Reaction-diffusion Neural Networks Research Articles

Full-Dimensional Proportional-Derivative Control Technique for Turing Pattern and Bifurcation of Delayed Reaction-Diffusion Bidirectional Ring Neural Networks

Abstract In neural networks, the states of neural networks often exhibit significant spatio-temporal heterogeneity due to the diffusion effect of electrons and differences in the concentration of neurotransmitters. One of the macroscopic reflections of this time-spatial inhomogeneity is Turing pattern. However, most current research in reaction-diffusion neural networks has focused only on one-dimensional location information, and the remaining results considering two-dimensional location information are still limited to the case of two neurons. In this paper, we conduct the dynamic analysis and optimal control of a delayed reaction-diffusion neural network model with bidirectional loop structure. First, several mathematical descriptions are given for the proposed neural network model and the full-dimensional partial differential proportional-derivative (PD) controller is introduced. Second, by analyzing the characteristic equation, the conditions for Hopf bifurcation and Turing instability of the controlled network model are obtained. Furthermore, the amplitude equation of the controlled neural network is obtained based on the multiscale analysis method. Subsequently, we determine the key parameters affecting the formation of Turing pattern depending on the amplitude equation. Finally, multiple sets of computer simulations are carried out to support our theoretical results. It is found that the diffusion coefficients and time delays have significant effects on spatio-temporal dynamics of neural networks. Moreover, after reasonable parameter proportioning, the full-dimensional PD control method can alleviate the spatial heterogeneity caused by diffusion projects and time delays.

Novel Caputo fractional differential approach and hybrid control scheme for finite‐time synchronization of nonidentical delayed reaction–diffusion neural networks

AbstractThis work focuses on the finite‐time synchronization (F‐TS) for nonidentical delayed neural networks (DNNs) including Caputo fractional partial differential operator and reaction–diffusion terms. A novel F‐TS lemma is proposed by constructing a new Caputo differential inequality. Applying the new lemma and designing two hybrid controllers with time delay and the sign function, the F‐TS criteria on the introduced Caputo reaction–diffusion nonidentical DNNs under the Neumann boundary condition are derived by making use of the Filippov differential inclusion, Green's theorem, and fractional Razumikhin theorem. The conditions are expressed through the algebraic inequality which can greatly decrease the computation in checking the F‐TS performance. Moreover, the validity and correctness of the F‐TS results are illustrated by selecting various orders, spatial positions, and diffusion parameters.

Observer-Based Boundary Stabilization of Coupled Semilinear Reaction-Diffusion Neural Networks With Spatially Varying Coefficients via Event-Triggered Controller.

The aim of this article is to propose an observer-based event-triggered Robin boundary control strategy for the exponential stabilization of the coupled semilinear reaction-diffusion neural networks with spatially varying coefficients. Toward this aim, we design an observer to estimate the value of system states by using some of these system values as the available measurement. An observer-based event-triggered boundary stabilizer is then presented to exponentially stabilize the considered systems with the Zeno behavior being excluded. Throughout this article, the main used method is backstepping, which yields an explicit expression of the control formulae. Moreover, we see that the proposed event-triggered boundary control scheme can ensure the desired level of control performance with fewer control law updates. A numerical example is finally given to illustrate the effectiveness of our proposed method.

Pattern Control of Neural Networks with Two-Dimensional Diffusion and Mixed Delays

In this paper, a two-neuron reaction–diffusion neural network with discrete and distributed delays is proposed, and the state feedback control strategy is adopted to achieve control of its spatiotemporal dynamical behaviours. Adding two virtual neurons, the original system is transformed into a neural network only containing the discrete delay. The conditions under which Hopf bifurcation and Turing instability arise are determined through analysis of the characteristic equation. Additionally, the amplitude equations are derived with the aid of weakly nonlinear analysis, and the selection of the Turing patterns is determined. The simulation results demonstrate that the state feedback controller can delay the onset of Hopf bifurcation and suppress the generation of Turing patterns.

Distributed adaptive robust containment control for reaction–diffusion neural networks with external disturbances under directed graphs

In this paper, the leader–follower robust synchronization issue is mainly addressed for reaction–diffusion neural networks (RDNNs) with multiple leaders and external disturbances under directed graphs. Based on the σ modification approach, we propose a novel distributed adaptive controller by adding a term −θi(ci−1)2 to avoid the phenomenon of parameter drift, that is, the adaptive parameters grow to infinity. Meanwhile, different from the adaptive control algorithm proposed in the undirected graph, we introduce a new function χi(t) to provide additional freedom for the design to achieve robust containment when confronted with external disturbances. Further, the robustness of tracking synchronization with one leader is guaranteed by the proposed adaptive controller when the external disturbances concerning L2 norm are bounded. Finally, relevant numerical simulation graphics are displayed separately to verify the correctness of the related theoretical results.

Quasisynchronization of reaction-diffusion neural networks with time-varying delays by static/dynamic event-triggered control and its application to secure communication

Quasisynchronization of reaction-diffusion neural networks with time-varying delays by static/dynamic event-triggered control and its application to secure communication

Event-triggered passivity and synchronization of multiple derivative coupled reaction–diffusion neural networks

The paper studies the event-triggered passivity and synchronization problems for multiple derivative coupled reaction–diffusion neural networks. Two types of multiple derivative coupled reaction–diffusion neural networks, with and without time-varying delay, are considered. According to the characteristics of multiple derivative coupled reaction–diffusion neural networks, an event-triggered condition is constructed. By imposing appropriate Lyapunov functionals and inequality techniques, several passivity conditions are established for these two networks under the designed event-triggered controllers. Then, synchronization criteria of these two networks are derived based on the relationship between output-strict passivity and synchronization. Moreover, the non-occurrence of the Zeno behavior in the proposed control is proved. Numerical example is illustrated to verify the effectiveness of the proposed criteria. Finally, theoretical results and the associated image encryption application are validated through numerical simulations and statistical analyses.

Passivity for Multiadaptive Coupled Fractional- Order Reaction-Diffusion Neural Networks

Passivity for Multiadaptive Coupled Fractional- Order Reaction-Diffusion Neural Networks

Fuzzy adaptive event-triggered synchronization control mechanism for T–S fuzzy RDNNs under deception attacks

In this paper, a fuzzy-dependent adaptive event-triggered mechanism (FAETM) for synchronizing Takagi–Sugeno (T–S) fuzzy reaction–diffusion neural networks (RDNNs) is developed while considering deception attacks. Firstly, a general neural network model considering both fuzzy logic rules and reaction–diffusion terms is established. Secondly, a FAETM based on an aperiodic sampling period is presented under deception attacks to alleviate the communication burden, wherein the adaptive threshold function is dependent on membership functions. Moreover, a membership-function-dependent asymmetric Lyapunov functional (LF) is constructed, and some positive-definite and symmetric constraints of matrices are removed as a result. Based on the LF method and integral inequality techniques, the H∞ synchronization criteria for the T–S fuzzy RDNNs are presented by linear matrix inequalities (LMIs). Finally, one simulation example is exploited to demonstrate the feasibility and validity of the established method, and the outcome is applied to image secure communication.

Adaptive Output Synchronization of Coupled Fractional-Order Memristive Reaction-Diffusion Neural Networks

This article discusses the adaptive output synchronization problem of coupled fractional-order memristive reaction-diffusion neural networks (CFOMRDNNs) with multiple output couplings or multiple output derivative couplings. Firstly, by using Lyapunov functional and inequality techniques, an adaptive output synchronization criterion for CFOMRDNNs with multiple output couplings is proposed. Then, an adaptive controller is designed for ensuring the output synchronization of CFOMRDNNs with multiple output derivative couplings. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.

Resilient fixed-time synchronization of delayed fuzzy memristive reaction-diffusion neural networks under DoS attacks

This work focuses on the resilient fixed-time synchronization of delayed fuzzy memristive reaction-diffusion neural networks under denial-of-service (DoS) attacks. To efficaciously tolerate the aperiodic DoS attacks, a new appropriate controller is designed to ensure the fixed-time resilient synchronization of the systems. Moreover, two mild sufficient conditions are first proposed and the constrained techniques of attacking intervals are employed to overcome the challenge of estimating the upper bound of the settling time under aperiodic DoS attacks. Lastly, an example is utilized to illustrate the effectiveness and feasibility of the theoretical result.

Finite-Time Output Synchronization of Multiple Weighted Reaction-Diffusion Neural Networks With Adaptive Output Couplings.

This article mainly considers the output synchronization (OS) problem of multiple weighted and adaptive output coupled reaction-diffusion neural networks (RDNNs) without and with coupling delays in finite time. Without coupling delays, an adaptive control law and an output feedback controller are, respectively, proposed to ensure that the multiple weighted and output coupled RDNNs are output synchronized and output synchronized in finite time. With coupling delays, an adaptive coupling weights control scheme and a novel feedback controller are put forward to make the multiple weighted RDNNs with output couplings achieve OS in finite time. Moreover, the finite-time OS is considered in the presence of external disturbances. By the Lyapunov approach, several finite-time OS and OS criteria are given. Finally, two simulation examples are presented to justify the effectiveness of the proposed adaptive control laws and controllers.

Adaptive Strategies and its Application in the Mittag-Leffler Synchronization of Delayed Fractional-Order Complex-Valued Reaction-Diffusion Neural Networks

Adaptive Strategies and its Application in the Mittag-Leffler Synchronization of Delayed Fractional-Order Complex-Valued Reaction-Diffusion Neural Networks

Event-triggered pinning passivity and synchronization of multiple spatial diffusion coupled reaction–diffusion neural networks

In this paper, the problem of the event-triggered pinning passivity and synchronization for multiple spatial diffusion coupled reaction–diffusion neural networks is investigated. Two types of multiple spatial diffusion coupled reaction–diffusion neural networks, with undirected and directed topologies, are considered. Combining the advantages of event-triggered and pinning control methods, event-triggered pinning controllers for these two networks are constructed. By imposing appropriate Lyapunov functionals and inequality techniques, several passivity and synchronization conditions are established for these two networks under the designed event-triggered pinning controllers. Moreover, the non-occurrence of the Zeno behavior in the proposed control is proved. Two numerical examples are illustrated to verify the effectiveness of the proposed criteria.

Event-triggered impulsive cluster synchronization of coupled reaction–diffusion neural networks and its application to image encryption

This paper investigates the cluster synchronization of coupled neural networks with reaction–diffusion terms. With the help of impulsive control strategies, some cluster synchronization criteria are proposed by an appropriate event-triggered mechanism. A numerical example is given to verify the validity of the theoretical results. Additionally, the proposed event-triggered impulsive synchronization is successfully applied to image encryption with encouraging cryptanalysis results demonstrating its strong ability to efficiently encrypt images.

Boundary stabilisation of fractional reaction-diffusion systems with time-varying delays

This paper examines the boundary stabilisation results for time fractional-order reaction-diffusion systems involving with time-varying delays. The main goal is to design the boundary control for the system by proving the well-posedness of the kernel function using the backstepping method. An invertible Volterra integral transformation is used to convert the considered system into a stable target system. Different from the existing results, the stability results for fractional RDEs are discussed in the sense of the Lyapunov–Krasovskii theory and sufficient conditions are derived with the help of the linear matrix inequality (LMI) approach. Finally, to show the application of the results, the proposed conditions are numerically validated over a time fractional-order reaction-diffusion cellular neural network model.

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.

Robust H∞ Pinning Synchronization for Multiweighted Coupled Reaction-Diffusion Neural Networks.

This article focuses on the robust H∞ synchronization of two types of coupled reaction-diffusion neural networks with multiple state and spatial diffusion couplings by utilizing pinning adaptive control strategies. First, based on the Lyapunov functional combined with inequality techniques, several sufficient conditions are formulated to ensure H∞ synchronization for these two networks with parameter uncertainties. Moreover, node-based pinning adaptive control strategies are devised to address the robust H∞ synchronization problem. In addition, some criteria of H∞ synchronization for these two networks under parameter uncertainties are developed via edge-based pinning adaptive controllers. Finally, two numerical examples are presented to verify our results.

Pinning Spatiotemporal Sampled-Data Synchronization of Coupled Reaction-Diffusion Neural Networks Under Deception Attacks.

In this article, we investigate the pinning spatiotemporal sampled-data (SD) synchronization of coupled reaction-diffusion neural networks (CRDNNs), which are directed networks with SD in time and space communications under random deception attacks. In order to handle with the random deception attacks, we establish a directed CRDNN model, which respects the impacts of variable sampling and random deception attacks within a unified framework. Through the designed pinning spatiotemporal SD controller, sufficient conditions are obtained by linear matrix inequalities (LMIs) that guarantee the mean square exponential stability of the synchronization error system (SES) derived by utilizing inequality techniques, the stochastic analysis technique, and Lyapunov-Krasovskii functional (LKF). Finally, a numerical example is utilized to support the presented pinning spatiotemporal SD synchronization method.

Stability of Delayed Reaction-Diffusion Neural-Network Models With Hybrid Impulses via Vector Lyapunov Function.

This article focuses on stability analysis of delayed reaction-diffusion neural-network models with hybrid impulses based on the vector Lyapunov function. First, several properties of a vector Halanay-type inequality are given to be the key ingredient for the stability analysis. Then, the Krasovskii-type theorems are established for sufficient conditions of exponential stability, which removes the common threshold of impulses in each neuron subsystem at every impulse time. It shows that the stability of neural networks can be retained with hybrid impulses involved in neural networks, and the synchronization of neural networks can be achieved by designing an impulsive controller, which allows the existence of impulsive perturbation in some nodes and time. Finally, the effectiveness of theoretical results is verified by numerical examples with a successful application to image encryption.