Université Libre de Bruxelles Faculté des Sciences
Service de Chimie Physique
Synchronization Phenomena in Light-Controlled Oscillators
Thèse présentée en vue de l’obtention du grade de Docteur en Sciences
Gonzalo Marcelo Ramírez Ávila Février 2004
SYNCHRONIZATION PHENOMENA IN LIGHT-CONTROLLED OSCILLATORS
By
Gonzalo Marcelo Ramírez Ávila
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY AT
FREE UNIVERSITY OF BRUSSELS BRUSSELS, BELGIUM
DECEMBER 2003
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Copyright by Gonzalo Marcelo Ramírez Ávila, 2004
FREE UNIVERSITY OF BRUSSELS
CENTRE FOR NONLINEAR PHENOMENA AND COMPLEX SYSTEMS The undersigned hereby certify that they have read and recommend to the Faculty of Sciences for acceptance a thesis entitled “Synchronization phenomena in light-controlled oscillators” by Gonzalo Marcelo Ramírez Ávila in partial fulfillment of the requirements for the degree ofDoctor of Philosophy.
Dated: December 2003
Examiner:
Christian Van Den Broeck Research Supervisors:
Jean-Louis Deneubourg
Jean-Luc Guisset Examining Committee:
Grégoire Nicolis
Pasquale Nardone
Léon Brenig
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FREE UNIVERSITY OF BRUSSELS
Date:December 2003 Author: Gonzalo Marcelo Ramírez Ávila
Title: Synchronization phenomena in light-controlled oscillators Department: Centre for Nonlinear Phenomena and Complex Systems Degree:Ph.D. Convocation:February Year:2004
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To my family.
To the Bolivian people massacred in October 2003.
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Table of Contents
Table of Contents vii
Abstract xi
Résumé xiii
Resumen xv
Acknowledgements xvii
I Introduction 1
I.1 Synchronization . . . . 1
I.2 Biological background . . . . 2
I.3 Relaxation oscillators . . . . 6
I.3.1 Examples of relaxation oscillators . . . . 6
I.3.1.a Van der Pol oscillator . . . . 7
I.3.1.b Integrate-and-fire oscillators . . . . 7
I.4 Goals and structure of the work . . . . 9
II Experimental setup and measurements 11 II.1 Presentation of an LCO . . . . 12
II.2 LCO functioning . . . . 13
II.3 Experimental measurements . . . . 16
II.3.1 Measurements on a two-LCO set . . . . 17
II.3.1.a An LCO driven by a short PG . . . . 17
II.3.1.b An LCO driven by a long PG . . . . 21
II.3.1.c An LCO driven by other LCO with the same characteristics . . 22
II.3.1.d Synchronization between two interacting LCOs. Measure and analysis . . . . 22
II.3.1.e Synchronization region and distance between LCOs . . . . 24
II.3.2 Other measurements . . . . 26
II.3.2.a Measurements on a three-LCO set . . . . 26
II.3.2.b Synchronization between several LCOs . . . . 27
II.4 Summary . . . . 27
III The model and its characteristics 29 III.1 Formulation of the model . . . . 29
III.2 Phase response curves . . . . 31
III.2.1 Definition of a PRC . . . . 31
III.2.1.a Example. PRCs for the integrate-and-fire oscillator . . . . 32 vii
III.2.1.a.1 Linear integrate-and-fire model. . . . 32
III.2.1.a.2 Quadratic integrate-and-fire model. . . . 34
III.2.1.b PRCs for the LCOs . . . . 35
III.2.1.b.1 Phase shift ratio. . . . 42
III.3 Model validation . . . . 45
III.3.1 Two-LCO set . . . . 45
III.3.1.a Master–slave LCOs, equal-width coupling-pulse . . . . 45
III.3.1.b Master (pulse generator)–Slave (LCO), narrow coupling-pulse 46 III.3.1.c Dual interaction setting . . . . 47
III.3.2 Three interacting LCOs in line . . . . 47
III.4 Summary . . . . 50
IV Synchronization of two LCOs 53 IV.1 Analysis of two identical LCOs . . . . 53
IV.1.1 Simplified model. Sawtooth waveform . . . . 53
IV.1.2 Our model . . . . 59
IV.1.2.a Overlapped short impulses . . . . 60
IV.1.2.b Non-overlapped short impulses . . . . 62
IV.1.2.b.1 Synchronization time. . . . 66
IV.2 Nonidentical LCOs . . . . 69
IV.2.1 Phase locking . . . . 69
IV.2.2 High order synchronization . . . . 70
IV.3 Summary . . . . 72
V Noise influence on LCOs 75 V.1 Uniform noise influence on two LCOs . . . . 76
V.1.1 Mathematical formulation . . . . 76
V.1.2 Uncoupled LCOs . . . . 77
V.1.3 Master–slave configuration . . . . 79
V.1.4 Mutual interaction . . . . 80
V.1.5 Arnold tongues . . . . 82
V.2 Gaussian noise influence on two coupled LCOs . . . . 85
V.3 Summary . . . . 92
VI Synchronization in locally coupled LCOs 95 VI.1 Coupled LCOs in a linear configuration . . . . 95
VI.1.1 Identical LCOs . . . . 97
VI.1.1.a Phase difference criterion . . . . 98
VI.1.1.b Period criterion . . . . 98
VI.1.2 Identical LCOs neglecting the influence on the discharge . . . 100
VI.1.2.a Phase difference criterion . . . 101
VI.1.2.b Period criterion . . . 101
VI.1.3 Nonidentical LCOs . . . 101
VI.1.3.a Phase difference criterion . . . 101
VI.1.3.b Period criterion . . . 103
VI.2 Coupled LCOs in a ring configuration . . . 103
VI.2.1 Identical LCOs . . . 105
VI.2.1.a Phase difference criterion . . . 105
VI.2.1.b Period criterion . . . 107 viii
VI.2.2 Identical LCOs neglecting the influence on the discharge . . . 108
VI.2.2.a Phase difference criterion . . . 108
VI.2.2.b Period criterion . . . 108
VI.2.3 Nonidentical LCOs . . . 108
VI.2.3.a Phase difference criterion . . . 108
VI.2.3.b Period criterion . . . 110
VI.3 Coupled LCOs in a lattice configuration . . . 112
VI.3.1 Clustering in lattices of LCOs . . . 115
VI.4 Summary . . . 120
VII Synchronization in globally coupled LCOs 123 VII.1 Static identical globally coupled LCOs . . . 124
VII.1.1 Clustering in static globally coupled LCOs . . . 128
VII.1.1.a Density and number of LCOs . . . 128
VII.1.1.b Nonidentical LCOs . . . 132
VII.2 Mobile identical globally coupled LCOs . . . 133
VII.2.1 Well-mixed MGC LCOs . . . 135
VII.2.2 Discrete movement and probability of motion . . . 136
VII.2.3 Limited movements and rest . . . 137
VII.2.3.a The same time of motion for all the LCOs . . . 138
VII.2.3.b Different times of motion for the LCOs . . . 138
VII.3 Summary . . . 138
VIII Conclusions and perspectives 143 VIII.1 The LCO a "real" integrate-and-fire oscillator . . . 143
VIII.2 Noise effects on LCOs synchronization . . . 144
VIII.3 Populations of LCOs and their implications . . . 145
VIII.4 Biological oscillators and LCOs . . . 147
VIII.5 Perspectives . . . 149
A Fitting results for mobile globally coupled LCOs 151 A.1 Well-mixed LCOs . . . 151
A.2 Discrete movements and probability of motion . . . 151
A.3 Limited movements and rest . . . 151
Bibliography 157
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Abstract
The aim of this work is to study both in experimental and theoretical ways the synchronous behavior in groups of light-controlled oscillators (LCOs) which are simple electronic devices that mimic firefly’s behavior in the sense that these circuits interact as fireflies do, i.e. by means of light-pulses coupling. At the same time, our LCOs constitute a good approach to study other systems that act as integrate-and-fire oscillators because an LCO is a typical relaxation oscillator with the presence of two time scales: a long charging stage and a very short discharging stage.
We performed several experimental measurements in order to understand the synchro- nization process in our LCOs. We found that synchronization is acquitted due to the effects (shortening of the charge or lengthening of the discharge) that are operated when an stimulus acts on it. The experimental work allows us to formulate a model that takes into account the effects mentioned above as well as the physics related to the LCO’s circuit. We firstly used the model in order to characterize our LCO by means of the Phase Response Curve (PRC) that we have obtained analytically.
The model has been validated by comparing the experimental measurements with the results obtained from the model. The fitting is excellent and the model reproduces even the bifurcation-like phenomenon observed in three LCOs in line due to the sensitivity to initial conditions that may lead the system to two different stable states. This fact enabled us to use the model for many different situations that are not easy to solve by experimental means.
We have analyzed two identical LCOs and even for this simple case, we assumed some simplifications in order to find exact solutions. Depending on the initial conditions, we found two possible states, the first one tends to in-phase synchronization and the second one tends to an anti-synchronous unstable state. We also found that the synchronization time rises with the coupling strength, i.e. with the distance between LCOs. We have built the Arnold tongues for two coupled LCOs founding regions with puren: msynchronization and regions in which synchronization and modulation overlap.
We have worked numerically with uniform and Gaussian noise acting on the voltage source. We have characterized the synchronous behavior of noisy LCOs using simple statistical parameters such as the mean value of the linear phase difference and the variance of the cyclic phase difference. We have demonstrated that noise does not only perturb synchronous states in LCO but it may enhance synchronization especially for Gaussian noise with unequal variances.
We studied by statistical means the synchronization for locally coupled LCOs arranged in line, in ring and in lattice. We found that LCOs in ring exhibit more easily total synchronization.
Concerning the synchronization time, it is unpredictable. The analytical and numerical results suggest that total synchronization is the most probable phenomenon when the number of oscillators is not very large.
We studied static and mobile globally coupled LCOs. In both cases we found that syn- chronization is less probable when the number of LCOs increases. Considering a mean field
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approach, the synchronization time tends to decrease with the number of LCOs. For realistic cases in which the LCO’s interaction depends on their distances, the determination of the synchronization time becomes unpredictable again. Finally, mobile LCOs in certain cases may improve synchronization. The latter could be useful in robotics but it requires further studies.
Our system is closer to the reality than those considered in literature because of its cha- racteristics and its experimental basis. The obtained results could be applied to biological systems such as fireflies, cardiac cells, neurones, and so on but also to robotics, where the long distance communication by light and the emergence of synchronization patterns may be useful to perform specific tasks.
Résumé
Le but de cette thèse est d’étudier d’une façon expérimentale et théorique le comporte- ment synchrone d’un groupe d’oscillateurs contrôlés par la lumière (LCOs). Ces LCOs sont très simples du point de vue électronique et ont la propriété d’imiter le comportement des lucioles puisqu’ils interagissent par des impulsions de lumière. En même temps, les LCOs sont une bonne approche pour étudier d’autres systèmes qui agissent comme des oscillateurs d’in- tégration et de tir car un LCO est un oscillateur de relaxation à deux échelles de temps : un long processus de charge alterné avec un très court processus de décharge. Une série d’expériences a été menée pour pouvoir comprendre le processus de synchronisation des LCOs. Nous avons trouvé que l’acquisition de la synchronisation est due aux effets de la perturbation à savoir : le raccourcissement de la charge et l’allongement de la décharge. Les mesures expérimentales ainsi que la physique liée aux LCOs nous ont permis de formuler un modèle qui a été utilisé pour trouver d’une façon analytique la courbe de réponse de phase (PRC) qui caractérise un LCO.
Le modèle a ensuite été validé en comparant les résultats expérimentaux et théoriques. Le modèle reproduit même, le phénomène de bifurcation qui apparaît lorsque trois LCOs sont couplés et disposés en ligne : deux états stables différents apparaissent selon les conditions initiales. L’accord trouvé entre théorie et expérience nous permet d’utiliser le modèle pour étudier d’autres situations qui ne sont pas facilement abordables du point de vue expérimental.
Nous avons étudié analytiquement deux LCOs identiques couplés. Même pour ce cas idéal, nous étions obligés de faire des simplifications pour pouvoir trouver des solutions exactes. On a trouvé pour ce système deux états possibles qui dépendent des conditions initiales, la syn- chronisation (stable) et l’anti-synchronisation (instable). Nous avons également montré que le temps de synchronisation augmente avec la distance entre LCOs. La construction des langues d’Arnold (régions de synchronisation) nous a permis de distinguer des régions de synchroni- sation pure d’ordre n:m et des régions de superposition synchronisation–modulation.
Nous avons travaillé numériquement avec des systèmes de LCOs affectés de bruits uni- forme et Gaussien. Le comportement synchrone de ce système a été caractérisé en utilisant des paramètres statistiques simples tels que la moyenne de la différence de phase linéaire et la variance de la différence de phase cyclique. Nous avons démontré que le bruit, bien qu’il puisse perturber la synchronisation, peut aussi la favoriser entre deux LCOs qui ne se synchroniseraient pas en conditions normales, surtout quand le bruit est Gaussien et que les variances du bruit ne sont pas égales.
Nous avons étudié en termes statistiques la synchronisation de LCOs couplés localement et arrangés en ligne, en anneau et en réseau. Nous avons montré que la synchronisation totale se produit plus facilement pour des LCOs disposés en anneau. Concernant le temps de synchronisation, il est imprédictible. Les résultats analytiques et numériques suggèrent que la synchronisation totale est le phénomène le plus probable quand le nombre d’oscillateurs n’est pas très grand.
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Finalement, nous avons étudié des LCOs statiques et mobiles couplés globalement. Dans les deux cas, nous avons trouvé que la synchronisation est moins probable quand le nombre d’oscillateurs augmente. Pour la condition statique, en considérant un couplage du type champ moyen, nous avons observé que le temps de synchronisation diminue avec le nombre de LCOs.
Cependant, pour la situation plus réaliste dans laquelle l’interaction entre LCOs dépend de la distance les séparant, le temps de synchronisation devient à nouveau imprédictible. Enfin, nous avons étudié l’influence de la mobilité sur la synchronisation, problème qui est important en biologie et en robotique.
Notre système, de par ses caractéristiques et sa base expérimentale, est beaucoup plus proche de la réalité que ceux considérés d’habitude dans la littérature. Les résultats obtenus peuvent s’appliquer à des systèmes biologiques (lucioles, cellules cardiaques, neurones, . . . ), mais également à la robotique, où la communication à longue portée par la lumière et l’émer- gence de patterns de synchronisation pourraient être très utiles dans le but d’effectuer des tâches spécifiques.
Resumen
El objetivo de esta tesis es de estudiar de forma tanto teórica como experimentalmente el comportamiento síncrono de un grupo de osciladores controlados por luz (LCOs). Electró- nicamente, estos LCOs son sencillos y tienen la propiedad de imitar el comportamiento de luciérnagas ya que interactúan mediante impulsos luminosos. Al mismo tiempo, los LCOs constituyen una buena aproximación para estudiar otros sistemas que actúan como osciladores de integración y disparo puesto que un LCO es un oscilador de relajación con dos escalas de tiempo: una larga para el proceso de carga seguida de una corta para el proceso de descarga.
Se realizaron varios experimentos con el fin de comprender el proceso de sincronización de los LCOs. Se encontró que la sincronización surge a partir del acortamiento y del alargamiento en los procesos de carga y descarga respectivamente, que son producidos debido a las pertur- baciones luminosas. Se formuló un modelo a partir de los resultados experimentales y de la física asociada a un LCO. Este modelo fue utilizado para encontrar la curva de respuesta de fase (PRC) del LCO.
El modelo fue validado comparando los resultados teóricos y experimentales. El modelo reproduce muy satisfactoriamente las mediciones experimentales, incluso el caso de la apari- ción de un fenómeno tipo bifurcación que fue observado en un grupo de tres LCOs dispuesto linealmente. La sensibilidad a las condiciones iniciales se evidencia. Por otra parte, el excelente acuerdo entre teoría y experiencia nos permite extender el modelo a situaciones que no son fáciles de alcanzar experimentalmente.
Se estudió analíticamente el caso de dos LCOs idénticos acoplados. Incluso para este caso ideal, se tuvieron que hacer simplificaciones de manera a poder resolver las ecuaciones correspondientes. Se determinaron dos estados posibles, un estable que corresponde con la sincronización en fase y otro inestable asociado a una situación de antifase. Se encontró también que el tiempo de sincronización aumenta con la distancia entre los LCOs. La construcción de las lenguas de Arnold (regiones de sincronización) permitió distinguir la sincronización pura de orden n:m de la sincronización acompañada de modulación.
Se trabajó numéricamente con un sistema de dos LCOs influenciado por ruido uniforme o gaussiano. El comportamiento síncrono de este sistema fue caracterizado utilizando paráme- tros estadísticos simples tales como el valor medio de la diferencia de fase lineal y la varianza de la diferencia de fase cíclica. Se demostró que si bien el ruido puede perturbar la sincro- nización, éste puede también inducir sincronización en situaciones en las que los LCOs no sincronizan de manera natural. Esto último fue observado trabajando con ruido gaussiano y en condiciones en las que las varianzas de ruido no son iguales.
Se estudió en términos estadísticos la sincronización de LCOs acoplados localmente y ordenados en configuraciones lineales, en anillo o en red. Se encontró que la sincronización total es más fácil de alcanzar en LCOs dispuestos en anillo. Aunque el tiempo de sincronización tiende a crecer con el número de LCOs, éste es impredecible. Los resultados analíticos y numéricos sugieren que la sincronización total es el fenómeno más probable cuando el número de osciladores no es muy grande.
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Finalmente se estudió el caso de acoplamiento global de LCOs tanto en la situación está- tica como en la situación en que los LCOs están dotados de movimiento. En ambos casos, la sincronización es menos probable cuando el número de LCOs aumenta. Para el caso estático y considerando un acoplamiento de tipo campo medio, se encontró que el tiempo de sin- cronización tiende a disminuir con el número de LCOs. Sin embargo, para la situación más realista en la cual la interacción entre los LCOs depende de la distancia que los separa, el tiempo de sincronización no es predecible. Con el fin de saber si el movimiento favorece o no la sincronización, se estudiaron diversas situaciones de LCOs en movimiento. Este último aspecto es de mucha importancia en biología y en robótica.
Nuestro sistema, tanto por sus características como por su base experimental es mucho más realista que los considerados habitualmente en la literatura. Los resultados obtenidos podrían aplicarse a otros sistemas biológicos (luciérnagas, grillos, células cardíacas, neuronas, etc.), así como en robótica, donde la comunicación luminosa de largo alcance y la formación de patrones de sincronización podrían ser útiles en el objetivo de realizar tareas específicas.
Acknowledgements
I would like to thank Professors Jean-Louis Deneubourg and Jean-Luc Guisset, my su- pervisors, for their many suggestions and constant support during this research. Despite to have very different approaches, both have the sparkle to trigger serious and rigorous works.
Jean-Luc with his meticulous and highly detailed behavior expressed in his "electronics art".
Jean-Louis with his enviable mind and smartness illuminates with his "firefly flashes" of clarity any doubt that I have. Of course when I write "firefly", I mean that due to be one of the most busy persons that I have ever met, his flashes, even though they are short, they are a delightful privilege. As we say in my country..."de lo bueno, poco"
I am also thankful to Professor Grégoire Nicolis for introducing me to the Nonlinear Science in a formal way and to receive me in his Department.
I am grateful to Professor Christian Van Den Broeck for accepting to be the external examiner of this thesis. I express as well my gratitude to Professors Léon Brenig and Pasquale Nardone for joining my examining committee.
Professor Frank Moss expressed his interest in my work and supplied me with some ideas to work with noisy LCOs; at the same time, I must mention that one of the motivations to this work was the lecture of a very nice paper appeared inScientific Americanunder the Moss’
signature.
I had the pleasure of meeting the Belgian Cooperation staff. They are wonderful people and their support makes research like this possible. The scholarship, which was awarded to me for the period 1999–2004, was crucial to the successful completion of this project.
I should also mention the institutions that have supported me for my participation in sum- mer Schools and Conferences: Belgian, Technical Cooperation, Belgian National Foundation for Scientific Research, North Atlantic Treaty Organization, Centre de Recerca Matemàtica (CRM-Barcelona), Universitat Politècnica de Catalunya (UPC), Centro Internacional de Cien- cias A.C. (México), and the European Project LEURRE.
A very special mention to the Fondation Universitaire David et Alice Van Buuren for the award that allowed me to finish my thesis in the best material conditions.
Of course, I am grateful to my parents and all my family for their love and their constant encouragement.
My little brother Luchifer (for staying all the time with me despite the distance); the people of our labo: the present (Ricardo, Etienne, Jean-Marc, Philippe, Arnaud, Jesús, Alexandre and José) and the past (Stamatios, Julien, Anne, Audrey, Johann and Christian); the people of the Department, in particular Lászlo, Anselmo, Vassilios, Sébastien, Massimiliano and Julie; all my friends during my staying in Brussels: Magaly, Manuel, Scarletz, Eva, Soumia, Walter, etc.; my friends Edwin Vargas and Gonzalo Ordoñez, and of course my Bolivian friends in
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Belgium, deserve my gratitude for their friendship, their good moon and for their helpful behavior.
Finally, I wish to thank Stéphanie for her invaluable help in the "life is hard" moments, for her infinite patience and for ALL she represents in MY LIFE. Yuspagara reinitawa qulilitay, Nayax qhumta suma muñequitawa quli nayra. Nayax jamp’atta suma warawarawa.
Brussels Gonzalo Marcelo Ramírez Ávila
December 17, 2003
Chapter I
Introduction
I.1 Synchronization
Etymologically, the word “synchronous” comes from the Greek words χρòνoς (chronos, meaning time) andσυν(syn, meaning the same, common). Therefore, the textual translation is "sharing the common time" or "occurring at the same time". This term and the related ones "synchronization" and "synchronized", refer to a variety of phenomena that appear to be rather different but nevertheless often obey universal laws and their occurrence is manifested in almost all branches of natural sciences, engineering and social life (Pikovsky et al., 2001).
Synchronization constitutes one of the most studied nonlinear phenomenon which was discovered at the beginning of the second half of the 17th century by the Dutch physicist Chris- tian Huygens who observed that a couple of pendulum clocks which ran with different rates became synchronized when attached to a light-weight beam instead of a wall. Huygens descri- bed in detail and with a remarkable correctness such coupled clocks (Huygens, 1986; Bennett et al., 2002), although, the basic laws of mechanics had not yet been rigorously formulated at that time. Obviously, Huygens could not perform a quantitative analysis of this phenomenon.
Only two centuries after did this phenomenon begin to be investigated systematically by scientists and engineers. Among them, Edward Appleton (Appleton, 1922) and Balthasar Van der Pol (Van der Pol, 1927) who observed synchronization in electrical generators that behave as oscillators. They especially showed that the frequency of a generator can be synchronized by a weak external signal of a slightly different frequency. Van der Pol’s oscillator constitu- ted the paradigmatic model to study synchronization. A further impulse to the development of the theory of synchronization was given by the representatives of the Soviet School. In Pikovsky and Rosenblum (2003) are cited the most important contributions of the Russian school, for instance, the development and generalization of Van der Pol methods performed by Andronov and Vit, the study of n:m external synchronization performed by Mandelshtam and Papaleksi, the analysis of mutual synchronization of two weakly nonlinear oscillators
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2 I.2. BIOLOGICAL BACKGROUND
treated by Mayer and Gaponov, the study of relaxation oscillators by Bremsen, Feinberg and Teodorchik, the development by Stratonovich of a theory of external synchronization of a weakly nonlinear oscillator in the presence of random noise. All these works made important advances on synchronization in theoretical and experimental aspects. Finally, the group of the Institute Mekhanobr headed by Blekhman (Blekhman, 1988) worked with electro-mechanical devices carrying out several experiments that enabled them to construct machines used in industrial applications. The results of the Soviet School became known only in the 1970s by
“occidental” scientists who took up this topic enthusiastically.
Synchronization is understood as an adjustment of rhythms of oscillating objects due to their interaction even if it is weak (Pikovsky et al., 2001). Recently, Strogatz (2003b) wrote a popularization book about synchronization, in which using metaphor and anecdotes he explains the different aspects of this phenomenon. This phenomenon is extremely widespread both in natural and man-made systems. Synchronization has been studied in systems of very different nature such as:
- Electronic devices (Van der Pol, 1927; Chua, 1993; Heagy et al., 1994; Brailove and Linsay, 1996; Pinto et al., 2000; Volkov and Volkov, 2002; Ramírez-Ávila et al., 2003b; Fortuna et al., 2003).
- Chemical systems (Marek and Stuchl, 1975; Neu, 1980; Yoshimoto et al., 1993; Wolf and Heinrich, 1997; Wang et al., 2000; Kiss et al., 2002a,b; Shabunin et al., 2003).
- Biological systems (Goss and Deneubourg, 1988; Tofts et al., 1992; Goldbeter, 1996;
Aizawa, 1998; Bonabeau et al., 1998; Delgado and Sole, 2000; Glass, 2001).
- Ecological systems (Greenfield, 1994; Trainer and McDonald, 1995; Weatherhead, 1997;
Weatherhead and Yezerinac, 1998; Blasius et al., 1999; Blasius and Stone, 2000).
Synchronization is characterized by the fact that various objects seek to achieve order and harmony in their behavior. The latter seems to be a general trend toward self-organization (Blekhman, 1988).
I.2 Biological background
Examples of temporal synchronization and rhythmicity of activities abound in group- living organisms. Different classifications have been suggested. In the case where we have no leader in the group, we distinguish two categories of rhythmicity: the first one involves activities for which the individuals show no intrinsic rhythmicity but are rhythmic as a group.
One example areLeptothoraxants in which colonies show synchronous rhythmic bursts of ac- tivity approximately every 20 minutes (Cole and Trampus, 1999). A second category involves
CHAPTER I. INTRODUCTION 3
activity in which each individual is rhythmic and as a group the activity pattern becomes syn- chronized. This category include the synchronized choruses of crickets (Walker, 1969), cicadas (Sueur, 2002), grasshoppers (Snedden et al., 1998) and katydid (Sismondo, 1990; Greenfield and Roizen, 1993; Snedden and Greenfield, 1998) among others, the bees respiration (Moritz and Southwick, 1992), the synchronized molting of springtails, the synchronization of human female menstrual cycles (McClintock, 1971), the activity in humans which can be performed in the classroom, the synchronous clapping in theaters (Neda et al., 2000b,a; Nikitin et al., 2001;
Maródi et al., 2002; Neda et al., 2003) and the synchronization of flashing among fireflies (Buck and Buck, 1968; Hanson et al., 1971; Buck and Buck, 1976; Buck, 1988; Mirollo and Strogatz, 1990; Ermentrout, 1991; Strogatz and Stewart, 1993; Copeland and Moiseff, 1997; Moiseffand Copeland, 2000). Thousands of fireflies gathered in certain swarm trees begin flashing soon after sunset and synchrony builds up slowly through the night. The synchronization favors the capture of additional individuals in a form of positive feedback (Buck and Buck, 1976;
Buck, 1988; Lloyd, 1973a,b). Recently, it has been reported that synchrony occurs in the North American Photuris frontalis. These results suggest that synchrony is common and pervasive rather than rare and sporadic (Moiseffand Copeland, 2000).
Several studies have been made dealing with physiological aspects of firefly flashing (Bagnoli et al., 1970, 1972), especially related to neural mechanisms (Buonomici and Magni, 1967). Experiments using ablation and local electrical excitation further supported the role of the brain as the central timer (Case and Buck, 1963; Buonomici and Magni, 1967; Bagnoli et al., 1976). Different authors reviewed the evidence suggesting that rhythmic flashing of male fireflies is controlled by a neural timing mechanism in the brain which oscillates at a regular frequency. Each flash is triggered by nerve impulses in the brain that travel down the ventral nerve cord and lanternal nerves to the firefly’s lantern.
A hypothesis to explain synchronization in fireflies is based on the fact that innate indi- vidual rhythmicity with phase-dependent sensitivity to mutual influences can give rise to synchronization (Winfree, 1967). Considering an individual firefly, flashing at its normal free run period (Fig. I.1(a)). The pacemaker resetting model assumes that some property of the oscillatory center in the firefly’s brain –called "excitation"– gradually changes over time. Excita- tion of the brain’s pacemaker rises from its baseline level to a threshold triggering level, which elicits a flash. It is assumed that once a flash is triggered, the excitatory state spontaneously falls back to the baseline level, and restarts the cycle of rising excitation.
The situation in the absence of any external stimuli results in a regular rhythmic flashing.
Now, let us suppose that the firefly receives a photic signal during the charging portion of its cycle (Fig. I.1(b)). A signal of sufficient intensity has the property of abruptly resetting the
4 I.2. BIOLOGICAL BACKGROUND
F. I.1 –Model considering a pacemaker with instantaneous reset to explain the flashing behavior of a firefly. (a) Signal form of an individual firefly flashing at its normal free run period. (b) Instantaneous reset of the signal in its charging stage due to the reception of a light pulse. (c) Same situation as in (b) but the signal receiving the pulse in the discharging part. (From Buck and Buck (1976))
excitation back to its zero baseline. If that happens, then the flash will not occur at the expected time, but will be delayed until the excitation has again built up from zero to the triggering level. The flash will be delayed by an amount of time equal to the time starting at the baseline level to the onset of the light signal. Finally, let us suppose that the firefly receives a signal shortly after the threshold level for triggering the flash has already been reached (Fig. I.1(c)).
The model assumes that the impulse to trigger the next flash has already left the brain, and is travelling down the ventral nerve cord. At this point, any light signal to the brain cannot affect the timing of the flash cycle in progress; it occurs as expected. However, the light signal does affect the oscillatory center in the brain. Instead of taking the normal 200 ms to reset (discharge) back to the zero baseline, the light signal causes an immediate reset. As a result the flash following the next flash occurs earlier by an amount of time equal to the time remaining for the normal reset.
Based upon this single process of photic resetting of the neural oscillator in the brain, self- organizing synchronized flashing emerges. Each firefly acts as an intrinsic oscillator flashing
CHAPTER I. INTRODUCTION 5
at its own characteristic frequency. But in addition, each firefly interacts with its neighbors.
Each firefly is coupled to its neighbors through light perceived from other flashing fireflies;
the sight of a neighbor’s flash shifts the individual’s rhythm.
A key feature of this self-organized system is that the pattern emerges as a result of multiple interactions among the fireflies. Synchronization is not imposed by any influence outside the system, such as a leader, a supervisor or external physical cue. Instead synchronization comes from within, based upon local interactions among fireflies, which take the form of a simple
"rule": A neighbor’s light emission shifts the timing of one’s own light emission. As we have discussed previously, the ability of a local group of synchronized individuals to build up a larger group is a form of positive feedback, a common feature of self-organizing systems. We also find negative feedback in the form of a physiological constraint that keeps the positive feedback from self-perpetuating out of control: A firefly can be stimulated to flash only within a fixed range of frequencies. Although its normal period is about 1000 ms, it can be paced only at a rhythm of between 800 and 1600 ms. Within a certain range the system is refractory, providing negative feedback that brakes and shapes the positive feedback, helping to create a precise temporal pattern.
Although we can now understand how an artificial pacing signal can control a single firefly, we have not shown that this mechanism is sufficient to explain what happens when a group of fireflies, all flashing out of phase, is brought together in synchrony. The collective situation is far more complicated. Many flashes are emitted concurrently, and there are the reciprocal effects of one firefly on another. We must also consider the unequal intensities of the flashes, a function of light intensity falling off inversely with the square of the distance from the source. In nature, during the incoherent initial stages of the process, each firefly sees a barrage of conflicting light emissions. Furthermore, each firefly has its own slightly different intrinsic rhythm and each firefly’s rhythm is somewhat variable on a flash to flash basis.
In this situation a theoretical approach becomes essential. The problem of synchronization among a population of oscillators has received considerable attention, in part because of its intrinsic mathematical interest and in part because of the importance and ubiquity of such processes in biology. Some of the works (Mirollo and Strogatz, 1990; Strogatz et al., 1992; Strogatz and Stewart, 1993), has been inspired by a subject of considerable medical importance, the origin of synchronicity in the heart’s natural pacemaker, a cluster of about 10,000 cells called the sinoatrial node (Peskin, 1975; Jalife, 1984; Michaels et al., 1987). Mirollo and Strogatz (1990) have mathematically analyzed a population of oscillators interacting by means of a mechanism similar to that found inPhotinus pyralis. However, in their model, they make a number of critical simplifying assumptions in the interest of mathematical tractability.
6 I.3. RELAXATION OSCILLATORS
They assume that all the oscillators in the population are identical, that the oscillator is sensitive to incoming light impulses throughout its charging cycle, and that the increase in excitation is concave downward, rather than linear as assumed by Buck and Buck (1976), Buck et al.
(1981) and Buck (1988) (see Fig. I.1). Under these assumptions, a population will become synchronized under almost all initial conditions. The system synchronizes rather slowly at first, but then builds up more rapidly (Mirollo and Strogatz, 1990). The analysis of a population of identical oscillators makes the problem more tractable mathematically.
Ermentrout (1991), and Strogatz and Stewart (1993) discuss a more complicated situation for which the oscillators are not identical. Their main conclusion is as follows: "The behavior of communities of oscillators whose members have differing frequencies depends on the strength of the coupling among them. If their interactions are too weak, the oscillators will be unable to achieve synchrony. The result is incoherence, a cacophony of oscillations." (Strogatz and Stewart, 1993). As the variation in the frequencies of the individual oscillators falls below a critical threshold, a portion of the system suddenly synchronizes. The combined signal of this synchronization cluster stands out above the background noise of random flashes and "captures" additional oscillators, further amplifying its collective signal. This infectious positive feedback results in an epidemic of synchrony.
I.3 Relaxation oscillators
These oscillators constitute one of the most useful models to study different systems which can synchronize. In particular, fireflies, cardiac cells and neurons. The essential feature of these relaxation oscillators is the presence of two time scales, within each cycle there is an integrating process followed by a fast firing process; it is why they are also calledintegrate- and-fireoscillators (Pikovsky et al., 2001). Each process ending at its own threshold. The form of the oscillation is very different from a sinusoidal wave; rather it resembles a sequence of pulses. As it was stated above, the general feature of relaxation oscillators is the slow growth (linear or not) of some quantity and its resetting at a threshold. There is no universal model for relaxation oscillators, so we present some examples hereafter.
I.3.1 Examples of relaxation oscillators
Numerous examples of relaxation oscillators may be found in literature, ranging from the electronic devices generating relaxation oscillations (Van der Pol, 1926; Brailove and Linsay, 1996; Hohl et al., 1997; Ruwisch et al., 1999; Volkov and Volkov, 2002), the relaxation oscillators based model applied to biology (Van der Pol and Van der Mark, 1928; Mirollo and Strogatz, 1990; Eck, 2000), specially in neurons (Coombes and Osbaldestin, 2000; Coombes, 2001b;
CHAPTER I. INTRODUCTION 7
Izhikevich, 2001); the application to the study of rhythmical phenomena (Eck, 2001b,a), or chemistry (Kiss et al., 2002b) or simply the mathematical (Pérez Pascual and Lomnitz-Adler, 1988; Izhikevich, 2000; Izhikevich and Hoppensteadt, 2003) or numerical analysis (Osipov and Sushchik, 1998). A good review of this type of oscillators and its implications can be found in Bottani (1995).
I.3.1.a Van der Pol oscillator
The most popular classical example of a self-oscillating system is the Van de Pol equation described by the equation of motion
¨
x−µ(1−x2) ˙x+x=0, (I.1)
that for largeµbehaves as a relaxation oscillator (Fig. I.2(a)). Eq. (I.1) may be cast into a set of first-order differential equations:
˙
x = µ
y−F(x)
(I.2a)
˙
y = − 1 µ
!
x, (I.2b)
that allows us to observe the following: the xnullcline given by the relation y = F(x), has a cubic form and the ynullcline, given by the expressionx = 0 both illustrated in Fig. I.2(c) as well as the corresponding limit cycle. This system has one fixed point, located at the origin, where the two nullclines cross one another. The motion along the limit cycle trajectory involves two time scales, a fast horizontal movement and slow vertical motion. When y is near thex nullcline, both dx/dtand dy/dtvary gradually, and the movement is slow. When the trajectory departs from the cubic nullcline dy/dtis large, and the horizontal movement is fast.
I.3.1.b Integrate-and-fire oscillators
Since the seminal work of Peskin (1975), integrate-and-fire oscillators were used extensi- vely to describe and model a great variety of phenomena such as synchronization in fireflies (Mirollo and Strogatz, 1990) and several aspects of neuronal systems (Ermentrout and Chow, 2002) among others, the self-sustained firing in populations of neurons (Van Vreeswijk and Abbott, 1993), the synchronization in neural networks (Hansel et al., 1995; Coombes and Bress- loff, 1999; Campbell et al., 1999; Bressloff and Coombes, 2000; Coombes, 2001b; Pakdaman, 2001), the intrinsic modulation in neurons (Coombes and Lord, 1997), the travelling waves (Bressloffand Coombes, 1999; Coombes, 2001a; Osan and Ermentrout, 2002). Integrate-and- fire models were also used to describe firing patterns (Goel and Ermentrout, 2002) and critical phenomena (Corral et al., 1995a; Mousseau, 1996) such as avalanches.
8 I.3. RELAXATION OSCILLATORS
F. I.2 –Van der Pol oscillator acting as a relaxation oscillator usingµ=10. Temporal evolution for the variable (a) x that shows the relaxation regime, and (b) y that shows a rotator regime. (c) Limit cycle trajectory and the accompanying fast and slow time scales.
Integrate-and-fire oscillators are principally used to describe collective behavior. In order to model self-synchronization of the cardiac pacemaker, Peskin considered a network of N integrate-and-fire oscillators, each characterized by a voltage-like state variableVi, subject to the dynamics
dVi
dt =I−ηVi, 0≤Vi ≤1, i=1, . . . ,N. (I.3) When the oscillator i reaches the threshold (Vi = 1), the oscillator "fires" and Vi is reset instantaneouslyto zero (Fig. I.3(a)). The oscillators interact by a simple form of pulse coupling:
when a given oscillator fires, all the other variablesVj, j ,iare increased by an amountβ/N (the quotient by N is made in order to get reasonable behavior in the limitN → ∞). That is,
If Vi(t)=1 =⇒ Vj(t+)=min(1,Vj(t)+β/N), ∀j,i. (I.4) Moreover, the oscillator at the state V = 0 (i.e. just after firing) cannot be affected by the others, so that the state V = 0 is absorbing. This latter property ensures the possibility of perfect synchronization. To illustrate how this model works, we have numerically solved (I.4) for two mutually coupled and for 500 globally coupled oscillators (see Fig. I.3). As we can observe from this figure, synchronization for small as well as for large populations of coupled integrate-and-fire oscillators is easily achieved. Nevertheless, we must point out that
CHAPTER I. INTRODUCTION 9
0 2 4 6 8 10
0 50 100 150 200 250 300 350 400 450 500
Time
Oscillator index
(b)
f i r e 2
f i r e 2 f i r e 1
f i r e 1
synchronization
(a) oscillator 1
oscillator 2
F. I.3 – (a) Synchronization mechanism in two coupled integrate-and-fire oscillators. (b) Points corresponding to firing times in order to represent the dynamics of a population of 500 coupled integrate- and-fire oscillators showing the tendency towards total synchronization. The parameter values used in both cases are I =2.5,η=1andβ=0.25.
the simplifications introduced in this model are very unrealistic from a physical and biological point of view, in particular the fact that the reset and the coupling are instantaneous as well as the mode of coupling that considers the same strength between all the oscillators. But despite this fact, it is found in many models dealing with synchronization.
To conclude this section, we would like to mention additional works of relaxation oscillator based models dealing with synchronization. One of them is the heartbeat: an isolated heart (ag- gregates of spontaneously beating heart cells) preserves the ability to rhythmic contractionin vitro(Glass and Mackey, 1988; Glass, 1997, 2001). The oscillations have the shape of a relaxation oscillator. Another example is the synchronization in the visual cortex and calcium waves and oscillations which can be modelled using coupled relaxation oscillators (Beckerman, 1997).
10 I.4. GOALS AND STRUCTURE OF THE WORK
I.4 Goals and structure of the work
The present work has been motivated by analogy with the phenomenon of synchronization that occurs in biological systems, in particular in groups of fireflies (see I.2). We will study experimentally and theoretically the properties of groups of firefly-type oscillators coupled by light pulses, whose intrinsic flash frequencies are similar, but not necessarily identical.
Many examples of synchrony and phase locking of coupled oscillators have been documented in the literature but very few deal with the case of realistic pulse coupling systems or with non identical oscillators. Our main concern is the synchronization between oscillators that in most cases are quantitatively different. This problem is very important in biological systems in which the oscillators in a group (e.g. fireflies of the same species, neighboring neurons) have intrinsic non-identical frequencies.
In this work, we study the phenomenon of synchronization in a real physical system com- posed of oscillators coupled by light pulses. In order to investigate deeply this phenomenon, we have proposed ourselves to accomplish the following goals:
– Development of a system of automata able to have interactions.
– Study of properties which can appear at the group level.
– Construction of a mathematical model reproducing the experimental results.
– Study of the oscillators’s behavior under different configurations.
– Investigation of the properties and the practical involvements.
The results are detailed in the following chapters. In chapter II we present the device we used and the principal experimental measurements. A model based on the physics of the used oscillators and in the experimental measurements as well as a characterization of the oscillator by phase response curves is detailed in chapter III. Analytical resolution of the model for the case of two identical oscillators is performed in chapter IV as well as other important features of two interacting oscillators such as the synchronization time and the Arnold tongues. The influence of uniform and Gaussian noise is shown in chapter V. The analysis of populations of locally coupled oscillators is developed in chapter VI. Finally, before to give the conclusions and perspectives of this research, in chapter VII we treat the case of static globally coupled oscillators and we begin the study of the complex phenomenon of mobile oscillators and the relationship between synchronization and motion.
Chapter II
Experimental setup and measurements
As it has been stated in the introductory chapter, there are numerous examples of theoreti- cal systems of coupled oscillators able to induce structured behaviors between the interacting oscillators. We can remark some of them that have become paradigms in synchronization of coupled oscillators (Winfree, 1967; Kuramoto and Nishikawa, 1987; Mirollo and Strogatz, 1990; Strogatz, 2001). In general these systems are composed of a large number of oscillating elements interacting in a complex way (see §I.3.1.b). Often, their study introduces simplifica- tions that are far from the experimental reality. Nevertheless, there are numerous experimental works dealing with synchronization in man-made devices such as lasers (Kapitaniak et al., 1994; Roy and Thornburg, 1994; Sugawara et al., 1994; Hohl et al., 1997; Uchida et al., 2003), electronic circuits (Chua, 1993; Murali et al., 1995; Brailove and Linsay, 1996; Delgado-Restituto et al., 1997; Tang and Heckenberg, 1997; Ruwisch et al., 1999; Pinto et al., 2000; Reddy et al., 2000; Fortuna et al., 2003; Roy et al., 2003; Uchida et al., 2003), chemical systems (Marek and Stuchl, 1975; Wang et al., 2001), and so on. Most of these experimental works consider simple interaction modes such as master–slave configurations. The system analyzed in this thesis consists of a small number of electronic oscillators that are easy to make and the oscillators can be coupled in a way easy to control. Despite this simplicity, our system has sufficient complexity to produce effects similar to those studied usually. On the other side, this great simplicity allows us to bring to the fore the dynamics of the system at the level of each of its components. These facts will permit us to construct a model closer to the reality.
Garver and Moss built a simple device that they called "electronic fireflies" (Garver and Moss, 1993) because these circuits interact in such a way that mimic fireflies and they may exhibit synchronous behavior as in the fireflies’ case. They reported collective behavior of these electronic fireflies in a smaller scale and in a qualitative way.
We constructed an “open” version of this electronic firefly, whose free-run duty cycle can be modified and adjusted manually on the spot, and on which quantitative measurements of periods and phase differences may be performed with the required precision. According to
11
12 II.1. PRESENTATION OF AN LCO
the properties of these type of relaxation oscillators, we call them Light Controlled Oscillators (LCOs).
II.1 Presentation of an LCO
Each LCO consists of an LM555 chip wired to function in its astable oscillating mode (Fig. II.1(a)) (National Semiconductor Corporation, 1982), the alternations of which are deter- mined by a dual RC circuit in parallel with four photo-sensors (Garver and Moss, 1993) that allow the LCO to interact with others by means of light pulses (Fig. II.1(b)). A square base
(a) (b)
F. II.1 –(a) Block diagram of an LCO with the LM555 in its astable functioning mode. (b) Simplified diagram of the LCO and schematic view of the coupling between LCOs.
(11 cm X11 cm) gives the over-all horizontal dimensions of each LCO in the global pattern (Fig. II.2(a)). Each base may sustain several printed circuits giving the possibility of vertical extension, but keeping the same over-all horizontal dimensions. For the time being, our LCOs have two levels (Fig. II.2(a)). The lower part consists of a 9 volt battery and its clamping sys- tem. The oscillator’s printed circuit with the variable resistors allowing the adjustment of the period’s two time intervals, makes the upper part of an LCO module. The circuit is square-like too but its size is smaller than the basis. Four printed circuits are fixed vertically on the sides of the upper part and each of one bears an infrared (IR) light emitting diode (LED) and a photo-sensor. We made provision for masking the sensors that allow the LCOs to oscillate "in the dark". In the aim of public presentations, the upper part bears a fifth LED flashing visible light in synchrony with the IR ones, just to produce a “firefly effect”.
CHAPTER II. EXPERIMENTAL SETUP AND MEASUREMENTS 13
The RC timing components of the LM555 consist of two resistors and a single capacitor (Fig. II.1). LetRλ,RγandCbe the values of those components responsible for the LCO’s timing with masked photo-sensors (timing “in the dark”).
In our LCOs, the resistorsRλ andRγare variable and the intervals of values that they can take are
Rλ = [68,118] kΩ Rγ = [1.2,2.2] kΩ.
We made nine LCOs (Fig. II.2(b)). With this number it is possible to obtain several different patterns when they are coupled by their IR beams. For example, they had much success when, disposed on a table, they went to synchronize like exotic fireflies which they are aimed to mimic
(a) (b)
F. II.2 –(a) View of a single LCO. (b) Group of nine LCOs.
II.2 LCO functioning
As such the LCO behaves as a relaxation oscillator whose period is related to the charge and discharge of the two external RC circuits with resistances Rλ +Rγ and Rγ. The astable mode in which the LM555 timer works is simple to describe. The LM555 combines three functions. Firstly, it measures the voltage across the capacitorC. Secondly, it may or may not establish a short-circuit across the RC components. Finally, it produces an on-offsignal at its output. When the short-circuit is established, the capacitor discharges through Rγ. Without the short-circuit, the capacitor charges throughRλ+Rγ. It switches from the discharging state
