一端受外力的交联振荡器链中的内部传输

本文研究了二类一端受外力的交联振荡器链：最邻近多相位交联振荡器链,以及多重交联振荡器链,讨论了它们产生内部传输,即各振荡器与外力具有相同频率的现象。文中近似相位差方程、指数二分性理论和中心流形理论被应用于系统的渐近近似。研究。本文得到了更符合于实际情况的神经网络CPG链动态特性分析结论。     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Mode analysis of a tubular structure of coupled non-linear oscillators for digestive-tract modelling

A commonly accepted mathematical model for the slow-wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected nonlinear oscillators. Using a van der Pol oscillator with third-power conductance characteristics as the unit oscillator a number of structures have been analysed using a matrix Krylov-Bogolioubov method linearisation. The mode analysis of one-dimensional chains and two-dimensional arrays has been reported. In this paper the method has been extended to consider a tubular structure which is relevant to modelling small-intestinal rhythms. It is shown that this structure is capable of producing stable single models, non-resonant double modes and degenerated modes. General expressions are obtained for anm×n structure and examples given of two special conditions of 3×4 (i.e. odd numbers of oscillators in a ring) and 4×3 cases. The analytical results obtained for these two cases have been vertified experimentally using an electronic implementation of coupled van der Pol oscillators. Results obtained using fifth-power non-linear oscillators are summarised.     

Interaction of mitotic oscillators as a source of variability of cell cycle duration

Interaction between membrane mitotic oscillators at the expense of exchange with the molecules of lipids (slow variable) and antioxidants (fast variable) was considered. Parameters of all the oscillators are equal, excluding a small noise added to the equation for lipids. These parameters are chosen in such a way that the oscillators are not far from the transition to the stable stationary state. The numerical modeling has shown that the exchange with lipids brings about the appearance of an additional limit cycle whose period is significantly greater than that of an autonomous oscillator. The addition of noise averages the behaviour of oscillators, and distribution according to cycle duration becomes broad and bimodal. Thus the exchange of the slow variable increases the dispersion of distribution of cell generation times. This conclusion seems to be true for any oscillator with similar dynamic properties.     

A Resonance Approach to Cochlear Mechanics

Background

How does the cochlea analyse sound into its component frequencies? In the 1850s Helmholtz thought it occurred by resonance, whereas a century later Békésy''s work indicated a travelling wave. The latter answer seemed to settle the question, but with the discovery in 1978 that the cochlea emits sound, the mechanics of the cochlea was back on the drawing board. Recent studies have raised questions about whether the travelling wave, as currently understood, is adequate to explain observations.

Approach

Applying basic resonance principles, this paper revisits the question. A graded bank of harmonic oscillators with cochlear-like frequencies and quality factors is simultaneously excited, and it is found that resonance gives rise to similar frequency responses, group delays, and travelling wave velocities as observed by experiment. The overall effect of the group delay gradient is to produce a decelerating wave of peak displacement moving from base to apex at characteristic travelling wave speeds. The extensive literature on chains of coupled oscillators is considered, and the occurrence of travelling waves, pseudowaves, phase plateaus, and forced resonance in such systems is noted.

Conclusion and significance

This alternative approach to cochlear mechanics shows that a travelling wave can simply arise as an apparently moving amplitude peak which passes along a bank of resonators without carrying energy. This highlights the possible role of the fast pressure wave and indicates how phase delays and group delays of a set of driven harmonic oscillators can generate an apparent travelling wave. It is possible to view the cochlea as a chain of globally forced coupled oscillators, and this model incorporates fundamental aspects of both the resonance and travelling wave theories.     

A Model Coupling Vibrational and Rotational Motion for the DNA Molecule

We investigate a mechanical model for the DNA molecule using an extension of the Peyrard and Bishop model. In the present model, there are two chains of oscillators linked by a Morse potential, which represent the hydrogen bonds. The rotational and vibrational motions of each base pair are considered and the coupling for these motions are introduced by a nonlinear combination of them in the Morse potential. In this context, thermodynamics and structural properties are discussed.     

Mammalian cultured cells as a model system of peripheral circadian clocks

The mammalian circadian system consists of multiple oscillators with basically hierarchical relationship, in which the hypothalamic suprachiasmatic nucleus (SCN) is the master pacemaker and the other oscillators in the periphery are subordinate. Although peripheral oscillators have been preceded by the SCN in circadian studies, accumulating data have revealed the importance and characteristics of peripheral oscillators. Cultured cell lines have also provided valuable information about intracellular mechanisms of circadian rhythms. This review outlines the properties of peripheral clocks in several perspectives such as the mechanisms of autonomous oscillations, the clock resetting, and the clock outputs, and describes the usefulness of immortalized cultured cells as a model system of mammalian circadian clocks by introducing some fruits of related works.     

A timely expression profile.

Molecular genetic analysis has yielded a detailed mechanistic understanding of invertebrate and vertebrate circadian oscillators, but we still know little about how such molecular oscillators are connected to rhythmic physiological processes. Two recent papers in Cell and Neuron now address this scientific issue through the use of gene chip technology to identify clock-regulated genes in an animal species.     

Deterministic characterization of phase noise in biomolecular oscillators

On top of the many external perturbations, cellular oscillators also face intrinsic perturbations due the randomness of chemical kinetics. Biomolecular oscillators, distinct in their parameter sets or distinct in their architecture, show different resilience with respect to such intrinsic perturbations. Assessing this resilience can be done by ensemble stochastic simulations. These are computationally costly and do not permit further insights into the mechanistic cause of the observed resilience. For reaction systems operating at a steady state, the linear noise approximation (LNA) can be used to determine the effect of molecular noise. Here we show that methods based on LNA fail for oscillatory systems and we propose an alternative ansatz. It yields an asymptotic expression for the phase diffusion coefficient of stochastic oscillators. Moreover, it allows us to single out the noise contribution of every reaction in an oscillatory system. We test the approach on the one-loop model of the Drosophila circadian clock. Our results are consistent with those obtained through stochastic simulations with a gain in computational efficiency of about three orders of magnitude.     

Phase computations and phase models for discrete molecular oscillators

Background

Biochemical oscillators perform crucial functions in cells, e.g., they set up circadian clocks. The dynamical behavior of oscillators is best described and analyzed in terms of the scalar quantity, phase. A rigorous and useful definition for phase is based on the so-called isochrons of oscillators. Phase computation techniques for continuous oscillators that are based on isochrons have been used for characterizing the behavior of various types of oscillators under the influence of perturbations such as noise.

Results

In this article, we extend the applicability of these phase computation methods to biochemical oscillators as discrete molecular systems, upon the information obtained from a continuous-state approximation of such oscillators. In particular, we describe techniques for computing the instantaneous phase of discrete, molecular oscillators for stochastic simulation algorithm generated sample paths. We comment on the accuracies and derive certain measures for assessing the feasibilities of the proposed phase computation methods. Phase computation experiments on the sample paths of well-known biological oscillators validate our analyses.

Conclusions

The impact of noise that arises from the discrete and random nature of the mechanisms that make up molecular oscillators can be characterized based on the phase computation techniques proposed in this article. The concept of isochrons is the natural choice upon which the phase notion of oscillators can be founded. The isochron-theoretic phase computation methods that we propose can be applied to discrete molecular oscillators of any dimension, provided that the oscillatory behavior observed in discrete-state does not vanish in a continuous-state approximation. Analysis of the full versatility of phase noise phenomena in molecular oscillators will be possible if a proper phase model theory is developed, without resorting to such approximations.     

Circadian rhythms in plants: Strategies for analysis

Molecular mechanism of the circadian clock which regulates the circadian rhythms has been believed to be common in different organisms. However, recent topic about multiple oscillators in a cell is thought to suggest other possibility. We may need to reconsider effectiveness of strategies for understanding molecular mechanism of the circadian clock.     

Intersegmental coordination in invertebrates and vertebrates

How does the CNS coordinate muscle contractions between different body segments during normal locomotion? Work on several preparations has shown that this coordination relies on excitability gradients and on differences between ascending and descending intersegmental coupling. Abstract models involving chains of coupled oscillators have defined properties of coordinating circuits that would permit them to establish a constant intersegmental phase in the face of changing periods. Analyses that combine computational and experimental strategies have led to new insights into the cellular organization of intersegmental coordinating circuits and the neural control of swimming in lamprey, tadpole, crayfish and leech.     

Populations of biochemical oscillators as circadian clocks

Various types of populations of interacting oscillators were analyzed and their synchronization states were determined. One of the systems involving biochemical oscillators was simulated on the computer and the occurence of rhythm splitting was observed. A comparison of its attributes with experimental results on circadian ryhythms showed good agreement. This allows us to distinguish between types of mechanisms held responsible for the splitting phenomenon in the past. The present model also offers a new explanation about the differences of light action on diurnal and nocturnal organisms.     

Synchronization of coupled nonidentical genetic oscillators

The study of the collective dynamics of synchronization among genetic oscillators is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. Genetic oscillators are biochemical networks, which can generally be modelled as nonlinear dynamic systems. We show in this paper that many genetic oscillators can be transformed into Lur'e form by exploiting the special structure of biological systems. By using a control theory approach, we provide a theoretical method for analysing the synchronization of coupled nonidentical genetic oscillators. Sufficient conditions for the synchronization as well as the estimation of the bound of the synchronization error are also obtained. To demonstrate the effectiveness of our theoretical results, a population of genetic oscillators based on the Goodwin model are adopted as numerical examples.     

视网膜生物钟研究进展

昼夜节律生物钟是以24h为周期的自主维持的振荡器。在高等的多细胞生物中,生物钟可以分为母钟和子钟。研究表明哺乳动物的母钟位于下丘脑视交叉上核(suprachiasmatic nucleus,SCN),由此发出信息控制全身的节律活动;子钟位于组织细胞内,调控效应器的节律。在分子水平上,生物钟的振荡由自身调控反馈环路的转录和翻译组成,并接受外界环境因素的影响,通过下丘脑视叉上核(Suprachiasmatic Nucleus,SCN)中枢震荡器的同步整和而产生作用。视网膜是一种十分节律性的组织,许多生化的、细胞的和生理的过程都是以节律的方式来进行的,如视觉灵敏度、视网膜杆细胞外片层脱落和视网膜色素上皮细胞的吞噬作用、光受体中的视觉色素基因的快速表达等。生物钟存在于很多脊椎动物的视网膜中,被认为是一种外周生物钟。本文综述了视网膜生物钟,生物钟信号传输以及生物钟网络等的最新研究进展。     

Many Circadian Oscillators Regulate Developmental and Behavioural Events in the Flesh-Fly, Sarcophaga Argyrostoma

In the flesh-fly, Sarcophaga argyrostoma, the initiation of larval wandering, pupal eclosion, and the induction of pupal diapause by seasonal changes in night length, are all regulated by circadian oscillators. They differ, however, in several respects. The rhythm of larval wandering shows a free-running period (S) of about 20 hr and a steady-state phase-relationship to the light cycle (±) in which maximum activity occurs at dusk or in the night; that for pupal eclosion shows ± close to 24 hr and ± close to dawn; and that for diapause induction ± longer than 24 hr and a photoinducible phase (φ_i) late in the subjective night. The three oscillators are, therefore, considered to be functionally separate. In addition, adult locomotor activity, the deposition of cuticular growth layers on thoracic apodemes, and the duration of larval wandering, are possibly regulated by further, distinct, oscillators. The circadian system in S. argyrostoma, therefore, contains at least three, and probably as many as six, known circadian pacemakers.     

The Goodwin Oscillator and its Legacy

In the 1960’s Brian Goodwin published a couple of mathematical models showing how feedback inhibition can lead to oscillations and discussed possible implications of this behaviour for the physiology of the cell. He also presented key ideas about the rich dynamics that may result from the coupling between such biochemical oscillators. Goodwin’s work motivated a series of theoretical investigations aiming at identifying minimal mechanisms to generate limit cycle oscillations and deciphering design principles of biological oscillators. The three-variable Goodwin model (adapted by Griffith) can be seen as a core model for a large class of biological systems, ranging from ultradian to circadian clocks. We summarize here main ideas and results brought by Goodwin and review a couple of modeling works directly or indirectly inspired by Goodwin’s findings.

     

Engineering entrainment and adaptation in limit cycle systems

Periodic behavior is key to life and is observed in multiple instances and at multiple time scales in our metabolism, our natural environment, and our engineered environment. A natural way of modeling or generating periodic behavior is done by using oscillators, i.e., dynamical systems that exhibit limit cycle behavior. While there is extensive literature on methods to analyze such dynamical systems, much less work has been done on methods to synthesize an oscillator to exhibit some specific desired characteristics. The goal of this article is twofold: (1) to provide a framework for characterizing and designing oscillators and (2) to review how classes of well-known oscillators can be understood and related to this framework. The basis of the framework is to characterize oscillators in terms of their fundamental temporal and spatial behavior and in terms of properties that these two behaviors can be designed to exhibit. This focus on fundamental properties is important because it allows us to systematically compare a large variety of oscillators that might at first sight appear very different from each other. We identify several specifications that are useful for design, such as frequency-locking behavior, phase-locking behavior, and specific output signal shape. We also identify two classes of design methods by which these specifications can be met, namely offline methods and online methods. By relating these specifications to our framework and by presenting several examples of how oscillators have been designed in the literature, this article provides a useful methodology and toolbox for designing oscillators for a wide range of purposes. In particular, the focus on synthesis of limit cycle dynamical systems should be useful both for engineering and for computational modeling of physical or biological phenomena.