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Hydrometric networks play a vital role in providing information for decision-making in water resource management. They should be set up optimally to provide as much information as possible that is as accurate as possible and, at the same time, be cost-effective. Although the design of hydrometric networks is a well-identified problem in hydrometeorology and has received considerable attention, there is still scope for further advancement. In this study, we use complex network analysis, defined as a collection of nodes interconnected by links, to propose a new measure that identifies critical nodes of station networks. The approach can support the design and redesign of hydrometric station networks. The science of complex networks is a relatively young field and has gained significant momentum over the last few years in different areas such as brain networks, social networks, technological networks, or climate networks. The identification of influential nodes in complex networks is an important field of research. We propose a new node-ranking measure – the weighted degree–betweenness (WDB) measure – to evaluate the importance of nodes in a network. It is compared to previously proposed measures used on synthetic sample networks and then applied to a real-world rain gauge network comprising 1229 stations across Germany to demonstrate its applicability. The proposed measure is evaluated using the decline rate of the network efficiency and the kriging error. The results suggest that WDB effectively quantifies the importance of rain gauges, although the benefits of the method need to be investigated in more detail.
Hydrometric networks play a vital role in providing information for decision-making in water resource management. They should be set up optimally to provide as much information as possible that is as accurate as possible and, at the same time, be cost-effective. Although the design of hydrometric networks is a well-identified problem in hydrometeorology and has received considerable attention, there is still scope for further advancement. In this study, we use complex network analysis, defined as a collection of nodes interconnected by links, to propose a new measure that identifies critical nodes of station networks. The approach can support the design and redesign of hydrometric station networks. The science of complex networks is a relatively young field and has gained significant momentum over the last few years in different areas such as brain networks, social networks, technological networks, or climate networks. The identification of influential nodes in complex networks is an important field of research. We propose a new node-ranking measure – the weighted degree–betweenness (WDB) measure – to evaluate the importance of nodes in a network. It is compared to previously proposed measures used on synthetic sample networks and then applied to a real-world rain gauge network comprising 1229 stations across Germany to demonstrate its applicability. The proposed measure is evaluated using the decline rate of the network efficiency and the kriging error. The results suggest that WDB effectively quantifies the importance of rain gauges, although the benefits of the method need to be investigated in more detail.
Subject of this work is the investigation of universal scaling laws which are observed in coupled chaotic systems. Progress is made by replacing the chaotic fluctuations in the perturbation dynamics by stochastic processes. First, a continuous-time stochastic model for weakly coupled chaotic systems is introduced to study the scaling of the Lyapunov exponents with the coupling strength (coupling sensitivity of chaos). By means of the the Fokker-Planck equation scaling relations are derived, which are confirmed by results of numerical simulations. Next, the new effect of avoided crossing of Lyapunov exponents of weakly coupled disordered chaotic systems is described, which is qualitatively similar to the energy level repulsion in quantum systems. Using the scaling relations obtained for the coupling sensitivity of chaos, an asymptotic expression for the distribution function of small spacings between Lyapunov exponents is derived and compared with results of numerical simulations. Finally, the synchronization transition in strongly coupled spatially extended chaotic systems is shown to resemble a continuous phase transition, with the coupling strength and the synchronization error as control and order parameter, respectively. Using results of numerical simulations and theoretical considerations in terms of a multiplicative noise partial differential equation, the universality classes of the observed two types of transition are determined (Kardar-Parisi-Zhang equation with saturating term, directed percolation).
The topic of synchronization forms a link between nonlinear dynamics and neuroscience. On the one hand, neurobiological research has shown that the synchronization of neuronal activity is an essential aspect of the working principle of the brain. On the other hand, recent advances in the physical theory have led to the discovery of the phenomenon of phase synchronization. A method of data analysis that is motivated by this finding - phase synchronization analysis - has already been successfully applied to empirical data. The present doctoral thesis ties up to these converging lines of research. Its subject are methodical contributions to the further development of phase synchronization analysis, as well as its application to event-related potentials, a form of EEG data that is especially important in the cognitive sciences. The methodical contributions of this work consist firstly in a number of specialized statistical tests for a difference in the synchronization strength in two different states of a system of two oscillators. Secondly, in regard of the many-channel character of EEG data an approach to multivariate phase synchronization analysis is presented. For the empirical investigation of neuronal synchronization a classic experiment on language processing was replicated, comparing the effect of a semantic violation in a sentence context with that of the manipulation of physical stimulus properties (font color). Here phase synchronization analysis detects a decrease of global synchronization for the semantic violation as well as an increase for the physical manipulation. In the latter case, by means of the multivariate analysis the global synchronization effect can be traced back to an interaction of symmetrically located brain areas.<BR> The findings presented show that the method of phase synchronization analysis motivated by physics is able to provide a relevant contribution to the investigation of event-related potentials in the cognitive sciences.
In the present work synchronization phenomena in complex dynamical systems exhibiting multiple time scales have been analyzed. Multiple time scales can be active in different manners. Three different systems have been analyzed with different methods from data analysis. The first system studied is a large heterogenous network of bursting neurons, that is a system with two predominant time scales, the fast firing of action potentials (spikes) and the burst of repetitive spikes followed by a quiescent phase. This system has been integrated numerically and analyzed with methods based on recurrence in phase space. An interesting result are the different transitions to synchrony found in the two distinct time scales. Moreover, an anomalous synchronization effect can be observed in the fast time scale, i.e. there is range of the coupling strength where desynchronization occurs. The second system analyzed, numerically as well as experimentally, is a pair of coupled CO₂ lasers in a chaotic bursting regime. This system is interesting due to its similarity with epidemic models. We explain the bursts by different time scales generated from unstable periodic orbits embedded in the chaotic attractor and perform a synchronization analysis of these different orbits utilizing the continuous wavelet transform. We find a diverse route to synchrony of these different observed time scales. The last system studied is a small network motif of limit cycle oscillators. Precisely, we have studied a hub motif, which serves as elementary building block for scale-free networks, a type of network found in many real world applications. These hubs are of special importance for communication and information transfer in complex networks. Here, a detailed study on the mechanism of synchronization in oscillatory networks with a broad frequency distribution has been carried out. In particular, we find a remote synchronization of nodes in the network which are not directly coupled. We also explain the responsible mechanism and its limitations and constraints. Further we derive an analytic expression for it and show that information transmission in pure phase oscillators, such as the Kuramoto type, is limited. In addition to the numerical and analytic analysis an experiment consisting of electrical circuits has been designed. The obtained results confirm the former findings.
We consider chimera states in a one-dimensional medium of nonlinear nonlocally coupled phase oscillators. Stationary inhomogeneous solutions of the Ott-Antonsen equation for a complex order parameter that correspond to fundamental chimeras have been constructed. Stability calculations reveal that only some of these states are stable. The direct numerical simulation has shown that these structures under certain conditions are transformed to breathing chimera regimes because of the development of instability. Further development of instability leads to turbulent chimeras.
Exploring one-sided communication and synchronization on a non-cache-coherent many-core architecture
(2017)
The ongoing many-core design aims at core counts where cache coherence becomes a serious challenge. Therefore, this paper discusses how one-sided communication and the required process synchronization can be realized on a non-cache-coherent many-core CPU. The Intel Single-chip Cloud Computer serves as an exemplary hardware architecture. The presented approach is based on software-managed cache coherence for MPI one-sided communication. The prototype implementation delivers a PUT performance of up to 5 times faster than the default message-based approach and reveals a reduction of the communication costs for the NAS Parallel Benchmarks 3-D fast Fourier Transform by a factor of 5. Further, the paper derives conclusions for future non-cache-coherent architectures.
We show that self-consistent partial synchrony in globally coupled oscillatory ensembles is a general phenomenon. We analyze in detail appearance and stability properties of this state in possibly the simplest setup of a biharmonic Kuramoto-Daido phase model as well as demonstrate the effect in limit-cycle relaxational Rayleigh oscillators. Such a regime extends the notion of splay state from a uniform distribution of phases to an oscillating one. Suitable collective observables such as the Kuramoto order parameter allow detecting the presence of an inhomogeneous distribution. The characteristic and most peculiar property of self-consistent partial synchrony is the difference between the frequency of single units and that of the macroscopic field.
We show that self-consistent partial synchrony in globally coupled oscillatory ensembles is a general phenomenon. We analyze in detail appearance and stability properties of this state in possibly the simplest setup of a biharmonic Kuramoto-Daido phase model as well as demonstrate the effect in limit-cycle relaxational Rayleigh oscillators. Such a regime extends the notion of splay state from a uniform distribution of phases to an oscillating one. Suitable collective observables such as the Kuramoto order parameter allow detecting the presence of an inhomogeneous distribution. The characteristic and most peculiar property of self-consistent partial synchrony is the difference between the frequency of single units and that of the macroscopic field.
Nested application conditions generalise the well-known negative application conditions and are important for several application domains. In this paper, we present Local Church-Rosser, Parallelism, Concurrency and Amalgamation Theorems for rules with nested application conditions in the framework of M-adhesive categories, where M-adhesive categories are slightly more general than weak adhesive high-level replacement categories. Most of the proofs are based on the corresponding statements for rules without application conditions and two shift lemmas stating that nested application conditions can be shifted over morphisms and rules.
In dieser Arbeit werden die Effekte der Synchronisation nichtlinearer, akustischer Oszillatoren am Beispiel zweier Orgelpfeifen untersucht. Aus vorhandenen, experimentellen Messdaten werden die typischen Merkmale der Synchronisation extrahiert und dargestellt. Es folgt eine detaillierte Analyse der Übergangsbereiche in das Synchronisationsplateau, der Phänomene während der Synchronisation, als auch das Austreten aus der Synchronisationsregion beider Orgelpfeifen, bei verschiedenen Kopplungsstärken. Die experimentellen Befunde werfen Fragestellungen nach der Kopplungsfunktion auf. Dazu wird die Tonentstehung in einer Orgelpfeife untersucht. Mit Hilfe von numerischen Simulationen der Tonentstehung wird der Frage nachgegangen, welche fluiddynamischen und aero-akustischen Ursachen die Tonentstehung in der Orgelpfeife hat und inwiefern sich die Mechanismen auf das Modell eines selbsterregten akustischen Oszillators abbilden lässt. Mit der Methode des Coarse Graining wird ein Modellansatz formuliert.
In dieser Arbeit werden nichtlineare Kopplungsmechanismen von akustischen Oszillatoren untersucht, die zu Synchronisation führen können. Aufbauend auf die Fragestellungen vorangegangener Arbeiten werden mit Hilfe theoretischer und experimenteller Studien sowie mit Hilfe numerischer Simulationen die Elemente der Tonentstehung in der Orgelpfeife und die Mechanismen der gegenseitigen Wechselwirkung von Orgelpfeifen identifiziert. Daraus wird erstmalig ein vollständig auf den aeroakustischen und fluiddynamischen Grundprinzipien basierendes nichtlinear gekoppeltes Modell selbst-erregter Oszillatoren für die Beschreibung des Verhaltens zweier wechselwirkender Orgelpfeifen entwickelt. Die durchgeführten Modellrechnungen werden mit den experimentellen Befunden verglichen. Es zeigt sich, dass die Tonentstehung und die Kopplungsmechanismen von Orgelpfeifen durch das entwickelte Oszillatormodell in weiten Teilen richtig beschrieben werden. Insbesondere kann damit die Ursache für den nichtlinearen Zusammenhang von Kopplungsstärke und Synchronisation des gekoppelten Zwei-Pfeifen Systems, welcher sich in einem nichtlinearen Verlauf der Arnoldzunge darstellt, geklärt werden. Mit den gewonnenen Erkenntnissen wird der Einfluss des Raumes auf die Tonentstehung bei Orgelpfeifen betrachtet. Dafür werden numerische Simulationen der Wechselwirkung einer Orgelpfeife mit verschiedenen Raumgeometrien, wie z. B. ebene, konvexe, konkave, und gezahnte Geometrien, exemplarisch untersucht. Auch der Einfluss von Schwellkästen auf die Tonentstehung und die Klangbildung der Orgelpfeife wird studiert. In weiteren, neuartigen Synchronisationsexperimenten mit identisch gestimmten Orgelpfeifen, sowie mit Mixturen wird die Synchronisation für verschiedene, horizontale und vertikale Pfeifenabstände in der Ebene der Schallabstrahlung, untersucht. Die dabei erstmalig beobachteten räumlich isotropen Unstetigkeiten im Schwingungsverhalten der gekoppelten Pfeifensysteme, deuten auf abstandsabhängige Wechsel zwischen gegen- und gleichphasigen Sychronisationsregimen hin. Abschließend wird die Möglichkeit dokumentiert, das Phänomen der Synchronisation zweier Orgelpfeifen durch numerische Simulationen, also der Behandlung der kompressiblen Navier-Stokes Gleichungen mit entsprechenden Rand- und Anfangsbedingungen, realitätsnah abzubilden. Auch dies stellt ein Novum dar.
In this paper, we analytically study a star motif of Stuart-Landau oscillators, derive the bifurcation diagram and discuss the different forms of synchronization arising in such a system. Despite the parameter mismatch between the central node and the peripheral ones, an analytical approach independent of the number of units in the system has been proposed. The approach allows to calculate the separatrices between the regions with distinct dynamical behavior and to determine the nature of the different transitions to synchronization appearing in the system. The theoretical analysis is supported by numerical results.
Although eye movements during reading are modulated by cognitive processing demands, they also reflect visual sampling of the input, and possibly preparation of output for speech or the inner voice. By simultaneously recording eye movements and the voice during reading aloud, we obtained an output measure that constrains the length of time spent on cognitive processing. Here we investigate the dynamics of the eye-voice span (EVS), the distance between eye and voice. We show that the EVS is regulated immediately during fixation of a word by either increasing fixation duration or programming a regressive eye movement against the reading direction. EVS size at the beginning of a fixation was positively correlated with the likelihood of regressions and refixations. Regression probability was further increased if the EVS was still large at the end of a fixation: if adjustment of fixation duration did not sufficiently reduce the EVS during a fixation, then a regression rather than a refixation followed with high probability. We further show that the EVS can help understand cognitive influences on fixation duration during reading: in mixed model analyses, the EVS was a stronger predictor of fixation durations than either word frequency or word length. The EVS modulated the influence of several other predictors on single fixation durations (SFDs). For example, word-N frequency effects were larger with a large EVS, especially when word N-1 frequency was low. Finally, a comparison of SFDs during oral and silent reading showed that reading is governed by similar principles in both reading modes, although EVS maintenance and articulatory processing also cause some differences. In summary, the EVS is regulated by adjusting fixation duration and/or by programming a regressive eye movement when the EVS gets too large. Overall, the EVS appears to be directly related to updating of the working memory buffer during reading.
Although eye movements during reading are modulated by cognitive processing demands, they also reflect visual sampling of the input, and possibly preparation of output for speech or the inner voice. By simultaneously recording eye movements and the voice during reading aloud, we obtained an output measure that constrains the length of time spent on cognitive processing. Here we investigate the dynamics of the eye-voice span (EVS), the distance between eye and voice. We show that the EVS is regulated immediately during fixation of a word by either increasing fixation duration or programming a regressive eye movement against the reading direction. EVS size at the beginning of a fixation was positively correlated with the likelihood of regressions and refixations. Regression probability was further increased if the EVS was still large at the end of a fixation: if adjustment of fixation duration did not sufficiently reduce the EVS during a fixation, then a regression rather than a refixation followed with high probability. We further show that the EVS can help understand cognitive influences on fixation duration during reading: in mixed model analyses, the EVS was a stronger predictor of fixation durations than either word frequency or word length. The EVS modulated the influence of several other predictors on single fixation durations (SFDs). For example, word-N frequency effects were larger with a large EVS, especially when word N-1 frequency was low. Finally, a comparison of SFDs during oral and silent reading showed that reading is governed by similar principles in both reading modes, although EVS maintenance and articulatory processing also cause some differences. In summary, the EVS is regulated by adjusting fixation duration and/or by programming a regressive eye movement when the EVS gets too large. Overall, the EVS appears to be directly related to updating of the working memory buffer during reading.
Although eye movements during reading are modulated by cognitive processing demands, they also reflect visual sampling of the input, and possibly preparation of output for speech or the inner voice. By simultaneously recording eye movements and the voice during reading aloud, we obtained an output measure that constrains the length of time spent on cognitive processing. Here we investigate the dynamics of the eye-voice span (EVS), the distance between eye and voice. We show that the EVS is regulated immediately during fixation of a word by either increasing fixation duration or programming a regressive eye movement against the reading direction. EVS size at the beginning of a fixation was positively correlated with the likelihood of regressions and refixations. Regression probability was further increased if the EVS was still large at the end of a fixation: if adjustment of fixation duration did not sufficiently reduce the EVS during a fixation, then a regression rather than a refixation followed with high probability. We further show that the EVS can help understand cognitive influences on fixation duration during reading: in mixed model analyses, the EVS was a stronger predictor of fixation durations than either word frequency or word length. The EVS modulated the influence of several other predictors on single fixation durations (SFDs). For example, word-N frequency effects were larger with a large EVS, especially when word N-1 frequency was low. Finally, a comparison of SFDs during oral and silent reading showed that reading is governed by similar principles in both reading modes, although EVS maintenance and articulatory processing also cause some differences. In summary, the EVS is regulated by adjusting fixation duration and/or by programming a regressive eye movement when the EVS gets too large. Overall, the EVS appears to be directly related to updating of the working memory buffer during reading.
The analysis of event time series is in general challenging. Most time series analysis tools are limited for the analysis of this kind of data. Recurrence analysis, a powerful concept from nonlinear time series analysis, provides several opportunities to work with event data and even for the most challenging task of comparing event time series with continuous time series. Here, the basic concept is introduced, the challenges are discussed, and the future perspectives are summarized.
Die Natur unterliegt ständigen Veränderungen und befindet sich nur vermeintlich in einem Gleichgewicht. Umweltparameter wie Temperatur, Luftfeuchtigkeit oder Sonneneinstrahlung schwanken auf einer Zeitskala von Sekunden bis Jahrmillionen und beinhalten teils beträchtliche Unterschiede. Mit diesen Umweltveränderungen müssen sich Arten als Teil eines Ökosystems auseinandersetzen. Für Ökologen ist interessant, wie sich individuelle Reaktionen auf die Umweltveränderungen im dynamischen Verhalten einer ganzen Population bemerkbar machen und ob deren Verhalten vorhersagbar ist. Der Demografie einer Population kommt hierbei eine entscheidende Rolle zu, da sie das Resultat von Wachstums- und Sterbeprozessen darstellt. Eben jene Prozesse werden von der Umwelt maßgeblich beeinflusst. Doch wie genau beeinflussen Umweltveränderungen das Verhalten ganzer Populationen? Wie sieht das vorübergehende, transiente Verhalten aus? Als Resultat von Umwelteinflüssen bilden sich in Populationen sogenannte Kohorten, hinsichtlich der Zahl an Individuen überproportional stark vertretene Alters- oder Größenklassen. Sterben z.B. aufgrund eines außergewöhnlich harten Winters, die alten und jungen Individuen einer Population, so besteht diese anschließend hauptsächlich aus Individuen mittleren Alters. Sie wurde sozusagen synchronisiert. Eine solche Populationen neigt zu regelmäßigen Schwankungen (Oszillationen) in ihrer Dichte, da die sich abwechselnden Phasen der individuellen Entwicklung und der Reproduktion nun von einem Großteil der Individuen synchron durchschritten werden. D.h., mal wächst die Population und mal nimmt sie entsprechend der Sterblichkeit ab. In Experimenten mit Phytoplankton-Populationen konnte ich zeigen, dass dieses oszillierende Verhalten mit dem in der Physik gebräuchlichen Konzept der Synchronisation beschrieben werden kann. Synchrones Verhalten ist eines der verbreitetsten Phänomene in der Natur und kann z.B. in synchron schwingenden Brücken, als auch bei der Erzeugung von Lasern oder in Form von rhythmischem Applaus auf einem Konzert beobachtet werden. Wie stark die Schwankungen sind, hängt dabei sowohl von der Stärke der Umweltveränderung als auch vom demografischen Zustand der Population vor der Veränderung ab. Zwei Populationen, die sich in verschiedenen Habitaten aufhalten, können zwar gleich stark von einer Umweltveränderung beeinflusst werden. Die Reaktionen im anschließenden Verhalten können jedoch äußerst unterschiedlich ausfallen, wenn sich die Populationen zuvor in stark unterschiedlichen demografischen Zuständen befanden. Darüber hinaus treten bestimmte, für das Verhalten einer Population relevante Mechanismen überhaupt erst in Erscheinung, wenn sich die Umweltbedingungen ändern. So fiel in Experimenten beispielsweise die Populationsdichte um rund 50 Prozent ab nachdem sich die Ressourcenverfügbarkeit verdoppelte. Der Grund für dieses gegenintuitive Verhalten konnte mit der erhöhten Aufnahme von Ressourcen erklärt werden. Damit verbessert eine Algenzelle zwar die eigene Konstitution, jedoch verzögert sich dadurch die auch die Reproduktion und die Populationsdichte nimmt gemäß ihrer Verluste bzw. Sterblichkeit ab. Zwei oder mehr räumlich getrennte Populationen können darüber hinaus durch Umwelteinflüsse synchronisiert werden. Dies wird als Moran-Effekt bezeichnet. Angenommen auf zwei weit voneinander entfernten Inseln lebt jeweils eine Population. Zwischen beiden findet kein Austausch statt – und doch zeigt sich beim Vergleich ihrer Zeitreihen eine große Ähnlichkeit. Das überregionale Klima synchronisiert hierbei die lokalen Umwelteinflüsse. Diese wiederum bestimmen das Verhalten der jeweiligen Population. Der Moran-Effekt besagt nun, dass die Ähnlichkeit zwischen den Populationen jener zwischen den Umwelteinflüssen entspricht, oder geringer ist. Meine Ergebnisse bestätigen dies und zeigen darüber hinaus, dass sich die Populationen sogar ähnlicher sein können als die Umwelteinflüsse, wenn man von unterschiedlich stark schwankenden Einflüssen ausgeht.
We performed numerical simulations with the Kuramoto model and experiments with oscillatory nickel electrodissolution to explore the dynamical features of the transients from random initial conditions to a fully synchronized (one-cluster) state. The numerical simulations revealed that certain networks (e.g., globally coupled or dense Erdos-Renyi random networks) showed relatively simple behavior with monotonic increase of the Kuramoto order parameter from the random initial condition to the fully synchronized state and that the transient times exhibited a unimodal distribution. However, some modular networks with bridge elements were identified which exhibited non-monotonic variation of the order parameter with local maximum and/or minimum. In these networks, the histogram of the transients times became bimodal and the mean transient time scaled well with inverse of the magnitude of the second largest eigenvalue of the network Laplacian matrix. The non-monotonic transients increase the relative standard deviations from about 0.3 to 0.5, i.e., the transient times became more diverse. The non-monotonic transients are related to generation of phase patterns where the modules are synchronized but approximately anti-phase to each other. The predictions of the numerical simulations were demonstrated in a population of coupled oscillatory electrochemical reactions in global, modular, and irregular tree networks. The findings clarify the role of network structure in generation of complex transients that can, for example, play a role in intermittent desynchronization of the circadian clock due to external cues or in deep brain stimulations where long transients are required after a desynchronization stimulus.
Synchronization of coupled oscillators manifests itself in many natural and man-made systems, including cyrcadian clocks, central pattern generators, laser arrays, power grids, chemical and electrochemical oscillators, only to name a few. The mathematical description of this phenomenon is often based on the paradigmatic Kuramoto model, which represents each oscillator by one scalar variable, its phase. When coupled, phase oscillators constitute a high-dimensional dynamical system, which exhibits complex behaviour, ranging from synchronized uniform oscillation to quasiperiodicity and chaos. The corresponding collective rhythms can be useful or harmful to the normal operation of various systems, therefore they have been the subject of much research.
Initially, synchronization phenomena have been studied in systems with all-to-all (global) and nearest-neighbour (local) coupling, or on random networks. However, in recent decades there has been a lot of interest in more complicated coupling structures, which take into account the spatially distributed nature of real-world oscillator systems and the distance-dependent nature of the interaction between their components. Examples of such systems are abound in biology and neuroscience. They include spatially distributed cell populations, cilia carpets and neural networks relevant to working memory. In many cases, these systems support a rich variety of patterns of synchrony and disorder with remarkable properties that have not been observed in other continuous media. Such patterns are usually referred to as the coherence-incoherence patterns, but in symmetrically coupled oscillator systems they are also known by the name chimera states.
The main goal of this work is to give an overview of different types of collective behaviour in large networks of spatially distributed phase oscillators and to develop mathematical methods for their analysis. We focus on the Kuramoto models for one-, two- and three-dimensional oscillator arrays with nonlocal coupling, where the coupling extends over a range wider than nearest neighbour coupling and depends on separation. We use the fact that, for a special (but still quite general) phase interaction function, the long-term coarse-grained dynamics of the above systems can be described by a certain integro-differential equation that follows from the mathematical approach called the Ott-Antonsen theory. We show that this equation adequately represents all relevant patterns of synchrony and disorder, including stationary, periodically breathing and moving coherence-incoherence patterns. Moreover, we show that this equation can be used to completely solve the existence and stability problem for each of these patterns and to reliably predict their main properties in many application relevant situations.