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Top 20 Most Cited Articles
The 20 most cited articles over time based on CrossRef data.
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Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series Chaos 5, 82 (1995); http://dx.doi.org/10.1063/1.166141 (6 pages) | Cited 587 times
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The healthy heartbeat is traditionally thought to be regulated according to the classical principle of homeostasis whereby physiologic systems operate to reduce variability and achieve an equilibrium‐like state [Physiol. Rev. 9, 399–431 (1929)]. However, recent studies [Phys. Rev. Lett. 70, 1343–1346 (1993); Fractals in Biology and Medicine (Birkhauser‐Verlag, Basel, 1994), pp. 55–65] reveal that under normal conditions, beat‐to‐beat fluctuations in heart rate display the kind of long‐range correlations typically exhibited by dynamical systems far from equilibrium [Phys. Rev. Lett. 59, 381–384 (1987)]. In contrast, heart rate time series from patients with severe congestive heart failure show a breakdown of this long‐range correlation behavior. We describe a new method—detrended fluctuation analysis (DFA)—for quantifying this correlation property in non‐stationary physiological time series. Application of this technique shows evidence for a crossover phenomenon associated with a change in short and long‐range scaling exponents. This method may be of use in distinguishing healthy from pathologic data sets based on differences in these scaling properties. © 1995 American Institute of Physics. |
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Practical implementation of nonlinear time series methods: The TISEAN package Chaos 9, 413 (1999); http://dx.doi.org/10.1063/1.166424 (23 pages) | Cited 331 times
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We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature. © 1999 American Institute of Physics. |
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Fundamentals of synchronization in chaotic systems, concepts, and applications Chaos 7, 520 (1997); http://dx.doi.org/10.1063/1.166278 (24 pages) | Cited 221 times
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The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and “cottage industries” have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution—a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (systems with more than one positive Lyapunov exponent) to be synchronized. Several proposals for “secure” communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases (short-wavelength bifurcations), and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization. © 1997 American Institute of Physics. |
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Varieties of spiral wave behavior: An experimentalist’s approach to the theory of excitable media Chaos 1, 303 (1991); http://dx.doi.org/10.1063/1.165844 (32 pages) | Cited 200 times
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Spiral waves in diverse excitable media exhibit strikingly variegated behavior. Mechanistic interpretations of excitability in laboratory systems are commonly tested by comparing the wavelength, period, and meander patterns of the model’s spiral waves with laboratory observations, but models seem seldom to be rejected by such tests. The reason may be that almost any excitable medium behaves in many respects like almost any other, if its parameters are properly adjusted within a reasonable range. What generalizations can be made about ‘‘excitable media’’ in the absence of more specifications? It would be useful to distinguish such generic features from idiosyncrasies of specific models. The range of behavioral flexibility of the FitzHugh–Nagumo excitable medium is explored by varying two of its parameters and comparing the results with other excitable media to suggest a generic pattern of parameter dependence. The results exhibit the remarkable diversity of rotor behavior in a single model and provide a database for quantitative testing of mathematical generalizations. |
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Chaos 8, 20 (1998); http://dx.doi.org/10.1063/1.166311 (28 pages) | Cited 180 times
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Wave propagation in ventricular muscle is rendered highly anisotropic by the intramural rotation of the fiber. This rotational anisotropy is especially important because it can produce a twist of electrical vortices, which measures the rate of rotation (in degree/mm) of activation wavefronts in successive planes perpendicular to a line of phase singularity, or filament. This twist can then significantly alter the dynamics of the filament. This paper explores this dynamics via numerical simulation. After a review of the literature, we present modeling tools that include: (i) a simplified ionic model with three membrane currents that approximates well the restitution properties and spiral wave behavior of more complex ionic models of cardiac action potential (Beeler-Reuter and others), and (ii) a semi-implicit algorithm for the fast solution of monodomain cable equations with rotational anisotropy. We then discuss selected results of a simulation study of vortex dynamics in a parallelepipedal slab of ventricular muscle of varying wall thickness (S) and fiber rotation rate (θz). The main finding is that rotational anisotropy generates a sufficiently large twist to destabilize a single transmural filament and cause a transition to a wave turbulent state characterized by a high density of chaotically moving filaments. This instability is manifested by the propagation of localized disturbances along the filament and has no previously known analog in isotropic excitable media. These disturbances correspond to highly twisted and distorted regions of filament, or “twistons,” that create vortex rings when colliding with the natural boundaries of the ventricle. Moreover, when sufficiently twisted, these rings expand and create additional filaments by further colliding with boundaries. This instability mechanism is distinct from the commonly invoked patchy failure or wave breakup that is not observed here during the initial instability. For modified Beeler-Reuter-like kinetics with stable reentry in two dimensions, decay into turbulence occurs in the left ventricle in about one second above a critical wall thickness in the range of 4–6 mm that matches experiment. However this decay is suppressed by uniformly decreasing excitability. Specific experiments to test these results, and a method to characterize the filament density during fibrillation are discussed. Results are contrasted with other mechanisms of fibrillation and future prospects are summarized. ©1998 American Institute of Physics. |
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Electrical alternans and spiral wave breakup in cardiac tissue Chaos 4, 461 (1994); http://dx.doi.org/10.1063/1.166024 (12 pages) | Cited 151 times
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This paper reports the results of a theoretical investigation of spiral wave breakup in model equations of action potential propagation in cardiac tissue. A general formulation of these equations is described in which arbitrary experimentally determined restitution and dispersion curves can in principle be fitted. Spiral wave behavior is studied in two‐dimension as a function of a parameter Re which controls the steepness of the restitution curve at short diastolic intervals. Spiral breakup is found to occur when the minimum period Tmin, below which a periodically stimulated tissue exhibits alternans in action potential duration, exceeds by a finite amount the spiral rotation period TS. At this point, oscillations in action potential duration are of sufficiently large amplitude to cause a spontaneous conduction block to form along the wavefront. The latter occurs closer to the initiation point of reentry (spiral tip) with increasing steepness and, hence, in smaller tissue sizes. Spiral breakup leads to a spatially disorganized wave activity which is always transient, except for tissues larger than some minimum size and within a very narrow range of Re which increases with dispersion. |
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Approximate entropy (ApEn) as a complexity measure Chaos 5, 110 (1995); http://dx.doi.org/10.1063/1.166092 (8 pages) | Cited 148 times
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Approximate entropy (ApEn) is a recently developed statistic quantifying regularity and complexity, which appears to have potential application to a wide variety of relatively short (greater than 100 points) and noisy time‐series data. The development of ApEn was motivated by data length constraints commonly encountered, e.g., in heart rate, EEG, and endocrine hormone secretion data sets. We describe ApEn implementation and interpretation, indicating its utility to distinguish correlated stochastic processes, and composite deterministic/ stochastic models. We discuss the key technical idea that motivates ApEn, that one need not fully reconstruct an attractor to discriminate in a statistically valid manner—marginal probability distributions often suffice for this purpose. Finally, we discuss why algorithms to compute, e.g., correlation dimension and the Kolmogorov–Sinai (KS) entropy, often work well for true dynamical systems, yet sometimes operationally confound for general models, with the aid of visual representations of reconstructed dynamics for two contrasting processes. © 1995 American Institute of Physics. |
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Quantum‐chaotic scattering effects in semiconductor microstructures Chaos 3, 665 (1993); http://dx.doi.org/10.1063/1.165928 (18 pages) | Cited 141 times
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We show that classical chaotic scattering has experimentally measurable consequences for the quantum conductance of semiconductor microstructures. These include the existence of conductance fluctuations—a sensitivity of the conductance to either Fermi energy or magnetic field—and weak‐localization—a change in the average conductance upon applying a magnetic field. We develop a semiclassical theory and present numerical results for these two effects in which we model the microstructures by billiards attached to leads. We find that the difference between chaotic and regular classical scattering produces a qualitative difference in the fluctuation spectrum and weak‐localization lineshape of chaotic and nonchaotic structures. While the semiclassical theory within the diagonal approximation accounts well for the weak‐localization lineshape and for the spectrum of the fluctuations, we uncover a surprising failure of the semiclassical diagonal‐approximation theory in describing the magnitude of these quantum transport effects. |
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Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity Chaos 12, 852 (2002); http://dx.doi.org/10.1063/1.1504242 (41 pages) | Cited 141 times Online Publication Date: 23 August 2002
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It has become widely accepted that the most dangerous cardiac arrhythmias are due to reentrant waves, i.e., electrical wave(s) that recirculate repeatedly throughout the tissue at a higher frequency than the waves produced by the heart’s natural pacemaker (sinoatrial node). However, the complicated structure of cardiac tissue, as well as the complex ionic currents in the cell, have made it extremely difficult to pinpoint the detailed dynamics of these life-threatening reentrant arrhythmias. A simplified ionic model of the cardiac action potential (AP), which can be fitted to a wide variety of experimentally and numerically obtained mesoscopic characteristics of cardiac tissue such as AP shape and restitution of AP duration and conduction velocity, is used to explain many different mechanisms of spiral wave breakup which in principle can occur in cardiac tissue. Some, but not all, of these mechanisms have been observed before using other models; therefore, the purpose of this paper is to demonstrate them using just one framework model and to explain the different parameter regimes or physiological properties necessary for each mechanism (such as high or low excitability, corresponding to normal or ischemic tissue, spiral tip trajectory types, and tissue structures such as rotational anisotropy and periodic boundary conditions). Each mechanism is compared with data from other ionic models or experiments to illustrate that they are not model-specific phenomena. Movies showing all the breakup mechanisms are available at http://arrhythmia.hofstra.edu/breakup and at ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-12-039203/ INDEX.html. The fact that many different breakup mechanisms exist has important implications for antiarrhythmic drug design and for comparisons of fibrillation experiments using different species, electromechanical uncoupling drugs, and initiation protocols. © 2002 American Institute of Physics. |
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Quantitative analysis of heart rate variability Chaos 5, 88 (1995); http://dx.doi.org/10.1063/1.166090 (7 pages) | Cited 121 times
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In the modern industrialized countries every year several hundred thousands of people die due to sudden cardiac death. The individual risk for this sudden cardiac death cannot be defined precisely by common available, noninvasive diagnostic tools like Holter monitoring, highly amplified ECG and traditional linear analysis of heart rate variability (HRV). Therefore, we apply some rather unconventional methods of nonlinear dynamics to analyze the HRV. Especially, some complexity measures that are based on symbolic dynamics as well as a new measure, the renormalized entropy, detect some abnormalities in the HRV of several patients who have been classified in the low risk group by traditional methods. A combination of these complexity measures with the parameters in the frequency domain seems to be a promising way to get a more precise definition of the individual risk. These findings have to be validated by a representative number of patients. © 1995 American Institute of Physics. |
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Nonlinear time series analysis of normal and pathological human walking Chaos 10, 848 (2000); http://dx.doi.org/10.1063/1.1324008 (16 pages) | Cited 116 times
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Characterizing locomotor dynamics is essential for understanding the neuromuscular control of locomotion. In particular, quantifying dynamic stability during walking is important for assessing people who have a greater risk of falling. However, traditional biomechanical methods of defining stability have not quantified the resistance of the neuromuscular system to perturbations, suggesting that more precise definitions are required. For the present study, average maximum finite-time Lyapunov exponents were estimated to quantify the local dynamic stability of human walking kinematics. Local scaling exponents, defined as the local slopes of the correlation sum curves, were also calculated to quantify the local scaling structure of each embedded time series. Comparisons were made between overground and motorized treadmill walking in young healthy subjects and between diabetic neuropathic (NP) patients and healthy controls (CO) during overground walking. A modification of the method of surrogate data was developed to examine the stochastic nature of the fluctuations overlying the nominally periodic patterns in these data sets. Results demonstrated that having subjects walk on a motorized treadmill artificially stabilized their natural locomotor kinematics by small but statistically significant amounts. Furthermore, a paradox previously present in the biomechanical literature that resulted from mistakenly equating variability with dynamic stability was resolved. By slowing their self-selected walking speeds, NP patients adopted more locally stable gait patterns, even though they simultaneously exhibited greater kinematic variability than CO subjects. Additionally, the loss of peripheral sensation in NP patients was associated with statistically significant differences in the local scaling structure of their walking kinematics at those length scales where it was anticipated that sensory feedback would play the greatest role. Lastly, stride-to-stride fluctuations in the walking patterns of all three subject groups were clearly distinguishable from linearly autocorrelated Gaussian noise. As a collateral benefit of the methodological approach taken in this study, some of the first steps at characterizing the underlying structure of human locomotor dynamics have been taken. Implications for understanding the neuromuscular control of locomotion are discussed. © 2000 American Institute of Physics. |
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Convection driven zonal flows and vortices in the major planets Chaos 4, 123 (1994); http://dx.doi.org/10.1063/1.165999 (12 pages) | Cited 111 times
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The dynamical properties of convection in rotating cylindrical annuli and spherical shells are reviewed. Simple theoretical models and experimental simulations of planetary convection through the use of the centrifugal force in the laboratory are emphasized. The model of columnar convection in a cylindrical annulus not only serves as a guide to the dynamical properties of convection in rotating sphere; it also is of interest as a basic physical system that exhibits several dynamical properties in their most simple form. The generation of zonal mean flows is discussed in some detail and examples of recent numerical computations are presented. The exploration of the parameter space for the annulus model is not yet complete and the theoretical exploration of convection in rotating spheres is still in the beginning phase. Quantitative comparisons with the observations of the dynamics of planetary atmospheres will have to await the consideration in the models of the effects of magnetic fields and the deviations from the Boussinesq approximation. |
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Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics Chaos 7, 159 (1997); http://dx.doi.org/10.1063/1.166252 (23 pages) | Cited 110 times
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A detailed description of fractional kinetics is given in connection to islands’ topology in the phase space of a system. The method of renormalization group is applied to the fractional kinetic equation in order to obtain characteristic exponents of the fractional space and time derivatives, and an analytic expression for the transport exponents. Numerous simulations for the web-map and standard map demonstrate different results of the theory. Special attention is applied to study the singular zone, a domain near the island boundary with a self-similar hierarchy of subislands. The birth and collapse of islands of different types are considered. © 1997 American Institute of Physics. |
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Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics Chaos 10, 427 (2000); http://dx.doi.org/10.1063/1.166509 (43 pages) | Cited 106 times
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In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the “interior” and “exterior” Hill’s regions and other resonant phenomena. © 2000 American Institute of Physics. |
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Adaptive synchronization of neural networks with or without time-varying delay Chaos 16, 013133 (2006); http://dx.doi.org/10.1063/1.2178448 (6 pages) | Cited 103 times Online Publication Date: 30 March 2006
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In this paper, based on the invariant principle of functional differential equations, a simple, analytical, and rigorous adaptive feedback scheme is proposed for the synchronization of almost all kinds of coupled identical neural networks with time-varying delay, which can be chaotic, periodic, etc. We do not assume that the concrete values of the connection weight matrix and the delayed connection weight matrix are known. We show that two coupled identical neural networks with or without time-varying delay can achieve synchronization by enhancing the coupling strength dynamically. The update gain of coupling strength can be properly chosen to adjust the speed of achieving synchronization. Also, it is quite robust against the effect of noise and simple to implement in practice. In addition, numerical simulations are given to show the effectiveness of the proposed synchronization method.
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Chaos 2, 19 (1992); http://dx.doi.org/10.1063/1.165920 (15 pages) | Cited 102 times
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Recent progress in the semiclassical description of two‐electron atoms is reported herein. It is shown that the classical dynamics for the helium atom is of mixed phase space structure, i.e., regular and chaotic motion coexists. Semiclassically, both types of motion require separate treatment. Stability islands are quantized via a torus–quantization‐type procedure, whereas a periodic‐orbit cycle expansion approach accounts for the states associated with hyperbolic electron pair motion. The results are compared with highly accurate ab initio quantum calculations, most of which are reported here for the first time. The results are discussed with an emphasis on previous interpretations of doubly excited electron states |
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A unifying definition of synchronization for dynamical systems Chaos 10, 344 (2000); http://dx.doi.org/10.1063/1.166500 (6 pages) | Cited 102 times
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We propose a unifying definition for synchronization between stationary finite dimensional deterministic dynamical systems. By example, we show that the synchronization phenomena discussed in the dynamical systems literature fits within the framework of this definition, and discuss problems with previous definitions of synchronization. We conclude with a discussion of possible extensions of the definition to infinite dimensional systems described by partial differential equations and/or systems where noise is present. © 2000 American Institute of Physics. |
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Transition to chemical turbulence Chaos 1, 411 (1991); http://dx.doi.org/10.1063/1.165851 (10 pages) | Cited 100 times
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Experiments have been conducted on Turing‐type chemical spatial patterns and their variants in a quasi‐two‐dimensional open spatial reactor with a chlorite–iodide–malonic acid reaction. A variety of stationary spatial structures−hexagons, stripes, and mixed states−were observed, and transitions to these states were studied. For conditions beyond those corresponding to the emergence of patterns, a transition was observed from stationary spatial patterns to chemical turbulence, which is marked by a continuous motion of the pattern within a domain and of the grain boundaries between domains. The transition to chemical turbulence was analyzed by measuring the correlation length, the average pattern speed, and the total length of the domain boundaries. The emergence of chemical turbulence is accompanied by a large increase in the defects in the pattern, which suggests that this is an example of defect‐mediated turbulence. |
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Velocity statistics in excited granular media Chaos 9, 682 (1999); http://dx.doi.org/10.1063/1.166442 (9 pages) | Cited 100 times
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We present an experimental study of velocity statistics for a partial layer of inelastic colliding beads driven by a vertically oscillating boundary. Over a wide range of parameters (accelerations 3–8 times the gravitational acceleration), the probability distribution P(v) deviates measurably from a Gaussian for the two horizontal velocity components. It can be described by P(v) ∼ exp(−∣v/vc∣1.5), in agreement with a recent theory. The characteristic velocity vc is proportional to the peak velocity of the boundary. The granular temperature, defined as the mean square particle velocity, varies with particle density and exhibits a maximum at intermediate densities. On the other hand, for free cooling in the absence of excitation, we find an exponential velocity distribution. Finally, we examine the sharing of energy between particles of different mass. The more massive particles are found to have greater kinetic energy. © 1999 American Institute of Physics. |
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Chaos 11, 227 (2001); http://dx.doi.org/10.1063/1.1349894 (10 pages) | Cited 97 times
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Xenopus oocyte maturation is an example of an all-or-none, irreversible cell fate induction process. In response to a submaximal concentration of the steroid hormone progesterone, a given oocyte may either mature or not mature, but it can exist in intermediate states only transiently. Moreover, once an oocyte has matured, it will remain arrested in the mature state even after the progesterone is removed. It has been hypothesized that the all-or-none character of oocyte maturation, and some aspects of the irreversibility of maturation, arise out of the bistability of the signal transduction system that triggers maturation. The bistability, in turn, is hypothesized to arise from the way the signal transducers are organized into a signaling circuit that includes positive feedback (which makes it so that the system cannot rest in intermediate states) and ultrasensitivity (which filters small stimuli out of the feedback loop, allowing the system to have a stable off-state). Here we review two simple graphical methods that are commonly used to analyze bistable systems, discuss the experimental evidence for bistability in oocyte maturation, and suggest that bistability may be a common means of producing all-or-none responses and a type of biochemical memory. © 2001 American Institute of Physics. |
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