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505
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 117 (49 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.
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 115 (22 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
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 79 (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 76 (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 ...
Theoretical reconstruction of field potentials and dendrodendritic synaptic interactions in olfactory bulb
 J Neurophysiol
, 1968
"... to apply a mathematical theory of generalized dendritic neurons (3639) to the interpretation and reconstruction of field potentials observed in the olfactory bulb of rabbit (32, 33). In the course of pursuing this objective, we were led to postulate dendrodendritic synaptic interactions which prob ..."
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Cited by 71 (6 self)
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to apply a mathematical theory of generalized dendritic neurons (3639) to the interpretation and reconstruction of field potentials observed in the olfactory bulb of rabbit (32, 33). In the course of pursuing this objective, we were led to postulate dendrodendritic synaptic interactions which probably play an important role in sensory discrimination and adaptation in the olfactory system (40). More specifically, our initial aim was to develop a computational model, based on the known anatomical organization of the olfactory bulb and on generally accepted properties of nerve membrane, that could reconstruct the distribution of electric potential as a function of two variables, time and depth in the bulbar layers, following a synchronous antidromic volley in the lateral olfactory tract. The experimental studies of Phillips, Powell, and Shepherd (32, 33) had previously established that the recorded potentials at successive bulbar depths are highly reproducible and correlated with the histological layers of the bulb. These authors recognized the importance of this finding in relation to the symmetry and synchrony of activity in the mitral cell population; they deferred the interpretation of these potentials, in terms of specific neuronal activity, with a view to the present theoretical study.
Stochastic resonance in neuron models
 J. Stat. Phys
, 1993
"... Periodically stimulated sensory neurons typically exhibit a kind of "statistical phase locking " to the stimulus: they tend to fire at a preferred phase of the stimulus cycle, but not at every cycle. Hence, the histogram of interspike intervals (ISIH), i.e., of times between successive fir ..."
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Cited by 60 (3 self)
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Periodically stimulated sensory neurons typically exhibit a kind of "statistical phase locking " to the stimulus: they tend to fire at a preferred phase of the stimulus cycle, but not at every cycle. Hence, the histogram of interspike intervals (ISIH), i.e., of times between successive firings, is multimodal for these neurons, with peaks centered at integer multiples of the driving period. A particular kind of residence time histogram for a large class of noisy bistable systems has recently been shown to exhibit he major features of the neural data. In the present paper, we show that an excitable cell model, the FitzhughNagumo equations, also reproduces these features when driven by additive periodic and stochastic forces. This model exhibits its own brand of stochastic resonance as the peaks of the ISIH successively go through a maximum when the noise intensity is increased. Further, the presence of a noiseinduced limit cycle introduces a third time scale in the problem. This limit cycle is found to modify qualitatively the phaselocking picture, e.g., by suppressing certain peaks in the IS1H. Finally, the role of noise and possibly of stochastic resonance (SR) in the neural encoding of sensory information is discussed. KEY WORDS: Stochastic resonance; neuron models. 1.
TOPOLOGICAL AND PHENOMENOLOGICAL CLASSIFICATION OF BURSTING OSCILLATIONS
, 1995
"... We describe a classification scheme for bursting oscillations which encompasses many of those found in the literature on bursting in excitable media. This is an extension of the scheme of Rinzel (in Mathematical Topics in Population Biology, Springer, Berlin, 1987), put in the context of a sequence ..."
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Cited by 57 (6 self)
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We describe a classification scheme for bursting oscillations which encompasses many of those found in the literature on bursting in excitable media. This is an extension of the scheme of Rinzel (in Mathematical Topics in Population Biology, Springer, Berlin, 1987), put in the context of a sequence of horizontal cuts through a twoparameter bifurcation diagram. We use this to describe the phenomenological haracter of different ypes of bursting, addressing the issue of how well the bursting can be characterized given the limited amount of information often available in experimental settings.
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 53 (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.