Asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons

We investigate theoretically the conditions for the emergence of synchronous activity in large networks, consisting of two populations of extensively connected neurons, one excitatory and one inhibitory. The neurons are modeled with quadratic integrate-and-fire dynamics, which provide a very good approximation for the subthreshold behavior of a large class of neurons. In addition to their synaptic recurrent inputs, the neurons receive a tonic external input that varies from neuron to neuron. Because of its relative simplicity, this model can be studied analytically. We investigate the stability of the asynchronous state (AS) of the network with given average firing rates of the two populations. First, we show that the AS can remain stable even if the synaptic couplings are strong. Then we investigate the conditions under which this state can be destabilized. We show that this can happen in four generic ways. The first is a saddle-node bifurcation, which leads to another state with different average firing rates. This bifurcation, which occurs for strong enough recurrent excitation, does not correspond to the emergence of synchrony. In contrast, in the three other instability mechanisms, Hopf bifurcations, which correspond to the emergence of oscillatory synchronous activity, occur. We show that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations. We also show that besides these codimension 1 bifurcations, the system can display several codimension 2 bifurcations: Takens-Bogdanov, Gavrielov-Guckenheimer, and double Hopf bifurcations.     

Bifurcation Control Of A Fractional‐Order Van Der Pol Oscillator Based On The State Feedback

In this paper, a dynamic state feedback is applied to control Hopf bifurcations arising from a fractional‐order Van Der Pol oscillator. The degree parameter indicating the strength of the nonlinear damping is chosen as the bifurcation parameter. It is shown that in the absences of the dynamic state feedback controller, the fractional‐order Van Der Pol oscillator loses the stability via the Hopf bifurcation early, and can maintain the stability only in a certain domain of the degree parameter. When applying the state feedback controller to the fractional‐order Van Der Pol oscillator, the onset of the undesirable Hopf bifurcation is postponed. Thus, the stability domain is extended, and the system possesses the stability in a larger parameter range. Numerical simulations are given to justify the validity of the dynamic state feedback controller in bifurcation controls.     

一个粘弹性振子的动力学与控制

本文详细分析了一个具有粘弹性项的非线性振子的动力学与控制.首先研究了系统平衡点的稳定性,表明系统存在复杂的无界动力学行为.然后引入时滞速度反馈对这个不稳定系统进行控制.研究结果表明速度反馈控制能镇定此不稳定的粘弹性系统.适当的选择控制增益和控制时滞,控制系统有稳定的平衡点,由Hopf分岔产生的周期解,拟周期解,并能展现出复杂的混沌解.数值模拟验证了结论的正确性.     

基于分岔理论的永磁同步电机有限时间混沌控制

永磁同步电机运行系统具有不稳定的分岔特性,随着系统参数的变化,系统会在平衡点处发生分岔行为.首先,基于分岔理论构建了永磁同步电机的混沌模型.其次,通过研究系统的分岔参数,分析了系统在平衡点处的分岔特性,发现系统在零平衡点处会产生静态分岔并出现新的平衡点,随着参数的继续变化,系统在新的平衡点处发生连续的Hopf分岔,而连...     

Bifurcation control in a delayed two-neuron fractional network

The issue of bifurcation control for a delayed fractional network involving two neurons is concerned. Delay-dependent stability conditions and the bifurcation point are established by discussing the associated characteristic equation of the proposed network. Then, a delayed feedback controller is firstly designed to stabilize the Hopf bifurcation, and desirable dynamics is achieved. It is indicated that the designed controller is extremely effective which can postpone the onset of bifurcation by carefully selecting the feedback gain. Finally, simulation results are given to verify the efficiency of the theoretical results.     

一类具有饱和发生率的时滞恶意病毒传播模型的分岔控制策略

考虑非线性的饱和发生率,建立一种刻画信息物理融合系统(cyber-physical systems, CPS)中恶意病毒传播的SIRS(susceptible-infected-recovered-susceptible)模型.为了避免因Hopf分岔的产生致使恶意病毒传播扩散,采用参数调节法和状态反馈法相结合的混合分岔控制策略,研究信息物理融合系统的Hopf分岔控制问题,建立受控系统的稳定性条件和分岔判据,探明控制增益参数对Hopf分岔点和分岔极限环幅值的影响规律,并给出分岔阈值与增益参数间的关系图.数值仿真结果表明,所提出的混合分岔控制策略不仅能够改变Hopf分岔点的位置,而且可以有效调节极限环幅值的大小,使得信息物理融合系统产生预期的动力学行为,有效降低恶意病毒传播的危害.     

An Implementation of Constructive Synchronous Programs in POLIS

Design tools for embedded reactive systems commonly use a model of computation that employs both synchronous and asynchronous communication styles. We form a junction between these two with an implementation of synchronous languages and circuits (Esterel) on asynchronous networks (POLIS). We implement fact propagation, the key concept of synchronous constructive semantics, on an asynchronous non-deterministic network: POLIS nodes (CFSMs) save state locally to deduce facts, and the network globally propagates facts between them. The result is a correct implementation of the synchronous input/output behavior of the program. Our model is compositional, and thus permits implementations at various levels of granularity from one CFSM per circuit gate to one CFSM per circuit. This allows one to explore various tradeoffs between synchronous and asynchronous implementations.     

高速列车转向架蛇行运动Hopf分岔控制研究

转向架蛇行运动稳定性直接决定高速列车能否安全和平稳运行,采用主动悬挂技术是提高稳定性的有效途径,但目前针对采用主动控制后的转向架蛇行运动Hopf分岔机理和稳定性演变规律的研究还十分缺乏.本文基于高速列车转向架摇头反馈控制下的蛇行运动Hopf分岔特性展开研究.建立包含转向架横移、摇头和车体横移的三自由度简化模型,采用作动器替代现有的抗蛇行减振器提供转向架摇头控制力矩.分别选取转向架摇头角位移和角速度作为状态反馈量,设计线性反馈控制器,并通过Matcont仿真环境实现Hopf 分岔和极限环计算.研究表明：对比被动悬挂的分岔特性,转向架摇头角位移和角速度两种主动控制方式均能延后蛇行运动Hopf分岔点,即提高系统线性临界速度;摇头角速度控制能降低蛇行运动频率和分岔后极限环幅值;摇头角位移控制会增大蛇行运动频率,而极限环幅值无明显改变.因此,通过引入转向架摇头控制并选取合适控制增益,能够有效提高高速转向架蛇行运动稳定性.     

Mean-field Theory and Synchronization in Random Recurrent Neural Networks

In this paper, we first present a new mathematical approach, based on large deviation techniques, for the study of a large random recurrent neural network with discrete time dynamics. In particular, we state a mean field property and a law of large numbers, in the most general case of random models with sparse connections and several populations. Our results are supported by rigorous proofs. Then, we focus our interest on large size dynamics, in the case of a model with excitatory and inhibitory populations. The study of the mean field system and of the divergence of individual trajectories allows to define different dynamical regimes in the macroscopic parameters space, which include chaos and collective synchronization phenomenons. At last, we look at the behavior of a particular finite-size system submitted to gaussian static inputs. The system adapts its dynamics to the input signal, and spontaneously produces dynamical transitions from asynchronous to synchronous behaviors, which correspond to the crossing of a bifurcation line in the macroscopic parameters space.     

Mechanisms for Frequency Control in Neuronal Competition Models

We investigate analytically a firing rate model for a two-population network based on mutual inhibition and slow negative feedback in the form of spike frequency adaptation. Both neuronal populations receive external constant input whose strength determines the system's dynamical state-a steady state of identical activity levels or periodic oscillations or a winner-take-all state of bistability. We prove that oscillations appear in the system through supercritical Hopf bifurcations and that they are antiphase. The period of oscillations depends on the input strength in a nonmonotonic fashion, and we show that the increasing branch of the period versus input curve corresponds to a release mechanism and the decreasing branch to an escape mechanism. In the limiting case of infinitely slow feedback we characterize the conditions for release, escape, and occurrence of the winner-take-all behavior. Some extensions of the model are also discussed.     

Bifurcation control of small‐world networks with delays VIA PID controller

In this paper, the problem of bifurcation control for a small‐world network model with time delay is studied. We first put forward a Proportional‐Integral‐Derivative (PID) feedback scheme to control the Hopf bifurcation of the network. The time delay is selected as the bifurcation parameter. The conditions of the stability and Hopf bifurcation are given for the controlled network. By using the center manifold theorem and the normal form theory, the direction and stability of bifurcating periodic solutions are confirmed. The feasible region of the parameters of the controller is determined. It is found that the bifurcation dynamics of the small‐world network are optimized by adjusting the parameters of the PID controller. Finally, a numerical example verifies the effectiveness of the designed PID controller, and the relationships between the onset of the Hopf bifurcation and the control parameters are obtained.     

Rapid temporal modulation of synchrony by competition in cortical interneuron networks

The synchrony of neurons in extrastriate visual cortex is modulated by selective attention even when there are only small changes in firing rate (Fries, Reynolds, Rorie, & Desimone, 2001). We used Hodgkin-Huxley type models of cortical neurons to investigate the mechanism by which the degree of synchrony can be modulated independently of changes in firing rates. The synchrony of local networks of model cortical interneurons interacting through GABA(A) synapses was modulated on a fast timescale by selectively activating a fraction of the interneurons. The activated interneurons became rapidly synchronized and suppressed the activity of the other neurons in the network but only if the network was in a restricted range of balanced synaptic background activity. During stronger background activity, the network did not synchronize, and for weaker background activity, the network synchronized but did not return to an asynchronous state after synchronizing. The inhibitory output of the network blocked the activity of pyramidal neurons during asynchronous network activity, and during synchronous network activity, it enhanced the impact of the stimulus-related activity of pyramidal cells on receiving cortical areas (Salinas & Sejnowski, 2001). Synchrony by competition provides a mechanism for controlling synchrony with minor alterations in rate, which could be useful for information processing. Because traditional methods such as cross-correlation and the spike field coherence require several hundred milliseconds of recordings and cannot measure rapid changes in the degree of synchrony, we introduced a new method to detect rapid changes in the degree of coincidence and precision of spike timing.     

Of Choices, Failures and Asynchrony: The Many Faces of Set Agreement

Set agreement is a fundamental problem in distributed computing in which processes collectively choose a small subset of values from a larger set of proposals. The impossibility of fault-tolerant set agreement in asynchronous networks is one of the seminal results in distributed computing. In synchronous networks, too, the complexity of set agreement has been a significant research challenge that has now been resolved. Real systems, however, are neither purely synchronous nor purely asynchronous. Rather, they tend to alternate between periods of synchrony and periods of asynchrony. Nothing specific is known about the complexity of set agreement in such a “partially synchronous” setting. In this paper, we address this challenge, presenting the first (asymptotically) tight bound on the complexity of set agreement in such systems. We introduce a novel technique for simulating, in a fault-prone asynchronous shared memory, executions of an asynchronous and failure-prone message-passing system in which some fragments appear synchronous to some processes. We use this simulation technique to derive a lower bound on the round complexity of set agreement in a partially synchronous system by a reduction from asynchronous wait-free set agreement. Specifically, we show that every set agreement protocol requires at least $\lfloor\frac{t}{k}\rfloor + 2$ synchronous rounds to decide. We present an (asymptotically) matching algorithm that relies on a distributed asynchrony detection mechanism to decide as soon as possible during periods of synchrony. From these two results, we derive the size of the minimal window of synchrony needed to solve set agreement. By relating synchronous, asynchronous and partially synchronous environments, our simulation technique is of independent interest. In particular, it allows us to obtain a new lower bound on the complexity of early deciding k-set agreement complementary to that of Gafni et al. (in SIAM J. Comput. 40(1):63–78, 2011), and to re-derive the combinatorial topology lower bound of Guerraoui et al. (in Theor. Comput. Sci. 410(6–7):570–580, 2009) in an algorithmic way.     

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part II: Hopf bifurcations

In this paper we derive necessary and sufficient conditions of stabilizability for multi‐input nonlinear systems possessing a Hopf bifurcation with the critical mode being linearly uncontrollable, under the non‐degeneracy assumption that stability can be determined by the third order term in the normal form of the dynamics on the centre manifold. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the Hopf bifurcation of the closed‐loop system is supercritical, which is equivalent to local asymptotic stability of the system at the bifurcation point. We prove that under the non‐degeneracy conditions, stabilizability is equivalent to the existence of solutions to a third order algebraic inequality of the feedback gains. Explicit conditions for the existence of solutions to the algebraic inequality are derived, and the stabilizing feedback laws are constructed. Part of the sufficient conditions are equivalent to the rank conditions of an augmented matrix which is a generalization of the Popov–Belevitch–Hautus (PBH) rank test of controllability for linear time invariant (LTI) systems. We also apply our theory to feedback control of rotating stall in axial compression systems using bleed valve as actuators. Copyright © 2006 John Wiley & Sons, Ltd.     

速度时滞反馈控制下磁浮系统的稳定性与Hopf分岔

The problem of time delay speed feedback in the control loop is considered here. Its effects on the linear stability and dynamic behavior of the maglev system are investigated. It is found that a Hopf bifurcation can take place when the time delay exceeds certain values. The stability condition of the maglev system with the time delay is acquired. The direction and stability of the Hopf bifurcation are determined by constructing a center manifold and by applying the normal form method. Finally, numerical simulations are performed to verify the analytical result.     

速度时滞反馈控制下磁浮系统的稳定性与Hopf分岔

The problem of time delay speed feedback in the control loop is considered here.Its effects on the linear stability and dynamic behavior of the maglev system are investigated.It is found that a Hopf bifurcation can take place when the time delay exceeds certain values.The stability condition of the maglev system with the time delay is acquired.The direction and stability of the Hopf bifurcation are determined by constructing a center manifold and by applying the normal form method.Finally,numerical simulations are performed to verify the analytical result.     

Consensus of multiple double‐integrator agents with intermittent measurement

This paper is concerned with consensus problems in directed networks of multiple agents with double‐integrator dynamics. It is assumed that each agent adjusts its state based on the information of its states relative to its neighbors at discrete times and the interaction topology among agents is time‐varying. Both synchronous and asynchronous cases are considered. The synchrony means that each agent's update times, at which it obtains new control signals, are the same as the others', and the asynchrony implies that each agent's update times are independent of the others'. In the synchronous case, the consensus problem is proved to be equivalent to the asymptotic stability problem of a discrete‐time switched system. By analyzing the asymptotic stability of the discrete‐time switched system, it is shown that consensus can be reached if the update time intervals are small sufficiently, and an allowable upper bound of update time intervals is obtained. In the asynchronous case, the consensus problem is transformed into the global asymptotic stability problem of a continuous‐time switched system with time‐varying delays. In virtue of a linear matrix inequality method, it is proved that consensus can be reached if the delays are small enough, and an admissible upper bound of delays is derived. Simulations are provided to illustrate the effectiveness of the theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

An experimental study of noise and asynchrony in elementary cellular automata with sampling compensation

This article focuses on the set of 32 legal elementary cellular automata (ECA). We perform an exhaustive study of the systems’ response under: (i) α-asynchronous dynamics, from full asynchronism to perfect synchrony, (ii) κ-scaling, which extends α-asynchrony to compensate for less cell activity, and (iii) ?-noise scheme, a perturbation that affects the local transition function and causes a cell to probabilistically miscalculate the new state when it is updated. We propose a new classification in three classes under asynchronous conditions: α-invariant, α-robust, and α-dependent. We classify the 32 legal ECA according to the degree of behavioural modification, and we show that our classifying scheme provides results coherent with the density-based classification. We also show that κ-scaling provides results comparable to synchronous systems, both quantitatively and qualitatively. Subsequently, we analyse the effects of including different levels of noise in synchronous systems. We identify different responses to noise, including systems that are robust to asynchrony and susceptible to noise. To conclude, we investigate the behavioural changes caused by simultaneous asynchrony and noise in models tolerant to both perturbations. We describe a number of effects caused by the interplay of noise and asynchrony, thus further reinforcing that both aspects are pertinent for future studies.     

Synchrony in a self-repair network with a simple lattice

We investigate a synchrony in a self-repairing network of autonomous agents capable of repairing mutually. In this paper, we define two models: a synchronous model and asynchronous one. They differ in the timing when the agents change their state. Computer simulations revealed that the synchronous model has a critical point, while the asynchronous one does not. We also studied a repair scheme in asynchronous model where the repaired agents in turn repair neighbor agents successively in a chain-reactive fashion. Performance of the scheme has been examined by computer simulations. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

机床无刷直流电机系统的分岔分析与控制

文章主要研究了机床无刷直流电机系统的Hopf分岔控制问题.首先,对系统进行分岔分析,通过计算极限环曲率系数判定系统的Hopf分岔类型;然后设计Washout滤波器对系统进行分岔控制,根据Hopf分岔理论给出使原系统Hopf分岔位置发生改变的参数条件,利用Normal Form方法计算出受控系统的Hopf分岔正规型,根据正规型的实部大小判定Hopf分岔类型,给出使原系统Hopf分岔类型发生改变的参数条件;并借助MATLAB软件对理论结果进行数值仿真,理论结果和数值仿真表明:控制器中的线性增益能使系统在所期望的参数值处发生Hopf分岔,甚至消除Hopf分岔,控制器中的非线性增益能改变原系统的Hopf分岔类型及极限环幅值的大小.研究结果对无刷直流电动机系统的工程实际具有一定的指导意义.