Switches of oscillations in coupled networks with multiple time delays
Introduction
Over the past decades, networks have attracted considerable attention since systems taking the form of networks abound in the world [1]. Examples include the Internet, the World Wide Web, neural networks, food webs, metabolic networks, epidemic networks, scientific citation webs, distribution networks, wireless communication networks, electrical power grids, social networks, etc. [1], [2], [3], [4]. However, many previous studies just considered individual networks, rather than coupled networks. Coupled networks are the ones which are combined by a number of individual sub-networks each with its own dynamical property and the interactions between the individual sub-networks. As a matter of fact, in recent years, great efforts have been devoted to studying coupled networks due to their extensive applications in real systems [5], [6], [7], [8], [9], [10], [11], [12], [13]. For instance, in order to describe the complicated interactions between billions of neurons in large-scale neural networks, the neurons are often lumped into highly connected sub-networks [5]. These sub-networks interactions (pathological synchronization) may induce serious problems such as Parkinson’s disease, essential tremor, and epilepsy [6], [9]. Another important application of coupled networks is the study of spread of infectious diseases among different species in epidemiology. For example, the fatal infectious disease, Acquired Immune Deficiency Syndrome (AIDS), spreads between two different mammalian communities (networks): gorillas and human beings. Moreover, in mad cow disease, cows and human beings can also be regarded as two different networks in terms of network language [10]. Coupled networks have also been applied to describe a wide variety of real systems, such as biological oscillators, genetic control networks, Josephson junction arrays, excitable media, self-organizing systems, etc [11], [12], [13].
Since the dynamical properties arising from the interactions of individual sub-networks are often different from the behaviors in isolation, coupled networks can exhibit rich and interesting dynamical phenomena, such as oscillation death, synchronization, quasi-periodic oscillations, chaotic attractors, etc [13], [14]. The oscillation death implies that the oscillations of the isolated sub-networks are suppressed and replaced by stable fixed-points. In fact, information is often stored as stable fixed-points of the system in some applications of artificial neural networks, such as content-addressable memories [15]. Synchronization is one of the significant and interesting features of coupled networks. Since the synchronization of coupled networks can well explain many natural phenomena observed, synchronization phenomena of coupled networks and the mechanisms behind them have become focal subjects [6], [9], [16], [17]. In particular, the study of synchronization of coupled neural networks is an important step for both understanding brain science and designing coupled neural networks for practical use. During the last decades, many kinds of synchronization have been proposed, such as complete synchronization, phase synchronization, lag synchronization, cluster synchronization, generalized synchronization, projective synchronization, and so on. Correspondingly, various schemes have been used to control these different kinds of synchronization, such as adaptive control, pinning control, impulsive control, etc. [3]. Because the extensive applications of coupled networks heavily depend on their dynamical behaviors, the investigation of dynamics of coupled networks is of both theoretical and practical importance.
Time delay is common and ubiquitous in real systems due to finite propagation speeds of signals and finite reaction times, such as controlled systems [18], networks [19], [20], laser systems [21], population dynamics [22], financial markets [23], chemical processes [24], etc. Unavoidably, neural networks incorporate time delays in the signal transmissions among neurons because of finite propagation velocity of action potentials and non-negligible time of a signal from a neuron to the receiving site of a postsynaptic neuron [15]. For example, the speed of signal conduction through unmyelinated axonal fibers is on the order of 1 m/s, resulting in time delays up to 80 ms for propagation through the cortical networks [25]. It is well known that time delay increases the dimensionality and complexity of systems. Time delays, albeit very short in most cases, often have a great effect upon the system performance. For instance, even small, comparing to the oscillation period, time delays may have a large impact on the dynamics of pulse-coupled integrate and fire oscillators and lead to the coexistence of regular and irregular dynamics [26]. Time delay is often regarded as a source of instability and oscillations of systems and can lead to rich and interesting dynamical behaviors, such as multi-stability, quasi-periodic responses, and even chaotic phenomena [13], [27]. For example, transmission time delay seems to play a significant role in integration of information arriving to a single neuron in different spatial and temporal windows and also at the network level in interneuron communication [7]. Different time delay lengths can change qualitative and quantitative properties of the dynamics, such as introducing or destroying stable oscillations, enhancing or suppressing synchronization between different neurons, as well as generating complex spatiotemporal patterns. Moreover, in practice, time-varying delay occurs commonly in most designs [28], [29], [30], [31], [32]. For instance, very recently, Li and Gao [29] studied model transformation and robust stability for discrete-time systems with time-varying delay and proposed fresh delay-range-dependent sufficient conditions in terms of a set of linear matrix inequalities by applying an input–output approach on the basis of scaled small gain theorem.
In the past few years, there has been an increasing activity and interest in the dynamics of coupled networks with time delays. For instance, in order to describe the interaction between parallel copies of a network through crosstalk, Hsu and Yang [33] presented a network consists of a pair of sub-networks with a single delayed coupling and built a general framework of local analysis for the sub-networks with an arbitrary number of neurons in contrast to the special case of three-neuron sub-networks in [34]. Song et al. [7], [14] investigated the stability switches, bifurcations and spatio-temporal patterns of nonlinear oscillations of coupled networks consisting of a pair of two-neuron sub-networks with two delayed couplings, rather than only one coupling between a single neuron of individual sub-networks in [33], [34], [35]. Wu et al. [36] studied the anti-synchronization of two general complex dynamical networks with non-delayed and delayed coupling by using pinning adaptive control scheme and derived a sufficient condition to guarantee the anti-synchronization between two networks based on Lyapunov stability theory and Barbalat lemma. For more related studies on the dynamics of coupled networks with time delays, readers are referred to the work in [37], [38], [39] and some references cited therein. However, the vast majority of these works is devoted to investigating the models of two coupled networks in which the number of sub-networks is minimal. From a practical point, coupled networks with multiple sub-networks should be considered since they have found applications in a variety of fields. For example, the brain organization can be viewed in gross sense as a number of local sub-networks (gray matter) coupled by long distance connections (white matter) [5]. Moreover, in the sea, the trophic web runs through plankton, fish, sharks up to whales and finally man, with the myriad of species in between. The species on one trophic level may predate several species below it [11]. To the best of the author’s knowledge, few results on nonlinear dynamics of coupled networks with multiple sub-networks in the presence of time delays can be found in literature, and the issue still remains challenging. Therefore, research in this area should be important and attractive.
Motivated by the above discussions, the purpose of this paper is to investigate the behaviors of a network consisting of four identical sub-networks and multiple couplings between the sub-networks, as shown in Fig. 1. Each sub-network has two neurons connected by bidirectional delayed connections. The whole network is constructed by adding two-way delayed couplings of the corresponding neurons between every two sub-networks. This network includes four sub-networks with multiple couplings, rather than a pair of sub-networks with only one coupling or a few couplings in previous studies [7], [14], [33], [34], [35], [37], [38]. The network involves two different time delays: internal time delay occurring among the internal neurons within the individual sub-networks and coupled time delay occurring in the interactions between the individual sub-networks. Internal time delay should be contained because the couplings between the individual sub-networks can be faster than the internal dynamics [13], [14], [34], but not be simply neglected or considered the same as coupled time delay in previous studies [7], [14], [33], [34], [38].
The rest of this paper is organized as follows. In Section 2, the stability of the network equilibrium and the existence of oscillations are discussed by analyzing the associated characteristic equation. In Section 3, case studies of numerical simulations are given to illustrate the effectiveness and usage of the proposed theoretical results and interesting dynamical phenomena of the coupled network are shown. Finally, conclusions are made and remarks are drawn in Section 4.
Section snippets
Stability analysis
As shown in Fig. 1, each sub-network involves two identical neurons with bidirectional delayed connections. The sub-networks X, Y, U, and V are described by the following nonlinear differential equations with time delays, respectivelywhere xi(t), yi(t), ui(t), and vi(t) are the states of
Numerical simulations
The nonlinear activation functions among the neurons within the sub-networks and nonlinear activation functions of the neurons between the sub-networks are taken as hyperbolic functions, i.e., .
To perform the numerical simulations, one can choose the internal connection weight and coupling strength as a=0.3 and b=−0.5, respectively. In this case, (α,β)∈N⊂M. From simple calculations, D1(υ)=0 has one positive root υ1=1.327. Then, one obtains τ1,n=1.55, 6.29,….. Similar arguments
Conclusions
This paper has presented a coupled network model consisting of four individual sub-networks and multiple couplings of neurons between the individual sub-networks. Time delays have been introduced into not only the internal connections within the individual sub-networks but also the couplings between the individual sub-networks since the couplings between the individual sub-networks can be faster than the internal dynamics [13], [14], [34]. The stability switches of network equilibrium and the
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant Nos. 11002047 and 11102058, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20100094120001, and the Open Funds from China State Key Laboratory of Structural Analysis for Industrial Equipment under Grant No. GZ1103.
The author would like to thank the associate editor and anonymous reviewers for their helpful comments and valuable suggestions that have helped
Xiaochen Mao received the B.S. degree in Mechanics from China University of Mining and Technology, Xuzhou, China, in 2003. He received the Ph.D. degree in Mechanics from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2009. In July 2009, he joined the Department of Engineering Mechanics, Hohai University, Nanjing, China. He is a reviewer of several journals. His current research interests include time-delay systems, coupled systems, and networks.
References (40)
- et al.
EEG-EMG, MEGEMG and EMG-EMG frequency analysis: physiological principles and clinical applications
Clin. Neurophysiol.
(2002) - et al.
Local bifurcation in symmetric coupled cell networks: linear theory
Physica. D
(2006) Stability switches, bifurcation, and multi-stability of coupled networks with time delays
Appl. Math. Comput.
(2012)- et al.
Analysis of bifurcation in a system of n coupled oscillators with delays
Appl. Math. Model
(2011) - et al.
Stability and Hopf bifurcation of a class of TCP/AQM networks
Nonlinear Anal: Real World Appl.
(2010) - et al.
Spatiotemporal dynamics on small-world neuronal networks: the roles of two types of time-delayed coupling
Chaos, Solitons Fractals
(2011) Induction of Hopf bifurcation and oscillation death by delays in coupled networks
Phys. Lett. A
(2009)- et al.
Pinning adaptive anti-synchronization between two general complex dynamical networks with non-delayed and delayed coupling
Appl. Math. Comput.
(2012) - et al.
Dynamics of a delayed two-coupled oscillator with excitatory-to-excitatory connection
Appl. Math. Comput.
(2010) - et al.
Stability switches and Hopf bifurcations in a pair of identical tri-neuron network loops
Phys. Lett. A
(2009)
Generalized projective synchronization between two different general complex dynamical networks with delayed coupling
Phys. Lett. A
Asynchronously switched control of switched linear systems with average dwell time
Automatica
The structure and function of complex networks
SIAM Rev.
An adaptive selection scheme of forwarding nodes in wireless sensor networks using a chaotic neural network
ICIC Express Lett.
Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling
Nonlinear Dyn.
Restoration of distribution networks with asynchronous wind generators
ICIC Express Lett.
Principles of Neural Science
Phase Resetting in Medicine and Biology: Stochastic Modeling and Data Analysis
Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks
Biol. Cybern.
Nonlinear dynamics of networks: the groupoid formalism
Bull. Am. Math. Soc.
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Xiaochen Mao received the B.S. degree in Mechanics from China University of Mining and Technology, Xuzhou, China, in 2003. He received the Ph.D. degree in Mechanics from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2009. In July 2009, he joined the Department of Engineering Mechanics, Hohai University, Nanjing, China. He is a reviewer of several journals. His current research interests include time-delay systems, coupled systems, and networks.