Multistability of Delayed Hybrid Impulsive Neural Networks With Application to Associative Memories

Coexistence and local stability of multiple equilibrium points for fractional-order state-dependent switched competitive neural networks with time-varying delays

This paper investigates the coexistence and local stability of multiple equilibrium points for a class of competitive neural networks with sigmoidal activation functions and time-varying delays, in which fractional-order derivative and state-dependent switching are involved at the same time. Some novel criteria are established to ensure that such n-neuron neural networks can have [Formula: see text] total equilibrium points and [Formula: see text] locally stable equilibrium points with m₁+m₂=n, based on the fixed-point theorem, the definition of equilibrium point in the sense of Filippov, the theory of fractional-order differential equation and Lyapunov function method. The investigation implies that the competitive neural networks with switching can possess greater storage capacity than the ones without switching. Moreover, the obtained results include the multistability results of both fractional-order switched Hopfield neural networks and integer-order switched Hopfield neural networks as special cases, thus generalizing and improving some existing works. Finally, four numerical examples are presented to substantiate the effectiveness of the theoretical analysis.

An Overview of the Stability Analysis of Recurrent Neural Networks With Multiple Equilibria

The stability analysis of recurrent neural networks (RNNs) with multiple equilibria has received extensive interest since it is a prerequisite for successful applications of RNNs. With the increasing theoretical results on this topic, it is desirable to review the results for a systematical understanding of the state of the art. This article provides an overview of the stability results of RNNs with multiple equilibria including complete stability and multistability. First, preliminaries on the complete stability and multistability analysis of RNNs are introduced. Second, the complete stability results of RNNs are summarized. Third, the multistability results of various RNNs are reviewed in detail. Finally, future directions in these interesting topics are suggested.

Analysis and Design of Multivalued High-Capacity Associative Memories Based on Delayed Recurrent Neural Networks

This article aims at analyzing and designing the multivalued high-capacity-associative memories based on recurrent neural networks with both asynchronous and distributed delays. In order to increase storage capacities, multivalued activation functions are introduced into associative memories. The stored patterns are retrieved by external input vectors instead of initial conditions, which can guarantee accurate associative memories by avoiding spurious equilibrium points. Some sufficient conditions are proposed to ensure the existence, uniqueness, and global exponential stability of the equilibrium point of neural networks with mixed delays. For neural networks with n neurons, m -dimensional input vectors, and 2k -valued activation functions, the autoassociative memories have (2k)ⁿ storage capacities and heteroassociative memories have min {(2k)ⁿ,(2k)^m} storage capacities. That is, the storage capacities of designed associative memories in this article are obviously higher than the 2ⁿ and min {2ⁿ,2^m} storage capacities of the conventional ones. Three examples are given to support the theoretical results.

Multimode function multistability for Cohen-Grossberg neural networks with mixed time delays

In this paper, we are concerned with the multimode function multistability for Cohen-Grossberg neural networks (CGNNs) with mixed time delays. It is introduced the multimode function multistability as well as its specific mathematical expression, which is a generalization of multiple exponential stability, multiple polynomial stability, multiple logarithmic stability, and asymptotic stability. Also, according to the neural network (NN) model and the maximum and minimum values of activation functions, n pairs of upper and lower boundary functions are obtained. Via the locations of the zeros of the n pairs of upper and lower boundary functions, the state space is divided into ∏_i=1ⁿ(2H_i+1) parts correspondingly. By virtue of the reduction to absurdity, continuity of function, Brouwer's fixed point theorem and Lyapunov stability theorem, the criteria for multimode function multistability are acquired. Multiple types of multistability, including multiple exponential stability, multiple polynomial stability, multiple logarithmic stability, and multiple asymptotic stability, can be achieved by selecting different types of function P(t). Two numerical examples are offered to substantiate the generality of the obtained criteria over the existing results.

Neuroscience and Network Dynamics Toward Brain-Inspired Intelligence

This article surveys the interdisciplinary research of neuroscience, network science, and dynamic systems, with emphasis on the emergence of brain-inspired intelligence. To replicate brain intelligence, a practical way is to reconstruct cortical networks with dynamic activities that nourish the brain functions, instead of using only artificial computing networks. The survey provides a complex network and spatiotemporal dynamics (abbr. network dynamics) perspective for understanding the brain and cortical networks and, furthermore, develops integrated approaches of neuroscience and network dynamics toward building brain-inspired intelligence with learning and resilience functions. Presented are fundamental concepts and principles of complex networks, neuroscience, and hybrid dynamic systems, as well as relevant studies about the brain and intelligence. Other promising research directions, such as brain science, data science, quantum information science, and machine behavior are also briefly discussed toward future applications.

Multiagent Meta-Reinforcement Learning for Adaptive Multipath Routing Optimization

In this article, we investigate the routing problem of packet networks through multiagent reinforcement learning (RL), which is a very challenging topic in distributed and autonomous networked systems. In specific, the routing problem is modeled as a networked multiagent partially observable Markov decision process (MDP). Since the MDP of a network node is not only affected by its neighboring nodes' policies but also the network traffic demand, it becomes a multitask learning problem. Inspired by recent success of RL and metalearning, we propose two novel model-free multiagent RL algorithms, named multiagent proximal policy optimization (MAPPO) and multiagent metaproximal policy optimization (meta-MAPPO), to optimize the network performances under fixed and time-varying traffic demand, respectively. A practicable distributed implementation framework is designed based on the separability of exploration and exploitation in training MAPPO. Compared with the existing routing optimization policies, our simulation results demonstrate the excellent performances of the proposed algorithms.

Uniform Stability of Complex-Valued Neural Networks of Fractional Order With Linear Impulses and Fixed Time Delays

As a generation of the real-valued neural network (RVNN), complex-valued neural network (CVNN) is based on the complex-valued (CV) parameters and variables. The fractional-order (FO) CVNN with linear impulses and fixed time delays is discussed. By using the sign function, the Banach fixed point theorem, and two classes of activation functions, some criteria of uniform stability for the solution and existence and uniqueness for equilibrium solution are derived. Finally, three experimental simulations are presented to illustrate the correctness and effectiveness of the obtained results.

Multistability and Stabilization of Fractional-Order Competitive Neural Networks With Unbounded Time-Varying Delays

This article investigates the multistability and stabilization of fractional-order competitive neural networks (FOCNNs) with unbounded time-varying delays. By utilizing the monotone operator, several sufficient conditions of the coexistence of equilibrium points (EPs) are obtained for FOCNNs with concave-convex activation functions. And then, the multiple μ -stability of delayed FOCNNs is derived by the analytical method. Meanwhile, several comparisons with existing work are shown, which implies that the derived results cover the inverse-power stability and Mittag-Leffler stability as special cases. Moreover, the criteria on the stabilization of FOCNNs with uncertainty are established by designing a controller. Compared with the results of fractional-order neural networks, the obtained results in this article enrich and improve the previous results. Finally, three numerical examples are provided to show the effectiveness of the presented results.

Delay-Dependent Stability Analysis for Switched Stochastic Networks With Proportional Delay

In this article, the issue of exponential stability (ES) is investigated for a class of switched stochastic neural networks (SSNNs) with proportional delay (PD). The key feature of PD is an unbounded time-varying delay. By considering the comparison principle and combining the extended formula for the variation of parameters, we conquer the difficulty in consideration of PD effects for such networks for the first time, where the subsystems addressed may be stable or unstable. New delay-dependent conditions with respect to the mean-square ES of systems are established by employing the average dwell-time (ADT) technique, stochastic analysis theory, and Lyapunov approach. It is shown that the acquired minimum average dwell time (MADT) is not only relevant to the stable subsystems (SSs) and unstable subsystems (USs) but also dependent on the decay ratio (DR), increasing ratio (IR), as well as PD. Finally, the availability of the derived results under an average dwell-time-switched regulation (ADTSR) is illustrated through two numerical simulation examples.

Stabilization of Positive Systems With Time Delay via the Takagi-Sugeno Fuzzy Impulsive Control

In this study, the Takagi-Sugeno (T-S) fuzzy impulsive control problem is investigated for a class of nonlinear positive systems with time delay. The time delay under consideration is both in the continuous-time dynamics and at the impulsive instants, which can model practical systems more accurately. An impulse-time-dependent copositive Lyapunov function (IDCLF) is constructed, and the Razumikhin technique is adopted to develop conditions that ensure the globally exponential stability of T-S fuzzy positive systems with delayed impulses. The size constraint between the impulse delay and the bound of impulsive intervals is removed. A T-S fuzzy impulsive controller is designed in terms of the solutions to certain vector inequalities that are readily solvable. Numerical examples and a practical example of lipoprotein metabolism and potassium ion transfer model are given to demonstrate the effectiveness, advantages, and practicality of the proposed results.

Multiple Mittag-Leffler Stability of Delayed Fractional-Order Cohen-Grossberg Neural Networks via Mixed Monotone Operator Pair

This article mainly investigates the multiple Mittag-Leffler stability of delayed fractional-order Cohen-Grossberg neural networks with time-varying delays. By using mixed monotone operator pair, the conditions of the coexistence of multiple equilibrium points are obtained for fractional-order Cohen-Grossberg neural networks, and these conditions are eventually transformed into algebraic inequalities based on the vertex of the divided region. In particular, when the symbols of these inequalities are determined by the dominant term, several verifiable corollaries are given. And then, the sufficient conditions of the Mittag-Leffler stability are derived for fractional-order Cohen-Grossberg neural networks with time-varying delays. In addition, two numerical examples are provided to illustrate the effectiveness of the theoretical results.

Exact coexistence and locally asymptotic stability of multiple equilibria for fractional-order delayed Hopfield neural networks with Gaussian activation function

This paper explores the multistability issue for fractional-order Hopfield neural networks with Gaussian activation function and multiple time delays. First, several sufficient criteria are presented for ensuring the exact coexistence of 3ⁿ equilibria, based on the geometric characteristics of Gaussian function, the fixed point theorem and the contraction mapping principle. Then, different from the existing methods used in the multistability analysis of fractional-order neural networks without time delays, it is shown that 2ⁿ of 3ⁿ total equilibria are locally asymptotically stable, by applying the theory of fractional-order linear delayed system and constructing suitable Lyapunov function. The obtained results improve and extend some existing multistability works for classical integer-order neural networks and fractional-order neural networks without time delays. Finally, an illustrative example with comprehensive computer simulations is given to demonstrate the theoretical results.

Dynamical and static multisynchronization analysis for coupled multistable memristive neural networks with hybrid control

This paper investigates the dynamical multisynchronization (DMS) and static multisynchronization (SMS) problems for a class of delayed coupled multistable memristive neural networks (DCMMNNs) via a novel hybrid controller which includes delayed impulsive control and state feedback control. Based on the state-space partition method and the geometrical properties of the activation function, each subnetwork has multiple locally exponential stable equilibrium states. By employing a new Halanay-type inequality and the impulsive control theory, some new linear matrix inequalities (LMIs)-based sufficient conditions are proposed. It is shown that the delayed impulsive control with suitable impulsive interval and allowable time-varying delay can still guarantee the DMS and SMS of DCMMNNs. Finally, a numerical example is presented to illustrate the effectiveness of the hybrid controller.

Multistability and associative memory of neural networks with Morita-like activation functions

This paper presents the multistability analysis and associative memory of neural networks (NNs) with Morita-like activation functions. In order to seek larger memory capacity, this paper proposes Morita-like activation functions. In a weakened condition, this paper shows that the NNs with n-neurons have (2m+1)ⁿ equilibrium points (Eps) and (m+1)ⁿ of them are locally exponentially stable, where the parameter m depends on the Morita-like activation functions, called Morita parameter. Also the attraction basins are estimated based on the state space partition. Moreover, this paper applies these NNs into associative memories (AMs). Compared with the previous related works, the number of Eps and AM's memory capacity are extensively increased. The simulation results are illustrated and some reliable associative memories examples are shown at the end of this paper.

Multistability of delayed fractional-order competitive neural networks

This paper is concerned with the multistability of fractional-order competitive neural networks (FCNNs) with time-varying delays. Based on the division of state space, the equilibrium points (EPs) of FCNNs are given. Several sufficient conditions and criteria are proposed to ascertain the multiple O(t^-α)-stability of delayed FCNNs. The O(t^-α)-stability is the extension of Mittag-Leffler stability of fractional-order neural networks, which contains monostability and multistability. Moreover, the attraction basins of the stable EPs of FCNNs are estimated, which shows the attraction basins of the stable EPs can be larger than the divided subsets. These conditions and criteria supplement and improve the previous results. Finally, the results are illustrated by the simulation examples.

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.

Multistability of Fractional-Order Neural Networks With Unbounded Time-Varying Delays

This article addresses the multistability and attraction of fractional-order neural networks (FONNs) with unbounded time-varying delays. Several sufficient conditions are given to ensure the coexistence of equilibrium points (EPs) of FONNs with concave-convex activation functions. Moreover, by exploiting the analytical method and the property of the Mittag-Leffler function, it is shown that the multiple Mittag-Leffler stability of delayed FONNs is derived and the obtained criteria do not depend on differentiable time-varying delays. In particular, the criterion of the Mittag-Leffler stability can be simplified to M-matrix. In addition, the estimation of attraction basin of delayed FONNs is studied, which implies that the extension of attraction basin is independent of the magnitude of delays. Finally, three numerical examples are given to show the validity of the theoretical results.

Monostability and Multistability for Almost-Periodic Solutions of Fractional-Order Neural Networks With Unsaturating Piecewise Linear Activation Functions

Since the unsaturating activation function is unbounded, more complex dynamics may exist in neural networks with this kind of activation function. In this article, monostability and multistability results of almost-periodic solutions are developed for fractional-order neural networks with unsaturating piecewise linear activation functions. Some globally Mittag-Leffler attractive sets are given, and the existence of globally Mittag-Leffler stable almost-periodic solution is demonstrated by using Ascoli-Arzela theorem. In particular, some sufficient conditions are provided to ascertain the multistability of almost-periodic solutions based on locally positively invariant set. It shows that there exists an almost-periodic solution in each positively invariant set, and all trajectories converge to this periodic trajectory in that rectangular area. Two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria.

Design and Practical Stability of a New Class of Impulsive Fractional-Like Neural Networks

In this paper, a new class of impulsive neural networks with fractional-like derivatives is defined, and the practical stability properties of the solutions are investigated. The stability analysis exploits a new type of Lyapunov-like functions and their derivatives. Furthermore, the obtained results are applied to a bidirectional associative memory (BAM) neural network model with fractional-like derivatives. Some new results for the introduced neural network models with uncertain values of the parameters are also obtained.