Adaptive Antisynchronization of Multilayer Reaction-Diffusion Neural Networks

Stability Analysis of Fractional Bidirectional Associative Memory Neural Networks With Multiple Proportional Delays and Distributed Delays

This article investigates the finite-time stability of a class of fractional-order bidirectional associative memory neural networks (FOBAMNNs) with multiple proportional and distributed delays. Different from the existing Gronwall integral inequality with single proportional delay ( ), we establish the Gronwall integral inequality with multiple proportional delays for the first time in the case of . Since the existing fractional-order single-constant delay Gronwall inequality with two different orders cannot be directly applied to the stability analysis of the aforementioned system, initially, we skillfully develop a novel one with generalized fractional multiproportional delays' Gronwall inequalities of different orders. Furthermore, combined with the newly constructed generalized inequality, the stability criteria of FOBAMNNs with fractional orders and under weaker conditions, i.e., at most linear growth and linear growth conditions rather than the global Lipschitz condition, are given respectively. Finally, numerical experiments verify the effectiveness of the proposed method.

Reachable Set Estimation for Memristive Complex-Valued Neural Networks With Disturbances

This brief focuses on reachable set estimation for memristive complex-valued neural networks (MCVNNs) with disturbances. Based on algebraic calculation and Gronwall-Bellman inequality, the states of MCVNNs with bounded input disturbances converge within a sphere. From this, the convergence speed is also obtained. In addition, an observer for MCVNNs is designed. Two illustrative simulations are also given to show the effectiveness of the obtained conclusions.

Bounded Antisynchronization of Multiple Neural Networks via Multilevel Hybrid Control

The bounded antisynchronization (AS) problem of multiple discrete-time neural networks (NNs) based on the fuzzy model is studied, in consideration of the differences in quantity and communication among different NN groups, the variabilities of dynamics, and communication topological affected by environments. To reduce the energy consumption of communication, a cluster pinning communication mechanism is proposed, and an impulsive observer is designed to estimate the state of target NN. Then, a multilevel hybrid controller based on the impulsive observer is built including the AS controller and the bounded synchronization (BS) controller. Sufficient conditions for bounded AS are obtained by analyzing the stability of the BS augmented error (BSAE) and the AS augmented error (ASAE) based on the fuzzy-based Lyapunov functional (FBLF). Finally, a numerical example and an application example are given to verify the validity of the obtained results.

Adaptive Synchronization of Reaction-Diffusion Neural Networks With Nondifferentiable Delay via State Coupling and Spatial Coupling

In this article, master-slave synchronization of reaction-diffusion neural networks (RDNNs) with nondifferentiable delay is investigated via the adaptive control method. First, centralized and decentralized adaptive controllers with state coupling are designed, respectively, and a new analytical method by discussing the size of adaptive gain is proposed to prove the convergence of the adaptively controlled error system with general delay. Then, spatial coupling with adaptive gains depending on the diffusion information of the state is first proposed to achieve the master-slave synchronization of delayed RDNNs, while this coupling structure was regarded as a negative effect in most of the existing works. Finally, numerical examples are given to show the effectiveness of the proposed adaptive controllers. In comparison with the existing adaptive controllers, the proposed adaptive controllers in this article are still effective even if the network parameters are unknown and the delay is nonsmooth, and thus have a wider range of applications.

Complete synchronization in fixed/preassigned time of multilayered heterogeneous networks

This paper is concentrated on the fixed/preassigned-time (FXT/PAT) synchronization of multilayered networks, in which the self-dynamics of nodes are heterogeneous and the synchronized state can be an arbitrary prescribed smooth orbit. Above all, the original network is augmented by involving the synchronized state as a virtual node, it is allowed to remove the topological connectivity limitations and reduce the conservatism of the synchronization conditions. Subsequently, several continuous control protocols have been developed to achieve FXT synchronization and some effective criteria are established by utilizing the theorem of FXT stability. Additionally, the relationship is revealed between the estimation of the synchronized time and the layer parameter. Moreover, the PAT synchronization is investigated for a preassigned synchronized time by proposing two control schemes with finite control gains. Eventually, the developed control designs and criteria are validated by some numerical simulations.

Interlayer antisynchronization in degree-biased duplex networks

With synchronization being one of nature's most ubiquitous collective behaviors, the field of network synchronization has experienced tremendous growth, leading to significant theoretical developments. However, most previous studies consider uniform connection weights and undirected networks with positive coupling. In the present article, we incorporate the asymmetry in a two-layer multiplex network by assigning the ratio of the adjacent nodes' degrees as the weights to the intralayer edges. Despite the presence of degree-biased weighting mechanism and attractive-repulsive coupling strengths, we are able to find the necessary conditions for intralayer synchronization and interlayer antisynchronization and test whether these two macroscopic states can withstand demultiplexing in a network. During the occurrence of these two states, we analytically calculate the oscillator's amplitude. In addition to deriving the local stability conditions for interlayer antisynchronization via the master stability function approach, we also construct a suitable Lyapunov function to determine a sufficient condition for global stability. We provide numerical evidence to show the necessity of negative interlayer coupling strength for the occurrence of antisynchronization, and such repulsive interlayer coupling coefficients cannot destroy intralayer synchronization.

Prespecified-time bipartite synchronization of coupled reaction-diffusion memristive neural networks with competitive interactions

In this paper, we investigate the prespecified-time bipartite synchronization (PTBS) of coupled reaction-diffusion memristive neural networks (CRDMNNs) with both competitive and cooperative interactions. Two types of bipartite synchronization are considered: leaderless PTBS and leader-following PTBS. With the help of a structural balance condition, the criteria for PTBS for CRDMNNs are derived by designing suitable Lyapunov functionals and novel control protocols. Different from the traditional finite-time or fixed-time synchronization, the settling time obtained in this paper is independent of control gains and initial values, which can be pre-set according to the task requirements. Lastly, numerical simulations are given to verify the obtained results.

Deep Learning Adapted to Differential Neural Networks Used as Pattern Classification of Electrophysiological Signals

This manuscript presents the design of a deep differential neural network (DDNN) for pattern classification. First, we proposed a DDNN topology with three layers, whose learning laws are derived from a Lyapunov analysis, justifying local asymptotic convergence of the classification error and the weights of the DDNN. Then, an extension to include an arbitrary number of hidden layers in the DDNN is analyzed. The learning laws for this general form of the DDNN offer a contribution to the deep learning framework for signal classification with biological nature and dynamic structures. The DDNN is used to classify electroencephalographic signals from volunteers that perform an identification graphical test. The classification results show exponential growth in the signal classification accuracy from 82 percent with one layer to 100 percent with three hidden layers. Working with DDNN instead of static deep neural networks (SDNN) represents a set of advantages, such as processing time and training period reduction up to almost 100 times, and the increment of the classification accuracy while working with less hidden layers than working with SDNN, which are highly dependent on their topology and the number of neurons in each layer. The DDNN employed fewer neurons due to the induced feedback characteristic.

Exponential Synchronization of Memristor-Based Competitive Neural Networks With Reaction-Diffusions and Infinite Distributed Delays

Taking into account the infinite distributed delays and reaction-diffusions, this article investigates the global exponential synchronization problem of a class of memristor-based competitive neural networks (MCNNs) with different time scales. Based on the Lyapunov-Krasovskii functional and inequality approach, an adaptive control approach is proposed to ensure the exponential synchronization of the addressed drive-response networks. The closed-loop system is a discontinuous and delayed partial differential system in a cascade form, involving the spatial diffusion, the infinite distributed delays, the parametric adaptive law, the state-dependent switching parameters, and the variable structure controllers. By combining the theories of nonsmooth analysis, partial differential equation (PDE) and adaptive control, we present a new analytical method for rigorously deriving the synchronization of the states of the complex system. The derived m-norm (m ≥ 2)-based synchronization criteria are easily verified and the theoretical results are easily extended to memristor-based neural networks (NNs) without different time scales and reaction-diffusions. Finally, numerical simulations are presented to verify the effectiveness of the theoretical results.

Adaptive Event-Triggered Synchronization of Reaction-Diffusion Neural Networks

This article focuses on the design of an adaptive event-triggered sampled-data control (ETSDC) mechanism for synchronization of reaction-diffusion neural networks (RDNNs) with random time-varying delays. Different from the existing ETSDC schemes with predetermined constant thresholds, an adaptive ETSDC mechanism is proposed for RDNNs. The adaptive ETSDC mechanism can be promptly adaptively adjusted since the threshold function is based on the current sampled and latest transmitted signals. Thus, the adaptive ETSDC mechanism can effectively save communication resources for RDNNs. By taking the influence of uncertain factors, the random time-varying delays are considered, which belongs to two intervals in a probabilistic way. Then, by constructing an appropriate Lyapunov-Krasovskii functional (LKF), new synchronization criteria are derived for RDNNs. By solving a set of linear matrix inequalities (LMIs), the desired adaptive ETSDC gain is obtained. Finally, the merits of the adaptive ETSDC mechanism and the effectiveness of the proposed results are verified by one numerical example.

QUAD-Condition, Synchronization, Consensus of Multiagents, and Anti-Synchronization of Complex Networks

In this article, we discuss quadratic condition (QUAD-condition) for general models of synchronization of complex networks and consensus of multiagents with or without pinning controller in detail. Synchronization analysis consists of two parts. One is connection structure, which is described with coupling matrix. The other one is the intrinsic property of the uncoupled system. QUAD-conditions play a key role in describing the intrinsic property of the uncoupled system. With QUAD-conditions, we unify synchronization and consensus of multiagents in a framework. It is interesting that anti-synchronization can be easily transformed to synchronization by introducing suitable QUAD-condition.

Bipartite Synchronization of Multiple Memristor-Based Neural Networks With Antagonistic Interactions

In this article, by introducing a signed graph to describe the coopetition interactions among network nodes, the mathematical model of multiple memristor-based neural networks (MMNNs) with antagonistic interactions is established. Since the cooperative and competitive interactions coexist, the states of MMNNs cannot reach complete synchronization. Instead, they will reach the bipartite synchronization: all nodes' states will reach an identical absolute value but opposite sign. To reach bipartite synchronization, two kinds of the novel node- and edge-based adaptive strategies are proposed, respectively. First, based on the global information of the network nodes, a node-based adaptive control strategy is constructed to solve the bipartite synchronization problem of MMNNs. Secondly, a local edge-based adaptive algorithm is proposed, where the weight values of edges between two nodes will change according to the designed adaptive law. Finally, two simulation examples validate the effectiveness of the proposed adaptive controllers and bipartite synchronization criteria.

Synchronization of Nonidentical Neural Networks With Unknown Parameters and Diffusion Effects via Robust Adaptive Control Techniques

This paper considers the self-synchronization and tracking synchronization issues for a class of nonidentically coupled neural networks model with unknown parameters and diffusion effects. Using the special structure of neural networks with global Lipschitz activation function, nonidentical terms are treated as external disturbances, which can then be compensated via robust adaptive control techniques. For the case where no common reference trajectory is given in advance, a distributed adaptive controller is proposed to drive the synchronization error to an adjustable bounded area. For the case where a reference trajectory is predesigned, two distributed adaptive controllers are proposed, respectively, to address the tracking synchronization problem with bounded and unbounded reference trajectories, different decomposition methods are given to extract the heterogeneous characteristics. To avoid the appearance of global information, such as the spectrum of the coupling matrix, corresponding adaptive designs on coupling strengths are also provided for both cases. Moreover, the upper bounds of the final synchronization errors can be gradually adjusted according to the parameters of the adaptive designs. Finally, numerical examples are given to test the effectiveness of the control algorithms.

Almost Periodicity in Impulsive Fractional-Order Reaction-Diffusion Neural Networks With Time-Varying Delays

A neural-network model of fractional order with impulsive perturbations, time-varying delays, and reaction-diffusion terms is investigated in this article. The focus is on investigating qualitative properties of the states and developing new almost periodicity and stability criteria. The uncertain case is also considered. Examples are established and the effectiveness of the obtained criteria is demonstrated.

Anti-Synchronization in Fixed Time for Discontinuous Reaction-Diffusion Neural Networks With Time-Varying Coefficients and Time Delay

This paper studies the fixed-time anti-synchronization (FTAS) of discontinuous reaction-diffusion neural networks (DRDNNs) with both time-varying coefficients and time delay. First, differential inclusion theory is used to deal with the influence caused by discontinuous activations. In addition, a new fixed-time convergence theorem is used to handle the time-varying coefficients. Second, a novel state-feedback control algorithm and integral state-feedback control algorithm are proposed to realize FTAS of DRDNNs. During the generalized (adaptive) pinning control strategy, a guideline is proposed to select neurons to pin the designed controller. Furthermore, we present several criteria on FTAS by using the generalized Lyapunov function method. Different from the traditional Lyapunov function with negative definite derivative, the derivative of the Lyapunov function can be positive in this paper. Finally, we give two numerical simulations to substantiate the merits of the obtained results.

Pattern formation in reaction-diffusion systems in the presence of non-Markovian diffusion

We study reaction-diffusion systems beyond the Markovian approximation to take into account the effect of memory on the formation of spatiotemporal patterns. Using a non-Markovian Brusselator model as a paradigmatic example, we show how to use reductive perturbation to investigate the formation and stability of patterns. Focusing in detail on the Hopf instability and short-term memory, we derive the corresponding complex Ginzburg-Landau equation that governs the amplitude of the critical mode and we establish the explicit dependence of its parameters on the memory properties. Numerical solution of this memory-dependent complex Ginzburg-Landau equation as well as direct numerical simulation of the non-Markovian Brusselator model illustrates that memory changes the properties of the spatiotemporal patterns. Our results indicate that going beyond the Markovian approximation might be necessary to study the formation of spatiotemporal patterns even in systems with short-term memory. At the same time, our work opens up a new window into the control of these patterns using memory.

Synchronization of Multiple Reaction-Diffusion Neural Networks With Heterogeneous and Unbounded Time-Varying Delays

The synchronization problem of multiple/coupled reaction-diffusion neural networks with time-varying delays is investigated. Differing from the existing considerations, state delays among distinct neurons and coupling delays among different subnetworks are included in the proposed model, the assumptions posed on the arisen delays are very weak, time-varying, heterogeneous, even unbounded delays are permitted. To overcome the difficulties from this kind of delay as well as diffusion effects, a comparison-based approach is applied to this model and a series of algebraic criteria are successfully obtained to verify the global asymptotical synchronization. By specifying the existing delays, some M -matrix-based criteria are derived to justify the power-rate synchronization and exponential synchronization. In addition, new criterion on synchronization of general connected neural networks without diffusion effects is also given. Finally, two simulation examples are given to verify the effectiveness of the obtained theoretical results and provide a comparison with the existing criterion.

Distributed Adaptive Tracking Synchronization for Coupled Reaction-Diffusion Neural Network

This paper considers the tracking synchronization problem for a class of coupled reaction-diffusion neural networks (CRDNNs) with undirected topology. For the case where the tracking trajectory has identical individual dynamic as that of the network nodes, the edge-based and vertex-based adaptive strategies on coupling strengths as well as adaptive controllers, which demand merely the local neighbor information, are proposed to synchronize the CRDNNs to the tracking trajectory. To reduce the control costs, an adaptive pinning control technique is employed. For the case where the tracking trajectory has different individual dynamic from that of the network nodes, the vertex-based adaptive strategy is proposed to drive the synchronization error to a relatively small area, which is adjustable according to the parameters of the adaptive strategy. This kind of adaptive design can enhance the robustness of the network against the external disturbance posed on the tracking trajectory. The obtained theoretical results are verified by two representative examples.

Nonlinear optimal control for the synchronization of biological neurons under time-delays

The article proposes a nonlinear optimal control method for synchronization of neurons that exhibit nonlinear dynamics and are subject to time-delays. The model of the Hindmarsh-Rose (HR) neurons is used as a case study. The dynamic model of the coupled HR neurons undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. The linearization procedure relies on Taylor series expansion of the model and on computation of the associated Jacobian matrices. For the approximately linearized model of the coupled HR neurons an H-infinity controller is designed. For the selection of the controller's feedback gain an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The stability properties of the control loop are proven through Lyapunov analysis. First, it is shown that the H-infinity tracking performance criterion is satisfied. Moreover, it is proven that the control loop is globally asymptotically stable.