Attractivity analysis of memristor-based cellular neural networks with time-varying delays

Multistability of Dynamic Memristor Delayed Cellular Neural Networks With Application to Associative Memories

Recently, dynamic memristor (DM)-cellular neural networks (CNNs) have received widespread attention due to their advantage of low power consumption. The previous works showed that DM-CNNs have at most 318 equilibrium points (EPs) with n=16 cells. Since time delay is unavoidable during the process of information transmission, the goal of this article is to research the multistability of DM-CNNs with time delay, and, meanwhile, to increase the storage capacity of DM-delay (D)CNNs. Depending on the different constitutive relations of memristors, two cases of the multistability for DM-DCNNs are discussed. After determining the constitutive relations, the number of EPs of DM-DCNNs is increased to 3ⁿ with n cells by means of the appropriate state-space decomposition and the Brouwer's fixed point theorem. Furthermore, the enlarged attraction domains of EPs can be obtained, and 2ⁿ of these EPs are locally exponentially stable in two cases. Compared with standard CNNs, the dynamic behavior of DM-DCNNs shows an outstanding merit. That is, the value of voltage and current approach to zero when the system becomes stable, and the memristor provides a nonvolatile memory to store the computation results. Finally, two numerical simulations are presented to illustrate the effectiveness of the theoretical results, and the applications of associative memories are shown at the end of this article.

Finite-/Fixed-Time Stability of Nonautonomous Functional Differential Inclusion: Lyapunov Approach Involving Indefinite Derivative

This article investigates a type of nonautonomous delayed differential equation (DDE) with discontinuity. Under the framework of the Filippov state solution, the finite-time stability (FNTS)/fixed-time stability (FXTS) problems of nonautonomous functional differential inclusion (FDI) are studied via the generalized Lyapunov functional method. The generalized Lyapunov functional used in this article is allowed to obtain an indefinite time derivative almost anywhere (a.a.) along the system's state solutions. Nevertheless, the classic Lyapunov functional requires that its time derivative retains seminegative/negative definiteness anywhere. As a result, novel FNTS and FXTS criteria of the trivial state solution for FDI are established. Moreover, the settling time (ST) of FNTS/FXTS is provided. Then, the developed Lyapunov functional approach is applied to realize the finite-/fixed-time stabilization control of delayed neuron networks (DNNs) possessing discontinuous activation and ball motion models. The proposed method and results concerning FNTS/FXTS are of great significance in neural network (NN)/mechanical control engineering applications.

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 7^p₁5^p₂3^p₃ equilibrium points, and 4^p₁3^p₂2^p₃ 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.

Proportional-Integral Observer-Based State Estimation for Markov Memristive Neural Networks With Sensor Saturations

This article investigates the resilient proportional-integral observer (PIO) problem for Markov switching memristive neural networks (MSMNNs) with randomly occurring sensor saturation within a finite-time interval. The Markov switching of memristive neural networks is regulated by a higher level deterministic switching signal, whose transition probabilities are piecewise time-varying and can be depicted by the average dwell-time strategy. Meanwhile, a Bernoulli stochastic process associated with an uncertain packet arriving rate is adopted to describe the randomly occurring sensor saturation. The aim is to design a resilient PIO such that the augmented dynamic has the property of stochastic finite-time boundedness while meeting the desired performance index. By applying the Lyapunov method and the average dwell-time scheme, sufficient criteria are established for MSMNNs, and a unified design method is presented for the existence of the PIO. Lastly, the attained theoretical results are validated via a numerical simulation.

Improved Results on Finite-Time Passivity and Synchronization Problem for Fractional-Order Memristor-Based Competitive Neural Networks: Interval Matrix Approach

This research paper deals with the passivity and synchronization problem of fractional-order memristor-based competitive neural networks (FOMBCNNs) for the first time. Since the FOMBCNNs’ parameters are state-dependent, FOMBCNNs may exhibit unexpected parameter mismatch when different initial conditions are chosen. Therefore, the conventional robust control scheme cannot guarantee the synchronization of FOMBCNNs. Under the framework of the Filippov solution, the drive and response FOMBCNNs are first transformed into systems with interval parameters. Then, the new sufficient criteria are obtained by linear matrix inequalities (LMIs) to ensure the passivity in finite-time criteria for FOMBCNNs with mismatched switching jumps. Further, a feedback control law is designed to ensure the finite-time synchronization of FOMBCNNs. Finally, three numerical cases are given to illustrate the usefulness of our passivity and synchronization results.

Global Exponential Stability of Memristive Neural Networks With Mixed Time-Varying Delays

This article investigates the Lagrange exponential stability and the Lyapunov exponential stability of memristive neural networks with discrete and distributed time-varying delays (DMNNs). By means of inequality techniques, theories of the M-matrix, and the comparison strategy, the Lagrange exponential stability of the underlying DMNNs is considered in the sense of Filippov, and the globally exponentially attractive set is estimated through employing the M-matrix and external input. Especially, when the external input is not concerned, the Lyapunov exponential stability of the corresponding DMNNs is developed immediately in the form of an M-matrix, which contains some published outcomes as special cases. Furthermore, by constructing an M-matrix-based differential system, the Lyapunov exponential stability of the DMNNs is studied, which is less conservative than some existing ones. Finally, three simulation examples are carried out to examine the validness of the theories.

Qualitative Analysis and Bifurcation in a Neuron System With Memristor Characteristics and Time Delay

This article focuses on the hybrid effects of memristor characteristics, time delay, and biochemical parameters on neural networks. First, we propose a novel neuron system with memristor and time delays in which the memristor is characterized by a smooth continuous cubic function. Second, the existence of equilibria of this type of neuron system is examined in the parameter space. Sufficient conditions that ensure the stability of equilibria and occurrence of pitchfork bifurcation are given for the memristor-based neuron system without delay. Third, some novel criteria of the addressed neuron system are constructed for guaranteeing the delay-dependent and delay-independent stability. The specific conditions are provided for Hopf bifurcations, and the properties of Hopf bifurcation are ascertained using the center manifold reduction and the normal form theory. Moreover, there exists a phenomenon of bistability for the delayed memristor-based neuron system having three equilibria. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

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.

Exponential Stabilization of Inertial Memristive Neural Networks With Multiple Time Delays

This article investigates the global exponential stabilization (GES) of inertial memristive neural networks with discrete and distributed time-varying delays (DIMNNs). By introducing the inertial term into memristive neural networks (MNNs), DIMNNs are formulated as the second-order differential equations with discontinuous right-hand sides. Via a variable transformation, the initial DIMNNs are rewritten as the first-order differential equations. By exploiting the theories of differential inclusion, inequality techniques, and the comparison strategy, the p th moment GES ( p ≥ 1 ) of the addressed DIMNNs is presented in terms of algebraic inequalities within the sense of Filippov, which enriches and extends some published results. In addition, the global exponential stability of MNNs is also performed in the form of an M-matrix, which contains some existing ones as special cases. Finally, two simulations are carried out to validate the correctness of the theories, and an application is developed in pseudorandom number generation.

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.

Nonlinear Networks With Mem-Elements: Complex Dynamics via Flux-Charge Analysis Method

Nonlinear dynamic memory elements, as memristors, memcapacitors, and meminductors (also known as mem-elements), are of paramount importance in conceiving the neural networks, mem-computing machines, and reservoir computing systems with advanced computational primitives. This paper aims to develop a systematic methodology for analyzing complex dynamics in nonlinear networks with such emerging nanoscale mem-elements. The technique extends the flux-charge analysis method (FCAM) for nonlinear circuits with memristors to a broader class of nonlinear networks N containing also memcapacitors and meminductors. After deriving the constitutive relation and equivalent circuit in the flux-charge domain of each two-terminal element in N , this paper focuses on relevant subclasses of N for which a state equation description can be obtained. On this basis, salient features of the dynamics are highlighted and studied analytically: 1) the presence of invariant manifolds in the autonomous networks; 2) the coexistence of infinitely many different reduced-order dynamics on manifolds; and 3) the presence of bifurcations due to changing the initial conditions for a fixed set of parameters (also known as bifurcations without parameters). Analytic formulas are also given to design nonautonomous networks subject to pulses that drive trajectories through different manifolds and nonlinear reduced-order dynamics. The results, in this paper, provide a method for a comprehensive understanding of complex dynamical features and computational capabilities in nonlinear networks with mem-elements, which is fundamental for a holistic approach in neuromorphic systems with such emerging nanoscale devices.

Exponential State Estimation for Stochastically Disturbed Discrete-Time Memristive Neural Networks: Multiobjective Approach

The state estimation of the discrete-time memristive model is studied in this article. By applying the stochastic analysis technique, sufficient formulas are established to ensure the exponentially mean-square stability of the error model. Moreover, the derived control gain matrix can be calculated via the linear matrix inequality (LMI). It should be mentioned that, by extending the derived conclusion to a multiobjective optimization problem, the maximum bound of the active function and the minimum bound of the disturbance attenuation are derived. The corresponding simulation figures are provided in the end.

New Criteria on Global Stabilization of Delayed Memristive Neural Networks With Inertial Item

In this paper, we are concerned with global stabilization for a kind of delayed memristive neural network with an inertial term. By building a new Lyapunov functional and designing a feedback controller, we obtain some new results on global stabilization of the addressed delayed memristive inertial neural networks (MINNs). An adaptive control strategy is also designed to realize the global stabilization. Compared with the reduced-order method used in the existing literature, we consider the stabilization directly from the MINNs themselves without a reduced-order method. In addition, the new results proposed here are shown as algebraic criteria, which are easy to test. At last, some simulations are given to show the validity of the derived criteria.

Global Stabilization of Fractional-Order Memristor-Based Neural Networks With Time Delay

This paper addresses the global stabilization of fractional-order memristor-based neural networks (FMNNs) with time delay. The voltage threshold type memristor model is considered, and the FMNNs are represented by fractional-order differential equations with discontinuous right-hand sides. Then, the problem is addressed based on fractional-order differential inclusions and set-valued maps, together with the aid of Lyapunov functions and the comparison principle. Two types of control laws (delayed state feedback control and coupling state feedback control) are designed. Accordingly, two types of stabilization criteria [algebraic form and linear matrix inequality (LMI) form] are established. There are two groups of adjustable parameters included in the delayed state feedback control, which can be selected flexibly to achieve the desired global asymptotic stabilization or global Mittag-Leffler stabilization. Since the existing LMI-based stability analysis techniques for fractional-order systems are not applicable to delayed fractional-order nonlinear systems, a fractional-order differential inequality is established to overcome this difficulty. Based on the coupling state feedback control, some LMI stabilization criteria are developed for the first time with the help of the newly established fractional-order differential inequality. The obtained LMI results provide new insights into the research of delayed fractional-order nonlinear systems. Finally, three numerical examples are presented to illustrate the effectiveness of the proposed theoretical results.

A Switched Operation Approach to Sampled-Data Control Stabilization of Fuzzy Memristive Neural Networks With Time-Varying Delay

This paper investigates the issue of sampled-data stabilization for Takagi-Sugeno fuzzy memristive neural networks (FMNNs) with time-varying delay. First, the concerned FMNNs are transformed into the tractable fuzzy NNs based on the excitatory and inhibitory of memristive synaptic weights using a new convex combination technique. Meanwhile, a switched fuzzy sampled-data controller is employed for the first time to tackle stability problems related to FMNNs. Then, the novel stabilization criteria of the FMNNs are established using the fuzzy membership functions (FMFs)-dependent Lyapunov-Krasovskii functional. This sufficiently utilizes information from not only the delayed state and the actual sampling pattern but also the FMFs. Two simulation examples are presented to demonstrate the feasibility and validity of the proposed method.

Passivity Analysis for Quaternion-Valued Memristor-Based Neural Networks With Time-Varying Delay

This paper is concerned with the problem of global exponential passivity for quaternion-valued memristor-based neural networks (QVMNNs) with time-varying delay. The QVMNNs can be seen as a switched system due to the memristor parameters are switching according to the states of the network. This is the first time that the global exponential passivity of QVMNNs with time-varying delay is investigated. By means of a nondecomposition method and structuring novel Lyapunov functional in form of quaternion self-conjugate matrices, the delay-dependent passivity criteria are derived in the forms of quaternion-valued linear matrix inequalities (LMIs) as well as complex-valued LMIs. Furthermore, the asymptotical stability criteria can be obtained from the proposed passivity criteria. Finally, a numerical example is presented to illustrate the effectiveness of the theoretical results.

Non-fragile state estimation for fractional-order delayed memristive BAM neural networks

This paper deals with the non-fragile state estimation problem for a class of fractional-order memristive BAM neural networks (FMBAMNNs) with and without time delays for the first time. By means of a novel transformation and interval matrix approach, non-fragile estimators are designed and parameter mismatch problem is averted. Sufficient criteria are established to ascertain the error system is asymptotically stable based on fractional-order Lyapunov functionals and linear matrix inequalities (LMIs). Two examples are put forward to show the effectiveness of the obtained results.

On the validity of memristor modeling in the neural network literature

An analysis of the literature shows that there are two types of non-memristive models that have been widely used in the modeling of so-called "memristive" neural networks. Here, we demonstrate that such models have nothing in common with the concept of memristive elements: they describe either non-linear resistors or certain bi-state systems, which all are devices without memory. Therefore, the results presented in a significant number of publications are at least questionable, if not completely irrelevant to the actual field of memristive neural networks.

Event-Based Synchronization Control for Memristive Neural Networks With Time-Varying Delay

In this paper, we investigate the global synchronization control problem for memristive neural networks (MNNs) with time-varying delay. A novel event-triggered controller is introduced with the linear diffusive term and discontinuous sign term. In order to greatly reduce the computation cost of the controller under certain event-triggering condition, two event-based control schemes are proposed with static event-triggering condition and dynamic event-triggering condition. Some sufficient conditions are derived by these control schemes to ensure the response MNN to be synchronized with the driving one. Furthermore, under certain event-triggering conditions, a positive lower bound is achieved for the interexecution time to guarantee that Zeno behavior cannot be executed. Finally, numerical simulations are provided to substantiate the effectiveness of the proposed theoretical results.

Global Exponential Stability and Synchronization for Discrete-Time Inertial Neural Networks With Time Delays: A Timescale Approach

This paper considers generalized discrete-time inertial neural network (GDINN). By timescale theory, the original network is rewritten as a timescale-type inertial NN. Two different scenarios are considered. In a first scenario, several criteria guaranteeing the global exponential stability for the addressed GDINN are obtained based on the generalized matrix measure concept. In this case, Lyapunov function or functional is not necessary. In a second scenario, some inequality analytical and scaling techniques are used to achieve the global exponential stability for the considered GDINN. The obtained criteria are also applied to the global exponential synchronization of drive-response GDINNs. Several illustrative examples, including applications to the pseudorandom number generator and encrypted image transmission, are given to show the effectiveness of the theoretical results.

Global Anti-Synchronization of Complex-Valued Memristive Neural Networks With Time Delays

This paper formulates a class of complex-valued memristive neural networks as well as investigates the problem of anti-synchronization for complex-valued memristive neural networks. Under the concept of drive-response, several sufficient conditions for guaranteeing the anti-synchronization are given by employing suitable Lyapunov functional and some inequality techniques. The proposed results of this paper are less conservative than existing literatures due to the characteristics of memristive complex-valued neural networks. Moreover, the proposed results are easy to be validated with the parameters of system itself. Finally, two examples with numerical simulations are showed to demonstrate the efficiency of our theoretical results.

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

The important topic of multistability of continuous-and discrete-time neural network (NN) models has been investigated rather extensively. Concerning the design of associative memories, multistability of delayed hybrid NNs is studied in this paper with an emphasis on the impulse effects. Arising from the spiking phenomenon in biological networks, impulsive NNs provide an efficient model for synaptic interconnections among neurons. Using state-space decomposition, the coexistence of multiple equilibria of hybrid impulsive NNs is analyzed. Multistability criteria are then established regrading delayed hybrid impulsive neurodynamics, for which both the impulse effects on the convergence rate and the basins of attraction of the equilibria are discussed. Illustrative examples are given to verify the theoretical results and demonstrate an application to the design of associative memories. It is shown by an experimental example that delayed hybrid impulsive NNs have the advantages of high storage capacity and high fault tolerance when used for associative memories.

Stability Analysis for Memristor-Based Complex-Valued Neural Networks with Time Delays

In this paper, the problem of stability analysis for memristor-based complex-valued neural networks (MCVNNs) with time-varying delays is investigated extensively. This paper focuses on the exponential stability of the MCVNNs with time-varying delays. By means of the Brouwer's fixed-point theorem and M-matrix, the existence, uniqueness, and exponential stability of the equilibrium point for MCVNNs are studied, and several sufficient conditions are obtained. In particular, these results can be applied to general MCVNNs whether the activation functions could be explicitly described by dividing into two parts of the real parts and imaginary parts or not. Two numerical simulation examples are provided to illustrate the effectiveness of the theoretical results.

Pseudo almost periodic solutions for quaternion-valued cellular neural networks with discrete and distributed delays

This paper is concerned with a class of quaternion-valued cellular neural networks with discrete and distributed delays. By using the exponential dichotomy of linear systems and a fixed point theorem, sufficient conditions are derived for the existence and global exponential stability of pseudo almost periodic solutions of this class of neural networks. Finally, a numerical example is given to illustrate the feasibility of the obtained results.

On the Topological Properties of the Certain Neural Networks

A topological index is a numeric quantity associated with a network or a graph that characterizes its whole structural properties. In [Javaid and Cao, Neural Computing and Applications, DOI 10.1007/s00521-017-2972-1], the various degree-based topological indices for the probabilistic neural networks are studied. We extend this study by considering the calculations of the other topological indices, and derive the analytical closed formulas for these new topological indices of the probabilistic neural network. Moreover, a comparative study using computer-based graphs has been carried out first time to clarify the nature of the computed topological descriptors for the probabilistic neural networks. Our results extend some known conclusions.

Finite-Time Stabilization of Delayed Memristive Neural Networks: Discontinuous State-Feedback and Adaptive Control Approach

In this paper, a general class of delayed memristive neural networks (DMNNs) system described by functional differential equation with discontinuous right-hand side is considered. Under the extended Filippov-framework, we investigate the finite-time stabilization problem for DMNNs by using the famous finite-time stability theorem and the generalized Lyapunov functional method. To do so, we design two classes of novel controllers including discontinuous state-feedback controller and discontinuous adaptive controller. Without assuming the boundedness and monotonicity of the activation functions, several sufficient conditions are given to stabilize the states of this class of DMNNs in finite time. Moreover, the upper bounds of the settling time for stabilization are estimated. Finally, numerical examples are provided to demonstrate the effectiveness of the developed method and the theoretical results.

Dissipativity Analysis for Stochastic Memristive Neural Networks With Time-Varying Delays: A Discrete-Time Case

In this paper, the dissipativity problem of discrete-time memristive neural networks (DMNNs) with time-varying delays and stochastic perturbation is investigated. A class of logical switched functions are put forward to reflect the memristor-based switched property of connection weights, and the DMNNs are then recast into a tractable model. Based on the tractable model, the robust analysis method and Refined Jensen-based inequalities are applied to establish some sufficient conditions that ensure the of DMNNs. Two numerical examples are presented to illustrate the effectiveness of the obtained results.

Synchronization of General Chaotic Neural Networks With Nonuniform Sampling and Packet Missing: A Switched System Approach

This paper is concerned with the exponential synchronization issue of general chaotic neural networks subject to nonuniform sampling and control packet missing in the frame of the zero-input strategy. Based on this strategy, we make use of the switched system model to describe the synchronization error system. First, when the missing of control packet does not occur, an exponential stability criterion with less conservatism is established for the resultant synchronization error systems via a superior time-dependent Lyapunov functional and the convex optimization approach. The characteristics induced by nonuniform sampling can be used to the full because of the structure and property of the constructed Lyapunov functional, that is not necessary to be positive definite except sampling times. Then, a criterion is obtained to guarantee that the general chaotic neural networks are synchronous exponentially when the missing of control packet occurs by means of the average dwell-time technique. An explicit expression of the sampled-data static output feedback controller is also gained. Finally, the effectiveness of the proposed new design methods is shown via two examples.