Author

# Thomas Erneux

Other affiliations: Vrije Universiteit Brussel, Free University of Brussels, Phillips Laboratory ...read more

Bio: Thomas Erneux is an academic researcher from Université libre de Bruxelles. The author has contributed to research in topics: Semiconductor laser theory & Laser. The author has an hindex of 40, co-authored 314 publications receiving 6459 citations. Previous affiliations of Thomas Erneux include Vrije Universiteit Brussel & Free University of Brussels.

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16 Apr 2009

TL;DR: In this article, Bernoulli's equation and phase equations are used to describe the phase equation of phase equation.Stability.Vibrations.Vectors. Lasers.

Abstract: Stability.- Biology.- Bernoulli's equation.- Chemistry.- Mechanical vibrations.- Lasers.- Phase equations.

502 citations

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TL;DR: In this article, the authors explored analytically and numerically in the context of the FitzHugh-Nagumo model of nerve membrane excitability, and found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects; the dynamic aspect of the parameter variation.

Abstract: This paper explores analytically and numerically, in the context of the FitzHugh–Nagumo model of nerve membrane excitability, an interesting phenomenon that has been described as a delay or memory effect. It can occur when a parameter passes slowly through a Hopf bifurcation point and the system's response changes from a slowly varying steady state to slowly varying oscillations. On quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects; the dynamic aspect of the parameter variation. This delay and its dependence on the speed of the parameter variation are described.The model involves several parameters and particular singular limits are investigated. One in particular is the slow passage through a low frequency Hopf bifurcation where the system's response changes from a slowly varying steady state to slowly varying relaxation oscillations. We find in this case the onset o...

265 citations

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TL;DR: In this article, a discrete bistable reaction-diffusion system modeled by N coupled Nagumo equations is considered and an asymptotic method is developed to describe the phenomenon of propagation failure.

Abstract: We consider a discrete bistable reaction-diffusion system modeled by N coupled Nagumo equations. We develop an asymptotic method to describe the phenomenon of propagation failure. The Nagumo model depends on two parameters: the coupling constant d and the bistability parameter a. We investigate the limit a→0 and d(a)→0 and construct traveling front solutions. We obtain the critical coupling constant d = d ∗ (a) above which propagation is possible and determine the propagation speed c = c(d) if d>d ∗ . We investigate two different cases for the initiation of a propagating front solution. Case 1 considers a uniform steady state distribution. A propagating front appears as the result of a fixed boundary condition. Case 2 also considers a uniform steady state distribution but a propagating front appears as the result of a localized perturbation.

233 citations

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TL;DR: In this article, a free viscous film is subject to van der Waals attractions that lead to film rupture and long wave asymptotics are used to derive approximate equations that govern the unstable evolution of the film.

Abstract: A free viscous film is subject to van der Waals attractions that lead to film rupture. Long‐wave asymptotics is used to derive approximate equations that govern the unstable evolution of the film. The solution of the nonlinear evolution equation is then considered using bifurcation techniques leading to an estimate for the nonlinear rupture time.

194 citations

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TL;DR: In this article, the dynamics of two mutually coupled but nonidentical semiconductor lasers are studied experimentally, numerically, and analytically for weak coupling, and it is shown that the coupled lasers exhibit a form of localized synchronization characterized by low amplitude oscillations in one laser, but large oscillations of the second laser.

Abstract: The dynamics of two mutually coupled but nonidentical semiconductor lasers are studied experimentally, numerically, and analytically for weak coupling. The lasers have dissimilar relaxation oscillation frequencies and intensities, and their mutual coupling strength may be asymmetric. We find that the coupled lasers exhibit a form of localized synchronization characterized by low amplitude oscillations in one laser, but large oscillations in the second laser.

152 citations

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TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.

Abstract: Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering. © 2005 Elsevier B.V. All rights reserved.

9,441 citations

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TL;DR: This work aims to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture.

Abstract: The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.

7,665 citations

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01 Oct 2006

TL;DR: This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition, providing a link between the two disciplines.

Abstract: This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems Each chapter proceeds from the simple to the complex, and provides sample problems at the end The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience

3,683 citations

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TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.

Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

3,015 citations

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University of Alcalá

^{1}, Charles III University of Madrid^{2}, University of Barcelona^{3}, Leonardo^{4}TL;DR: In this paper, a review of the Kuramoto model of coupled phase oscillators is presented, with a rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years.

Abstract: Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.

2,864 citations