111 research outputs found

    Fundamentals and applications of spatial dissipative solitons in photonic devices : [Chapter 6]

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    We review the properties of optical spatial dissipative solitons (SDS). These are stable, self‐localized optical excitations sitting on a uniform, or quasi‐uniform, background in a dissipative environment like a nonlinear optical cavity. Indeed, in optics they are often termed “cavity solitons.” We discuss their dynamics and interactions in both ideal and imperfect systems, making comparison with experiments. SDS in lasers offer important advantages for applications. We review candidate schemes and the tremendous recent progress in semiconductor‐based cavity soliton lasers. We examine SDS in periodic structures, and we show how SDS can be quantitatively related to the locking of fronts. We conclude with an assessment of potential applications of SDS in photonics, arguing that best use of their particular features is made by exploiting their mobility, for example in all‐optical delay lines

    Processing spatial and temporal information in cells using protein-based pattern formation

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    Fourth SIAM Conference on Applications of Dynamical Systems

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    Principles and theory of protein-based pattern formation

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    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 fĂŒhren Funktionen durch das orchestrierte Zusammenspiel vieler kleiner Komponenten ohne einen "Dirigenten" aus. Solche Selbstorganisation durchdringt das Leben auf vielen Skalen, von der subzellulĂ€ren Ebene bis zu Populationen vieler Organismen und ganzen Ökosystemen. Auf der intrazellulĂ€ren Ebene koordiniert und instruieren proteinbasierte Muster Funktionen wie Zellteilung, Differenzierung und MotilitĂ€t. Ein wesentliches Merkmal der proteinbasierten Musterbildung ist, dass die Gesamtzahl der beteiligten Proteine auf der Zeitskala der Musterbildung konstant bleibt. Das ĂŒbergreifende 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. FĂŒr 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. DarĂŒber hinaus fĂŒhren wir den Begriff der regionalen (In-)stabilitĂ€t ein, der es erlaubt, die Dynamik im hochgradig nichtlinearen Regime zu charakterisieren und einen inhĂ€renten Zusammenhang zwischen Turing-InstabilitĂ€t und stimulusinduzierter Musterbildung aufzuzeigen. Die fĂŒr 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 FlĂ€chen lokaler reaktiver Gleichgewichte organisiert wird. Aufbauend auf der Phasenraumanalyse fĂŒr Zweikomponentensysteme entwickeln wir einen neuen Ansatz fĂŒr die wichtige offene Fragestellung der WellenĂ€ngenselektion im hochgradig nichtlinearen Regime. Wir zeigen, dass "coarsening" (das stetige wachsen der charakteristischen LĂ€ngenskala) von Mustern in Zweikomponentensystemen nie stoppt, wenn sie exakt massenerhaltend sind. Die Selektion einer endlichen WellenlĂ€nge entsteht durch schwach gebrochene Massenerhaltung oder durch Kopplung an zusĂ€tzliche Komponenten. Diese Prozesse wirken der Masseumverteilung, die coarsening treibt, entgegen und stoppen so das coarsening. Bei komplexen dynamischen PhĂ€nomenen wie Wellenmustern und dem Übergang zu raumzeitlichen Chaos bietet eine Analyse in Bezug auf lokale Gleichgewichte und deren StabilitĂ€tseigenschaften 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 OberflĂ€che zu einer starken AbhĂ€ngigkeit der Musterbildung von der rĂ€umlichen Geometrie fĂŒhrt. Das erklĂ€rt 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 abhĂ€ngen. Zusammengenommen beleuchtet unsere Arbeit die vereinheitlichenden Prinzipien, die der intrazellulĂ€ren Selbstorganisation zugrunde liegen, und zeigt, wie mikroskopische Interaktionsregeln und physikalische Bedingungen gemeinsam zu spezifischen biologischen Funktionen fĂŒhren

    Mathematical frameworks for oscillatory network dynamics in neuroscience

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    The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience

    Neural networks as spatio-temporal pattern-forming systems

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

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    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

    All-optical spiking neurons integrated on a photonic chip

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    Control of pattern formation in excitable systems

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    Pattern formation embodies the beauty and complexity of nature. Some patterns like traveling and rotating waves are dynamic, while others such as dots and stripes are static. Both dynamic and static patterns have been observed in a variety of physiological and biological processes such as rotating action potential waves in the brain during sleep, traveling calcium waves in the cardiac muscle, static patterns on the skins of animals, and self-regulated patterns in the animal embryo. Excitable systems represent a class of ultrasensitive systems that are capable of generating different kinds of patterns depending on the interplay between activator and inhibitor dynamics. Through manipulation of different excitable parameters, a diverse array of traveling wave and standing wave patterns can be obtained. In this thesis, I use pattern formation theory to control the excitable systems involved in cell migration and neuroscience to alter the observed phenotype, in an attempt to affect the underlying biological process. Cell migration is critical in many processes such as cancer metastasis and wound healing. Cells move by extending periodic protrusions of their cortex, and recent years have shown that the cellular cortex is an excitable medium where waves of biochemical species organize the cellular protrusion. Altering the protrusive phenotype can drastically alter cell migration — that can potentially affect critical physiological processes. In the first part of this thesis, I use excitable wave theory to model and predict wave pattern changes in amoeboid cells. Using theories of pattern formation, key nodes of the underlying excitable network governing cell migration are altered — to drastically change the cellular migratory phenotype, moving from amoeboid cells to oscillatory cells and from cells that extend long finger-like protrusions to cells that sustain stable rings on the cortex, potentially uncovering a novel method of pattern formation. Excitable systems originated in neuroscience, where different patterns of activity reflect different brain states. Sleep is associated with slow waves, while repeated high-frequency waves are associated with epileptic seizures. These patterns arise from the interplay between the cerebral cortex and the thalamus, which form a closed-loop architecture. In the second part of this thesis, I use a three-layer two-dimensional thalamocortical model, to explore the different parameters that may influence different spatio-temporal dynamics on the cortex. This study reveals that inter- and intra-cortical connectivity, excitation-inhibition balance and synaptic strengths can influence the wave activity patterns, to recreate different dynamic patterns observed in different brain states

    Discrete Breathers

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    Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices - quantum breathers. Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm
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