 2 times than for a fully connected network (see Figure 16a).The maximum of solutions at ηth = 30% and Un = 0mV falls on the "16 of 16" task.Another important conclusion is that the input signal can be represented not by the levels of the ION-IOFF currents, but by signals with a certain CH0, i.In this way, the input and output of the network will be signals of the same nature -signals with a certain synchronization (see Figure 19c).This feature is important for designing large-scale networks, when the output of one network can become the input for another network, as in TTL logic circuits.Next, we summarize the concept of devices that can be implemented on the studied scheme.

Technological concepts of neural network converters
This section provides options for the technological concepts of neural network converters.Figure 19a shows a diagram based on 6 oscillators.In this scheme, current levels are applied to the input, and the output is a signal with a certain chimera synchronization.The circuit requires the generation mode of one of the input oscillators, therefore, the input currents must belong to areas 1 or 2 (see Figure 11).The scheme in Figure 19b consists of only two oscillators, and the input signal is given by the currents of the thermistors.In fact, this is an imitation of the operation of the circuit on Figure 19a in the area 3 of current parameters.Thermistors set the offset of the threshold characteristics of oscillator No.5 and by that change the output synchronization.The advantages of the scheme are a lower number of oscillators and increased resistance to noise.
The scheme in Figure 19c differres from the previous schemes, and principle of its operation is analyzed in Section 3.4.At the input and output of the network there are signals of the same nature -signals with a certain synchronization.This is important for designing large-scale networks, when the output of one network becomes the input for another network, similar to TTL logic circuits.
The coupling forces Δi,j determine the input weights and largely determine the functionality of all devices.

Figure 1 .
Figure 1.An example of a chimera states of oscillators with an alternating synchronization pattern of current pulses in an oscillogram.The dotted line marks synchronization moments of individual pulses.Isw is the oscillator current.

Figure 2 .
Figure 2. Oscillogram of oscillator current and the corresponding array of positions of the leading edges of the current pulse LE[i][t].(where i -the oscillator number, time t=nt, nnumber of the calculation step of the model oscillogram, t -calculation time interval).

Figure 4 .
Figure 4. Arrays LE[i][t] and LE[j][t] for two oscillators with non-constant period of synchronization T z s.

Figure 5 .
Figure 5. Histogram of probabilities distribution P(Mj, Mi), calculated by using formula (2) for signals LE, shown in Figure 5.

Figure 7
Figure 7. a) Diagram of a single oscillator based on a VO2 structure.i-number of the oscillator, Ip-current source, C-capacitance, Un-noise source, Isw -current passing through the VO2 switch, U -voltage on the switch.b) Model I-V characteristic of a separate switch.

Figure 9 .
Figure 9. Model scheme of ONN with an example of converting binary numbers to CH values of the output oscillator's synchronous state.

Figure 12 .Figure 13 .
Figure 12.Electrical circuit of thermally coupled VO2-oscillators located on the substrate.The numbers of the VO2 structures correspond to the numbers of the oscillators.Double arrows indicate thermal bond.

Figure 14 .
Figure 14.Schematic mapping of two coupled oscillators with one-way coupling (a), distribution of their synchronization in the space of supply currents without considering chimera states (b) and considering the chimera states (c).Colors are calculated by the method presented in paragraph 2.2 with SHRmax = 4.The dotted line indicates the nonlinear frequency domain of the oscillator, and the arrows indicate the sample values of SHR0,5 and CH0,5.The values of the parameters are th= 90%, Un=1 mV, Δ0,5=0.2V.

Figure 15 .
Figure 15.Distribution of the number of NS solutions to the "x of x" problem, for different regions of the ION-IOFF current space without accounting for chimera synchronization at Un = 0 V (a) and Un = 1 mV (b).The numbers of the regions correspond to Figure 11.The dotted line highlights the area of the solutions to the "16 of 16" problem.

Figure 17
Figure 17.a) The density of distribution of solutions to the "16 of 16" task in the ION and IOFF current space, dotted lines indicate the boundaries of the areas (area 1, area 2, area 3 in accordance with Figure 11); b) The dependence of the synchronization efficiency η on the input data (in decimal format) at two amplitudes of internal noise Un.

Figure 18
Figure 18.a) Model scheme of ONN containing only unidirectional couplings of oscillators No.0-4 with output oscillator No.5, and b) distribution of the number of NS solutions to the "x of x" problem.

Figure 19
Figure 19.a) A fully connected network, where the input signal is determined by oscillator currents Ip and the output signal is determined by synchronization value CH b) A simplified two-oscillator circuit, where the input signal is given by the currents of thermistors c) A simplified two-oscillator circuit, where the input and output signals consist of signals of a certain synchronization.
Examples of the calculated oscillogram sections for voltage U and current Isw for the reference oscillator are shown in Figure