400 research outputs found
Global analysis of a continuum model for monotone pulse-coupled oscillators
We consider a continuum of phase oscillators on the circle interacting
through an impulsive instantaneous coupling. In contrast with previous studies
on related pulse-coupled models, the stability results obtained in the
continuum limit are global. For the nonlinear transport equation governing the
evolution of the oscillators, we propose (under technical assumptions) a global
Lyapunov function which is induced by a total variation distance between
quantile densities. The monotone time evolution of the Lyapunov function
completely characterizes the dichotomic behavior of the oscillators: either the
oscillators converge in finite time to a synchronous state or they
asymptotically converge to an asynchronous state uniformly spread on the
circle. The results of the present paper apply to popular phase oscillators
models (e.g. the well-known leaky integrate-and-fire model) and draw a strong
parallel between the analysis of finite and infinite populations. In addition,
they provide a novel approach for the (global) analysis of pulse-coupled
oscillators.Comment: 33 page
Kick synchronization versus diffusive synchronization
The paper provides an introductory discussion about two fundamental models of oscillator synchronization: the (continuous-time) diffusive model, that dominates the mathematical literature on synchronization, and the (hybrid) kick model, that accounts for most popular examples of synchronization, but for which only few theoretical results exist. The paper stresses fundamental differences between the two models, such as the different contraction measures underlying the analysis, as well as important analogies that can be drawn in the limit of weak coupling.Peer reviewe
Contraction of monotone phase-coupled oscillators
This paper establishes a global contraction property for networks of
phase-coupled oscillators characterized by a monotone coupling function. The
contraction measure is a total variation distance. The contraction property
determines the asymptotic behavior of the network, which is either finite-time
synchronization or asymptotic convergence to a splay state.Comment: 10 page
On the dichotomic collective behaviors of large populations of pulse-coupled firing oscillators
The study of populations of pulse-coupled firing oscillators is a general and simple paradigm to investigate a wealth of natural phenomena, including the collective behaviors of neurons, the synchronization of cardiac pacemaker cells, or the dynamics of earthquakes. In this framework, the oscillators of the network interact through an instantaneous impulsive coupling: whenever an oscillator fires, it sends out a pulse which instantaneously increments the state of the other oscillators by a constant value. There is an extensive literature on the subject, which investigates various model extensions, but only in the case of leaky integrate-and-fire oscillators. In contrast, the present dissertation addresses the study of other integrate-and-fire dynamics: general monotone integrate-and-fire dynamics and quadratic integrate-and-fire dynamics. The main contribution of the thesis highlights that the populations of oscillators exhibit a dichotomic collective behavior: either the oscillators achieve perfect synchrony (slow firing frequency) or the oscillators converge toward a phase-locked clustering configuration (fast firing frequency). The dichotomic behavior is established both for finite and infinite populations of oscillators, drawing a strong parallel between discrete-time systems in finite-dimensional spaces and continuous-time systems in infinite-dimensional spaces. The first part of the dissertation is dedicated to the study of monotone integrate-and-fire dynamics. We show that the dichotomic behavior of the oscillators results from the monotonicity property of the dynamics: the monotonicity property induces a global contraction property of the network, that forces the dichotomic behavior. Interestingly, the analysis emphasizes that the contraction property is captured through a 1-norm, instead of a (more common) quadratic norm. In the second part of the dissertation, we investigate the collective behavior of quadratic integrate-and-fire oscillators. Although the dynamics is not monotone, an “average” monotonicity property ensures that the collective behavior is still dichotomic. However, a global analysis of the dichotomic behavior is elusive and leads to a standing conjecture. A local stability analysis circumvents this issue and proves the dichotomic behavior in particular situations (small networks, weak coupling, etc.). Surprisingly, the local stability analysis shows that specific integrate-and-fire oscillators exhibit a non-dichotomic behavior, thereby suggesting that the dichotomic behavior is not a general feature of every network of pulse-coupled oscillators. The present thesis investigates the remarkable dichotomic behavior that emerges from networks of pulse-coupled integrate-and-fire oscillators, putting emphasis on the stability properties of these particular networks and developing theoretical results for the analysis of the corresponding dynamical systems.Les populations d’oscillateurs impulsivement couplés constituent un paradigme simple et
général pour étudier une multitude de phénomènes naturels, tels que les comportements collectifs
des neurones, la synchronisation des cellules pacemaker du coeur, ou encore la dynamique
des tremblements de terre. Dans ce contexte, les oscillateurs interagissent au sein du réseau
par le biais d’un couplage instantané: quand un oscillateur décharge, il envoie vers les autres
oscillateurs une impulsion qui incrémente instantanément leur état par une valeur constante.
Diverses extensions du modèle ont été intensément étudiées dans la littérature, mais seulement
dans le cas d’oscillateurs leaky integrate-and-fire. Afin de pallier cette restriction, le présent manuscrit traite de l’étude d’autres dynamiques integrate-and-fire: les dynamiques générales integrate-and-fire monotones et les dynamiques integrate-and-fire quadratiques.
La contribution principale de la thèse met en évidence le comportement d’ensemble dichotomique
selon lequel s’organisent les populations d’oscillateurs: soit les oscillateurs atteignent
un état de synchronisation parfaite (taux de décharge lent), soit ils convergent vers
une configuration de clustering en blocage de phase (taux de décharge rapide). Ce comportement
dichotomique est Ă©tabli aussi bien pour des populations finies que pour des populations
infinies, ce qui démontre un parallèle élégant entre des systèmes en temps-discret dans des
espaces de dimension finie et des systèmes en temps-continu dans des espaces de dimension
infinie.
La première partie du manuscrit se concentre sur l’étude des dynamiques integrate-and-fire
monotones. Dans ce cadre, nous montrons que le comportement dichotomique résulte de la
propriété de monotonicité des oscillateurs. Cette dernière induit une propriété de contraction
globale, elle-même engendrant le comportement dichotomique. En outre, l’analyse révèle que
la propriété de contraction est capturée par une norme 1, au lieu d’une norme quadratique
(plus usuelle).
Dans la seconde partie de la thèse, nous étudions le comportement d’ensemble d’oscillateurs
integrate-and-fire quadratiques. Bien que la dynamique ne soit plus monotone, une propriété
de monotonicité “en moyenne” implique que le comportement collectif est encore dichotomique.
Alors qu’une analyse de stabilité globale s’avère être difficile et conduit à plusieurs
conjectures, une analyse locale permet de prouver le comportement dichomique dans certaines
situations (réseaux de petite taille, couplage faible, etc.). De plus, l’analyse locale prouve que
des oscillateurs integrate-and-fire particuliers ne s’organisent pas suivant un comportement
dichotomique, ce qui suggère que ce dernier n’est pas une caractéristique générale de tous les
réseaux d’oscillateurs impulsivement couplés.
En résumé, la thèse étudie le remarquable comportement dichotomique qui émerge des
réseaux d’oscillateurs integrate-and-fire impulsivement couplés, mettant ainsi l’emphase sur
les propriĂ©tĂ©s de stabilitĂ© desdits rĂ©seaux et dĂ©veloppant les rĂ©sultats thĂ©oriques nĂ©cessaires Ă
l’étude mathématique des systèmes dynamiques correspondants
On the dichotomic collective behaviors of large populations of pulse-coupled firing oscillators
The study of populations of pulse-coupled firing oscillators is a general and simple paradigm to investigate a wealth of natural phenomena, including the collective behaviors of neurons, the synchronization of cardiac pacemaker cells, or the dynamics of earthquakes. In this framework, the oscillators of the network interact through an instantaneous impulsive coupling: whenever an oscillator fires, it sends out a pulse which instantaneously increments the state of the other oscillators by a constant value. There is an extensive literature on the subject, which investigates various model extensions, but only in the case of leaky integrate-and-fire oscillators. In contrast, the present dissertation addresses the study of other integrate-and-fire dynamics: general monotone integrate-and-fire dynamics and quadratic integrate-and-fire dynamics. The main contribution of the thesis highlights that the populations of oscillators exhibit a dichotomic collective behavior: either the oscillators achieve perfect synchrony (slow firing frequency) or the oscillators converge toward a phase-locked clustering configuration (fast firing frequency). The dichotomic behavior is established both for finite and infinite populations of oscillators, drawing a strong parallel between discrete-time systems in finite-dimensional spaces and continuous-time systems in infinite-dimensional spaces. The first part of the dissertation is dedicated to the study of monotone integrate-and-fire dynamics. We show that the dichotomic behavior of the oscillators results from the monotonicity property of the dynamics: the monotonicity property induces a global contraction property of the network, that forces the dichotomic behavior. Interestingly, the analysis emphasizes that the contraction property is captured through a 1-norm, instead of a (more common) quadratic norm. In the second part of the dissertation, we investigate the collective behavior of quadratic integrate-and-fire oscillators. Although the dynamics is not monotone, an “average” monotonicity property ensures that the collective behavior is still dichotomic. However, a global analysis of the dichotomic behavior is elusive and leads to a standing conjecture. A local stability analysis circumvents this issue and proves the dichotomic behavior in particular situations (small networks, weak coupling, etc.). Surprisingly, the local stability analysis shows that specific integrate-and-fire oscillators exhibit a non-dichotomic behavior, thereby suggesting that the dichotomic behavior is not a general feature of every network of pulse-coupled oscillators. The present thesis investigates the remarkable dichotomic behavior that emerges from networks of pulse-coupled integrate-and-fire oscillators, putting emphasis on the stability properties of these particular networks and developing theoretical results for the analysis of the corresponding dynamical systems.Les populations d’oscillateurs impulsivement couplés constituent un paradigme simple et
général pour étudier une multitude de phénomènes naturels, tels que les comportements collectifs
des neurones, la synchronisation des cellules pacemaker du coeur, ou encore la dynamique
des tremblements de terre. Dans ce contexte, les oscillateurs interagissent au sein du réseau
par le biais d’un couplage instantané: quand un oscillateur décharge, il envoie vers les autres
oscillateurs une impulsion qui incrémente instantanément leur état par une valeur constante.
Diverses extensions du modèle ont été intensément étudiées dans la littérature, mais seulement
dans le cas d’oscillateurs leaky integrate-and-fire. Afin de pallier cette restriction, le présent manuscrit traite de l’étude d’autres dynamiques integrate-and-fire: les dynamiques générales integrate-and-fire monotones et les dynamiques integrate-and-fire quadratiques.
La contribution principale de la thèse met en évidence le comportement d’ensemble dichotomique
selon lequel s’organisent les populations d’oscillateurs: soit les oscillateurs atteignent
un état de synchronisation parfaite (taux de décharge lent), soit ils convergent vers
une configuration de clustering en blocage de phase (taux de décharge rapide). Ce comportement
dichotomique est Ă©tabli aussi bien pour des populations finies que pour des populations
infinies, ce qui démontre un parallèle élégant entre des systèmes en temps-discret dans des
espaces de dimension finie et des systèmes en temps-continu dans des espaces de dimension
infinie.
La première partie du manuscrit se concentre sur l’étude des dynamiques integrate-and-fire
monotones. Dans ce cadre, nous montrons que le comportement dichotomique résulte de la
propriété de monotonicité des oscillateurs. Cette dernière induit une propriété de contraction
globale, elle-même engendrant le comportement dichotomique. En outre, l’analyse révèle que
la propriété de contraction est capturée par une norme 1, au lieu d’une norme quadratique
(plus usuelle).
Dans la seconde partie de la thèse, nous étudions le comportement d’ensemble d’oscillateurs
integrate-and-fire quadratiques. Bien que la dynamique ne soit plus monotone, une propriété
de monotonicité “en moyenne” implique que le comportement collectif est encore dichotomique.
Alors qu’une analyse de stabilité globale s’avère être difficile et conduit à plusieurs
conjectures, une analyse locale permet de prouver le comportement dichomique dans certaines
situations (réseaux de petite taille, couplage faible, etc.). De plus, l’analyse locale prouve que
des oscillateurs integrate-and-fire particuliers ne s’organisent pas suivant un comportement
dichotomique, ce qui suggère que ce dernier n’est pas une caractéristique générale de tous les
réseaux d’oscillateurs impulsivement couplés.
En résumé, la thèse étudie le remarquable comportement dichotomique qui émerge des
réseaux d’oscillateurs integrate-and-fire impulsivement couplés, mettant ainsi l’emphase sur
les propriĂ©tĂ©s de stabilitĂ© desdits rĂ©seaux et dĂ©veloppant les rĂ©sultats thĂ©oriques nĂ©cessaires Ă
l’étude mathématique des systèmes dynamiques correspondants
Population Dynamics of Globally Coupled Degrade-and-Fire Oscillators
This paper reports the analysis of the dynamics of a model of pulse-coupled
oscillators with global inhibitory coupling. The model is inspired by
experiments on colonies of bacteria-embedded synthetic genetic circuits. The
total population can be either of finite (arbitrary) size or infinite, and is
represented by a one-dimensional profile. Profiles can be discontinuous,
possibly with infinitely many jumps. Their time evolution is governed by a
singular differential equation. We address the corresponding initial value
problem and characterize the dynamics' main features. In particular, we prove
that trajectory behaviors are asymptotically periodic, with period only
depending on the profile (and on the model parameters). A criterion is obtained
for the existence of the corresponding periodic orbits, which reveals the
existence of a sharp transition as the coupling parameter is increased. The
transition separates a regime where any profile can be obtained in the limit of
large times, to a situation where only trajectories with sufficiently large
groups of synchronized oscillators perdure
The link between coherence echoes and mode locking
We investigate the appearance of sharp pulses in the mean field of Kuramoto-type globally- coupled phase oscillator systems. In systems with exactly equidistant natural frequencies self- organized periodic pulsations of the mean field, called mode locking, have been described re- cently as a new collective dynamics below the synchronization threshold. We show here that mode locking can appear also for frequency combs with modes of finite width, where the natu- ral frequencies are randomly chosen from equidistant frequency intervals. In contrast to that, so called coherence echoes, which manifest themselves also as pulses in the mean field, have been found in systems with completely disordered natural frequencies as the result of two consecutive stimulations applied to the system. We show that such echo pulses can be explained by a stimula- tion induced mode locking of a subpopulation representing a frequency comb. Moreover, we find that the presence of a second harmonic in the interaction function, which can lead to the global stability of the mode-locking regime for equidistant natural frequencies, can enhance the echo phenomenon significantly. The non-monotonous behavior of echo amplitudes can be explained as a result of the linear dispersion within the self-organized mode-locked frequency comb. Fi- nally we investigate the effect of small periodic stimulations on oscillator systems with disordered natural frequencies and show how the global coupling can support the stimulated pulsation by increasing the width of locking plateaus
Optimal Control and Synchronization of Dynamic Ensemble Systems
Ensemble control involves the manipulation of an uncountably infinite collection of structurally identical or similar dynamical systems, which are indexed by a parameter set, by applying a common control without using feedback. This subject is motivated by compelling problems in quantum control, sensorless robotic manipulation, and neural engineering, which involve ensembles of linear, bilinear, or nonlinear oscillating systems, for which analytical control laws are infeasible or absent. The focus of this dissertation is on novel analytical paradigms and constructive control design methods for practical ensemble control problems. The first result is a computational method %based on the singular value decomposition (SVD) for the synthesis of minimum-norm ensemble controls for time-varying linear systems. This method is extended to iterative techniques to accommodate bounds on the control amplitude, and to synthesize ensemble controls for bilinear systems. Example ensemble systems include harmonic oscillators, quantum transport, and quantum spin transfers on the Bloch system. To move towards the control of complex ensembles of nonlinear oscillators, which occur in neuroscience, circadian biology, electrochemistry, and many other fields, ideas from synchronization engineering are incorporated. The focus is placed on the phenomenon of entrainment, which refers to the dynamic synchronization of an oscillating system to a periodic input. Phase coordinate transformation, formal averaging, and the calculus of variations are used to derive minimum energy and minimum mean time controls that entrain ensembles of non-interacting oscillators to a harmonic or subharmonic target frequency. In addition, a novel technique for taking advantage of nonlinearity and heterogeneity to establish desired dynamical structures in collections of inhomogeneous rhythmic systems is derived
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