This paper considers one of the problems that arise in the developing of the ergatic brain-computer interfaces. This technology allows a person to control various mechatronic systems through the "power of thought", i.e. based on the registration of electrical activity of the brain. The problem is the complexity and poor knowledge of the brain. To describe the electrical activity of the brain, various models of neural ensembles are used, one of which is the neural mass model proposed by Jansen and Rit in 1995. To tune the parameters of this model according to real data, it is proposed to use an adaptive parameter identifier. An important condition for the synthesis of an adaptive identifier is that only the system output, which is the potential difference between two points of the head, can be measured. At the beginning, it is assumed that the entire state vector of the neural mass model is available for measurement. An identifier is synthesized to tune the parameters of such a system and its convergence is proved using the Lyapunov function method. Further, the obtained identifier is refined in such a way that it uses only the output of the system. To do this, using the finite difference method, the output derivative of the neural mass model is approximately calculated, which is used to make several replacements of the unknown components of the state vector. It is very difficult to analytically prove the convergence of the obtained adaptive parameter identifier, therefore, the possibility of using it to estimate the parameters of a neural mass model is checked using simulation. The synthesized identifier uses only the system output to tune the parameters, which in the future will allow us to consider real data instead of the system output. Thus, this identifier can be used to tune the parameters of the neural mass model based on real data.

This paper explores the concept of Yakubovich oscillatory behavior. The existing result on this topic has been extended, and sufficient conditions for Yakubovich oscillatory behavior in nonlinear systems with bounded delay have been established and proven. Estimates of the oscillation range for nonlinear dynamic systems are provided. To illustrate the theory, the oscillatory nonlinear system with cubic nonlinearity and bounded delay has been considered.

Study of dynamics of complex networked systems is one of the relevant problems. Networked systems can be in various states, ranging from complete synchronization, when all systems in the network are coherent, to complete desynchronization, i.e. complete incoherence in the functioning of systems. Synchronization phenomenon has already been well studied, namely, the mathematical definitions of synchronization are introduced, algorithms of studying synchronization are proposed, and synchronization conditions of various types of networked systems are established. Whereas a few works are devoted to the study of desynchronization nowadays. This paper introduces output desynchronization notion for networks of nonlinear systems. The definitions about Yakubovich oscillatority are considered and the link between oscillatority and desynchronization in networks of excitable nonlinear systems is established. Excitable systems are stable; therefore, they do not generate oscillations. Adding couplings between such systems can lead to occurrence of oscillations. The conditions about oscillatority in diffusively coupled networks of FitzHugh-Nagumo systems, which are the simplest neuron models, are derived. Firstly, the case of the simplest network of two coupled systems is considered, and afterwards, obtained result is generalized for the case of several systems. Laplace matrix spectrum plays crucial role in dynamics of such networks. The condition that connects the parameters of the uncoupled system in the network and the eigenvalues of the Laplace matrix, is obtained which determines whether the network is oscillatory or not. The number of systems that generate oscillations in such a network depends on the number of eigenvalues of the Laplace matrix that satisfy the obtained conditions. Obtained analytical results are confirmed by simulation. The results of simulation of complete desynchronization in the network, when all systems begin to oscillate, as well as a chimera-like state, in which only a part of the systems oscillates, while the other part are rest, are presented.

The problem of synchronization in networks of neural mass model populations with discrete couplings is consid-ered. The considered network is hybrid one, therefore Mikheev approach is applied to transform it to the network with time-varying delayed couplings. Thus the problem of hybrid network synchronization is reduced to the studying of synchronization in networks with delayed couplings, which was previously solved by analytical means. It is showed that the Laplace matrix spectrum and maximum sampling interval are defining for networks dynamics. The dynamics of 5 neural mass model populations with discrete couplings was simulated for 3 different situations. The first case deal with the asymptotic synchronization, when both maximum eigenvalue of Laplacian and maximum sampling interval are small enough. The second case is about $\varepsilon$ -synchronization, which is achieved for small enough maximum eigenvalue of Laplacian and big sampling intervals. And the last case is desynchronization of oscillations, which has been observed for big values of Laplacian eigenvalues and sampling intervals.

The problem of synchronization in heterogeneous networks of linear systems with nonlinear delayed diffusive coupling is considered. The network is presented in new coordinates mean-field dynamics and synchronization errors. Thus the problem of network synchronization is reduced to the studying of synchronization-error system stability. The circle criterion for time-delay systems is used to derive the stability conditions of synchronization-error system. Obtained results are applied to a network of neural mass model populations, and the synchronization conditions are established. Simulation results are provided to illustrate the obtained analytical results.

Neural oscillations are electrical activities of the brain measurable at different frequencies. This paper studies the interaction between the fast and slow processes in the brain. We recorded signals intracranially from the simple Wistar rats, performed the signal processing, and computed the correlation between envelopes of the high-frequency gamma rhythm and a low-frequency signal. The analysis shows that the low-frequency signal (delta rhythm) modulates the gamma rhythm with a small time delay. Further, we used simple excitable neuron models, namely FitzHugh-Nagumo and Hindmarsh-Rose, to simulate the gamma rhythm. The low-frequency signal delta rhythm can be used as the input to affect the threshold and simulate gamma rhythm using these neuron models.

The paper studies controlled synchronization in regular delay-coupled Hindmarsh-Rose network with a constant delay. It is the fact that signal propagation delays between nodes can hinder their synchronization. This investigation introduces a controller that can ensure the asymptotic synchronization between neurons in the network under study. The provided analysis is based on the Lyapunov-Krasovskii method.

Neural oscillations are electrical activities of the brain measurable at different frequencies. This paper studies the interaction between the fast and slow processes in the brain. We analyze electrocorticogram (ECoG) signals from the simple Wistar rats with developed tools application: low-pass and band- pass zero-phase filters to separate slow and fast components and envelope extraction to determine gamma-amplitude bursts and attenuations. Then we compute the correlation between the gamma rhythm envelope and a low-frequency ECoG signal. The analysis shows that the low-frequency signal (identified as delta rhythm) modulates the gamma rhythm with an approximately half-second time delay.

The algebraic connectivity is crucial parameter in studying of synchronization of diffusively coupled networks. This paper studies the synchronization in networks of Hindmarsh-Rose systems, which is one of the most used neuron models. It presents sufficient condition for synchronization in these networks using the Lyapunov function method. This is a simple condition which depends on the algebraic connectivity and on the parameters of the individual system. Numerical examples are presented to illustrate the obtained results.

The problem of synchronization in networks of linear systems with nonlinear diffusive coupling and a connected undirected graph is studied. By means of a coordinate transformation, the system is reduced to the form of mean-field dynamics and a synchronization-error system. The network synchronization conditions are established based on the stability conditions of the synchronization-error system obtained using the circle criterion, and the results are used to derive the condition for synchronization in a network of neural-mass-model populations with a connected undirected graph. Simulation examples are presented to illustrate the obtained results.

The brain is processing information 24 hours a day. There are millions of processes proceeding in it accompanied by various spectra of rhythms. This paper tests the hypothesis that the slow delta rhythm excites the gamma rhythm oscillations. Unlike other papers, we determine the slow rhythm spectrum not at the hypothesis stage but during the experiment. We design algorithms of filtering, envelope extraction, and correlation coefficient calculation for signal processing. Moreover, we examine the data on all electroencephalogram channels, which allows us to make a more reasonable conclusion. We confirm that a slow delta rhythm excites a fast gamma rhythm with an amplitude-phase type of interaction and calculate a delay between these two signals equal to about half a second.

The desynchronization problems in oscillatory networks is considered. A new desynchronization notion is introduced and desynchronization conditions are provided. The desynchronization notion is formulated in terms of Yakubovich oscillatority of the auxiliary synchronization error system. As an example, the network of diffusively coupled FitzHugh-Nagumo systems with undirected graph is considered. The simple inequality guaranteeing network desynchronization is derived. The simulation results confirm the validity of the obtained analytical results.

The paper provides a historical overview of the Speed-gradient method and its applications to adaptive control and identification problems since mid-1970-th, when the method was originated, till the present days. It is demonstrated that it is an efficient and an useful tool for solving a wide range of engineering problems.

In this paper, a method for forming the neuro-feedback (NFB) signal is suggested. The implementation of the method is based on the design of an adaptive model of subjects’ neural activity in the form of a vector autoregression (VAR) model for capturing the spatio-temporal dynamics of an electroencephalogram (EEG). The adaptive model parameter adjustment algorithm is iterative and is designed using gradient methods for solving goal inequalities. At each stage of training, the control signal is first calculated. Then, based on the control signal, the model coefficients are modified. With the improvement of the EEG recording in the sense of its proximity to the desired one, the coefficients of the EEG model are recalculated. The proposed scheme has physiological analogues and is not directly related to the EEG model used in calculating the control effect. The proposed algorithm has advantages over other known algorithms because of its adaptability to each specific subject.

This paper studies the influence of the coupling type on the network desynchro-nization and proposes the algorithms to control it. We consider the FitzHugh-Nagumo network with random topology and different coupling values for different nodes. The simple mean field control which is similar for all nodes can be applicable for desynchronization of FitzHugh-Nagumo network with direct coupling. However, the network with diffusive coupling is much harder to desynchronize. In this case for large enough coupling values one should use different control algorithms for different nodes to desynchronize the network.

A new approach to the design of neurofeedback systems based on using Artificial Intelligence (AI) tools is proposed. The concept of control models of biological neural networks, and the set-up including equipment and software tools developed in IPME RAS in order to implement the proposed concept is described. as well as the AI methods and programs proposed for use.

article studies the influence of the small delays on the FitzHugh-Nagumo network synchronization. It is widely known that high delays in signal propagation between the nodes make synchronization difficult or even impossible. The sufficient conditions of the linearized network synchronization for the case of the small delay are obtained. This problem is successfully reduced to the feasibility of the LMIs. The simulation results confirm the efficiency of the obtained conditions. We suppose that the similar conditions can be applicable even to the nonlinear FitzHugh-Nagumo network.

The problem of control of the number of cycle slippings of an electric machine rotor by means of an external moment is considered by the example of a simple mathematical model. The speed-gradient method with the objective function determined by the oscillation energy function is applied to solve this problem. The use of quite a small control is a feature of this approach, which helps to save energy. We have developed an algorithm to control oscillations of an electric machine rotor, so that the rotor performs a predetermined number of cycle slippings. The simulation results illustrate the efficiency of the suggested algorithm.

This paper formulates a new inverse Stoker problem: to design the control algorithm for performing the desired number of cycle slippings under arbitrary initital conditions. To solve the posed problem two control algorithms suggested. The first algorithm is based on the speed-gradient algorithm, while the second one is a simple relay algorithm. For simulation the problem to perform a desired number of cycle slippings at the begining and then to make rotor oscillate with a constant amplitude is posed. The results of simulation showed the efficiency of proposed algorithms.

The problem of pendulum’s energy control in presence of an irregular input disturbance is considered. A feedback control law is chosen based on the speed gradient method. The main contribution of the paper is in studying the complex behavior of the previously designed system under irregular disturbances. The main result is precise estimates for an initial set and a limit set (attractor) as well as the conditions guaranteeing the following: all the solutions starting in the initial set will enter the limit set in a finite time.