A Distributed k-Winners-Take-All Model With Binary Consensus Protocols.

This article concentrates on solving the k -winners-take-all (k WTA) problem with large-scale inputs in a distributed setting. We propose a multiagent system with a relatively simple structure, in which each agent is equipped with a 1-D system and interacts with others via binary consensus protocols. That is, only the signs of the relative state information between neighbors are required. By virtue of differential inclusion theory, we prove that the system converges from arbitrary initial states. In addition, we derive the convergence rate as O(1/t) . Furthermore, in comparison to the existing models, we introduce a novel comparison filter to eliminate the resolution ratio requirement on the input signal, that is, the difference between the k th and (k+1) th largest inputs must be larger than a positive threshold. As a result, the proposed distributed k WTA model is capable of solving the k WTA problem, even when more than two elements of the input signal share the same value. Finally, we validate the effectiveness of the theoretical results through two simulation examples.

Synchronization of coupled switched neural networks subject to hybrid stochastic disturbances.

In this paper, the theoretical analysis on exponential synchronization of a class of coupled switched neural networks suffering from stochastic disturbances and impulses is presented. A control law is developed and two sets of sufficient conditions are derived for the synchronization of coupled switched neural networks. First, for desynchronizing stochastic impulses, the synchronization of coupled switched neural networks is analyzed by Lyapunov function method, the comparison principle and a impulsive delay differential inequality. Then, for general stochastic impulses, by partitioning impulse interval and using the convex combination technique, a set of sufficient condition on the basis of linear matrix inequalities (LMIs) is derived for the synchronization of coupled switched neural networks. Eventually, two numerical examples and a practical application are elaborated to illustrate the effectiveness of the theoretical results.

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.

A Second-Order Projected Primal-Dual Dynamical System for Distributed Optimization and Learning.

This article focuses on developing distributed optimization strategies for a class of machine learning problems over a directed network of computing agents. In these problems, the global objective function is an addition function, which is composed of local objective functions. Such local objective functions are convex and only endowed by the corresponding computing agent. A second-order Nesterov accelerated dynamical system with time-varying damping coefficient is developed to address such problems. To effectively deal with the constraints in the problems, the projected primal-dual method is carried out in the Nesterov accelerated system. By means of the cocoercive maximal monotone operator, it is shown that the trajectories of the Nesterov accelerated dynamical system can reach consensus at the optimal solution, provided that the damping coefficient and gains meet technical conditions. In the end, the validation of the theoretical results is demonstrated by the email classification problem and the logistic regression problem in machine learning.

Adaptive Exact Penalty Design for Optimal Resource Allocation.

In this article, a distributed adaptive continuous-time optimization algorithm based on the Laplacian-gradient method and adaptive control is designed for resource allocation problem with the resource constraint and the local convex set constraints. In order to deal with local convex sets, a distance-based exact penalty function method is adopted to reformulate the resource allocation problem instead of the widely used projection operator method. By using the nonsmooth analysis and set-valued LaSalle invariance principle, it is proven that the proposed algorithm is capable of solving the nonsmooth resource allocation problem. Finally, two simulation examples are presented to substantiate the theoretical results.

Multiple asymptotical ω-periodicity of fractional-order delayed neural networks under state-dependent switching.

This paper presents theoretical results on multiple asymptotical ω-periodicity of a state-dependent switching fractional-order neural network with time delays and sigmoidal activation functions. Firstly, by combining the geometrical properties of activation functions with the range of switching threshold, a partition of state space is given. Then, the conditions guaranteeing that the solutions can approach each other infinitely in each positive invariant set are derived. Furthermore, the S-asymptotical ω-periodicity and the convergence of solutions in positive invariant sets are discussed. It is worth noting that the number of attractors increases to 3n from 2n in a neural network without switching. Finally, three numerical examples are given to substantiate the theoretical results.

Lag H∞ synchronization in coupled reaction-diffusion neural networks with multiple state or derivative couplings.

This paper mainly attempts to discuss lag H∞ synchronization in multiple state or derivative coupled reaction-diffusion neural networks without and with parameter uncertainties. Firstly, we respectively propose two types of reaction-diffusion neural networks with multiple state and derivative couplings subject to parameter uncertainties. Secondly, by exploiting designed state feedback controllers, several criteria of the lag H∞ synchronization for these two networks are developed based on Lyapunov functional and inequality techniques. Thirdly, lag H∞ synchronization issues of these two networks are also coped with by virtue of devised adaptive control strategies. Finally, we provide two numerical examples to verify the obtained lag H∞ synchronization criteria.

Distributed k-winners-take-all via multiple neural networks with inertia.

This paper is dedicated to solving the k-winners-take-all problem with large-scale input signals in a distributed manner. According to the decomposition of global input signals, a novel dynamical system consisting of multiple coordinated neural networks is proposed for finding the k largest inputs. In the system, each neural network is designed to tackle its available partial inputs only for a local objective ki (ki≤k). Simultaneously, a consensus-based approach is adopted to coordinate multiple neural networks for achieving the global objective k. In addition, an inertial term is introduced in each neural network for regulating its transient behavior, which has the potential of accelerating the convergence. By developing a cocoercive operator, we theoretically prove that the multiple neural networks with inertial terms converge asymptotically/exponentially to the k-winners-take-all solution exactly from arbitrary initial states for whatever decomposition of inputs and objective. Furthermore, some extensions to distributed constrained k-winners-take-all are also investigated. Finally, simulation results are presented to substantiate the effectiveness of the proposed system as well as its superior performance over existing distributed networks.

Multistability of Switched Neural Networks With Gaussian Activation Functions Under State-Dependent Switching.

This article presents theoretical results on the multistability of switched neural networks with Gaussian activation functions under state-dependent switching. It is shown herein that the number and location of the equilibrium points of the switched neural networks can be characterized by making use of the geometrical properties of Gaussian functions and local linearization based on the Brouwer fixed-point theorem. Four sets of sufficient conditions are derived to ascertain the existence of 7p15p23p3 equilibrium points, and 4p13p22p3 of them are locally stable, wherein p1 , p2 , and p3 are nonnegative integers satisfying 0 ≤ p1+p2+p3 ≤ n and n is the number of neurons. It implies that there exist up to 7n equilibria, and up to 4n of them are locally stable when p1=n . It also implies that properly selecting p1 , p2 , and p3 can engender a desirable number of stable equilibria. Two numerical examples are elaborated to substantiate the theoretical results.

Stability behavior of a two-susceptibility SHIR epidemic model with time delay in complex networks.

Taking two susceptible groups into account, we formulate a modified subhealthy-healthy-infected-recovered (SHIR) model with time delay and nonlinear incidence rate in networks with different topologies. Concretely, two dynamical systems are designed in homogeneous and heterogeneous networks by utilizing mean field equations. Based on the next-generation matrix and the existence of a positive equilibrium point, we derive the basic reproduction numbers R 0 1 and R 0 2 which depend on the model parameters and network structure. In virtue of linearized systems and Lyapunov functions, the local and global stabilities of the disease-free equilibrium points are, respectively, analyzed when R 0 1 < 1 in homogeneous networks and R 0 2 < 1 in heterogeneous networks. Besides, we demonstrate that the endemic equilibrium point is locally asymptotically stable in homogeneous networks in the condition of R 0 1 > 1 . Finally, numerical simulations are performed to conduct sensitivity analysis and confirm theoretical results. Moreover, some conjectures are proposed to complement dynamical behavior of two systems.

Multi-periodicity of switched neural networks with time delays and periodic external inputs under stochastic disturbances.

This paper presents new theoretical results on the multi-periodicity of recurrent neural networks with time delays evoked by periodic inputs under stochastic disturbances and state-dependent switching. Based on the geometric properties of activation function and switching threshold, the neuronal state space is partitioned into 5n regions in which 3n ones are shown to be positively invariant with probability one. Furthermore, by using Itô's formula, Lyapunov functional method, and the contraction mapping theorem, two criteria are proposed to ascertain the existence and mean-square exponential stability of a periodic orbit in every positive invariant set. As a result, the number of mean-square exponentially stable periodic orbits increases to 3n from 2n in a neural network without switching. Two illustrative examples are elaborated to substantiate the efficacy and characteristics of the theoretical results.

Periodic Event-Triggered Synchronization of Multiple Memristive Neural Networks With Switching Topologies and Parameter Mismatch.

This article investigates the synchronization problem of multiple memristive neural networks (MMNNs) in the case of switching communication topologies and parameter mismatch. First, the distributed event-triggered control under continuous sampling conditions is studied. Then, a periodic event-triggered control (PETC) model is proposed to substantially reduce control consumption. Using the Lyapunov method, the properties of M -matrix, and some inequalities, the sufficient criteria of synchronous control are derived. The results can be used in the analysis of other multiagent nonlinear systems. A norm-based threshold function is given to determine the update time of the controller, and it is proved that the trigger condition excludes the Zeno behavior. Subject to parameter mismatch, a quasisynchronous control strategy is proposed, which can be extended to complete synchronization provided that the system mismatch or disturbance disappears. It is worth mentioning that this article introduces the signal function into the controller, so that the theoretical error can be limited to an arbitrarily small range. Furthermore, this new controller is used in the PETC strategy which automatically avoids the Zeno behavior. Finally, one example is given to illustrate our results.

Global Exponential Synchronization of Coupled Delayed Memristive Neural Networks With Reaction-Diffusion Terms via Distributed Pinning Controls.

This article presents new theoretical results on global exponential synchronization of nonlinear coupled delayed memristive neural networks with reaction-diffusion terms and Dirichlet boundary conditions. First, a state-dependent memristive neural network model is introduced in terms of coupled partial differential equations. Next, two control schemes are introduced: distributed state feedback pinning control and distributed impulsive pinning control. A salient feature of these two pinning control schemes is that only partial information on the neighbors of pinned nodes is needed. By utilizing the Lyapunov stability theorem and Divergence theorem, sufficient criteria are derived to ascertain the global exponential synchronization of coupled neural networks via the two pining control schemes. Finally, two illustrative examples are elaborated to substantiate the theoretical results and demonstrate the advantages and disadvantages of the two control schemes.

Finite-Time and Fixed-Time Synchronization of Coupled Memristive Neural Networks With Time Delay.

This article is devoted to analyzing the finite-time and fixed-time synchronization of coupled memristive neural networks with time delays. The synchronization is leaderless rather than leader-follower as the tracking targets are uncertain. By designing a proper controller and using the Lyapunov method, several sufficient conditions are obtained to achieve the finite-time and fixed-time synchronization of coupled memristive neural networks by introducing a class of special auxiliary matrices. Moreover, the settling times can be estimated for finite-time synchronization that depends on the initial values as well as fixed-time synchronization that is uniformly bounded for any initial values. Finally, two examples are presented to substantiate the effectiveness of the theoretical results.

Sliding Mode Stabilization of Memristive Neural Networks With Leakage Delays and Control Disturbance.

In this article, we investigate a class of memristive neural networks (MNNs) with time-varying delays and leakage delays via sliding mode control (SMC) with and without control disturbance. SMC is used to ensure MNNs' stability. According to the characteristics of the MNNs, we consider the following three models: the first is the MNNs with time-varying delays, the second is the MNNs with time-varying delays and the control disturbance, and the third is the MNNs with time-varying delays, leakage delays, and the control disturbance. We quote some assumptions and lemmas to ensure that our main results are true. The sliding surface, the corresponding sliding mode controller, and the Lyapunov functions are constructed in different models to ensure MNNs' stability. Finally, some examples and simulations verify the validity of our main results by solving linear matrix inequality (LMI), and the conclusions and analysis of the results are given.

Multiple and Complete Stability of Recurrent Neural Networks With Sinusoidal Activation Function.

This article presents new theoretical results on multistability and complete stability of recurrent neural networks with a sinusoidal activation function. Sufficient criteria are provided for ascertaining the stability of recurrent neural networks with various numbers of equilibria, such as a unique equilibrium, finite, and countably infinite numbers of equilibria. Multiple exponential stability criteria of equilibria are derived, and the attraction basins of equilibria are estimated. Furthermore, criteria for complete stability and instability of equilibria are derived for recurrent neural networks without time delay. In contrast to the existing stability results with a finite number of equilibria, the new criteria, herein, are applicable for both finite and countably infinite numbers of equilibria. Two illustrative examples with finite and countably infinite numbers of equilibria are elaborated to substantiate the results.

Observer-Based Quasi-Synchronization of Delayed Dynamical Networks With Parameter Mismatch Under Impulsive Effect.

This article focuses on the observer-based quasi-synchronization problem of delayed dynamical networks with parameter mismatch under impulsive effect. First, since the state of each node is unknown in the real situation, the state estimation strategy is proposed to estimate the state of each node, so as to design an appropriate synchronization controller. Then, the corresponding controller is constructed to synchronize the slave nodes with their leader node. In this article, we take the impulsive effect into consideration, which means that an impulsive signal will be applied to the system every so often. Due to the existence of parameter mismatch and time-varying delay, by constructing an appropriate Lyapunouv function, we will eventually obtain a differential equation with constant and time-varying delay terms. Then, we analyze its trajectory by introducing the Cauchy matrix and prove its boundedness by contradiction. Finally, a numerical simulation is presented to illustrate the validness of obtained results.

Exponential synchronization of memristive neural networks with time-varying delays via quantized sliding-mode control.

In the paper, exponential synchronization issue is considered for memristive neural networks (MNNs) with time-varying delays via quantized sliding-mode algorithm. Quantized Sliding-mode controller is introduced to ensure the slave system can be exponentially synchronized with the host system via the super-twisting algorithm, which has been proved in the main results. Quantization function consists of uniform quantizer and logarithmic quantizer. Simulation results are given with comparisons between two quantizers in the end.

Global synchronization of coupled delayed memristive reaction-diffusion neural networks.

This paper focuses on the global exponential synchronization of multiple memristive reaction-diffusion neural networks (MRDNNs) with time delay. Due to introducing the influences of space as well as time on state variables and replacing resistors with memristors in circuit realization, the state-dependent partial differential mathematical model of MRDNN is more general and realistic than traditional neural network model. Based on Lyapunov functional theory, Divergence theorem and inequality techniques, global exponential synchronization criteria of coupled delayed MRDNNs are derived via directed and undirected nonlinear coupling. Finally, three numerical simulation examples are presented to verify the feasibility of our main results.

Global exponential synchronization of delayed memristive neural networks with reaction-diffusion terms.

This paper investigates the global exponential synchronization problem of delayed memristive neural networks (MNNs) with reaction-diffusion terms. First, by utilizing the pinning control technique, two novel kinds of control methods are introduced to achieve synchronization of delayed MNNs with reaction-diffusion terms. Then, with the help of inequality techniques, pinning control technique, the drive-response concept and Lyapunov functional method, two sufficient conditions are obtained in the form of algebraic inequalities, which can be used for ensuring the exponential synchronization of the proposed delayed MNNs with reaction-diffusion terms. Moreover, the obtained results based on algebraic inequality complement and improve the previously known results. Finally, two illustrative examples are given to support the effectiveness and validity of the obtained theoretical results.