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# Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation

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## Abstract

In this paper, the generalized outer synchronization between two different delay-coupled complex dynamical networks with noise perturbation is investigated. With a nonlinear control scheme, the sufficient condition for almost sure generalized outer synchronization is developed based on the LaSalle-type invariance principle for stochastic differential equations. Numerical examples are examined to illustrate the effectiveness of the analytical results. The theoretic result is also applied to investigate the outer synchronization between two delay-coupled Hindmarsh–Rose neuronal networks with noise perturbation.

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... n (x z (t))] T and f (2) ...
... Noting the facts thatf (1) (0) = 0 andf (2) (0) = 0, the trivial solution of the system (7) exists. Moreover, under Assumption 2, we have the following: ...
... where g (1) z (y z (t)) =f (1) (e(t))U z and g (2) z (y z (t − τ (t))) =f (2) (e(t − τ (t)))U z . So far, we transformed the stability problem of the complex dynamical networks (2) into the stability problem of the N pieces of the corresponding dynamical networks (11). ...
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This study examines the problem of exponential stability of complex dynamical networks with impulse control and semi-Markovian switching parameters. By utilizing a supplementary variable technique and a plant transformation, the semi-Markovian switching complex dynamical networks can be equivalently expressed as its associated Markovian switching complex dynamical networks. By applying the Lyapunov stability theory, Jensen’s inequality, Dynkins formula, Schur complement and linear matrix inequality technique, some new delay-dependent conditions are derived to guarantee the exponential stability of the equilibrium point. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the results obtained.
... In order to achieve this purpose, a large number of approaches have been proposed for the synchronization of chaotic systems in the literature, [1][2][3][4][5][6][7][8][9] such as adaptive control method, 1,2,9 impulsive control method, 3 feedback control method, 4 and sliding mode control method. 5 Recently, since synchronization of stochastic neural networks plays an important role in fundamental science and technological practice, it has attracted much attention of researchers and many results of research have been reported in the literature (see other works [9][10][11][12][13][14][15][16][17][18][19] and the references therein). Wang et al 10 and Zhang et al 11,12 considered the problem of exponential synchronization for stochastic neural networks with time-varying delays. ...
... Remark 2. Assumption 3 is weaker than those investigated in other works. 2,[9][10][11][12]15,16,19,24,25 In reality, if we take R 4 ≡ 0, then Assumption 3 is the same as in the works of Zhu and Cao, 9 Sun et al, 19 and Dai et al. 25 Assumption 4. (t, 0, 0, 0, 0) ≡ 0. ...
... Remark 2. Assumption 3 is weaker than those investigated in other works. 2,[9][10][11][12]15,16,19,24,25 In reality, if we take R 4 ≡ 0, then Assumption 3 is the same as in the works of Zhu and Cao, 9 Sun et al, 19 and Dai et al. 25 Assumption 4. (t, 0, 0, 0, 0) ≡ 0. ...
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... or consensus of multi-agent systems in a network has been intensively investigated by many researchers for more than one decade [5][6][7][8][9][10][11][12][13][14][15][16]. ...
... Theorem 1 Suppose that the communication topology G contains a directed spanning tree. Consider the CDN model (9) with the trigger function (11). Under Assumption 1 and 2, if the coupling strength ...
... Based on trigger function (11) and inequality (35), we have ...
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... In addition to applications in physics [1,2], biology [3,4], and chemistry [5] etc., continuous systems are widely used in nonlinear dynamics studies as test systems [6][7][8][9]. As three-dimensional dissipative continuous systems can exhibit chaotic behavior according to their parameter values, these systems have become more popular recently [10][11][12][13]. The period-doubling bifurcations that are seen in model systems which exhibit a transition from periodic to chaotic behavior [14,15] can also be seen in nature, and in experiments involving for example semiconductors [16] and RLC circuits [17]. ...
... We use line segments of length since the Poincaré section points of the Rössler system construct a one-dimensional phase space and we fill these line segments with the Poincaré points. The box-counting fractal dimension is defined as (13) where N box is the number of line segments needed to include all the points of geometric object for chosen length. ...
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... In [18], the generalized outer synchronization between two delay-coupled complex dynamical networks was investigated, which may have nonidentical topological structures and different node dynamics. The complex networks are described bẏ ...
... Furthermore, the authors in [18] consider the dynamics of neural networks with delays. It is well known that the delays in neural networks are usually time-varying in the real world. ...
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... Guo et al. [25] investigated the global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling via pinning control. In Ref. [26], we investigated the generalized outer synchronization between two complex dynamical networks with time delay and noise perturbation. In Ref. [27], the synchronization of general complex networks with time-varying delays and impulsive effects was investigated. ...
... Assumption 3 in this paper is extensively used in Refs. [19,26,28,29,46,47]. The requirements of the local Lipschitz condition and the inequality (7) guarantee the existence and uniqueness of the solution of the error systems. ...
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In this paper, we investigate the outer synchronization between two uncertain complex delayed networks with noise coupling. With an adaptive control scheme, sufficient conditions for the stochastic outer synchronization are developed based on the LaSalle invariance principle for stochastic differential equations. Different from existing results, our method is valid for networks with linear or nonlinear inner coupling. Finally, numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.
... In general, time delays occur commonly in networks because of the network traffic congestion as well as the finite speed of signal transmission over the links. Hence, the synchronization study of CDNs with coupling time delays is quite important [21][22][23][24][25]. Exponential synchronization in CDNs with time-varying delay and hybrid coupling is investigated in [23]. ...
... -(13), (18)-(22), it is straightforward to show that ...
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... Therefore, studying synchronization of complex networks with time delay has become an active research topic and some results have proposed [14][15][16]. Most of the existing results in synchronization have been concerned on continuous dynamics and deterministic complex networks with or without delays [15,[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] rather than discrete dynamics. In fact, many mathematical models related to nonlinear phenomena are defined as discrete dynamical system in biological process, population growth, and neural networks. ...
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... In recent decades, considerable attention has been devoted to the time delay systems due to their extensive applications in practical systems including circuit theory, chemical processing, bio engineering, complex dynamical networks, automatic control and so on. In the implementation of complex dynamical networks, time delay is unavoidably encountered due to the finite speed of signal transmission over the link and the network traffic congestions [20][21][22][23]. Neural networks with leakage delay are a class of one type of important neural networks. ...
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... Study of hybrid synchronization involves multiple chaotic systems like complete, adaptive, global, projective, and antisynchronization systems [17,[20][21][22][23][24][25][26][27], and synchronization systems in multiple coupled complex networks [28,29]. Currently, hybrid synchronization of several connected chaotic systems is a hot topic of research and the work includes investigation of complex network synchronization for perturbations and delays [30] and a study of neural networks for synchronization [31]. ...
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... By investigating the existing references, it is seen that the earlier studies seldom took into account the effect of noise perturbation on complex networks. Recently, synchronization problems of complex networks with noise perturbation were studied in some references, such as [25,26]. However, it is found that the designed controllers in these references were not added as noise perturbation. ...
... In recent decades, considerable attention has been devoted to the time delay systems due to their extensive applications in practical systems including circuit theory, chemical processing, bio engineering, complex dynamical networks, automatic control and so on. In the implementation of complex dynamical networks, time delay is unavoidably encountered due to the finite speed of signal transmission over the link and the network traffic congestions [23][24][25][26] . ...
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... However, for outer-synchronization, all individuals in two networks will achieve identical behaviors. In many application fields, outer-synchronization may seem practical [16][17][18][19][20][21][22][23]. For example, in heuristic computational intelligence, it is known that outer-synchronization is rooted in brain-inspired computing from evolutionary strategies to cognitive tasks. ...
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... Author Name: Preparation of Papers for IEEE Access (February 2017) 2 synchronization for two fractional-order complex dynamical networks with unknown parameters and unknown bounded external disturbances. Considering that time delays often occur in dynamical networks due to the finite speed of information propagation or processing, it is imperative to incorporate time delays into the models of complex dynamical networks [27]- [29]. However, the absolute constant delay may be scarce in the practical networks, it makes more sense to study complex networks with multiple time-varying delay couplings [30]- [33]. ...
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... It has been demonstrated that many real-world problems have close relationships with network synchronization. Various synchronization notations have been extended for its potential applications in various fields, such as complete synchronization [18], generalized synchronization [19], phase synchronization [20], projective synchronization [21], lag synchronization [22], cluster synchronization [23], and adaptive synchronization [24]. To date, most of the existing synchronization results for dynamic networks are asymptotic synchronization, which means that the convergence rate is at best exponential with infinite timesetting. ...
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We study a large population of globally coupled phase oscillators subject to common white Gaussian noise and find analytically that the critical coupling strength between oscillators for synchronization transition decreases with an increase in the intensity of common noise. Thus, common noise promotes the onset of synchronization. Our prediction is confirmed by numerical simulations of the phase oscillators as well as of limit-cycle oscillators.
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Complete chaotic synchronization of end lasers has been observed in a line of mutually coupled, time-delayed system of three lasers, with no direct communication between the end lasers. The present paper uses ideas from generalized synchronization to explain the complete synchronization in the presence of long coupling delays, applied to a model of mutually coupled semiconductor lasers in a line. These ideas significantly simplify the analysis by casting the stability in terms of the local dynamics of each laser. The variational equations near the synchronization manifold are analyzed, and used to derive the synchronization condition that is a function of parameters. The results explain and predict the dependence of synchronization on various parameters, such as time delays, strength of coupling and dissipation. The ideas can be applied to understand complete synchronization in other chaotic systems with coupling delays and no direct communication between synchronized subsystems.
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We study synchronization for two unidirectionally coupled networks. This is a substantial generalization of several recent papers investigating synchronization inside a network. We derive analytically a criterion for the synchronization of two networks which have the same (inside) topological connectivity. Then numerical examples are given which fit the theoretical analysis. In addition, numerical calculations for two networks with different topological connections are presented and interesting synchronization and desynchronization alternately appear with increasing value of the coupling strength.
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Recently, it has been demonstrated that many large complex networks display a scale-free feature, that is, their connectivity distributions are in the power-law form. In this paper, we investigate the synchronization phenomenon in scale-free dynamical networks. We show that if the coupling strength of a scale-free dynamical network is greater than a positive threshold, then the network will synchronize no matter how large it is. We show that the synchronizability of a scale-free dynamical network is robust against random removal of nodes, but is fragile to specific removal of the most highly connected nodes
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This article offers a survey of the recent research advances in pinning control and pinning synchronization on complex dynamical networks. The emphasis is on research ideas and theoretical developments. Some technical details, if deemed necessary for clarity, will be outlined as well.
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The primary objective of this paper is to propose a new approach for analyzing pinning stability in a complex dynamical network via impulsive control. A simple yet generic criterion of impulsive pinning synchronization for such coupled oscillator network is derived analytically. It is shown that a single impulsive controller can always pin a given complex dynamical network to a homogeneous solution. Subsequently, the theoretic result is applied to a small-world (SW) neuronal network comprised of the Hindmarsh–Rose oscillators. It turns out that the firing activities of a single neuron can induce synchronization of the underlying neuronal networks. This conclusion is obviously in consistence with empirical evidence from the biological experiments, which plays a significant role in neural signal encoding and transduction of information processing for neuronal activity. Finally, simulations are provided to demonstrate the practical nature of the theoretical results.
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Chaotically-spiking dynamics of Hindmarsh–Rose neurons are discussed based on a flexible definition of phase for chaotic flow. The phase synchronization of two coupled chaotic neurons is in fact the spike synchronization. As a multiple time-scale model, the coupled HR neurons have quite different behaviors from the Rössler oscillators only having single time-scale mechanism. Using such a multiple time-scale model, the phase function can detect synchronization dynamics that cannot be distinguished by cross-correlation. Moreover, simulation results show that the Lyapunov exponents cannot be used as a definite criterion for the occurrence of chaotic phase synchronization for such a system. Evaluation of the phase function shows its utility in analyzing nonlinear neural systems.
Book
1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of Stochastic Differential Equations.- 8. Time Discrete Approximation of Deterministic Differential Equations.- 9. Introduction to Stochastic Time Discrete Approximation.- 10. Strong Taylor Approximations.- 11. Explicit Strong Approximations.- 12. Implicit Strong Approximations.- 13. Selected Applications of Strong Approximations.- 14. Weak Taylor Approximations.- 15. Explicit and Implicit Weak Approximations.- 16. Variance Reduction Methods.- 17. Selected Applications of Weak Approximations.- Solutions of Exercises.- Bibliographical Notes.
Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes is investigated in this paper. Based on Barbalat’s lemma, some sufficient synchronization criteria are derived by applying the nonlinear feedback control. Although previous work studied function projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In our work, the dynamics of the nodes of the complex networks are any chaotic systems without the limitation of the partial linearity. In addition, each network can be undirected or directed, connected or disconnected, and nodes in either network may have identical or different dynamics. The proposed strategy is applicable to almost all kinds of complex networks. Numerical simulations further verify the effectiveness and feasibility of the proposed synchronization method. Numeric evidence shows that the synchronization rate is sensitively influenced by the feedback strength, the time delay, the network size and the network topological structure.
The synchronization stability problem of general complex dynamical networks with non-delayed and delayed coupling is investigated based on a piecewise analysis method, the variation interval of the time delay is firstly divided into several subintervals, by checking the variation of derivative of a Lyapunov functional in every subinterval, several new delay-dependent synchronization stability conditions are derived in the form of linear matrix inequalities, which are easy to be verified by the LMI toolbox. Some numerical examples show that, when the number of the divided subintervals increases, the corresponding criteria can provide much less conservative results.
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The main aim of this paper is to establish the LaSalle-type asymptotic convergence theorems for the solutions of stochastic differential delay equations. These stochastic versions are then applied to establish sufficient criteria for the stochastically asymptotic stability of the delay equations. Several examples are also given for illustration.
In this paper, generalized synchronization (GS) between two coupled complex networks is theoretically and numerically studied, where the node vectors in different networks are not the same, and the numbers of nodes of both networks are not necessarily equal. First, a sufficient criterion for GS, one kind of outer synchronizations, of two coupled networks is established based on the auxiliary system method and the Lyapunov stability theory. Numerical examples are also included which coincide with the theoretical analysis.
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This paper regards the outer synchronization between two delay-coupled complex dynamical networks with nonidentical topological structures and a noise perturbation. Considering one network as the drive network and the other one as the response network, the drive–response system achieves synchronous states through a suitably designed adaptive controller. The stochastic LaSalle invariance principle is employed to theoretically prove the almost sure synchronization between two networks. Finally, two numerical examples are examined in order to illustrate the proposed synchronization scheme.
This paper investigates the adaptive synchronization between two nonlinearly delay-coupled complex networks with the bidirectional actions and nonidentical topological structures. Based on LaSalle’s invariance principle, some criteria for the synchronization between two coupled complex networks are achieved via adaptive control. To validate the proposed methods, the unified chaotic system as the nodes of the networks are analyzed in detail, and numerical simulations are given to illustrate the theoretical results.Research highlights► We investigate the outer synchronization of two nonlinear coupled complex network. ► We not only consider their own network of coupling, but also take into account the bidirectional actions and time delays. ► Nonidentical topological structures are also considered.
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In this paper, the finite-time stochastic outer synchronization between two different complex dynamical networks with noise perturbation is investigated. By using suitable controllers, sufficient conditions for finite-time stochastic outer synchronization are derived based on the finite-time stability theory of stochastic differential equations. It is noticed that the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results. The effect of control parameters on the settling time is also numerically demonstrated.
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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.
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In this paper, synchronization control of stochastic neural networks with time-varying delays has been considered. A novel control method is given using the Lyapunov functional method and linear matrix inequality (LMI) approach. Several sufficient conditions have been derived to ensure the global asymptotical stability in mean square for the error system, and thus the drive system synchronize with the response system. Also, the estimation gains can be obtained. With these new and effective methods, synchronization can be achieved. Simulation results are given to verify the theoretical analysis in this paper.
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Recently, it has been demonstrated that many large-scale complex dynamical networks display a collective synchronization motion. Here, we introduce a time-varying complex dynamical network model and further investigate its synchronization phenomenon. Based on this new complex network model, two network chaos synchronization theorems are proved. We show that the chaos synchronization of a time-varying complex network is determined by means of the inner coupled link matrix, the eigenvalues and the corresponding eigenvectors of the coupled configuration matrix, rather than the conventional eigenvalues of the coupled configuration matrix for a uniform network. Especially, we do not assume that the coupled configuration matrix is symmetric and its off-diagonal elements are nonnegative, which in a way generalizes the related results existing in the literature.
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A new method of controlling arbitrary nonlinear dynamic systems, , is presented. It is proved that there exists solutions, x(t), in the neighborhood of any arbitrary‘goal’ dynamics g(t) that are entrained to g(t), through the use of an additive controlling action, , which is the sum of the open-loop (Hübler) action, , and a suitable linear closed-loop (feedback) action C(g,t). Examples of some newly obtained entrainment capabilities are given for the Duffing and Van der Pol systems. For these and the Lorenz, and Rössler systems proofs are given for global basins of entrainment for all goal dynamics that can be exponentially bounded in time. The basin of entrainment is also established for the Chua system, as well as the possibility of a coexisting basin of attraction to another fixed point.
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This paper addresses the theoretical analysis of synchronization between two complex networks with nonidentical topological structures. By designing effective adaptive controllers, we achieve synchronization between two complex networks. Both the cases of identical and nonidentical network topological structures are considered and several useful criteria for synchronization are given. Illustrative examples are presented to demonstrate the application of the theoretical results.
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There exist some fundamental and yet challenging problems in pinning control of complex networks: (1) What types of pinning schemes may be chosen for a given complex network to realize synchronization? (2) What kinds of controllers may be designed to ensure the network synchronization? (3) How large should the coupling strength be used in a given complex network to achieve synchronization? This paper addresses these technique questions. Surprisingly, it is found that a network under a typical framework can realize synchronization subject to any linear feedback pinning scheme by using adaptive tuning of the coupling strength. In addition, it is found that the nodes with low degrees should be pinned first when the coupling strength is small, which is contrary to the common view that the most-highly-connected nodes should be pinned first. Furthermore, it is interesting to find that the derived pinning condition with controllers given in a high-dimensional setting can be reduced to a low-dimensional condition without the pinning controllers involved. Finally, simulation examples of scale-free networks are given to verify the theoretical results.
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In this paper, the complete synchronization problem is investigated in an array of linearly stochastically coupled identical networks with time delays. The stochastic coupling term, which can reflect a more realistic dynamical behavior of coupled systems in practice, is introduced to model a coupled system, and the influence from the stochastic noises on the array of coupled delayed neural networks is studied thoroughly. Based on a simple adaptive feedback control scheme and some stochastic analysis techniques, several sufficient conditions are developed to guarantee the synchronization in an array of linearly stochastically coupled neural networks with time delays. Finally, an illustrate example with numerical simulations is exploited to show the effectiveness of the theoretical results.
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We observed the arising of a new slow regular rhythm along the chain of unidirectional coupled neurons whose individual dynamics is periodic spiking. In this study we use the Hindmarsh–Rose type neurons which are potentially able to produce several modes of behavior: periodic spiking, periodic spiking-bursting and chaotic spiking-bursting activities. Several spatial bifurcations take place along the chain: the bifurcation from periodic spiking regime to chaotic spiking-bursting, transformation corresponding to the developing chaos, and finally, the transition from a irregular spiking-bursting regime to a regime with regular bursts. The calculation of the Kolmogorov–Sinai entropy indicates that the periodic oscillations of some neurons at the beginning of the chain are transformed into spiking-bursting chaos that is localized along the network, becoming later regular slow oscillations in spite of the chaoticity of the spiking pulsations.
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In this paper, a design of coupling and effective sufficient condition for stable complete synchronization and antisynchronization of a class of coupled time-delayed systems with parameter mismatch and noise perturbation are established. Based on the LaSalle-type invariance principle for stochastic differential equations, sufficient conditions guaranteeing complete synchronization and antisynchronization with constant time delay are developed. Also delay-dependent sufficient conditions for the case of time-varying delay are derived by using the Lyapunov approach for stochastic differential equations. Numerical examples fully support the analytical results.
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In this paper, the problem of generalized outer synchronization between two completely different complex dynamical networks is investigated. With a nonlinear control scheme, a sufficient criterion for this generalized outer synchronization is derived based on Barbalat's lemma. Two corollaries are also obtained, which contains the situations studied in two lately published papers as special cases. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results.
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The Hodgkin-Huxley model1 of the nerve impulse consists of four coupled nonlinear differential equations, six functions and seven constants. Because of the complexity of these equations and the necessity for numerical solution, it is difficult to use them in simulations of interactions in small neural networks. Thus, it would be useful to have a second-order differential equation which predicted correctly properties such as the frequency-current relationship. Fitzhugh2 introduced a second-order model of the nerve impulse, but his equations predict an action potential duration which is similar to the inter-spike interval3 and they do not give a reasonable frequency-current relationship. To develop a second-order model having few parameters but which does not have these disadvantages, we have generalized the second-order Fitzhugh equations2, and based the form of the functions in the new equations on voltage-clamp data obtained from a snail neurone. We report here an unexpected property of the resulting equations-the x and y null clines in the phase plane lie close together when the phase point is on the recovery side of the phase plane. The resulting slow movement along the phase path gives a long inter-spike interval, a property not shown clearly by previous models2,4. The model also predicts the linearity of the frequency-current relationship, and may be useful for studying detailed interactions in networks containing small numbers of neurones.
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Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
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We report experimental observation of phase synchronization in an array of nonidentical noncoupled noisy neuronal oscillators, due to stimulation with external noise. The synchronization derives from a noise-induced qualitative change in the firing pattern of single neurons, which changes from a quasiperiodic to a bursting mode. We show that at a certain noise intensity the onsets of bursts in different neurons become synchronized, even though the number of spikes inside the bursts may vary for different neurons. We demonstrate this effect both experimentally for the electroreceptor afferents of paddlefish, and numerically for a canonical phase model, and characterize it in terms of stochastic synchronization.
Article
In this paper different topologies of populations of FitzHugh-Nagumo neurons have been introduced in order to investigate the role played by the noise in the network. Each neuron is subjected to an independent source of noise. In these conditions the behavior of the population depends on the connection among the elements. By analyzing several kinds of topology (ranging from regular to random) different behaviors have been observed. Several topologies behave in an optimal way with respect to the range of noise level leading to an improvement in the stimulus response coherence, while others with respect to the maximum values of the performance index. However, the best results in terms of both the suitable noise level and high stimulus response coherence have been obtained when a diversity in neuron characteristic parameters has been introduced and the neurons have been connected in a small-world topology.
Article
Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences. Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com/