Results 1  10
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189
AUTO97: Continuation and bifurcation software for ordinary differential equations (with HomCont
, 1998
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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 62 (33 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.
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 35 (0 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 ...
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 33 (6 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.
The dynamics of legged locomotion: Models, analyses, and challenges
 SIAM Review
, 2006
"... Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we ..."
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Cited by 30 (2 self)
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Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal locomotion, focusing on rapidly running insects, and highlighting achievements and challenges that remain. Newtonian bodylimb dynamics are most naturally formulated as piecewiseholonomic rigid body mechanical systems, whose constraints change as legs touch down or lift off. Central pattern generators and proprioceptive sensing require models of spiking neurons, and simplified phase oscillator descriptions of ensembles of them. A full neuromechanical model of a running animal requires integration of these elements, along with proprioceptive feedback and models of goaloriented sensing, planning and learning. We outline relevant background material from neurobiology and biomechanics, explain key properties of the hybrid dynamical systems that 1 underlie legged locomotion models, and provide numerous examples of such models, from the simplest, completely soluble ‘pegleg walker ’ to complex neuromuscular subsystems that are yet to be assembled into models of behaving animals. 1
Stable concurrent synchronization in dynamic system networks
 Neural Networks
, 2007
"... In a network of dynamical systems, concurrent synchronization is a regime where multiple groups of fully synchronized elements coexist. In the brain, concurrent synchronization may occur at several scales, with multiple “rhythms ” interacting and functional assemblies combining neural oscillators of ..."
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Cited by 28 (17 self)
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In a network of dynamical systems, concurrent synchronization is a regime where multiple groups of fully synchronized elements coexist. In the brain, concurrent synchronization may occur at several scales, with multiple “rhythms ” interacting and functional assemblies combining neural oscillators of many different types. Mathematically, stable concurrent synchronization corresponds to convergence to a flowinvariant linear subspace of the global state space. We derive a general condition for such convergence to occur globally and exponentially. We also show that, under mild conditions, global convergence to a concurrently synchronized regime is preserved under basic system combinations such as negative feedback or hierarchies, so that stable concurrently synchronized aggregates of arbitrary size can be constructed. Simple applications of these results to classical questions in systems neuroscience and robotics are discussed. 1
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 22 (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.
Finding Downbeats with a Relaxation Oscillator
 Psychological Research
, 2001
"... A relaxation oscillator model of neural spiking dynamics is applied to the task of finding downbeats in rhythmical patterns. The importance of downbeat discovery or beat induction is discussed, and the relaxation oscillator model is compared to other oscillator models. In a set of computer simulatio ..."
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Cited by 19 (7 self)
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A relaxation oscillator model of neural spiking dynamics is applied to the task of finding downbeats in rhythmical patterns. The importance of downbeat discovery or beat induction is discussed, and the relaxation oscillator model is compared to other oscillator models. In a set of computer simulations the model is tested on 35 rhythmical patterns from Povel and Essens (1985). The model performs well, making good predictions in 34 of 35 cases. In an analysis we identify some shortcomings of the model and relate model behavior to dynamical properties of relaxation oscillators.
Motor skill acquisition under environmental perturbations: on the necessity of alternate freezing and freeing of . . .
 ADAPTIVE BEHAVIOR
, 2004
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Stability of Traveling Waves: Dichotomies and Eigenvalue Conditions on Finite Intervals
, 1999
"... If a traveling wave is stable or unstable depends essentially on the spectrum of a differential operator P obtained by linearization. We investigate how spectral properties of the allline operator P are related to eigenvalues of finiteinterval BVPs Pu(x) = su(x); x \Gamma x x+ ; Ru = 0. Here R ..."
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Cited by 18 (2 self)
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If a traveling wave is stable or unstable depends essentially on the spectrum of a differential operator P obtained by linearization. We investigate how spectral properties of the allline operator P are related to eigenvalues of finiteinterval BVPs Pu(x) = su(x); x \Gamma x x+ ; Ru = 0. Here R is a linear boundary operator, for which we will derive determinant conditions, and the xinterval is assumed to be sufficiently large. Under suitable assumptions, we show (a) resolvent estimates for large s; (b) if s is in the resolvent of the allline operator P , then s is also in the resolvent for finiteinterval BVPs; (c) eigenvalues of P lead to approximating eigenvalues on finite intervals. These results allow to study the stability question for traveling waves by investigating eigenvalues of finiteinterval problems. We give applications to the FitzHughNagumo system with small diffusion and to the complex GinzburgLandau equations.