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      Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators

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          Abstract

          A chimera state is a spatio-temporal pattern in a network of identical coupled oscillators in which synchronous and asynchronous oscillation coexist. This state of broken symmetry, which usually coexists with a stable spatially symmetric state, has intrigued the nonlinear dynamics community since its discovery in the early 2000s. Recent experiments have led to increasing interest in the origin and dynamics of these states. Here we review the history of research on chimera states and highlight major advances in understanding their behaviour.

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          Low dimensional behavior of large systems of globally coupled oscillators.

          It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered. (c) 2008 American Institute of Physics.
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            Synaptic mechanisms and network dynamics underlying spatial working memory in a cortical network model.

            Single-neuron recordings from behaving primates have established a link between working memory processes and information-specific neuronal persistent activity in the prefrontal cortex. Using a network model endowed with a columnar architecture and based on the physiological properties of cortical neurons and synapses, we have examined the synaptic mechanisms of selective persistent activity underlying spatial working memory in the prefrontal cortex. Our model reproduces the phenomenology of the oculomotor delayed-response experiment of Funahashi et al. (S. Funahashi, C.J. Bruce and P.S. Goldman-Rakic, Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex. J Neurophysiol 61:331-349, 1989). To observe stable spontaneous and persistent activity, we find that recurrent synaptic excitation should be primarily mediated by NMDA receptors, and that overall recurrent synaptic interactions should be dominated by inhibition. Isodirectional tuning of adjacent pyramidal cells and interneurons can be accounted for by a structured pyramid-to-interneuron connectivity. Robust memory storage against random drift of the tuned persistent activity and against distractors (intervening stimuli during the delay period) may be enhanced by neuromodulation of recurrent synapses. Experimentally testable predictions concerning the neural basis of working memory are discussed.
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              Synchronization in Complex Oscillator Networks and Smart Grids

              The emergence of synchronization in a network of coupled oscillators is a fascinating topic in various scientific disciplines. A coupled oscillator network is characterized by a population of heterogeneous oscillators and a graph describing the interaction among them. It is known that a strongly coupled and sufficiently homogeneous network synchronizes, but the exact threshold from incoherence to synchrony is unknown. Here we present a novel, concise, and closed-form condition for synchronization of the fully nonlinear, non-equilibrium, and dynamic network. Our synchronization condition can be stated elegantly in terms of the network topology and parameters, or equivalently in terms of an intuitive, linear, and static auxiliary system. Our results significantly improve upon the existing conditions advocated thus far, they are provably exact for various interesting network topologies and parameters, they are statistically correct for almost all networks, and they can be applied equally to synchronization phenomena arising in physics and biology as well as in engineered oscillator networks such as electric power networks. We illustrate the validity, the accuracy, and the practical applicability of our results in complex networks scenarios and in smart grid applications.
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                Author and article information

                Journal
                2014-03-24
                2015-02-18
                Article
                10.1088/0951-7715/28/3/R67
                1403.6204

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                34C15, 34C23, 35B36, 82C22, 82C44
                Nonlinearity 28, R67-R87 (2015)
                26 pages, 3 figures
                nlin.CD math-ph math.MP nlin.PS

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