Synchronization of oscillators coupled through narrow-band networks

The ability of two coupled oscillators to synchronize depends critically on the coupling network. Previous analyses have accurately predicted the performance of quasi-optical microwave oscillator arrays for both weak and strong coupling, but have been limited to coupling networks with bandwidths considerably larger than the locking bandwidths of the oscillators. In this paper, the authors develop a method for deriving a suitable system of nonlinear differential equations describing the oscillator amplitude and phase dynamics using a generalization of Kurokawa's method. The method is applied to the case of two Van der Pol oscillators coupled through a resonant network for a wide range of coupling strengths and bandwidths. Simple approximate formulas are developed for the size of the frequency locking region as functions of the basic circuit parameters     

Pulse Synchronization of Intestinal Myoelectrical Models

Electrical stimulation of the stomach and small intestine in animals has been shown to give a "pacing" effect, whereby the normal frequency of the myoelectrical slow-waves can be altered to that of the pulse stimulus if it has a frequency close to that of the natural rhythm. In this paper the effect of pulse stimulation on three different types of models used for gastrointestinal studies is investigated. Each of the models is an electronic implementation comprising coupled oscillators, where the unit oscillators are either based on van der Pol's equation, Hodgkin-Huxley type equations, or a relaxation switching circuit. The synchronization range is investigated for each model for variations in stimulus pulse height and width. The effect of the number of oscillators in a chain, the waveshape of the individual oscillators, and the coupling between oscillators are also studied. It is shown that the relaxation model has different synchronization characteristics than the other two models. These differences are that a weakly coupled system is easier to synchronize than a strongly coupled system, that increasing pulsewidth does not always increase the pacing band, phase lead cannot be induced, and the synchronization-band can be entirely above the unpaced system frequency.     

Control of state transitions in an in silico model of epilepsy using small perturbations

We propose the use of artificial neural networks in an in silico epilepsy model of biological neural networks: 1) to predict the onset of state transitions from higher complexities, possibly chaotic to lower complexity possibly rhythmic activities; and 2) to restore the original higher complexity activity. A coupled nonlinear oscillators model (Bardakjian and Diamant, 1994) was used to represent the spontaneous seizure-like oscillations of CA3 hippocampal neurons (Bardakjian and Aschebrenner-Scheibe, 1995) to illustrate the prediction and control schemes of these state transition onsets. Our prediction scheme consists of a recurrent neural network having Gaussian nonlinearities. When the onset of lower complexity activity is predicted in the in silico model, then our control scheme consists of applying a small perturbation to a system variable (i.e., the transmembrane voltage) when it is sufficiently close to the unstable higher complexity manifold. The system state can be restored back to its higher complexity mode utilizing the forces of the system's vector field.     

Analysis of interinjection-locked oscillators for integrated phased arrays

The behavior of a system of coupled oscillators is shown to have potential applications in the generation of power for integrated phased antenna arrays. Nonlinear differential equations are derived to describe a system of oscillators coupled by an arbitrary frequency-dependent network. State-variable analysis of the linearized equations leads to closed-form solutions for one- and two-dimensional phased array systems. Experimental data for a VHF prototype system is presented, and practical considerations in system design are discussed.     

Modeling of a Neural Pattern Generator with Coupled nonlinear Oscillators

A set of van der Pol oscillators is arranged in a network in which each oscillator is coupled to each other oscillator. Through the selection of coupling coefficients, the network is made to appear as a ring and as a chain of coupled oscillators. Each oscillator is provided with amplitude, frequency, and offset parameters which have analytically indeterminable effects on the output waves. These systems are simulated on the digital computer in order to study the amplitude, frequency, offset, and phase relationships of the waves versus parameter changes. Based on the simulations, systems of coupled oscillators are configured so that they exhibit stable patterns of signals which can be used to model the central pattern generator (CPG) of living organisms. Using a simple biped as an example locomotory system, the CPG model generates control signals for simulated walking and jumping maneuvers. It is shown that with parameter adjustments, as guided by the simulations, the model can be made to generate kinematic trajectories which closely resemble those for the human walking gait. Further-more, minor tuning of these parameters along with some algebraic sign changes of coupling coefficients can effect a transition in the trajectories to those of a two-legged hopping gait. The generalized CPG model is shown to be versatile enough that it can also generate various n-legged gaits and spinal undulatory motions, as in the swimming motions of a fish.     

时间离散神经振子中的多吸引子共存

本文给出一类耦合时间离散神经网络振子动力学的理论分析和数值模拟结果,重点分析了多吸引子共存的震子及其吸引域的边界（不稳定的周期的周期不变坏）,研究了多上子共趣及耦合强度变化对于神经信号处理保,如受激盛同步和对模式形成有重要影响的锁频现象。     

Multimodal behavior in a four neuron ring circuit: mode switching

We study a four-neuron ring circuit comprised of oscillating burst-type neurons unidirectionally coupled via inhibitory synapses. Simple circuits of this type have been used previously to study gait patterns. The ring circuit itself is a variant of the basic reciprocal inhibition network, and it exhibits the property of multistability (multiple stable modes of behavior). That is, different gait modes can be achieved via appropriate initialization of and parameterization of this self-excited oscillatory network. We demonstrate three common gait modes with this circuit: the walk, the bound, and a slightly rotated trot mode. Attention is focused mainly on the mechanisms of rapidly and effectively switching between these modes. Our simulations suggest that neuron membrane dynamics, as well as synaptic junctional properties, strongly influence phase sensitivity in the network; each synapse is a combination of both and can be characterized by a transient phase response curve (PRC). We use the same bursting neuron model to characterize all network neurons, and shape different transient PRCs by using different synaptic properties. The characteristics of these PRCs determine the gait modes sustained in any network configuration, as well as, the ability to switch between modes. The mechanisms explored in this simple circuit, may find application in the switching of more complicated gait pattern networks, as well as, in the design of neuromorphic gait pattern circuits.     

Continuum modeling of the dynamics of externally injection-lockedcoupled oscillator arrays

Mutually injection-locked arrays of electronic oscillators provide a novel means of controlling the aperture phase of a phased-array antenna, thus achieving the advantages of spatial power combining while retaining the ability to steer the radiated beam. In a number of design concepts, one or more of the oscillators are injection locked to a signal from an external master oscillator. The behavior of such a system has been analyzed by numerical solution of a system of nonlinear differential equations which, due to its complexity, yields limited insight into the relationship between the injection signals and the aperture phase. In this paper, we develop a continuum model, which results in a single partial differential equation for the aperture phase as a function of time. Solution of the equation is effected by means of the Laplace transform and yields detailed information concerning the dynamics of the array under the influence of the external injection signals     

A multiconductance silicon neuron with biologically matched dynamics

We have designed, fabricated, and tested an analog integrated-circuit architecture to implement the conductance-based dynamics that model the electrical activity of neurons. The dynamics of this architecture are in accordance with the Hodgkin-Huxley formalism, a widely exploited, biophysically plausible model of the dynamics of living neurons. Furthermore the architecture is modular and compact in size so that we can implement networks of silicon neurons, each of desired complexity, on a single integrated circuit. We present in this paper a six-conductance silicon-neuron implementation, and characterize it in relation to the Hodgkin-Huxley formalism. This silicon neuron incorporates both fast and slow ionic conductances, which are required to model complex oscillatory behaviors (spiking, bursting, subthreshold oscillations).     

Modeling and Analyzing Biological Oscillations in Molecular Networks

One of the major challenges for postgenomic biology is to understand how genes, proteins, and small molecules dynamically interact to form molecular networks which facilitate sophisticated biological functions. In this paper, we present a survey on recent developments on modelling molecular networks and analyzing synchronization of bio-oscillators in multicellular systems from the viewpoint of systems biology. Attention will be focused on deriving general theoretical results to understand the dynamical behaviors of biological systems based on nonlinear dynamical and control theory. Specifically, we first describe the stochastic and deterministic approaches to model molecular networks and give a brief comparison between them. Then, we explain how to construct a molecular network, in particular, a gene regulatory network with specific functions, e.g., switches and oscillators, in individual cells at the molecular level by using feedback systems, and how to model a general multicellular system with the consideration of external fluctuations and intercellular coupling to study the general cooperative behaviors for a population of bio-oscillators. Finally, as an illustrative example, a synthetic multicellular system is designed to show how synchronization is effectively achieved and how dynamics of individual cells is efficiently controlled. Some recent developments and perspectives of analysis on biological oscillations in future are also discussed.     

Measurement and modelling of radiative coupling in oscillatorarrays

Arrays of coupled oscillators can be used for power-combining at microwave and millimeter-wave frequencies, and have been successfully demonstrated with a variety of devices. Such arrays have also recently been mode-locked for pulse generation, and can be configured for phase-shifterless beam scanning. The nonlinear theory of coupled oscillator phase dynamics depends crucially on the parameters describing the coupled between oscillators. Methods for experimental characterization of these parameters are described here, and simple models which reproduce the measurements quite well are developed. The models apply to radiative coupling and the effects of external reflectors which are sometimes used for stabilization. The theory is verified with a two-oscillator system     

Spiking and Bursting in Josephson Junction

This brief reports spiking and bursting in numerical simulations of the resistive-capacitive-inductive shunted Josephson junction model. Regular spiking, intrinsic bursting, and fast spiking, which are usually seen in the mammalian neocortex, are observed in the junction dynamics under external dc bias. The junction voltage is amplitude and frequency modulated when forced by weaker sinusoidal forcing of frequency much lower than the junction resonant frequency. For stronger forcing, bursting is observed. The autonomous junction also shows bursting in the high inductive regime. Bifurcation scenarios of bursting are discussed for both autonomous and nonautonomous cases     

A Simple Programmable Autowave Generator Network for Wave Computing Applications

Spatiotemporal-wave-based computing on an active medium, which is known as wave computing paradigm, mimics the way biological systems process information. Recent anatomic and physiological studies and technological developments in very large-scale integration technology have encouraged the engineers to brace up with this new paradigm. In this paper, a simple network model that is essential in order to explore spatiotemporal behavior and to develop new algorithms based on wave computing technique has been presented. The presented network is a 2-D reaction-diffusion cellular neural network consisting of relaxation oscillators. The network can be programmed as an autowave generator and sources of the wave can be placed at any place on the network. The network has the simplest model that exhibits autowaves. Furthermore, the model is suitable for the implementation of a larger network. The propagation of autowaves between the cells whose dynamics is fixed shows that the network can be adapted to already developed wave computing algorithms in literature.      

A 10-GHz global clock distribution using coupled standing-wave oscillators

In this paper, a global clock network that incorporates standing waves and coupled oscillators to distribute a high-frequency clock signal with low skew and low jitter is described. The key design issues involved in generating standing waves on a chip are discussed, including minimizing wire loss within an available technology. A standing-wave oscillator, which is a distributed oscillator that sustains ideal standing waves on lossy wires, is introduced. A clock grid architecture comprised of coupled standing-wave oscillators and differential low-swing clock buffers is presented, along with a compact circuit model for networks of oscillators. The measured results for a prototyped standing-wave clock grid operating at 10 GHz and fabricated in a 0.18-/spl mu/m 6M CMOS logic process are presented. A technique is proposed for on-chip skew measurements with subpicosecond precision.     

Dynamics of two neuronlike elements with inhibitory feedback

An analog electronic model is proposed to qualitatively reproduce the main dynamic regimes of inferior-olive (IO) neurons. An analog simulation is used to study the dynamics of a single IO neuron and a pair of neurons involved in inhibitory feedback. The experimental results that demonstrate that such feedback leads to a partial spike synchronization of the interacting neurons are presented. It is shown that the electronic model can be used for the simulation of the burst activity of oscillatory neurons.     

基于耦合振荡器阵列的非线性有源天线阵

利用耦合振荡器阵列实现有源集成天线阵电扫描,可以实现相控阵天线高效率、低成本、小型化。文中利用Y-参数对利用任意耦合网络实现的耦合振荡器阵列进行了分析,得到了系统幅度及相位非线性动力系统的微分方程组,然后针对利用耦合振荡器阵列实现非线性有源天线阵列这一实际应用,提出了一种不用移相器实现有源天线阵列波束扫描的方法,并进行了稳定性分析。最后利用计算机仿真验证了该方法的可行性。     

Synchronization in Networks of Hindmarsh–Rose Neurons

Synchronization is deemed to play an important role in information processing in many neuronal systems. In this work, using a well known technique due to Pecora and Carroll, we investigate the existence of a synchronous state and the bifurcation diagram of a network of synaptically coupled neurons described by the Hindmarsh–Rose model. Through the analysis of the bifurcation diagram, the different dynamics of the possible synchronous states are evidenced. Furthermore, the influence of the topology on the synchronization properties of the network is shown through an example.      

Waveform generation by enzymatic oscillators

The control of cell chemistry is being investigated through a multipronged approach combining the techniques of physics, chemistry, and biology with the development of electronic instrumentation and the application of analog and digital computers. An example of such metabolic control phenomena is provided by biological oscillators involving enzymatic reactions. These oscillators, which may exist in nearly every kind of cell and even in several forms in a single cell, reveal basic instabilities in biochemical reactions and metabolic control that may be of significance to health and disease. In addition, the high-frequency oscillations observed in simple enzyme systems may be models for the longer-period rhythms that regulate the activities of nearly all biological systems.     

Role of elastic dissipation in the formation of the resonant properties of magnetization precession in the magnetoelastic environment

A set of coupled magnetoelastic equations is used to examine the problem concerning the establishment of free magnetization oscillations under the condition that dissipation is lacking in the magnetic system. The critical relationship between elastic damping parameters and the magnetoelasticity constant, which corresponds to the minimum in the dependence of magnetic damping on elastic one, is found. The bellshaped peak, on both sides of which the influence of an elastic system on the magnetic one diminishes, is detected in the dependence between the effective magnetic oscillation damping parameter and the elastic oscillation damping parameter. The observed phenomena is suggested to interpret using the model of two (magnetic and elastic) oscillators coupled through magnetoelastic interaction. The given model makes it possible to reveal four regimes of steady-state oscillations: weak damping with beats, strong damping without beats, weak damping without beats, and supercritical exponential growth. The mechanical similarity of observed phenomena is discussed, and recommendations needed to organize experiments are proposed.