How single node dynamics enhances synchronization in neural networks with electrical coupling

The stability of the completely synchronous state in neural networks with electrical coupling is analytically investigated applying both the Master Stability Function approach (MSF), developed by Pecora and Carroll (1998), and the Connection Graph Stability method (CGS) proposed by Belykh et al. (2004). The local dynamics is described by Morris–Lecar model for spiking neurons and by Hindmarsh–Rose model in spike, burst, irregular spike and irregular burst regimes. The combined application of both CGS and MSF methods provides an efficient estimate of the synchronization thresholds, namely bounds for the coupling strength ranges in which the synchronous state is stable. In all the considered cases, we observe that high values of coupling strength tend to synchronize the system. Furthermore, we observe a correlation between the single node attractor and the local stability properties given by MSF. The analytical results are compared with numerical simulations on a sample network, with excellent agreement.     

Synchronization analysis through coupling mechanism in realistic neural models

Synchronization which relates to the system’s stability is important to many engineering and neural applications. In this paper, an attempt has been made to implement response synchronization using coupling mechanism for a class of nonlinear neural systems. We propose an OPCL (open-plus-closed-loop) coupling method to investigate the synchronization state of driver-response neural systems, and to understand how the behavior of these coupled systems depend on their inner dynamics. We have investigated a general method of coupling for generalized synchronization (GS) in 3D modified spiking and bursting Morris–Lecar (M-L) neural models. We have also presented the synchronized behavior of a network of four bursting Hindmarsh–Rose (H-R) neural oscillators using a bidirectional coupling mechanism. We can extend the coupling scheme to a network of N neural oscillators to reach the desired synchronous state. To make the investigations more promising, we consider another coupling method to a network of H-R oscillators using bidirectional ring type connections and present the effectiveness of the coupling scheme.     

Dynamics of chaotic maps for modelling the multifractal spectrum of human brain Diffusion Tensor Images

The multifractal spectra of 3d Diffusion Tensor Images (DTI) obtained by magnetic resonance imaging of the human brain are studied. They are shown to deviate substantially from artificial brain images with the same white matter intensity. All spectra, obtained from 12 healthy subjects, show common characteristics indicating non-trivial moments of the intensity. To model the spectra the dynamics of the chaotic Ikeda map are used. The DTI multifractal spectra for positive q are best approximated by 3d coupled Ikeda maps in the fully developed chaotic regime. The coupling constants are as small as α = 0.01. These results reflect not only the white tissue non-trivial architectural complexity in the human brain, but also demonstrate the presence and importance of coupling between neuron axons. The architectural complexity is also mirrored by the deviations in the negative q-spectra, where the rare events dominate. To obtain a good agreement in the DTI negative q-spectrum of the brain with the Ikeda dynamics, it is enough to slightly modify the most rare events of the coupled Ikeda distributions. The representation of Diffusion Tensor Images with coupled Ikeda maps is not unique: similar conclusions are drawn when other chaotic maps (Tent, Logistic or Henon maps) are employed in the modelling of the neuron axons network.     

Synchronization of heteroclinic circuits through learning in coupled neural networks

The synchronization of oscillatory activity in neural networks is usually implemented by coupling the state variables describing neuronal dynamics. Here we study another, but complementary mechanism based on a learning process with memory. A driver network, acting as a teacher, exhibits winner-less competition (WLC) dynamics, while a driven network, a learner, tunes its internal couplings according to the oscillations observed in the teacher. We show that under appropriate training the learner can “copy” the coupling structure and thus synchronize oscillations with the teacher. The replication of the WLC dynamics occurs for intermediate memory lengths only, consequently, the learner network exhibits a phenomenon of learning resonance.     

Winding Numbers and Average Frequencies in Phase Oscillator Networks

We study networks of coupled phase oscillators and show that network architecture can force relations between average frequencies of the oscillators. The main tool of our analysis is the coupled cell theory developed by Stewart, Golubitsky, Pivato, and Torok, which provides precise relations between network architecture and the corresponding class of ODEs in R^M and gives conditions for the flow-invariance of certain polydiagonal subspaces for all coupled systems with a given network architecture. The theory generalizes the notion of fixed-point subspaces for subgroups of network symmetries and directly extends to networks of coupled phase oscillators. For systems of coupled phase oscillators (but not generally for ODEs in R^M, where M ≥ 2), invariant polydiagonal subsets of codimension one arise naturally and strongly restrict the network dynamics. We say that two oscillators i and j coevolve if the polydiagonal θ_i = θ_j is flow-invariant, and show that the average frequencies of these oscillators must be equal. Given a network architecture, it is shown that coupled cell theory provides a direct way of testing how coevolving oscillators form collections with closely related dynamics. We give a generalization of these results to synchronous clusters of phase oscillators using quotient networks, and discuss implications for networks of spiking cells and those connected through buffers that implement coupling dynamics.     

Adaptive coupled synchronization among multi-Lorenz systems family

In this paper, the adaptive synchronization method of coupled system is proposed for multi-Lorenz systems family. This method can avoid estimating the value of coupling coefficient. Strict theoretical proofs are given. And we derived a sufficient condition of synchronization for a general unidirectional coupling ring network with N identical Lorenz systems. The network is coupled through the first state variable of each equation. In fact, the whole unidirectional coupling network will synchronize by adding only one adaptive feedback gain equation. Numerical simulations show the effectiveness of the methods.     

Analysis and stability of consensus in networked control systems

In this paper, the consensus problem in networks of integrators is investigated. After recalling the classical diffusive protocol, we present in a unified framework some results on the rate of convergence previously presented in the literature. Then, we introduce two switching communication protocols, one based on a switching coupling law between neighboring nodes, the other on the conditional activation of links in the network. We show that the former protocol induces the monotonicity of each system in the network, enhancing the speed of convergence to consensus. Moreover, adopting this novel protocol, we are able to control the network, steering the nodes’ dynamics to a desired consensus value. The aim of the latter protocol is instead to select adaptively the activation of the edges of the network, in accordance with the dynamics of the network. After showing the effectiveness of both approaches through numerical simulations, the stability properties of these protocols are discussed.     

Generalized outer synchronization between non-dissipatively coupled complex networks with different-dimensional nodes

This paper investigates the generalized outer synchronization (GOS) between two non-dissipatively coupled complex dynamical networks (CDNs) with different time-varying coupling delays. Our drive-response networks also possess nonlinear inner coupling functions and time-varying outer coupling configuration matrices. Besides, in our network models, the nodes in the same network are nonidentical and the nodes in different networks have different state dimensions. Asymptotic generalized outer synchronization (AGOS) and exponential generalized outer synchronization (EGOS) are defined for our CDNs. Our main objective in this paper is to design AGOS and EGOS controllers for our drive-response networks via the open-plus-closed-loop control technique. Distinguished from most existing literatures, it is the partial intrinsic dynamics of each node in response network that is restricted by the QUAD condition, which is easy to be satisfied. Representative simulation examples are given to verify the effectiveness and feasibility of our GOS theoretical results in this paper.     

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.     

Representations and derivations of quasi ∗-algebras induced by local modifications of states

The relationship between the GNS representations associated to states on a quasi ∗-algebra, which are local modifications of each other (in a sense which we will discuss) is examined. The role of local modifications on the spatiality of the corresponding induced derivations describing the dynamics of a given quantum system with infinite degrees of freedom is discussed.     

爆炸冲击载荷下航空铝合金平板动态响应数值分析方法

为得到适用于爆炸冲击载荷下航空铝合金平板动态响应的数值分析方法,采用LS-DYNA显式动力学分析软件对爆炸冲击载荷下的铝合金平板进行数值仿真计算.主要研究了不同的任意Lagrange-Euler(拉格朗日-欧拉)网格（ALE）输运步算法、流固耦合方式、流固耦合点数量、网格尺寸、有限元单元类型对计算结果的影响.通过计算结果与实验结果的分析对比,表明采用van Leer+HIS输运步算法、罚函数耦合方式、在流体网格与结构网格之间采用3个耦合点、结构网格尺寸与空气域网格尺寸比例设为2∶1、结构单元采用163号壳单元时可以较为准确地计算航空铝合金平板在爆炸冲击载荷下的动态响应,并且能提高计算效率,节约计算时间.     

The rigorous derivation of the T2 focusing cubic NLS from 3D

We derive rigorously the 2D periodic focusing cubic NLS as the mean-field limit of the 3D focusing quantum many-body dynamics describing a dilute Bose gas with periodic boundary condition in the x-direction and a well of infinite-depth in the z-direction. Physical experiments for these systems are scarce. We find that, to fulfill the empirical requirement for observing NLS dynamics in experiments, namely, that the kinetic energy dominates the potential energy, it is necessary to impose an extra restriction on the system parameters. This restriction gives rise to an unusual coupling constant.     

The dynamics ofn weakly coupled identical oscillators

Summary We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.     

A novel method of control using chaotic dynamics in systems having many degrees-of-freedom

Chaotic dynamics in systems having many degrees of freedom are investigated from the viewpoint of harnessing chaos and is applied to complex control problems to indicate that chaotic dynamics has potential capabilities for complex control functions by simple rule(s). An important idea is that chaotic dynamics generated in these systems give us autonomous complex pattern dynamics itinerating through intermediate state points between embedded designed attractors in high-dimensional state space. A key point is that, with the use of simple adaptive switching between a weakly chaotic regime and a strongly chaotic regime, complex problems can be solved. As an actual example, a two-dimensional maze, where it should be noted that the set context is one of typical ill-posed problems, is solved with the use of chaos in a recurrent neural network model. Our computer experiments show that the success rate over several hundreds trials is much better, at least, than that of a random number generator. Our functional simulations indicate that harnessing of chaos is one of essential ideas to approach mechanisms of brain functions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The map with no predetermined firing order for the network of oscillators with time-delayed pulsatile coupling

The network of the pulse-coupled oscillators is studied in the presence of coupling delays. Because of the delays the past activity of the network is capable to influence the future network dynamics. In general case this leads to the infinite dimension of the corresponding dynamical system. We prove the Theorem that states that under certain conditions (weak coupling and appropriate initial conditions) the network can be fully characterized by a finite dimensional state vector. We construct the return map describing the evolution of this state vector over time. This map does not need any presupposed activity pattern in the network and works for any initial conditions.     

Numerical implementation and applications of a coupling algorithm for structural–acoustic models with unequal discretization and partially interfacing surfaces

The numerical implementation of a coupled finite element–boundary element algorithm for computing simultaneously the structural vibration and the associated acoustic field and a complete set of validation and application data is presented. The new developments in the coupling algorithm presented in this paper are associated with a capability for unequal mesh density between the structural and the acoustic model, division of both models into interfacing and non-interfacing zones, efficient computation of the coupling matrices, and incorporation of acoustic multiple connection constraints in the coupling computations. Applications are identified in the area of launch vehicle dynamics where a reverberant acoustic environment provides the excitation for computing either noise transmitted through flexible structures, or noise-induced vibration due to acoustic loads. Results and correlation to test data are presented for a fairing and an expansion nozzle of a rocket. Numerical results are also compared to an analytical solution for noise transmitted through a flexible cavity backed plate.     

Heteroclinic Cycles in Hopfield Networks

It is widely believed that information is stored in the brain by means of the varying strength of synaptic connections between neurons. Stored patterns can be replayed upon the arrival of an appropriate stimulus. Hence, it is interesting to understand how an information pattern can be represented by the dynamics of the system. In this work, we consider a class of network neuron models, known as Hopfield networks, with a learning rule which consists of transforming an information string to a coupling pattern. Within this class of models, we study dynamic patterns, known as robust heteroclinic cycles, and establish a tight connection between their existence and the structure of the coupling.     

Emergence of a multilayer structure in adaptive networks of phase oscillators

We report on self-organization of adaptive networks, where topology and dynamics evolve in accordance to a competition between homophilic and homeostatic mechanisms, and where links are associated to a vector of weights. Under an appropriate balance between the intra- and inter- layer coupling strengths, we show that a multilayer structure emerges due to the adaptive evolution, resulting in different link weights at each layer, i.e. different components of the weights’ vector. In parallel, synchronized clusters at each layer are formed, which may overlap or not, depending on the values of the coupling strengths. Only when intra- and inter- layer coupling strengths are high enough, all layers reach identical final topologies, collapsing the system into, in fact, a monolayer network. The relationships between such steady state topologies and a set of dynamical network’s properties are discussed.     

Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling

In this paper, network of stochastic van der Pol oscillators with time-varying delayed coupling is considered. By using graph theory and Lyapunov functional method, the asymptotic boundedness in pth moment of the network is investigated. Moreover, by constructing an appropriate Lyapunov function, sufficient principle in the form of coefficients of network which ensures the asymptotic boundedness is established. Finally, a numerical example is given to show the effectiveness of the proposed criteria.