We implement a capacitorless model of a VO2 oscillator by introducing into the circuit of a field-effect transistor and a VO2 thermal sensor, which provide negative current feedback with a time delay. We compare the dynamics of current and voltage oscillations on a switch in a circuit with a capacitor and without a capacitor. The oscillation period in the capacitorless model is controlled in a narrow range by changing the distance between the switch and the sensor. The capacitorless model provides the possibility of significant miniaturization of the oscillator circuit, and it is important for the implementation of large arrays of oscillators in oscillatory neural networks to solve the problem of classification and pattern recognition.

by Petr Boriskov and Andrei Velichko Electronics2019, 8(9), 922; 

https://doi.org/10.3390/electronics8090922 – 22 Aug 2019

The circuits, presented in the article, are implemented in the MathCAD and LTSpice software packages, and source files are available in Supplementary Materials.

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 VO₂-oscillator circuit, the network setup is demonstrated through the selection of coupling strength, power supply levels, and the synchronization efficiency 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 effects.

In 2018, we conducted theoretical and experimental research activities on the methods development for oscillatory neural networks’ operations (networks training), for such applications as pattern storing and pattern recognition, information coding. The synchronization effects in coupled oscillators’ arrays were investigated in details, using a special family of synchronization estimation metrics; and the effects of thermal coupling delay in 3D integration of oscillators were demonstrated. The results were published in the ranked journals, including journals with open access, a number of publications are in review process; intellectual property rights’ applications were submitted and intellectual property rights were obtained. VO2 switches (oscillators) are model objects, which manufacturing technology is widely used all over the world. The described effects are of a general nature and can be used to create networks based on a wide variety of physical oscillators – electric, magnetic, and optical, which industrial manufacturing technology have been developed. In addition, the results form a new direction in technology development for the oscillatory neural networks’ operations on high-performance computing platforms: video cards and programmable logic integrated circuits.

1. A technology has been developed for creating a coupled oscillators’ array based on switching structures of transition metal oxides. The main elements of the oscillator network are VO2 switches, which constitute two-electrode planar structures with a functional layer of vanadium dioxide and two metal contacts. The connection between the oscillators is carried out by heat flows, propagating through the substrate and resulting from the Joule heating when the switches are turned on. Synthetic corundum (Al2O3) is used as a substrate; however, any other dielectric material (quartz, sitall, etc.) can be used. The substrates underwent a standard cleaning procedure from organic impurities using acetone, methanol and isopropanol. A film of amorphous vanadium oxide, which is close in stoichiometry to VO2, was deposited onto the substrate using a magnetron sputtering of a metal target in an oxygen-argon atmosphere at room temperature. The VO2 film thickness varied from 100 to 200 nm, depending on the requirements for the switch parameters. At the second stage of oscillators’ array fabrication, the square-shaped vanadium dioxide regions were formed using optical lithography methods. After the production of a lithographic mask, an amorphous oxide was etched in a 3N solution of nitric acid for 2 minutes. Then, a photoresist mask was removed using acetone, and the substrate was annealed again at 480 °C for 30-60 minutes (depending on the required switching parameters) at a pressure of 10E-2 mm Hg. and flow rate of 10 cm3/min. The final stage of the coupled oscillators’ array formation was the deposition of gold contacts.
2. A method for considering the interaction time delay of the thermally coupled oscillators has been proposed. The heat signal can propagate with a significant delay relative to the magnitude of the oscillation period, and the standard algorithm for determining the synchronization requires clarification. The modelling of the two oscillators operation at 3D integration was performed using the COMSOL computing platform. In the case of the presence of the time delay  in the interaction of oscillators, there are relative temporal displacements  of the current pulses on the oscillograms of currents. To determine the basic synchronization parameters of high order (SHR and ) in the presence of interaction time shift , it is required to determine the set of time shifts of current peaks 1, 2…i…p, choose from this set the most common value i within the duration of the processed oscillogram Tall, shift the oscillograms relative to each other by this value, and then apply standard calculation procedures. Analysing η() graph, the curve has a number of periodic minima and, in general, the efficiency of η decreases with increasing of . Designing 3D ONS, the distance l between the oscillators can be varied, and because not only the value of τ changes, but also the amount of thermal interaction changes, we observe a rather complex dependence η(l) with present minima. Apparently, the appearance of minima and maxima of the synchronization parameters, when the time delay of oscillator interaction in the network varies, is a general effect.
3. The modelling of the sandwich switch was performed. It was demonstrated that the modelled I-V characteristics had a region with negative differential resistance. The resistance of the high-resistance branch was Roff ~ 15 kΩ, and the low-resistance brach was Ron ~ 1 kΩ, the turn-on voltage was Vth ~ 1.55 V, and the switch-off voltage was Vth ~ 0.88 V, which corresponded to the order of magnitudes observed in the experiment with anode films. The temperature distribution at the turning-on moment of switch was calculated. The maximum temperature was detected at the centre of the channel and reached a value of ~ 370 K, corresponding to the phase transition temperature in vanadium dioxide. With a channel diameter of 200 nm and a film thickness of 100 nm, the effective thermal interaction radius was RT ~ 4.5 µm, and the interaction region was hemispherical.
4. The influence of a bipolar memory element on oscillatory circuits was investigated. We modified the model circuit of the memristor by adding two diodes to it, which allowed us to simulate different rates of generation and recombination of vacancies, which concentration determined the resistances of the memristor RM. In addition, the circuit had the ability to supply a negative voltage to the VM memristor, which transferred it into a closed state. The circuit of two neuron-oscillators, connected by a memristor RM and a serial capacitance Cc, was investigated. Timing diagrams of the memristor resistance RM(t) and the output voltage Vo (t) demonstrated the bursting activity of the output neuron.
5. The oscillations dynamics of an oscillators array and its application to the pattern recognition and logic problems was studied. A new direction was highlighted in this area, and the methods we propose can be successfully used in coding and recognition tasks along with already known techniques developed for neural networks. A special feature of the method is the use of the family of metrics, currently consisting of two parameters – this is the high order synchronization value SHR and the synchronization efficiency value (in the future we plan to increase their number). Another feature is that these parameters are integral over time values, which are not measured after a single impulse arrival, but the state is determined (set) after a significant number of oscillations, so, it is a more complex system where it is necessary to consider the dynamics of all associated oscillators at the same time. Two approaches to solving the recognition problem, “The Integral Approach” and “The Differential Approach”, were highlighted. To implement the former, we define a vector image, memorised or recognized as parameters of individual oscillators in the system, and study the complex synchronization of the entire system, including the input neurons. Hence, the concept of the input and output layers is absent, as they can be combined. In this case, the high order synchronization parameter has a complicated form of the type SHR=k1:k2:k3:..kN, consisting of the numbers of harmonics of several (N) oscillators. We presented this idea in a paper published in the journal “Electronics”. In the latter approach, called “The Differential Approach”, we separate the input and output layers of neurons, and synchronization is measured only between two oscillators — the output oscillator and the master (reference) oscillator, which has a constant frequency. This approach is similar to the methods applied to standard neural networks. The difference is that a single neuron at the output can have multiple states (a multilevel neuron), allowing the number of neurons to be reduced to solve certain tasks. The influence of noise on the network was studied, and the effects of stochastic resonance were detected.
6. Using numerical simulation of two- and three-oscillator circuits with thermally coupled VO2 switches, a new method of pattern (vector) storing and recognition in an impulse oscillatory neural network (ONN) based on a high-order synchronization effect was demonstrated. The method allows the storing of many images, which amount is determined by the number of synchronous states Ns. Each state of the system is characterized by a synchronization order, defined as the ratio of the harmonic numbers at the total synchronization frequency. The model achieved Ns values of several orders of magnitude, Ns~650 for three oscillatory circuits and Ns~260 for two oscillatory circuits. A number of regularities were discovered, in particular, the presence of an optimal coupling force of oscillators was found, at which Ns reached a maximum. In addition, a general tendency towards a decrease in the information capacity with an increase in the coupling force and the amplitude of the internal noise of the switches was demonstrated. An algorithm for vectors storing and test vector recognition was proposed, and it was demonstrated that to reduce recognition uncertainty, it is necessary to use the number of coordinates in each vector by one less than the number of oscillators.
7. Applying “The Differential Approach”, a new method of information coding using an impulse oscillatory neural network based on the high order synchronization effect was presented. In the proposed neural network scheme, the data is fed to the input layer in the form of supply current levels of the oscillators and converted into a set of non-repeating synchronous states of the output oscillator. Using the example of modelling a thermally coupled VO2-oscillator circuit, the network setup is demonstrated through the selection of coupling forces, power supply levels and the synchronization efficiency parameter.
8. A problem of pattern recognition, representing the figures in a 3×3 matrix, was posed and solved. The input was a set of figures in the form of a 3×3 matrix, having 512 different combinations, divided into 102 classes Cm on the basis of symmetry. The output consisted of a single oscillator. The network was trained using the model annealing method, with the values of Un and ηth being recorded, while the currents (ION, IOFF, I0, I10) and the coupling strengths between oscillators (sr, so, sm) varied randomly in certain ranges. The problem was divided into the following subproblems. Problem I: It is required to find a solution when the network recognizes one class (P = 1, where P is the number of recognizable classes) out of 102 possible Cm, with a certain value of m. Problem II: The network recognizes P> 1 classes, with assigning an original SHR synchronization value to each class. Problem III: The network recognizes all classes P = 102. Results indicated that after training, the probability of finding any solution with P = 1 (Problem I) is ~ 10%, and this is the highest probability compared to other P>1. The highest probability (~ 4%) of the solution for the set of C1 and C102 appears when all cells of the input pattern are either empty or painted over. Solutions for other m appear much less frequently, with a probability two orders of magnitude lower (~ 0.03%). However, this histogram shows that it is possible to train the network in solving Problem I, with a certain, given in advance, value m. After all the stages of training, the maximum value of P reached P = 14 that currently constitutes the best result for solving Problem II. Clearly, it is not the limit, and the configuration and the training algorithm can be developed further; however, the purpose of this task was to show the possibility of implementing a multi-level neural network and its application for pattern recognition. Currently, Problem III has no solution that represents the goal for the future research. There may be number of solutions Np of the same problem, and the probability of finding a solution can be expressed in percent; this is another argument in favour of the validity of the solution search algorithm we used by random searching (as a type of method of model annealing). In addition, we studied the noise influence on solutions search, and there is an optimal amount of noise when both NP and P reach a maximum. Therefore, there is the presence of an effect similar to the effect of stochastic resonance, which should be studied in the future. Varying the value of synchronization efficiency, which is a parameter of the processing algorithm rather than a network parameter, also showed the presence of a maximum.
9. An experiment was conducted on the additive effect of a harmonic signal and noise on a VO2 oscillator with a planar switch. In the self-oscillatory mode, we demonstrated the effect of stochastic synchronization with the presence of a resonance of signal-to-noise ratio, estimated by the originally developed spectral method.
10. It was demonstrated that photo-resistive transformation of oscillator parameters by the elements of their external circuit (photodiodes) is a convenient way of translating visual images into ONN. Since the simplest option for adjusting the oscillation frequency of oscillators is a variation of their supply currents, photodiodes comprising load elements in the EMF circuit effectively solve this problem, unlike the complex and ambiguous electron beam modification of VO2 switches.

Electronics 2019, 8(1), 75;

https://doi.org/10.3390/electronics8010075

Abstract

The current study uses a novel method of multilevel neurons and high order synchronization effects described by a family of special metrics, for pattern recognition in an oscillatory neural network (ONN). The output oscillator (neuron) of the network has multilevel variations in its synchronization value with the reference oscillator, and allows classification of an input pattern into a set of classes. The ONN model is implemented on thermally-coupled vanadium dioxide oscillators. The ONN is trained by the simulated annealing algorithm for selection of the network parameters. The results demonstrate that ONN is capable of classifying 512 visual patterns (as a cell array 3 × 3, distributed by symmetry into 102 classes) into a set of classes with a maximum number of elements up to fourteen. The classification capability of the network depends on the interior noise level and synchronization effectiveness parameter. The model allows for designing multilevel output cascades of neural networks with high net data throughput. The presented method can be applied in ONNs with various coupling mechanisms and oscillator topology.

 

The effect of the time delay of thermal coupling on the synchronization of two VO2-oscillators has been studied for the first time. A modification of the methods for determining high order synchronization parameters has been proposed. The dependence of these parameters on the magnitude of the time delay and
distance between the oscillators in the presence of internal noise has been studied. The results will contribute to the development of technology for creating three-dimensional neural networks.

Technical Physics Letters, 2019, Vol. 45, No. 2, pp. 61–64.

Please follow the link and read the full text of the article published in the journal Electronics. Development of neuromorphic systems based on new nanoelectronics materials and devices is of immediate interest for solving the problems of cognitive technology and cybernetics. Computational modelling of two- and three-oscillator schemes with thermally coupled VO₂-switches is used to demonstrate a novel method of pattern storage and recognition in an impulse oscillator neural network (ONN) based on the high-order synchronization effect. The method allows storage of many patterns and their number depends on the number of synchronous states N_s.  The modelling demonstrates attainment of N_s of several orders both for a three-oscillator scheme N_s~650 and for a two-oscillator scheme N_s~260. A number of regularities are obtained, in particular, an optimal strength of oscillator coupling is revealed when N_s has a maximum. Algorithms of vectors storage, network training and test vector recognition are suggested, where the parameter of synchronization effectiveness is used as a degree of match. It is shown that  to reduce the ambiguity of recognition the number of coordinated in each vector should be at least by one unit less than the number of oscillators. The demonstrated результаты is a general one and it may be applied in ONNs with various mechanisms and oscillator coupling topology.

The advantage of the thermal mode of interaction of oscillators for 3D integration of VO2 switches is that thermal waves propagate radially from the switch in all directions over the substrate, and therefore allow thermally coupling oscillators located on different layers.

1. Velichko, A.; Belyaev, M.; Putrolaynen, V.; Perminov, V.; Pergament, A. Thermal coupling and effect of subharmonic synchronization in a system of two VO2 based oscillators. Solid. State. Electron. 2018, 141, 40–49, https://doi.org/10.1016/J.SSE.2017.12.003
2. Velichko, A.; Belyaev, M.; Putrolaynen, V.; Perminov, V.; Pergament, A. Modeling of thermal coupling in VO 2 -based oscillatory neural networks. Solid. State. Electron. 2018, 139, 8–14, https://doi.org/10.1016/j.sse.2017.09.014
3. Patent for invention №2663546 from 07.08.2018 A method of interaction in the system of coupled oscillators based on oxide structures with the effect of an electric switch, authors V. Putrolainen , Velichko A.A. , Belyaev M.A. , Pergament A.L. (priority date 05/31/2017)

In the course of research during 2017 we have studied theoretically and experimentally new synchronization effects of oscillators including memristors and specific features of electrical switching for planar and sandwich switches at scaling. We have proposed a type of thermal coupling that is alternative to electrical coupling and is promising for 3D oscillator networks integration and all-to-all coupling. We have performed research of synchronization effect on subharmonics to increase classification capacity of an oscillator system and pattern recognition as well as long-distance synchronization. 
The described findings are mostly of a universal character and may be used in other switching structures including the ones based on semiconducting components that are mass produced. 
1. We fabricated planar switching structures with interelectrode spacing a from 500 to 3000 nm using electron and laser lithography with magnetron sputtering. We found out experimentally that threshold currents and voltages, time of switch-on and off, and threshold  power decreased with the decrease  of a value. Generalized dependences of threshold characteristics on switching area scale and film features were revealed using numerical modeling. 
Evaluation of maximal oscillation frequencies seems to be the most interesting thing so we simulated the operation of a switch included into the oscillator scheme. Evaluation of oscillator self-frequency F0 at specific capacity (C=10 nF) shows that F0 falls insignificantly when the dimensions of a switching structure decrease despite the reduction of switching time characteristics. Moreover, with the decrease of а it is possible to decrease minimal capacity at which oscillations still exist. Thus, when a~100 nm, maximum oscillation frequency may reach F0~20 MHz and F0~300 MHz in some cases. 
We have shown that sandwich structures are easy to use when studying the physics of single nanostructures switching, for instance, the structures based on amorphous anode film VO2 are good model objects. Therefore we have developed a stand including an atomic-force microscope SMM-2000 to obtain topographical maps and capacity distribution map for the studied surface before and after electrical formation (in situ). 
We used the stand to study the process of electrical formation in a vanadium oxide film ~50 nm thick. We showed that after electrical formation a modified area with increased conductivity was formed; this area corresponded to the formed channel of ~500 nm in diameter that in its turn consisted of nanochannels of 10-100 nm in diameter. 
To conclude, it is preferable to use planar structures at this stage because sandwich technology lacks channel electrical formation. Nevertheless, amorphous VO2 sandwich structures are promising in developing 3D technology of elements integration and miniaturization. 
2. It is shown that there are some possibilities to use memristors of both types in oscillation circuits; however, transition between states for an unipolar resistive memory requires more complex schemes that include a current controlled stopper. For a bipolar memory the transition scheme is much simpler as it is possible to create alternative impulses of positive and negative polarity on certain branches of an oscillation circuit. 
It is shown that bipolar memory displays a multistability character when low resistance state may vary in a wide range. It is appropriate to speak about a quasi-stable state because cell resistance depends not only on the value of applied voltage but on the time as well. Such behavior stems from field control of oxygen vacancies concentration in the interelectrode space. At positive polarity their concentration increases; when this happens back diffusion is observed eventually resulting in recovery of resistance equilibrium values. 
Two-electrode Ti, Ta, Nb, V oxide-based MOS (MOM) sandwich structures were formed using reactive magnetron sputtering and anodic oxidation. A bipolar memory was obtained without preliminary formation in most cases which is a definite advantage because formation leads to the increase of switch-on current and multistability suppression. . 
Niobium and vanadium anodic amorphous films seem to be good candidates to perform the role of a multistable bipolar memory but for one disadvantage – liquid electrolyte usage. Therefore it must be concluded that a multistable bipolar memory on the basis of the mentioned oxides may become the element for modeling a synaptic plasticity of a neural network. 
3.  We have developed a schematic model of a bipolar memory that resembled quasi-stable behavior of real cells in its I-V characteristics dynamics. Here we used an analogy of vacancies concentration to the capacitor voltage that depended not only on the input voltage amplitude but on the time as well because of resistive charging and discharging. 
A model cell (memristor) was used in schemes of single and double coupled circuits. In the first scheme we observed a very interesting mode when the pattern of the voltage oscillogram on the memristors reminds a burst neural activity and is caused by sequential change of a single oscillator operation mode. 
The scheme of coupled oscillators with a memory cell was created to simulate the education process. A memristor is often used in simulating organism’s neural systems as an element that allows for simulation of a synaptic plasticity effect expressed in the increase of synaptic coupling force when postsynaptic receptors are activated. To simulate the effect of synaptic plasticity we used two neurons-oscillators linked by a memristor and sequential capacity. The capacity was necessary to realize unipolar voltage impulses in a memory cell during oscillations. The first oscillator operating at a normal mode excited the second oscillator in a subthreshold mode when the value of supply current was not enough to cause oscillations. Memristor’s resistance started to decrease under the influence of voltage impulses, i.e. the electric coupling between circuits became stronger. When memristor’s resistance fell down to some threshold level the second oscillator switched on. 
Thus, we have shown the dynamics of a single and two coupled oscillators with a memristor included into the scheme that simulates the effects of neurons’ train activity and synaptic plasticity and associated educational process.
4. We have developed a scheme of two thermally coupled oscillators with load resistances (Rx, Ry) represented as 3×3 matrix of photoconductive converters. Two galvanically isolated photoconductors in one field of the matrix are illuminated simultaneously and the matrix itself is a sensor of illuminated images (figures). A specific pattern of light and dark fields and certain values of Rx and Ry and therefore its own SHR (if the static parameters of the scheme are adequate) correspond to each figure. Thus this system consisting of two oscillators is able to classify and recognize patterns with regard to figure symmetry. 
5. We have performed experimental research and numerical modeling of a switching channel dynamics in planar vanadium dioxide-based structures and have established some regularities of behavior for transition time from high to low impedance states and back depending on the impulse amplitude and duration and the value of base voltage. The obtained data together with the calculated ones and the results of temperature measurements in the switching channel indicate that in vanadium dioxide “metal-semiconductor” phase transition induced by Joule heating plays an essential role. 
We have proposed a type of thermal coupling that is alternative to electrical coupling and is promising for 3D oscillator networks integration, all-to-all coupling and the effect of subharmonic synchronization. 
Thermal coupling between oscillators is realized because of heat transfer through the substrate where the oscillator elements are located. To do this circuit elements should act as heat sources of variable power during oscillations generation (it could be any resistive element suitable for current flow including a switching element itself) and the parameter of the surrounding oscillators should be highly dependent on temperature (for instance, threshold voltage of a switch depends heavily on temperature). 
We have obtained the calculated dependences of thermal coupling radii RTC corresponding ΔT ~ 0.2 K on the capacity value and limit resistor. This proves that the force and radius of coupling could be controlled not only by the static values such as the space between the structures and substrate heat constants, but also by the dynamic values through varying the scheme parameters. 
Experimental and numerical studies of oscillations of two thermally coupled oscillators led us to the phenomenon of subharmonic synchronization (partial synchronization). This phenomenon indicates that synchronization may occur on nonbasic (divisible) harmonics of an oscillation spectrum. 
We have developed a rather simple but effective algorithm based on phase transition that determines the presence of synchronization effect, synchronization effectiveness µ and SHR value (ratio of harmonic order at synchronization frequency). 
A map of the form of Arnold’s tongue was calculated for synchronous states in the area of power currents. Twelve various states were obtained for the cases of strong coupling that corresponded to experimental parameters and noise level. For other system parameters with lower noise level the number of possible states could increase significantly and could reach 200 allowing for 20 first harmonics. 
Thus, the subharminic synchronization effect possesses a large potential for classification and may be used for pattern recognition problems. 
6.  It has been shown that the synchronization effect at long distances may be observed in the chain of thermally coupled oscillators when the synchronization parameter (SHR) between the outermost chain elements is expressed by product of neighboring oscillators. Therefore the sunharmonics synchronization effect enables us to realize synchronization at larger distances through pairwise interaction of oscillators comprising the coupling chain. It has been shown that when power current for two first oscillators (I1, I2) in the chain of N=100 oscillators is varied then the distribution SHR1,100  in the area of controlling current has the gradient only along I1 axes in contrast to two-oscillators scheme. At the same time the symmetry of synchronous states remains diagonal thus highlighting the role of mutual current ratio of neighboring oscillators for the physics of synchronism spreading along the chain. However, the synchronization areas are distributed unevenly and do not form a striking system of Arnold’s tongues. 
When power currents of central oscillators (I51, I52) are varied we obtain the distribution of only one single state SHR1,100=1/1, this seems to be caused by unchanged frequencies of outermost oscillators. Thus, the ratio I51 и I52 performs the role of a trigger that switches on and off the synchronization process of the outermost oscillators. 
7.  We consider that the phenomenon of subharmonic synchronization should be used as the main method of pattern recognition that increases the classification capacity of an oscillator system. We have studied thermal coupling in the course of our work, however, analogous effects of partial synchronization may be obtained for other coupling types (R- and C-type), because the physical mechanism of subharmonic synchronization is universal.

1) A technology has been developed for the fabrication of planar switching elements based on VO2 films with the metal–insulator phase transition (MIPT). Thin (~ 200 nm) vanadium dioxide films are obtained by DC magnetron sputtering followed by annealing in an oxygen atmosphere (10 mTorr, 480°C, t = 40 min.). The best for the electrical switching are the non-stoichiometric VO2 films with the resistivity jump at MIPT of up to two orders of magnitude, the phase transition temperature of ~ 60°C (as compared with Tt ≈ 70°C in stoichiometric VO2) and with an increased conductivity. By optical lithography and lift-off process, planar switching structures are formed with two electrodes (50-nm thick V-Au bilayer metallic contacts) and an interelectrode gap of about 2.5 microns; the electrode width is ~ 10 microns. The I-V characteristics of the switch are S-shaped with a threshold voltage of 1 to 15 Volts; a current jump at the switching threshold voltage Vth is about 10, and at the switching-off voltage Vh it is ~5; the dynamic resistances in the ON and OFF states are Ron ~ 40 Ohm and Roff  ~ 1.1 kOhm. Static resistance in the ON state is nonlinear and varies from 40 to 200 Ohms in the voltage range from Vh to Vth.
This technology has allowed us to obtain two-electrode structures, with the stable switching effect, that can withstand more than 10E8 switching cycles, as well as to install the switches onto breadboards for further study in vacuum.

2) To observe the auto-oscillation, a circuit has been assembled on the basis of a single VO2 switch. The series resistance Rs and the power supply voltage Vdd (in the range of 20 to 100 V) are chosen so that the load line passes across the negative differential resistance (NDR) region, which is a condition for the oscillation to occur. By variation of the conditions and regimes of film preparation (in particular, the annealing time t), the optimum parameters are selected for which the minimum value of Vth and the maximum ratio of the holding current to the threshold current Ih/Ith are reached. This provides the maximum slope of the NDR region and hence a better stability of the resulting oscillations.
Using the LTSpice software, the model of a VO2-switch is built, which describes the switching structure behavior. Experimental waveforms of oscillation of the voltage drop across the switch, as well as and the oscillation spectrum, coincide, with a high degree of accuracy, with the corresponding theoretical curves obtained within this model. A broadening of spectral peaks associated with the presence of noise in the subthreshold region is found, which leads to a weak frequency modulation of auto-oscillations by a chaotic signal; also, the stochastic resonance effects are observed.
When studying a single circuit, the phenomenon of bistability is observed, which consists in a controlled oscillation switching “ON” and “OFF” by an external pulse. This effect is caused by the total structure heating due to the fact that, during one oscillation period, the switching channel does not have time to cool down to the equilibrium temperature. To terminate the auto-oscillations, a pulse of reverse polarity is applied with an amplitude sufficient to switch-off the structure and duration equal to the time of the channel cooling to the equilibrium temperature. One can utilize this effect to create oscillatory memory cells, as well as in order to implement and a pulse communication mode in artificial neural networks.

3) It is shown that electron-beam modification (EBM) substantially changes the MIPT parameters, lowers the transition temperature Tt and results in a general decrease in resistance both in metallic and in semiconducting phase. This decrease in the Roff and Tt values, in turn, causes a reduction of Vth.  
A quantum-chemical calculation of the energies of formation of oxygen vacancies and oxygen migration in the monoclinic and tetragonal phases of vanadium dioxide has been performed. It is shown that diffusion in both phases of vanadium dioxide has a preferential direction of migration of oxygen along the crystallographic axis a in the monoclinic phase (T < Tt), while it does preferentially along the axis c in the tetragonal rutile phase (T > Tt). The difference in the rate of generation of oxygen vacancies under electron-beam exposure above and below the MIPT temperature is due to a jump (by a factor of ~1.5) of the oxygen diffusion activation energy at the structural phase transition, which changes the mobility of oxygen (oxygen vacancies) by more than an order of magnitude. Calculations of the electronic structure of the crystal have been carried out using the Quantum Espresso software package.
Next we have studied the EBM of the I-V curve threshold parameters of a VO2-switch and oscillation frequency in a circuit based on it. It is shown that EBM conducted in vacuum possesses a reversible nature, and the parameters can be recovered in air at a pressure of 150 Pa. At EBM with a dose of 3 C/cm2, the voltages Vth and Vh, as well as the OFF state resistance Roff  decrease down to 50% of the initial values. The oscillation frequency increases by 30% at an EBM dose of 0.7 C/cm2. The physical mechanism of the observed processes is associated with redox reactions and formation of oxygen vacancies as a result of EBM, taking into account the above-described differences in contributions of metallic and semiconducting phases of the switching channel.
Controllable EBM allows fulfillment of electron-beam forming of switches with preset parameters, and EBM can be used in artificial neural networks for pattern recognition based on frequency shift keying.

4) Numerical simulation of the dynamics of oscillations, taking into account the switch parameters change at EBM, as well as calculation of the temporal characteristics and temperature distribution in the channel switch, have been carried out. For this, the two approaches are utilized: 1) the use of standard mathematical complexes that can predict the operation of a circuit (LTSpice package) and to simulate physical processes in microstructures (COMSOL Multiphysics package); 2) the creation of the own software systems using ab initio methods and analytical evaluation of the parameters of simplest oscillation circuits.
It is shown that the theoretical (obtained by numerical simulation) curves accurately fit the experimental data, in particular, such as the dependence of the oscillator frequency on the EBM radiation dose, temporary characteristics of recovery after EBM in the metal and semiconductor phases, as well as the stationary current-voltage curves and temperature distribution in a planar switch.

5) The switching dynamics of two coupled VO2-based oscillators with resistive and capacitive coupling has been studied, and the capability of their application in oscillatory neural networks has been explored. An adequate SPICE model to describe the modes of operation of coupled oscillator circuits is proposed. Physical mechanisms influencing the time of forward and reverse electrical switching, that determine the applicability limits of the proposed model, are identified.
For the resistive coupling, it is shown that synchronization takes place at a certain value of the coupling resistance, though it is unstable and a synchronization failure occurs periodically. At the minimum coupling strength value, is neither frequency nor phase synchronization of oscillator circuits is observed. When resistive coupling lies in the range 3 kOhm < Rcoup < 10 kOhm, there is a gradual irregular distortion of the voltage oscillation waveforms. This change is due to a bypass of one switch by another at their mutual switching into a high-conductivity phase. In this range of the resistive coupling, spectra distortion is observe accompanied by the appearance of new harmonics at mutual modulation, and oscillations in general have a quasi-periodic character.
For the capacitive coupling, two synchronization modes, with weak and strong coupling, are found. The transition between these modes is accompanied by chaotic oscillations, and the evolution to chaos occurs via the period doubling. A decrease in the width of the spectrum harmonics in the weak-coupling mode, and its increase in the strong-coupling one, is detected. Thus, oscillators coupled via capacitance shows a greater variety of operation modes as compared to the resistive coupling. At the capacitive coupling, oscillations are observed in the entire range of the Ccoup values. In the weak-coupling mode, with an increase in the coupling capacity, the oscillators’ frequencies tend to converge, and beyond the threshold Ccoup ~ 2.5 nF, the oscillators are synchronized. Unlike the resistive coupling, the phase difference after synchronization can be controllably varied over a wide range (from 30º to 170º), so the use of this type of coupling is more preferable in the pattern recognition systems using phase shift based detection schemes (phase shift keying).
As an application of the considered systems of coupled oscillators, a joint operation of R- and C-couplings in the ensemble of four oscillator, to simulate the central pattern generator (CPG), is demonstrated.

6) The effect of electron irradiation on the synchronization of coupled oscillators has been studied. Initial frequencies of oscillators are 850 Hz and 770 Hz, which do not provide synchronization at the capacitive coupling of Ccoup = 10 nF. To get synchronization, the frequency of one of the oscillators has been changed using EBM. It is shown that synchronization is established at a radiation dose of ~1 C/cm2, and upon further irradiation, desynchronization occurs.
The response of the oscillatory system to the electron beam exposure is thus demonstrated, and an important aspect here is the direction of the frequency change under EBM. Also, we note the fact that even in the presence of a noisiness of the oscillation frequency parameters, one can fairly accurately predict the occurrence of synchronization, basing upon the behavior of the phase difference. Therefore, the method of phase detection of synchronization may be a key method for the analysis of oscillatory neural networks.

7) A photoresistor connected in series has been added to the auto-oscillation circuit. It is shown that the illumination of this photoresistor affects the oscillation frequency. The illumination values (1.25 - 4.1 lux, for the photoresistor type LDR GL5516), at which stable oscillations commence, are found. When the illumination is I < 1.25 lux, the switch is in a steady off state, and when I > 4.1 lux, it is in a steady on state. This method of controlling the parameters of oscillation in circuits based on VO2-switch is similar to the above-described method with EBM. However, the need of e-beam modification of the switch may not always be appropriate because of the complexity of the electron exposure system and because of the vacuum conditions requirement.

8) The simulation of the recognition of a vector image function, using the method of the oscillation phase synchronization, has been carried out. The recognition technique when using such a system consists in the establishment of an initial (input) phase difference Δφt (input phase vector) and identification, after some time Δts, a finite phase difference Δφij, which is one of the stored eigenvector images. In the model experiment, we have used three single oscillator circuit based on VO2-switches connected, via coupling capacities, in the “asterisk” topology. For modeling, the NGSpice medium, equipped with a software for the model experiment automation (written in C++) for calculations in conditions of variation of the circuit parameter values, is used. The simulation results have shown that, the smaller the difference between the input and eigen phase vectors, the smaller is phase stabilization time, and the act of recognition consists in the identification of the "winner" (the least time criterion) from the set of input vectors.