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380
The approach of solutions of nonlinear diffusion equations to travelling wave solutions
, 1975
"... The paper is concerned with the asymptotic behavior as t, oo of solutions u(x, t) of the equation in the case utuxxf(u)=O, xe( ~, oo), f(0) =f(1) =0, f'(0)<0, f'(1)<0. Commonly, a travelling front solution u=U(xct), U(oo)=0, U(oo)=l, exists. The following types of global stab ..."
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Cited by 240 (4 self)
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The paper is concerned with the asymptotic behavior as t, oo of solutions u(x, t) of the equation in the case utuxxf(u)=O, xe( ~, oo), f(0) =f(1) =0, f'(0)<0, f'(1)<0. Commonly, a travelling front solution u=U(xct), U(oo)=0, U(oo)=l, exists. The following types of global stability results for fronts and various combinations of them will be given. 1. Let u(x,O)=uo(X) satisfy 0<u0<l. Let a _ = lim sup Uo(X), a + = lira infuo(X). Then u approaches a translate of U uniformly in x and exponentially in time, if a is not too far from 0, and a+ not too far from 1. 1 2. Suppose ~f(u)du>O. Ifa and a+ are not too far from 0, but u o exceeds a o certain threshold level for a sufficiently large xinterval, then u approaches a pair of diverging travelling fronts. 3. Under certain circumstances, u approaches a "stacked " combination of wave fronts, with differing ranges. 1.
A tutorial on the crossentropy method
 Annals of Operations Research
, 2005
"... Abstract: The crossentropy method is a recent versatile Monte Carlo technique. This article provides a brief introduction to the crossentropy method and discusses how it can be used for rareevent probability estimation and for solving combinatorial, continuous, constrained and noisy optimization ..."
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Cited by 172 (18 self)
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Abstract: The crossentropy method is a recent versatile Monte Carlo technique. This article provides a brief introduction to the crossentropy method and discusses how it can be used for rareevent probability estimation and for solving combinatorial, continuous, constrained and noisy optimization problems. A comprehensive list of references on crossentropy methods and applications is included.
On partial contraction analysis for coupled nonlinear oscillators
 technical Report, Nonlinear Systems Laboratory, MIT
, 2003
"... We describe a simple but general method to analyze networks of coupled identical nonlinear oscillators, and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized)results on synchroni ..."
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Cited by 115 (48 self)
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We describe a simple but general method to analyze networks of coupled identical nonlinear oscillators, and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized)results on synchronization, antisynchronization and oscillatordeath. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positivedefinite diffusion coupling, it can be shown that synchronization always occur globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in “flocks ” of oscillators or dynamic elements.
Parameter estimation for differential equations: A generalized smoothing approach
 JOURNAL OF THE ROYAL STATISTICAL SOCIETY, SERIES B
, 2007
"... We propose a new method for estimating parameters in nonlinear differential equations. These models represent change in a system by linking the behavior of a derivative of a process to the behavior of the process itself. Current methods for estimating parameters in differential equations from noi ..."
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Cited by 110 (11 self)
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We propose a new method for estimating parameters in nonlinear differential equations. These models represent change in a system by linking the behavior of a derivative of a process to the behavior of the process itself. Current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to statistical techniques such as inference and interval estimation. This paper describes a new method that uses noisy data to estimate the parameters defining a system of nonlinear differential equations. The approach is based on a modification of data smoothing methods along with a generalization of profiled estimation. We derive interval estimates and show that these have good coverage properties on data simulated from chemical engineering and neurobiology. The method is demonstrated using realworld data from chemistry and from the progress of the autoimmune disease lupus.
Image segmentation based on oscillatory correlation
 Neural Computation
, 1997
"... We study image segmentation on the basis of locally excitatory globally inhibitory oscillator networks (LEGION), whereby the phases of oscillators encode the binding of pixels. We introduce a potential for each oscillator so that only those oscillators with strong connections from their neighborhood ..."
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Cited by 98 (23 self)
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We study image segmentation on the basis of locally excitatory globally inhibitory oscillator networks (LEGION), whereby the phases of oscillators encode the binding of pixels. We introduce a potential for each oscillator so that only those oscillators with strong connections from their neighborhood can develop high potentials. Based on the concept of potential, a solution to remove noisy regions in an image is proposed for LEGION, so that it suppresses the oscillators corresponding to noisy regions, without affecting those corresponding to major regions. We show analytically that the resulting oscillator network separates an image into several major regions, plus a background consisting of all noisy regions, and illustrate network properties by computer simulation. The network exhibits a natural capacity in segmenting images. The oscillatory dynamics leads to a computer algorithm, which is applied successfully to segmenting real graylevel images. A number of issues regarding biological plausibility and perceptual organization are discussed. We argue that LEGION provides a novel and effective framework for image segmentation and figureground segregation. DeLiang Wang and David Terman Image Segmentation 1.
Generalized IntegrateandFire Models of Neuronal Activity Approximate Spike Trains of a . . .
"... We demonstrate that singlevariable integrateandfire models can quantitatively capture the dynamics of a physiologicallydetailed model for fastspiking cortical neurons. Through a systematic set of approximations, we reduce the conductance based model to two variants of integrateandfire mode ..."
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Cited by 84 (16 self)
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We demonstrate that singlevariable integrateandfire models can quantitatively capture the dynamics of a physiologicallydetailed model for fastspiking cortical neurons. Through a systematic set of approximations, we reduce the conductance based model to two variants of integrateandfire models. In the first variant (nonlinear integrateandfire model), parameters depend on the instantaneous membrane potential whereas in the second variant, they depend on the time elapsed since the last spike (Spike Response Model). The direct reduction links features of the simple models to biophysical features of the full conductance based model. To quantitatively
Nonlinear dynamics of networks: the groupoid formalism
 Bull. Amer. Math. Soc
, 2006
"... Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which ..."
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Cited by 76 (13 self)
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Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the grouptheoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the ‘input sets’. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend grouptheoretic methods to more general networks, and in particular it leads to a complete classification of ‘robust ’ patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a highdimensional phase space. It is also equipped with a canonical set of observables—the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology—which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood. Contents 1.
A Simple TwoVariable Model of Cardiac Excitation
, 1996
"... We modified the FitzHughNagumo model of an excitable medium so that it describes adequately the dynamics of pulse propagation in the canine myocardium. The modified model is simple enough to be used for intensive threedimensional computations of the whole heart. It simulates the pulse shape and th ..."
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Cited by 75 (2 self)
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We modified the FitzHughNagumo model of an excitable medium so that it describes adequately the dynamics of pulse propagation in the canine myocardium. The modified model is simple enough to be used for intensive threedimensional computations of the whole heart. It simulates the pulse shape and the restitution property of the canine myocardium with good precision. In 1952 Hodgkin and Huxley proposed the first quantitative mathematical model of wave propagation in squid nerve [1]. This work has had a great impact on modeling of various nonlinear phenomena in biology. On the basis of this model Noble in 1962 developed the first physiological model of cardiac Email: rubin@wave.biol.ruu.nl; permanent address: Institute of Theoretical and Experimental Biophysics, Puschino, Moscow Region, 142292 Russia A simple model of cardiac excitation 2 tissue [2]. Further studies in this field resulted in the development of several realistic ionic models of cardiac tissue which were derived from ...
Is there chaos in the brain? II. Experimental evidence and related models
 C. R. Biol
, 2003
"... The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The meth ..."
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Cited by 53 (0 self)
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The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysis of experimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/or of the Hindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have prevailed over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions. Yet, and although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single and coupled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributed according to chaotic patterns at all levels of its various forms of hierarchy. This chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn, C. R. Acad. Sci. Paris, Ser. III 324 (2001) 773–793] that was focused on the basic concepts of nonlinear dynamics and methods of investigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioning their control using external perturbations. Here we present the data and main arguments that support the existence of chaos at all levels from the simplest to the most complex forms of organization of the nervous system.
The slow passage through a Hopf bifurcation: Delay, memory effects and resonance
 SIAM J. Appl. Math
, 1989
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at. ..."
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Cited by 52 (1 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at.