Intermediate Stable Phase Locked States In Oscillator Networks

The study of nonlinear oscillations is important in a variety of physical and biological contexts (especially in neuroscience). Synchronization of oscillators has been a problem of interest in recent years. In networks of nearest neighbor coupled oscillators it is possible to obtain synchrony between oscillators, but also a variety of constant phase shifts between 0 and pi. We coin these phase shifts intermediate stable phase-locked states. In neuroscience, both individual neurons and populations of neurons can behave as complex nonlinear oscillators. Intermediate stable phase-locked states are shown to be obtainable between individual oscillators and populations of identical oscillators.These intermediate stable phase-locked states may be useful in the construction of central pattern generators: autonomous neural cicuits responsible for motor behavior. In large chains and two-dimenional arrays of oscillators, intermediate stable phase-locked states provide a mechanism to produce waves and patterns that cannot be obtained in traditional network models. A particular pattern of interest is known as an anti-wave. This pattern corresponds to the collision of two waves from opposite ends of an oscillator chain. This wave may be relevant in the spinal central pattern generators of various fish. Anti-wave solutions in both conductance based neuron models and phase oscillator models are analyzed. It is shown that such solutions arise in phase oscillator models in which the nonlinearity (interaction function) contains both higher order odd and even Fourier modes. These modes are prominent in pairs of synchronous oscillators which lose stability in a supercritical pitchfork bifurcation

Mechanical Control of Sensory Hair-Bundle Function

Hair bundles detect sound in the auditory system, head position and rotation in the vestibular system, and fluid flow in the lateral-­‐‑line system. To do so, bundles respond to periodic, static, and hydrodynamic forces contingent upon the receptor organs in which they are situated. As the mechanosensory function of a hair bundle varies, so too do the mechanical properties of the bundle and its microenvironment. Hair bundles range in height from 1 μμm to 100 μμm and in stiffness from 100 μμN·∙m-­‐‑1 to 10,000 μμN·∙m-­‐‑1. They are composed of actin-­‐‑filled, hypertrophic microvilli—stereocilia—that number from fewer than 20 through more than 300 per bundle. In addition, bundles may or may not possess one true cilium, the kinocilium. Hair bundles differ in shape across organs and organisms: they may be isodiametric, fan-­‐‑shaped, or V-­‐‑shaped. Depending on the organ in which they occur, bundles may be free-­‐‑standing or they may be coupled to a tectorial membrane, otolithic membrane, cupula, or sallet. Because all hair bundles are comprised of similar molecular components, their distinct mechanosensory functions may instead be regulated by their mechanical loads. Dynamical-­‐‑systems analysis provides mathematical predictions of hair-­‐‑bundle behavior. One such model captures the effects of mechanical loading on bundle function in a state diagram. A mechanical-­‐‑load clamp permits exploration of this state diagram by robustly controlling the loads—constant force, load stiffness, virtual drag, and virtual mass—imposed on a hair bundle. Upon changes in these mechanical parameters, the bundle’s response characteristics alter. Subjected to particular control parameters, a bundle may oscillate spontaneously or remain quiescent. It may respond nonlinearly to periodic stimuli with high sensitivity, sharp frequency tuning, and easy entrainment; or it may respond linearly with low sensitivity, broad tuning, and reluctant entrainment. The bundle’s response to a force pulse may resemble that of an edge-­‐‑detection system or a low-­‐‑pass filter. Finally, a bundle from an amphibian vestibular organ can operate in a manner qualitatively similar to that from a mammalian auditory organ, implying an essential similarity between hair bundles. The bifurcation near which a bundle’s operating point resides controls its function: the state diagram provides a functional map of mechanosensory modalities. Auditory function is best tuned near a supercritical Hopf bifurcation, whereas vestibular function is captured by a subcritical Hopf bifurcation and a cusp bifurcation. Within the proposed region vestibular responsiveness, a hair bundle exhibits mechanical excitability analogous to the electrical excitability of neurons. This behavior implies for the first time a direct relationship between the mechanical behaviors of sensory organelles and the electrical behaviors of afferent neurons. Man-­‐‑made detectors function in limited capacities, each designed for a unique purpose. A single hair bundle, on the other hand, evolved to serve multiple purposes with the requirement of only two functional traits: adaptation and nonlinear channel gating. The remarkable conservation of these capabilities thus provides unique insight into the evolution of sensory systems

Principles and theory of protein-based pattern formation

Biological systems perform functions by the orchestrated interplay of many small components without a "conductor." Such self-organization pervades life on many scales, from the subcellular level to populations of many organisms and whole ecosystems. On the intracellular level, protein-based pattern formation coordinates and instructs functions like cell division, differentiation and motility. A key feature of protein-based pattern formation is that the total numbers of the involved proteins remain constant on the timescale of pattern formation. The overarching theme of this thesis is the profound impact of this mass-conservation property on pattern formation and how one can harness mass conservation to understand the underlying physical principles. The central insight is that changes in local densities shift local reactive equilibria, and thus induce concentration gradients which, in turn, drive diffusive transport of mass. For two-component systems, this dynamic interplay can be captured by simple geometric objects in the (low-dimensional) phase space of chemical concentrations. On this phase-space level, physical insight can be gained from geometric criteria and graphical constructions. Moreover, we introduce the notion of regional (in)stabilities, which allows one to characterize the dynamics in the highly nonlinear regime reveals an inherent connection between Turing instability and stimulus-induced pattern formation. The insights gained for conceptual two-component systems can be generalized to systems with more components and several conserved masses. In the minimal setting of two diffusively coupled "reactors," the full dynamics can be embedded in the phase-space of redistributed masses where the phase space flow is organized by surfaces of local reactive equilibria. Building on the phase-space analysis for two component systems, we develop a new approach to the important open problem of wavelength selection in the highly nonlinear regime. We show that two-component reaction–diffusion systems always exhibit uninterrupted coarsening (the continual growth of the characteristic length scale) of patterns if they are strictly mass conserving. Selection of a finite wavelength emerges due to weakly broken mass-conservation, or coupling to additional components, which counteract and stop the competition instability that drives coarsening. For complex dynamical phenomena like wave patterns and the transition to spatiotemporal chaos, an analysis in terms of local equilibria and their stability properties provides a powerful tool to interpret data from numerical simulations and experiments, and to reveal the underlying physical mechanisms. In collaborations with different experimental labs, we studied the Min system of Escherichia coli. A central insight from these investigations is that bulk-surface coupling imparts a strong dependence of pattern formation on the geometry of the spatial confinement, which explains the qualitatively different dynamics observed inside cells compared to in vitro reconstitutions. By theoretically studying the polarization machinery in budding yeast and testing predictions in collaboration with experimentalists, we found that this functional module implements several redundant polarization mechanisms that depend on different subsets of proteins. Taken together, our work reveals unifying principles underlying biological self-organization and elucidates how microscopic interaction rules and physical constraints collectively lead to specific biological functions.Biologische Systeme führen Funktionen durch das orchestrierte Zusammenspiel vieler kleiner Komponenten ohne einen "Dirigenten" aus. Solche Selbstorganisation durchdringt das Leben auf vielen Skalen, von der subzellulären Ebene bis zu Populationen vieler Organismen und ganzen Ökosystemen. Auf der intrazellulären Ebene koordiniert und instruieren proteinbasierte Muster Funktionen wie Zellteilung, Differenzierung und Motilität. Ein wesentliches Merkmal der proteinbasierten Musterbildung ist, dass die Gesamtzahl der beteiligten Proteine auf der Zeitskala der Musterbildung konstant bleibt. Das übergreifende Thema dieser Arbeit ist es, den tiefgreifenden Einfluss dieser Massenerhaltung auf die Musterbildung zu untersuchen und Methoden zu entwickeln, die Massenerhaltung nutzen, um die zugrunde liegenden physikalischen Prinzipien von proteinbasierter Musterbildung zu verstehen. Die zentrale Erkenntnis ist, dass Änderungen der lokalen Dichten lokale reaktive Gleichgewichte verschieben und somit Konzentrationsgradienten induzieren, die wiederum den diffusiven Transport von Masse antreiben. Für Zweikomponentensysteme kann dieses dynamische Wechselspiel durch einfache geometrische Objekte im (niedrigdimensionalen) Phasenraum der chemischen Konzentrationen erfasst werden. Auf dieser Phasenraumebene können physikalische Erkenntnisse durch geometrische Kriterien und grafische Konstruktionen gewonnen werden. Darüber hinaus führen wir den Begriff der regionalen (In-)stabilität ein, der es erlaubt, die Dynamik im hochgradig nichtlinearen Regime zu charakterisieren und einen inhärenten Zusammenhang zwischen Turing-Instabilität und stimulusinduzierter Musterbildung aufzuzeigen. Die für konzeptionelle Zweikomponentensysteme gewonnenen Erkenntnisse können auf Systeme mit mehr Komponenten und mehreren erhaltenen Massen verallgemeinert werden. In der minimalen Fassung von zwei diffusiv gekoppelten "Reaktoren" kann die gesamte Dynamik in den Phasenraum umverteilter Massen eingebettet werden, wobei der Phasenraumfluss durch Flächen lokaler reaktiver Gleichgewichte organisiert wird. Aufbauend auf der Phasenraumanalyse für Zweikomponentensysteme entwickeln wir einen neuen Ansatz für die wichtige offene Fragestellung der Wellenängenselektion im hochgradig nichtlinearen Regime. Wir zeigen, dass "coarsening" (das stetige wachsen der charakteristischen Längenskala) von Mustern in Zweikomponentensystemen nie stoppt, wenn sie exakt massenerhaltend sind. Die Selektion einer endlichen Wellenlänge entsteht durch schwach gebrochene Massenerhaltung oder durch Kopplung an zusätzliche Komponenten. Diese Prozesse wirken der Masseumverteilung, die coarsening treibt, entgegen und stoppen so das coarsening. Bei komplexen dynamischen Phänomenen wie Wellenmustern und dem Übergang zu raumzeitlichen Chaos bietet eine Analyse in Bezug auf lokale Gleichgewichte und deren Stabilitätseigenschaften ein leistungsstarkes Werkzeug, um Daten aus numerischen Simulationen und Experimenten zu interpretieren und die zugrunde liegenden physikalischen Mechanismen aufzudecken. In Zusammenarbeit mit verschiedenen experimentellen Labors haben wir das Min-System von Escherichia coli untersucht. Eine zentrale Erkenntnis aus diesen Untersuchungen ist, dass die Kopplung zwischen Volumen und Oberfläche zu einer starken Abhängigkeit der Musterbildung von der räumlichen Geometrie führt. Das erklärt die qualitativ unterschiedliche Dynamik, die in Zellen im Vergleich zu in vitro Rekonstitutionen beobachtet wird. Durch die theoretische Untersuchung der Polarisationsmaschinerie in Hefezellen, kombiniert mit experimentellen Tests theoretischer Vorhersagen, haben wir herausgefunden, dass dieses Funktionsmodul mehrere redundante Polarisationsmechanismen implementiert, die von verschiedenen Untergruppen von Proteinen abhängen. Zusammengenommen beleuchtet unsere Arbeit die vereinheitlichenden Prinzipien, die der intrazellulären Selbstorganisation zugrunde liegen, und zeigt, wie mikroskopische Interaktionsregeln und physikalische Bedingungen gemeinsam zu spezifischen biologischen Funktionen führen

Dynamics of delay-coupled semiconductor laser systems

Nonlinear laser dynamics has received considerable attention because of possible applications, but also fundamental physical and mathematical aspects are of great interest. This thesis is concerned with the dynamical behavior of semiconductor lasers subject to external delayed perturbations. In particular the time delay in the coupling to external elements is of importance, because it substantially complicates the dynamical behavior. This time delay arises from finite signal propagation times and, hence, is large compared to the laser internal time scales so that it cannot be neglected. Specifically, the thesis investigates two different delay-coupled semiconductor laser systems: (I) a semiconductor laser subject to delayed filtered optical feedback, where a part of the laser emission is filtered by a Fabry-Perot filter and then feed back into the laser, and (II) two semiconductor lasers that are mutually delay-coupled via their optical fields. With concepts and tools from dynamical systems theory a comprehensive study of the underlying bifurcation structure of two systems is presented. Knowledge of this underlying structure is the key to understanding complicated laser dynamics. The results from the bifurcation analysis are interpreted in terms of the dynamics of the real laser system and compared with experiments.Krauskopf, B. [Promotor]Lenstra, D. [Promotor

Controlling turbulence and pattern formation in chemical reactions

Räumlich ausgedehnte Systeme fern des thermodynamischen Gleichgewichts zeichnen sich durch die Fähigkeit aus, spontan raumzeitliche Strukturen und Turbulenz auszubilden. Die vorliegende Arbeit beschäftigt sich theoretisch und experimentell mit der Steuerung und Kontrolle derartiger Phänomene. Als Beispiel wird die katalytische Oxidationsreaktion von Kohlenmonoxid auf einer Platin-Einkristalloberfläche untersucht. Um Turbulenz zu unterdrücken sowie um neuartige Muster in dieses System zu induzieren werden zwei verschiedene Steuerungsverfahren, globale verzögerte Rückkopplung und periodische Forcierung, eingesetzt. Die Effekte einer künstlich implementierten globalen Rückkopplungsschleife werden zunächst in einem mathematischen Reaktions-Diffusions-Modell der CO-Oxidation auf Pt(110) mit Hilfe numerischer Simulationen untersucht. Durch Variation eines globalen Kontrollparameters in Abhängigkeit einer räumlich gemittelten Systemgröße lässt sich chemische Turbulenz in dem Modell unterdrücken und ein homogen oszillierender Zustand stabilisieren. Weiterhin kann eine Vielzahl komplexer raumzeitlicher Strukturen, beispielsweise "phase flips", asynchrone Oszillationen, intermittente Turbulenz in Form chaotischer Kaskaden von Blasen und Ringstrukturen, zelluläre Strukturen und verschiedene Arten von Domänenmustern induziert werden. Die simulierten raumzeitlichen Muster werden mit Hilfe einer zuvor entwickelten Transformation zu Phasen- und Amplitudenvariablen charakterisiert und analysiert. Es zeigt sich, daß die erhaltenen Strukturen große Ähnlichkeit mit dem Verhalten eines generischen Modells, der komplexen Ginzburg-Landau-Gleichung mit globaler Kopplung, aufweisen. Eine globale verzögerte Rückkopplung kann in Experimenten mit der CO-Oxidation auf Pt(110) durch eine externe, zustandsabhängige Variation des CO-Partialdrucks in der Reaktionskammer realisiert werden. Die sich auf der Platinoberfläche ausbildenden Bedeckungsmuster werden dabei mit Hilfe von Photoemissions-Elektronenmikroskopie sichtbar gemacht. In solchen Experimenten kann chemische Spiralwellenturbulenz erstmals unterdrückt und ein Großteil der vorhergesagten Muster - unter anderem intermittente Turbulenz, Domänenmuster und zelluläre Strukturen - tatsächlich nachgewiesen werden. Die experimentell beobachteten Muster werden ebenfalls durch eine Phasen- und Amplitudendarstellung charakterisiert. In weiteren Experimenten wird die Wirkung periodischer Partialdruckmodulationen auf chemische Turbulenz untersucht. Auch mittels dieser Methode läßt sich Spiralwellenturbulenz unterdrücken und eine Vielfalt komplexer Muster induzieren. Als resonante Strukturen sind irreguläre Streifenmuster in subharmonischer Resonanz sowie Domänenmuster mit koexistenten Resonanzen zu nennen. Zudem treten auch nichtresonante Muster in Form intermittenter Turbulenz und ungeordneter zellulärer Strukturen auf. Die Resultate dieser Arbeit zeigen somit, daß sich mit Hilfe globaler Rückkopplung und periodischer Forcierung Turbulenz und Strukturbildung in der betrachteten Oberflächenreaktion wirkungsvoll kontrollieren und manipulieren lassen. Ähnliche Phänomene können auch in anderen Reaktions-Diffusions-Systemen erwartet werden.Spontaneous pattern formation and spatiotemporal chaos (turbulence) are common features of spatially extended nonlinear systems maintained far from equilibrium. The aim of this work is to control and engineer such phenomena. As an example, the catalytic oxidation of carbon monoxide on a platinum (110) single crystal surface is considered. In order to control turbulence and to manipulate pattern formation in this reaction, two different control methods, global delayed feedback and periodic forcing, are employed. The effects of a global delayed feedback on the self-organized behavior of the system are first studied numerically in a reaction-diffusion model of CO oxidation on Pt(110). By applying a global control force generated by the spatially averaged state of one of the system variables, turbulence can be suppressed and uniform oscillations can be stabilized. Moreover, global delayed feedback can be used as a tool to produce a variety of complex spatiotemporal patterns, including phase flips, asynchronous oscillations, intermittent turbulence represented by irregular cascades of ring-shaped objects on a uniformly oscillating background, cellular structures, and different types of cluster patterns. The simulated structures are analyzed using a newly developed transformation to phase and amplitude variables designed for non-harmonic oscillations. The obtained patterns resemble the structures exhibited by a general model, the complex Ginzburg-Landau equation with global feedback. The simulated phenomena of pattern formation are then tested in laboratory experiments with CO oxidation on Pt(110). Global delayed feedback is introduced into the system via a controlled state-dependent variation of the CO partial pressure in the reaction chamber. The spatiotemporal patterns developing on the catalytic surface are imaged by means of photoemission electron microscopy. In such experiments, it is shown that chemical turbulence can be suppressed and a large part of the predicted patterns, including intermittent turbulence, clusters, and cellular structures, can be indeed observed. The experimentally obtained patterns are also transformed into the corresponding spatial distributions of oscillation phase and amplitude. In a further set of experimental investigations, the effects of periodic external forcing on chemical turbulence in CO oxidation on Pt(110) are studied. Using this method, turbulence can be also suppressed and several complex patterns can be induced. The observed frequency locked structures are represented by irregular stripes in subharmonic resonance with the forcing and cluster patterns with coexistent resonances. In addition, non-resonant patterns such as intermittent turbulence and disordered cellular structures are found. Thus, the results of this work demonstrate that by means of global delayed feedback and periodic forcing, turbulence and pattern formation can be effectively controlled and manipulated in the considered surface reaction. Similar phenomena are expected to arise also in other reaction-diffusion systems of various origins

Modellierung und Analyse des Thalamokortischen Systems

Physiological evidence localizes the thalamocortical system as the functional unit being responsible for the perception of sensory input. In this thesis the dynamical processes in the thalamus during sleep are reduced to their bare bones. For this purpose the dynamical behavior of conductance based neuron models, which describe biophysical details with high accuracy, is investigated and reduced models of this behavior are derived. The simplified models derived in this thesis allow an explanation of how sensory perception is strongly decreased during sleep within the framework of nonlinear dynamics. A minimal model for such a mechanism is derived, coarse graining out details but preserving most salient dynamical features. If several of these models are coupled in a network the experimental observed influence of cortical slow-wave oscillations on thalamic spindle oscillations during deep sleep can be reproduced. In particular the influence of cortical oscillations on the synchrony in a thalamic network is studied and the underlying control mechanism is uncovered, leading to a control method which might be applicable for several types of oscillations in the central nervous system.Physiologisch betrachtet ist das thalamokortische System für die Verarbeitung und Wahrnehmung von sensorischen Reizen zuständig. In dieser Arbeit werden die dynamischen Vorgänge im Thalamus während des Schlafes auf ihre grundlegenden Eigenschaften reduziert. Dazu wird das dynamische Verhalten von komplexen Neuronenmodellen untersucht, die biophysikalische Details mit hoher Genauigkeit wiedergeben und vereinfachte Modelle dieses Verhaltens eingeführt. Diese vereinfachten Modelle erlauben es, mit Hilfe der nichtlinearen Dynamik den Rückgang der sensorischen Wahrnehmung im Schlaf zu erklären. Dazu wird ein minimales Modell für den zugrunde liegenden Mechanismus abgeleitet, in dem Details vernachlässigt werden, ohne dass jedoch die wichtigsten dynamischen Eigenschaften verloren gehen. Koppelt man viele dieser Modelle in einem Netzwerk, so lässt sich der experimentell beobachtete Einfluss kortikaler Oszillationen auf thalamische Oszillationen reproduzieren. Ein besonderes Augenmerk liegt dabei auf der Synchronisation der thalamischen Oszillationen und dem zugrunde liegenden Mechanismus, welcher möglicherweise auch in anderen neuronalen Systemen anwendbar ist

Electrostatic MEMS Bifurcation Sensors

We report experimental evidence of a new instability in electrostatic sensors, dubbed quasi-static pull-in, in two types of micro-sensors operating in ambient air. We find that the underlying mechanism and features of this instability are distinct from those characterizing hitherto known static and dynamic pull-in instabilities. Specifically, the mechanism instigating quasi-static pull-in is a global Shilnikov homoclinic bifurcation where a slow-varying waveform drives the sensor periodically through a saddle-node bifurcation. Based on these findings, we propose a new taxonomy of pull-in instabilities in electrostatic sensors. Experimental evidence of nonlinear chaotic behaviors were observed in an electrostatic MEMS sensor. Period doubling bifurcation (P-2), period three (P-3), and period six (P-6) were observed. A new class of intermittency subsequent to homoclinic bifurcation in addition to the traditional intermittencies of type-I and type-II were demonstrated. Quasiperiodicity and homoclinic tangles leading to chaos were also reported. All of these nonlinear phenomena instigate either banded chaos or full chaos and both are observed in this work. Based on our knowledge, this is the first observation such chaotic behaviors in electrostatic MEMS sensors. All of the experimental observations have been measured optically via a laser Doppler-vibrometer (LDV) in ambient pressure. Also, a new class of intermittencies was found in the oscillations of an electrostatic sensor. These intermittencies involve a dynamic system spending irregular time intervals in the vicinity of the ghost of an orbit before undergoing bursts that are arrested by landing on a larger attractor. Re-injection into the vicinity of the ghost orbit is noise induced. As a control parameter is increased, switching intermittency of type-I leads to a stable periodic orbit, whereas switching intermittency of type-II leads to a chaotic attractor. These significant findings in nonlinear dynamic were used to develop novel MEMS sensors. An electrostatic MEMS gas sensor is demonstrated. It employs a dynamic-bifurcation detection technique. In contrast to traditional gas or chemical sensors that measure (quantify) the concentration of an analyte in analog mode, this class of sensors does not seek to quantify the concentration. Rather, it detects the analyte's concentration in binary mode, reporting ON-state (1) for concentrations above a preset threshold and OFF-state (0) for concentrations below the threshold. The sensing mechanism exploits the qualitative difference between the sensor state before and after the dynamic pull-in bifurcation. Experimental demonstration was carried out using a laser-Doppler vibrometer to measure the sensor response before and after detection. The sensor was able to detect ethanol vapor concentrations as 100\,ppb in dry nitrogen. A closed-form expression for the sensitivity of dynamic bifurcation sensors was derived. It captured the dependence of sensitivity on the sensor dimensions, material properties, and electrostatic field. An analog dynamic bifurcation mass sensor is developed to demonstrate a sensing mechanism that exploits a quantitative change in the sensor state before and after depositing added mass. A polymeric material was deposited on the top surface of the sensor plate to represent added mass. A variation in the frequency and current amplitude were utilized to demarcate the added mass optically and electrically. A chemical sensor was also developed to detect mercury in deionized-water in a fashion of analog detection. A polymeric sensing material that has high selectivity to mercury was utilized to captured mercury molecules in water. The sensor was submerged completely in water with a pre-defined flow-rate. The sensor was excited electrostatically. A variation in the frequency response due to added mass was measured electrically using a lock-in amplifier. A frequency-shift was observed while releasing the mercury to the water

Modeling phase synchronization of interacting neuronal populations:from phase reductions to collective behavior of oscillatory neural networks

Synchronous, coherent interaction is key for the functioning of our brain. The coordinated interplay between neurons and neural circuits allows to perceive, process and transmit information in the brain. As such, synchronization phenomena occur across all scales. The coordination of oscillatory activity between cortical regions is hypothesized to underlie the concept of phase synchronization. Accordingly, phase models have found their way into neuroscience. The concepts of neural synchrony and oscillations are introduced in Chapter 1 and linked to phase synchronization phenomena in oscillatory neural networks. Chapter 2 provides the necessary mathematical theory upon which a sound phase description builds. I outline phase reduction techniques to distill the phase dynamics from complex oscillatory networks. In Chapter 3 I apply them to networks of weakly coupled Brusselators and of Wilson-Cowan neural masses. Numerical and analytical approaches are compared against each other and their sensitivity to parameter regions and nonlinear coupling schemes is analysed. In Chapters 4 and 5 I investigate synchronization phenomena of complex phase oscillator networks. First, I study the effects of network-network interactions on the macroscopic dynamics when coupling two symmetric populations of phase oscillators. This setup is compared against a single network of oscillators whose frequencies are distributed according to a symmetric bimodal Lorentzian. Subsequently, I extend the applicability of the Ott-Antonsen ansatz to parameterdependent oscillatory systems. This allows for capturing the collective dynamics of coupled oscillators when additional parameters influence the individual dynamics. Chapter 6 draws the line to experimental data. The phase time series of resting state MEG data display large-scale brain activity at the edge of criticality. After reducing neurophysiological phase models from the underlying dynamics of Wilson-Cowan and Freeman neural masses, they are analyzed with respect to two complementary notions of critical dynamics. A general discussion and an outlook of future work are provided in the final Chapter 7

Nonlinear and Stochastic Analysis of Miniature Optoelectronic Oscillators based on Whispering-Gallery Mode Modulators

Optoelectronic oscillators are nonlinear closed-loop systems that convert optical energy into electrical energy. We investigate the nonlinear dynamics of miniature optoelectronic oscillators (OEOs) based on whispering-gallery mode resonators. In these systems, the whispering-gallery mode resonator features a quadratic nonlinearity and operates as an electrooptical modulator, thereby eliminating the need for an integrated Mach-Zehnder modulator. The narrow optical resonances eliminate as well the need for both an optical fiber delay line and an electric bandpass filter in the optoelectronic feedback loop. The architecture of miniature OEOs therefore appears as significantly simpler than the one of their traditional counterparts, and permits to achieve competitive metrics in terms of size, weight, and power (SWAP). Our theoretical approach is based on the closed-loop coupling between the optical intracavity modes and the microwave signal generated via the photodetection of the output electrooptical comb. In the first part of our investigation, we use a slowly-varying envelope approach to propose a time-domain model to analyze the dynamical behavior of miniature OEOs. This model takes into account the interactions among the intracavity modes, as well as the coupled interactions with the radiofrequency (RF) microstrip. The stability analysis allows us to determine analytically and optimize the critical value of the feedback gain needed to trigger self-sustained oscillations. It also allows us to understand how key parameters of the system such as cavity detuning or coupling efficiency influence the onset of the radiofrequency oscillation. Furthermore, we determine the threshold laser power needed to trigger oscillations in amplifierless miniature OEOs based on WGM modulators. This latter architecture, while also improving on the size, weight, performance and cost (SWAP-C) constraints, is intended to reduce noise in the system. In the second part of our investigation, we use a Langevin approach to perform a stochastic analysis of our miniature OEO. We propose a stochastic mathematical model to describe the system dynamics and analyze the stochastic behavior below threshold. We also propose a normal form approach for the noise power density and the phase noise spectrum. Our study is complemented by time-domain simulations for the microwave and optical signals, which are in excellent agreement with the analytical predictions. In the third part of our study, we discuss our preliminary results in the analysis of the effects of dispersion in a microcomb oscillator with optical feedback. For this purpose, we propose a closed-loop miniature optical oscillator. The output signal is optically amplified before being coupled back into the cavity using a prism coupling. Using a Lugiato-Lefever approach, we propose a spatiotemporal nonlinear partial differential equation to describe the dynamics of the total intracavity field. We perform temporal and spatial analysis and derive the bifurcation maps in anomalous and normal dispersion regimes