Finite-time uniform stability of functional differential equations with applications in network synchronization control

In this paper, we investigate finite-time uniform stability of functional differential equations with applications in network synchronization control. First, a Razumikhin-type theorem is derived to ensure finite-time uniform stability of functional differential equations. Based on the theoretical results, finite-time uniform synchronization is proposed for a class of delayed neural networks and delayed complex dynamical networks by designing nontrivial and simple control strategies and some novel criteria are established. Especially, a feasible region of the control parameters for each neuron is derived for the realization of finite-time uniform synchronization of the addressed neural networks, which provide a great convenience for the application of the theoretical results. Finally, two numerical examples with numerical simulations are provided to show the effectiveness and feasibility of the theoretical results.     

Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field

Fractional order quaternion-valued neural networks are a type of fractional order neural networks for which neuron state, synaptic connection strengths, and neuron activation functions are quaternion. This paper is dealing with the Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field. The fractional order quaternion-valued neural networks are separated into four real-valued systems forming an equivalent four real-valued fractional order neural networks, which decreases the computational complexity by avoiding the noncommutativity of quaternion multiplication. Via some fractional inequality techniques and suitable Lyapunov functional, a brand new criterion is proposed first to ensure the Mittag-Leffler stability for the addressed neural networks. Besides, the combination of quaternion-valued adaptive and impulsive control is intended to realize the asymptotically synchronization between two fractional order quaternion-valued neural networks. Ultimately, two numerical simulations are provided to check the accuracy and validity of our obtained theoretical results.     

Chaotic synchronization of a fractional-order system based on washout filter control

In this paper, chaos in a fractional-order neural network system with varying time delays is presented, and chaotic synchronization system with varying time delays is constructed. The stability of constructed synchronization system is analyzed by Laplace transformation theory. In addition, the bifurcation graph of the chaotic system is illustrated. The study results show that the chaos in such fractional-order neural networks with varying time delay can be synchronized, and Washout filter control can be used to reduce the range of coupled parameter.     

Chaotic synchronization of nearest-neighbor diffusive coupling Hindmarsh–Rose neural networks in noisy environments

The chaotic synchronization of Hindmarsh–Rose neural networks linked by a nonlinear coupling function is discussed. The HR neural networks with nearest-neighbor diffusive coupling form are treated as numerical examples. By the construction of a special nonlinear-coupled term, the chaotic system is coupled symmetrically. For three and four neurons network, a certain region of coupling strength corresponding to full synchronization is given, and the effect of network structure and noise position are analyzed. For five and more neurons network, the full synchronization is very difficult to realize. All the results have been proved by the calculation of the maximum conditional Lyapunov exponent.     

Delay-induced diversity of firing behavior and ordered chaotic firing in adaptive neuronal networks

In this paper, we study the effect of time delay on the firing behavior and temporal coherence and synchronization in Newman–Watts thermosensitive neuron networks with adaptive coupling. At beginning, the firing exhibit disordered spiking in absence of time delay. As time delay is increased, the neurons exhibit diversity of firing behaviors including bursting with multiple spikes in a burst, spiking, bursting with four, three and two spikes, firing death, and bursting with increasing amplitude. The spiking is the most ordered, exhibiting coherence resonance (CR)-like behavior, and the firing synchronization becomes enhanced with the increase of time delay. As growth rate of coupling strength or network randomness increases, CR-like behavior shifts to smaller time delay and the synchronization of firing increases. These results show that time delay can induce diversity of firing behaviors in adaptive neuronal networks, and can order the chaotic firing by enhancing and optimizing the temporal coherence and enhancing the synchronization of firing. However, the phenomenon of firing death shows that time delay may inhibit the firing of adaptive neuronal networks. These findings provide new insight into the role of time delay in the firing activity of adaptive neuronal networks, and can help to better understand the complex firing phenomena in neural networks.     

Sliding mode synchronization controller design with neural network for uncertain chaotic systems

A sliding mode synchronization controller is presented with RBF neural network for two chaotic systems in this paper. The compound disturbance of the synchronization error system consists of nonlinear uncertainties and exterior disturbances of chaotic systems. Based on RBF neural networks, a compound disturbance observer is proposed and the update law of parameters is given to monitor the compound disturbance. The synchronization controller is given based on the output of the compound disturbance observer. The designed controller can make the synchronization error convergent to zero and overcome the disruption of the uncertainty and the exterior disturbance of the system. Finally, an example is given to demonstrate the availability of the proposed synchronization control method.     

Vibrational resonance in neuron populations with hybrid synapses

In this paper, vibrational resonance in excitable neuron populations with synapses is investigated by numerical simulation. In particular, the effect of the hybrid synapses on the signal detection and transmission in neural system is studied. Different topologies from regular and random networks to small-world networks are considered to analyze the dependence of vibrational resonance on the network structure and parameters. It is shown that there exists an optimal amplitude of high-frequency driving, enhancing the response of coupled neuron populations to a subthreshold signal. We find that chemical synaptic coupling is more efficient than the electrical coupling in signal detection and electrical synaptic coupling is better in signal transmission. Neuron populations with hybrid synapses compromise the merits of the two types of coupling and have an advantage in information communication.     

Dynamical behavior of delayed Hopfield neural networks with discontinuous activations

In this paper, a general class of neural networks with arbitrary constant delays is studied, whose neuron activations are discontinuous and may be unbounded or nonmonotonic. Based on the Leray–Schauder alternative principle and generalized Lyapunov approach, conditions are given under which there is a unique equilibrium of the neural network, which is globally asymptotically stable. Moreover, the existence and global asymptotic stability of periodic solutions are derived, where the neuron inputs are periodic. The obtained results extend previous works not only on delayed neural networks with Lipschitz continuous neuron activations, but also on delayed neural networks with discontinuous neuron activations.     

Impulsive synchronization seeking in general complex delayed dynamical networks

The present paper investigates the issues of impulsive synchronization seeking in general complex delayed dynamical networks with nonsymmetrical coupling. By establishing the extended Halanay differential inequality on impulsive delayed dynamical systems, some simple yet generic sufficient conditions for global exponential synchronization of the impulsive controlled delayed dynamical networks are derived analytically. Compared with some existing works, the distinctive features of these sufficient conditions indicate two aspects: on the one hand, these sufficient conditions can provide an effective impulsive control scheme to synchronize an arbitrary given delayed dynamical network to a desired synchronization state even if the original given network may be asynchronous itself. On the other hand, the controlled synchronization state can be selected as a weighted average of all the states in the network for the purpose of practical control strategy, which reveals the contributions and influences of various nodes in synchronization seeking processes of the dynamical networks. It is shown that impulses play an important role in making the delayed dynamical networks globally exponentially synchronized. Furthermore, the results are applied to a typical nearest-neighbor unidirectional time-delay coupled networks composed of chaotic FHN neuron oscillators, and numerical simulations are given to demonstrate the effectiveness of the proposed control methodology.     

Collective behavior of interacting locally synchronized oscillations in neuronal networks

Local circuits in the cortex and hippocampus are endowed with resonant, oscillatory firing properties which underlie oscillations in various frequency ranges (e.g. gamma range) frequently observed in the local field potentials, and in electroencephalography. Synchronized oscillations are thought to play important roles in information binding in the brain. This paper addresses the collective behavior of interacting locally synchronized oscillations in realistic neural networks. A network of five neurons is proposed in order to produce locally synchronized oscillations. The neuron models are Hindmarsh–Rose type with electrical and/or chemical couplings. We construct large-scale models using networks of such units which capture the essential features of the dynamics of cells and their connectivity patterns. The profile of the spike synchronization is then investigated considering different model parameters such as strength and ratio of excitatory/inhibitory connections. We also show that transmission time-delay might enhance the spike synchrony. The influence of spike-timing-dependence-plasticity is also studies on the spike synchronization.     

Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays

This paper investigates delay-dependent robust exponential state estimation of Markovian jumping fuzzy neural networks with mixed random time-varying delay. In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the robust exponential state estimation of Markovian jumping Hopfield neural networks with mixed random time-varying delays. Moreover probabilistic delay satisfies a certain probability-distribution. By introducing a stochastic variable with a Bernoulli distribution, the neural networks with random time delays is transformed into one with deterministic delays and stochastic parameters. The main purpose is to estimate the neuron states, through available output measurements such that for all admissible time delays, the dynamics of the estimation error is globally exponentially stable in the mean square. Based on the Lyapunov–Krasovskii functional and stochastic analysis approach, several delay-dependent robust state estimators for such T–S fuzzy Markovian jumping Hopfield neural networks can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using some standard numerical packages. The unknown gain matrix is determined by solving a delay-dependent LMI. Finally some numerical examples are provided to demonstrate the effectiveness of the proposed method.     

ADAPTIVE PINNING SYNCHRONIZATION OF COUPLED NEURAL NETWORKS WITH MIXED DELAYS AND VECTOR-FORM STOCHASTIC PERTURBATIONS＊

In this article,we consider the global chaotic synchronization of general coupled neural networks,in which subsystems have both discrete and distributed delays.Stochastic perturbations between subsyste...     

G-networks: a unifying model for neural and queueing networks

We survey results concerning a new stochastic network we have developed [1–7], which was initially motivated by neural network modelling [1], or — as we called it — by queueing networks with positive and negative customers [2, 3]. Indeed, it is well known that signals in neural networks are formed by impulses or action potentials, traveling much like customers in a queueing network. We call this model a G-network because it serves as a unifying basis for diverse areas of stochastic modelling in queueing networks, computer networks, computer system performance and neural networks. In its simplest version, negative and positive signals or customers circulate among a finite set of units, modelling inhibitory and excitatory signals of a neural network, or negative and positive customers of a queueing network. Signals can arrive either from other units or from the outside world. Positive signals are accumulated at the input of each unit, and constitute its signal potential. The state of each unit or neuron is its signal potential (which is equivalent to the queue length), while the network state is the vector of signal potentials at each neuron. If its potential is positive, a unit or neuron fires, and sends out signals to the other neurons or to the outside world. As it does so, its signal potential is depleted. In the Markovian case, this model has product form, i.e. the steady-state probability distribution of its potential vector is the product of the marginal probabilities of the potential at each neuron. The signal flow equations of the network, which describe the rate at which positive or negative signals arrive to each neuron, are non-linear. We discuss the relationship between this model and the usual connectionist (formal) model of neural networks, and present applications to combinatorial optimization and to image texture processing. Extensions of the model to the case of multiple signal classes, and to networks with triggered customer motion are presented. We also examine the general stability conditions which guarantee that the network has a well-defined steady-state behaviour.     

Bursting synchronization in networks with long-range coupling mediated by a diffusing chemical substance

Many networks of physical and biological interest are characterized by a long-range coupling mediated by a chemical which diffuses through a medium in which oscillators are embedded. We considered a one-dimensional model for this effect for which the diffusion is fast enough so as to be implemented through a coupling whose intensity decays exponentially with the lattice distance. In particular, we analyzed the bursting synchronization of neurons described by two timescales (spiking and bursting activity), and coupled through such a long-range interaction network. One of the advantages of the model is that one can pass from a local (Laplacian) type of coupling to a global (all-to-all) one by varying a single parameter in the interaction term. We characterized bursting synchronization using an order parameter which undergoes a transition as the coupling parameters are changed through a critical value. We also investigated the role of an external time-periodic signal on the bursting synchronization properties of the network. We show potential applications in the control of pathological rhythms in biological neural networks.     

Projective Synchronization of Fractional Quaternion Neural Networks With Time-Varying Delays北大核心CSCD

研究了具有时变时滞的分数阶四元数神经网络的投影同步问题.该文不将分数阶四元数神经网络系统转化成两个复值系统或四个实值系统,而是将四元数系统当做一个整体进行处理.在合适的控制器下,通过构造合适的Lyapunov函数,并利用一些不等式技巧,得到了具有时变时滞分数阶四元数时滞神经网络投影同步的充分性判据.最后,通过数值仿真实例验证了所得结论的有效性和可行性.     

Synchronization of a growing chaotic network model

Synchronization in large ensembles of coupled interacting units is a fundamental phenomenon, which is helpful for the understanding of working mechanisms in neuronal networks, social network, etc. In this paper, we will investigate the synchronization phenomenon in a network model. A feedback control scheme is proposed for the synchronization of the given complex networks. The obtained result indicates that synchronization can be achieved for growing chaotic network model. Method enhance the synchronizability of the given model are given at the same time. Finally, numerical simulations are given to show the effectiveness of obtained results.     

Adaptive synchronization in an array of chaotic neural networks with mixed delays and jumping stochastically hybrid coupling

In this paper, a general model of an array of N linearly coupled delayed neural networks with Markovian jumping hybrid coupling is introduced. The hybrid coupling consists of constant coupling, discrete and distributed time-varying delay coupling. The complex dynamical network jumps from one mode to another according to a Markovian chain, where all the coupling configurations are also dependent on mode switching. Meanwhile, all the coupling terms are subjected to stochastic disturbances which are described in terms of a Brownian motion. By adaptive approach, some sufficient criteria have been derived to ensure the synchronization in an array of jump neural networks with mixed delays and hybrid coupling in mean square. Surprisingly, it is found that complex networks with two different structure can also be synchronized according to known probability matrix. Finally, an example illustrated by switching between small-world networks and nearest-neighbor networks is given to show the effectiveness of the proposed criteria.     

Robust global synchronization of complex networks with neutral-type delayed nodes

In this paper, the problems of robust global exponential synchronization for a class of complex networks with time-varying delayed couplings are considered. Each node in the network is composed of a class of delayed neural networks described by a nonlinear delay differential equation of neutral-type. Since model errors commonly exist in practical applications, the parameter uncertainties are involved in the considered model. Sufficient conditions that ensure the complex networks to be robustly globally exponentially synchronized are obtained by using the Lyapunov functional method and some properties of Kronecker product. An illustrative example is presented to show the effectiveness of the proposed approach.     

A neural network system for solving an assortment problem in the steel industry

A neural network model for solving an assortment problem found in the iron and steel industry is discussed in this paper. The problem arises in the yard where steel plate is cut into rectangular pieces. The neural network model can be categorized as a Hopfield model, but the model is expanded to handle inequality constraints. The idea of a penalty function is used. A large penalty is applied to the network if a constraint is not satisfied. The weights are updated based on the penalty values. A special term is added to the energy function of the network to guarantee the convergence of the neural network which has this feature. The performance of the neural network was evaluated by comparison with an existing expert system. The results showed that the neural network has the potential to identify in a short time near-optimal solutions to the assortment problem. The neural network is used as the core of a system for dealing with the assortment problem. In building the neural networks system for practical use, there were many implementation issues. Some of them are presented here, and the fundamental ideas are explained. The performance of the neural network system is compared to that of the expert system and evaluated from the practical viewpoint. The results show that the neural network system is useful in handling the assortment problem.     

Stochastic resonance in small-world neuronal networks with hybrid electrical–chemical synapses

The dependence of stochastic resonance in small-world neuronal networks with hybrid electrical–chemical synapses on the probability of chemical synapse and the rewiring probability is investigated. A subthreshold periodic signal is imposed on one single neuron within the neuronal network as a pacemaker. It is shown that, irrespective of the probability of chemical synapse, there exists a moderate intensity of external noise optimizing the response of neuronal networks to the pacemaker. Moreover, the effect of pacemaker driven stochastic resonance of the system depends largely on the probability of chemical synapse. A high probability of chemical synapse will need lower noise intensity to evoke the phenomenon of stochastic resonance in the networked neuronal systems. In addition, for fixed noise intensity, there is an optimal chemical synapse probability, which can promote the propagation of the localized subthreshold pacemaker across neural networks. And the optimal chemical synapses probability turns even larger as the coupling strength decreases. Furthermore, the small-world topology has a significant impact on the stochastic resonance in hybrid neuronal networks. It is found that increasing the rewiring probability can always enhance the stochastic resonance until it approaches the random network limit.