Amplitude Death: The emergence of stationarity in coupled nonlinear systems

When nonlinear dynamical systems are coupled, depending on the intrinsic dynamics and the manner in which the coupling is organized, a host of novel phenomena can arise. In this context, an important emergent phenomenon is the complete suppression of oscillations, formally termed amplitude death (AD). Oscillations of the entire system cease as a consequence of the interaction, leading to stationary behavior. The fixed points that the coupling stabilizes can be the otherwise unstable fixed points of the uncoupled system or can correspond to novel stationary points. Such behaviour is of relevance in areas ranging from laser physics to the dynamics of biological systems. In this review we discuss the characteristics of the different coupling strategies and scenarios that lead to AD in a variety of different situations, and draw attention to several open issues and challenging problems for further study.Comment: Physics Reports (2012

Synchronous behavior in networks of coupled systems : with applications to neuronal dynamics

Synchronization in networks of interacting dynamical systems is an interesting phenomenon that arises in nature, science and engineering. Examples include the simultaneous flashing of thousands of fireflies, the synchronous firing of action potentials by groups of neurons, cooperative behavior of robots and synchronization of chaotic systems with applications to secure communication. How is it possible that systems in a network synchronize? A key ingredient is that the systems in the network "communicate" information about their state to the systems they are connected to. This exchange of information ultimately results in synchronization of the systems in the network. The question is how the systems in the network should be connected and respond to the received information to achieve synchronization. In other words, which network structures and what kind of coupling functions lead to synchronization of the systems? In addition, since the exchange of information is likely to take some time, can systems in networks show synchronous behavior in presence of time-delays? The first part of this thesis focusses on synchronization of identical systems that interact via diffusive coupling, that is a coupling defined through the weighted difference of the output signals of the systems. The coupling might contain timedelays. In particular, two types of diffusive time-delay coupling are considered: coupling type I is diffusive coupling in which only the transmitted signals contain a time-delay, and coupling type II is diffusive coupling in which every signal is timedelayed. It is proven that networks of diffusive time-delay coupled systems that satisfy a strict semipassivity property have solutions that are ultimately bounded. This means that the solutions of the interconnected systems always enter some compact set in finite time and remain in that set ever after. Moreover, it is proven that nonlinear minimum-phase strictly semipassive systems that interact via diffusive coupling always synchronize provided the interaction is sufficiently strong. If the coupling functions contain time-delays, then these systems synchronize if, in addition to the sufficiently strong interaction, the product of the time-delay and the coupling strength is sufficiently small. Next, the specific role of the topology of the network in relation to synchronization is discussed. First, using symmetries in the network, linear invariant manifolds for networks of the diffusively time-delayed coupled systems are identified. If such a linear invariant manifold is also attracting, then the network possibly shows partial synchronization. Partial synchronization is the phenomenon that some, at least two, systems in the network synchronize with each other but not with every system in the network. It is proven that a linear invariant manifold defined by a symmetry in a network of strictly semipassive systems is attracting if the coupling strength is sufficiently large and the product of the coupling strength and the time-delay is sufficiently small. The network shows partial synchronization if the values of the coupling strength and time-delay for which this manifold is attracting differ from those for which all systems in the network synchronize. Next, for systems that interact via symmetric coupling type II, it is shown that the values of the coupling strength and time-delay for which any network synchronizes can be determined from the structure of that network and the values of the coupling strength and time-delay for which two systems synchronize. In the second part of the thesis the theory presented in the first part is used to explain synchronization in networks of neurons that interact via electrical synapses. In particular, it is proven that four important models for neuronal activity, namely the Hodgkin-Huxley model, the Morris-Lecar model, the Hindmarsh-Rose model and the FitzHugh-Nagumo model, all have the semipassivity property. Since electrical synapses can be modeled by diffusive coupling, and all these neuronal models are nonlinear minimum-phase, synchronization in networks of these neurons happens if the interaction is sufficiently strong and the product of the time-delay and the coupling strength is sufficiently small. Numerical simulations with various networks of Hindmarsh-Rose neurons support this result. In addition to the results of numerical simulations, synchronization and partial synchronization is witnessed in an experimental setup with type II coupled electronic realizations of Hindmarsh-Rose neurons. These experimental results can be fully explained by the theoretical findings that are presented in the first part of the thesis. The thesis continues with a study of a network of pancreatic -cells. There is evidence that these beta-cells are diffusively coupled and that the synchronous bursting activity of the network is related to the secretion of insulin. However, if the network consists of active (oscillatory) beta-cells and inactive (dead) beta-cells, it might happen that, due to the interaction between the active and inactive cells, the activity of the network dies out which results in a inhibition of the insulin secretion. This problem is related to Diabetes Mellitus type 1. Whether the activity dies out or not depends on the number of cells that are active relative to the number of inactive cells. A bifurcation analysis gives estimates of the number of active cells relative to the number of inactive cells for which the network remains active. At last the controlled synchronization problem for all-to-all coupled strictly semipassive systems is considered. In particular, a systematic design procedure is presented which gives (nonlinear) coupling functions that guarantee synchronization of the systems. The coupling functions have the form of a definite integral of a scalar weight function on a interval defined by the outputs of the systems. The advantage of these coupling functions over linear diffusive coupling is that they provide high gain only when necessary, i.e. at those parts of the state space of the network where nonlinearities need to be suppressed. Numerical simulations in networks of Hindmarsh-Rose neurons support the theoretical results

Complex partial synchronization patterns in networks of delay-coupled neurons

We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh–Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept

Leaders do not look back, or do they?

We study the effect of adding to a directed chain of interconnected systems a directed feedback from the last element in the chain to the first. The problem is closely related to the fundamental question of how a change in network topology may influence the behavior of coupled systems. We begin the analysis by investigating a simple linear system. The matrix that specifies the system dynamics is the transpose of the network Laplacian matrix, which codes the connectivity of the network. Our analysis shows that for any nonzero complex eigenvalue $\lambda$ of this matrix, the following inequality holds: $\frac{|\Im \lambda |}{|\Re \lambda |} \leq \cot\frac{\pi}{n}$. This bound is sharp, as it becomes an equality for an eigenvalue of a simple directed cycle with uniform interaction weights. The latter has the slowest decay of oscillations among all other network configurations with the same number of states. The result is generalized to directed rings and chains of identical nonlinear oscillators. For directed rings, a lower bound $\sigma_c$ for the connection strengths that guarantees asymptotic synchronization is found to follow a similar pattern: $\sigma_c=\frac{1}{1-\cos\left( 2\pi /n\right)}$. Numerical analysis revealed that, depending on the network size $n$, multiple dynamic regimes co-exist in the state space of the system. In addition to the fully synchronous state a rotating wave solution occurs. The effect is observed in networks exceeding a certain critical size. The emergence of a rotating wave highlights the importance of long chains and loops in networks of oscillators: the larger the size of chains and loops, the more sensitive the network dynamics becomes to removal or addition of a single connection

Amplitude and phase effects on the synchronization of delay-coupled oscillators

We consider the behavior of Stuart–Landau oscillators as generic limit-cycle oscillators when they are interacting with delay. We investigate the role of amplitude and phase instabilities in producing symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no chaotic dynamics is observed, and the stability of the identical synchronization solution is the same in each of the three studied networks of delay-coupled elements. When allowing for a variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability always results in antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding selffeedback with the same strength and delay as the coupling stabilizes the system in the transverse direction and, thus, promotes the onset of identically synchronized behaviorWe would like to thank T. Erneux, E. Schöll, S. Yanchuk, and P. Perlikowski for helpful discussions. O.D. acknowledges the Research Foundation Flanders FWO-Vlaanderen for a fellowship and for project support. This work was partially supported by the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under Grant No. IAP VI-10 “photonics@be,” by MICINN Spain under project DeCoDicA Grant No. TEC2009- 14101 ,, and by the project PHOCUS EU FET Open Grant No. 240763 .Peer reviewe

Synchrony and bifurcations in coupled dynamical systems and effects of time delay

Dynamik auf Netzwerken ist ein mathematisches Feld, das in den letzten Jahrzehnten schnell gewachsen ist und Anwendungen in zahlreichen Disziplinen wie z.B. Physik, Biologie und Soziologie findet. Die Funktion vieler Netzwerke hängt von der Fähigkeit ab, die Elemente des Netzwerkes zu synchronisieren. Mit anderen Worten, die Existenz und die transversale Stabilität der synchronen Mannigfaltigkeit sind zentrale Eigenschaften. Erst seit einigen Jahren wird versucht, den verwickelten Zusammenhang zwischen der Kopplungsstruktur und den Stabilitätseigenschaften synchroner Zustände zu verstehen. Genau das ist das zentrale Thema dieser Arbeit. Zunächst präsentiere ich erste Ergebnisse zur Klassifizierung der Kanten eines gerichteten Netzwerks bezüglich ihrer Bedeutung für die Stabilität des synchronen Zustands. Folgend untersuche ich ein komplexes Verzweigungsszenario in einem gerichteten Ring von Stuart-Landau Oszillatoren und zeige, dass das Szenario persistent ist, wenn dem Netzwerk eine schwach gewichtete Kante hinzugefügt wird. Daraufhin untersuche ich synchrone Zustände in Ringen von Phasenoszillatoren die mit Zeitverzögerung gekoppelt sind. Ich bespreche die Koexistenz synchroner Lösungen und analysiere deren Stabilität und Verzweigungen. Weiter zeige ich, dass eine Zeitverschiebung genutzt werden kann, um Muster im Ring zu speichern und wiederzuerkennen. Diese Zeitverschiebung untersuche ich daraufhin für beliebige Kopplungsstrukturen. Ich zeige, dass invariante Mannigfaltigkeiten des Flusses sowie ihre Stabilität unter der Zeitverschiebung erhalten bleiben. Darüber hinaus bestimme ich die minimale Anzahl von Zeitverzögerungen, die gebraucht werden, um das System äquivalent zu beschreiben. Schließlich untersuche ich das auffällige Phänomen eines nichtstetigen Übergangs zu Synchronizität in Klassen großer Zufallsnetzwerke indem ich einen kürzlich eingeführten Zugang zur Beschreibung großer Zufallsnetzwerke auf den Fall zeitverzögerter Kopplungen verallgemeinere.Since a couple of decades, dynamics on networks is a rapidly growing branch of mathematics with applications in various disciplines such as physics, biology or sociology. The functioning of many networks heavily relies on the ability to synchronize the network’s nodes. More precisely, the existence and the transverse stability of the synchronous manifold are essential properties. It was only in the last few years that people tried to understand the entangled relation between the coupling structure of a network, given by a (di-)graph, and the stability properties of synchronous states. This is the central theme of this dissertation. I first present results towards a classification of the links in a directed, diffusive network according to their impact on the stability of synchronization. Then I investigate a complex bifurcation scenario observed in a directed ring of Stuart-Landau oscillators. I show that under the addition of a single weak link, this scenario is persistent. Subsequently, I investigate synchronous patterns in a directed ring of phase oscillators coupled with time delay. I discuss the coexistence of multiple of synchronous solutions and investigate their stability and bifurcations. I apply these results by showing that a certain time-shift transformation can be used in order to employ the ring as a pattern recognition device. Next, I investigate the same time-shift transformation for arbitrary coupling structures in a very general setting. I show that invariant manifolds of the flow together with their stability properties are conserved under the time-shift transformation. Furthermore, I determine the minimal number of delays needed to equivalently describe the system’s dynamics. Finally, I investigate a peculiar phenomenon of non-continuous transition to synchrony observed in certain classes of large random networks, generalizing a recently introduced approach for the description of large random networks to the case of delayed couplings

Normalized Connectomes Show Increased Synchronizability with Age through Their Second Largest Eigenvalue

The synchronization of different brain regions is widely observed under both normal and pathological conditions, such as epilepsy. However, the relationship between the dynamics of these brain regions, the connectivity between them, and the ability to synchronize remains an open question. We investigate the problem of inter-region synchronization in networks of Wilson--Cowan/neural field equations with homeostatic plasticity, each of which acts as a model for an isolated brain region. We consider arbitrary connection profiles with only one constraint: the rows of the connection matrices are all identically normalized. We found that these systems often synchronize to the solution obtained from a single, self-coupled neural region. We analyze the stability of this solution through a straightforward modification of the master stability function (MSF) approach and found that synchronized solutions lose stability for connectivity matrices when the second largest positive eigenvalue is sufficiently large for values of the global coupling parameter that are not too large. This result was numerically confirmed for ring systems and lattices and was also robust to small amounts of heterogeneity in the homeostatic set points in each node. Finally, we tested this result on connectomes obtained from 196 subjects over a broad age range (4--85 years) from the Human Connectome Project. We found that the second largest eigenvalue tended to decrease with age, indicating an increase in synchronizability that may be related to the increased prevalence of epilepsy with advancing age.Natural Sciences and Engineering Research Council, Discovery grant