Waves and propagation failure in discrete space models with nonlinear coupling and feedback

Many developmental processes involve a wave of initiation of pattern formation, behind which a uniform layer of discrete cells develops a regular pattern that determines cell fates. This paper focuses on the initiation of such waves, and then on the emergence of patterns behind the wavefront. I study waves in discrete space differential equation models where the coupling between sites is nonlinear. Such systems represent juxtacrine cell signalling, where cells communicate via membrane bound molecules binding to their receptors. In this way, the signal at cell j is a nonlinear function of the average signal on neighbouring cells. Whilst considerable progress has been made in the analysis of discrete reaction-diffusion systems, this paper presents a novel and detailed study of waves in juxtacrine systems. I analyse travelling wave solutions in such systems with a single variable representing activity in each cell. When there is a single stable homogeneous steady state, the wave speed is governed by the linearisation ahead of the wave front. Wave propagation (and failure) is studied when the homogeneous dynamics are bistable. Simulations show that waves may propagate in either direction, or may be pinned. A Lyapunov function is used to determine the direction of propagation of travelling waves. Pinning is studied by calculating the boundaries for propagation failure for sigmoidal and piecewise linear feedback functions, using analysis of 2 active sites and exact stationary solutions respectively. I then explore the calculation of travelling waves as the solution of an associated n-dimensional boundary value problem posed on [0, 1], using continuation to determine the dependence of speed on model parameters. This method is shown to be very accurate, by comparison with numerical simulations. Furthermore, the method is also applicable to other discrete systems on a regular lattice, such as the discrete bistable reaction-diffusion equation. Finally, I extend the study to more detailed models including the reaction kinetics of signalling, and demonstrate the same features of wave propagation. I discuss how such waves may initiate pattern formation, and the role of such mechanisms in morphogenesis

Traveling and pinned fronts in bistable reaction-diffusion systems on network

Traveling fronts and stationary localized patterns in bistable reaction-diffusion systems have been broadly studied for classical continuous media and regular lattices. Analogs of such non-equilibrium patterns are also possible in networks. Here, we consider traveling and stationary patterns in bistable one-component systems on random Erd\"os-R\'enyi, scale-free and hierarchical tree networks. As revealed through numerical simulations, traveling fronts exist in network-organized systems. They represent waves of transition from one stable state into another, spreading over the entire network. The fronts can furthermore be pinned, thus forming stationary structures. While pinning of fronts has previously been considered for chains of diffusively coupled bistable elements, the network architecture brings about significant differences. An important role is played by the degree (the number of connections) of a node. For regular trees with a fixed branching factor, the pinning conditions are analytically determined. For large Erd\"os-R\'enyi and scale-free networks, the mean-field theory for stationary patterns is constructed

Dissipative solitons in pattern-forming nonlinear optical systems : cavity solitons and feedback solitons

Many dissipative optical systems support patterns. Dissipative solitons are generally found where a pattern coexists with a stable unpatterned state. We consider such phenomena in driven optical cavities containing a nonlinear medium (cavity solitons) and rather similar phenomena (feedback solitons) where a driven nonlinear optical medium is in front of a single feedback mirror. The history, theory, experimental status, and potential application of such solitons is reviewed

Transverse Patterns in Nonlinear Optical Resonators

The book is devoted to the formation and dynamics of localized structures (vortices, solitons) and extended patterns (stripes, hexagons, tilted waves) in nonlinear optical resonators such as lasers, optical parametric oscillators, and photorefractive oscillators. The theoretical analysis is performed by deriving order parameter equations, and also through numerical integration of microscopic models of the systems under investigation. Experimental observations, and possible technological implementations of transverse optical patterns are also discussed. A comparison with patterns found in other nonlinear systems, i.e. chemical, biological, and hydrodynamical systems, is given. This article contains the table of contents and the introductory chapter of the book.Comment: 37 pages, 14 figures. Table of contents and introductory chapter of the boo

Time Delay Effects on Coupled Limit Cycle Oscillators at Hopf Bifurcation

We present a detailed study of the effect of time delay on the collective dynamics of coupled limit cycle oscillators at Hopf bifurcation. For a simple model consisting of just two oscillators with a time delayed coupling, the bifurcation diagram obtained by numerical and analytical solutions shows significant changes in the stability boundaries of the amplitude death, phase locked and incoherent regions. A novel result is the occurrence of amplitude death even in the absence of a frequency mismatch between the two oscillators. Similar results are obtained for an array of N oscillators with a delayed mean field coupling and the regions of such amplitude death in the parameter space of the coupling strength and time delay are quantified. Some general analytic results for the N tending to infinity (thermodynamic) limit are also obtained and the implications of the time delay effects for physical applications are discussed.Comment: 20 aps formatted revtex pages (including 13 PS figures); Minor changes over the previous version; To be published in Physica

Switch and template pattern formation in a discrete reaction diffusion system inspired by the Drosophila eye

We examine a spatially discrete reaction diffusion model based on the interactions that create a periodic pattern in the Drosophila eye imaginal disc. This model is capable of generating a regular hexagonal pattern of gene expression behind a moving front, as observed in the fly system. In order to better understand the novel switch and template mechanism behind this pattern formation, we present here a detailed study of the model's behavior in one dimension, using a combination of analytic methods and numerical searches of parameter space. We find that patterns are created robustly provided that there is an appropriate separation of timescales and that self-activation is sufficiently strong, and we derive expressions in this limit for the front speed and the pattern wavelength. Moving fronts in pattern-forming systems near an initial linear instability generically select a unique pattern, but our model operates in a strongly nonlinear regime where the final pattern depends on the initial conditions as well as on parameter values. Our work highlights the important role that cellularization and cell-autonomous feedback can play in biological pattern formation

The Turing bifurcation in network systems: Collective patterns and single differentiated nodes

We study the emergence of patterns in a diffusively coupled network that undergoes a Turing instability. Our main focus is the emergence of stable solutions with a single differentiated node in systems with large and possibly irregular network topology. Based on a mean-field approach, we study the bifurcations of such solutions for varying system parameters and varying degree of the differentiated node. Such solutions appear typically before the onset of Turing instability and provide the basis for the complex scenario of multistability and hysteresis that can be observed in such systems. Moreover, we discuss the appearance of stable collective patterns and present a codimension-two bifurcation that organizes the interplay between collective patterns and patterns with single differentiated nodes

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