We investigate chimera states in a ring of identical phase oscillators coupled in a time-delayed and spatially nonlocal fashion. We find novel clustered chimera states that have spatially distributed phase coherence separated by incoherence with adjacent coherent regions in antiphase. The existence of such time-delay induced phase clustering is further supported through solutions of a generalized functional self-consistency equation of the mean field. Our results highlight an additional mechanism for cluster formation that may find wider practical applications.

Coupled oscillators are shown to experience amplitude death for a much larger set of parameter values when they are connected with time delays distributed over an interval rather than concentrated at a point. Distributed delays enlarge and merge death islands in the parameter space. Furthermore, when the variance of the distribution is larger than a threshold the death region becomes unbounded and amplitude death can occur for any average value of delay. These phenomena are observed even with a small spread of delays, for different distribution functions, and an arbitrary number of oscillators. (http://link.aps.org/abstract/PRL/v91/e094101) Coupled oscillators constitute an effective and popular paradigm for the study of interacting oscillatory processes in the physical and biological sciences [1]. The rich dynamics arising from the interaction of simple units have been a source of interest for scientists modeling the collective behavior of real-life systems. Among the most widely studied phenomena is synchronization, where individual units oscillate at a common frequency and phase when coupled [2]. Synchronization may be observed even under weak coupling; so it has usually been studied through reduced models that retain only the phase information along the limit cycles. With stronger coupling further interesting behavior is possible, whose investigation requires the use of full models that include the amplitudes of the oscillators. An example is amplitude death, which refers to the quenching of oscillations under coupling, as the system evolves to a stable equilibrium [3,4]. If the information flow through the coupled system is instantaneous, amplitude death occurs when the individual oscillators have sufficiently different frequencies. [5,6,7]. On the other hand, if the information from one oscillator reaches the others after a certain time delay, which may be due to finite propagation or information processing speeds, then even identical oscillators can experience amplitude death when coupled [8]. Recent experimental and theoretical studies have confirmed the role of delays in inducing amplitude death [9,10,11].While the importance of time delays in amplitude death is now clear, studies in this area have so far been confined only to discrete, or constant, delays. In other words, it has been assumed that information reaches from one unit to another after a fixed time τ which is unchanging as the system evolves, and moreover, the units act only on the instantaneous value of the received information and forget any previous values. Such discrete-delay models often fail to adequately describe physical systems by neglecting the possibilities that (a) the quantity τ may only be approximately known, (b) it may only represent an average value of a quantity that varies between pairs of oscillators in a network or (c) varies in time through a process involving unmodelled factors, and (d) the oscillators may incorporate "memory" effects by using the past history of the received information. The first possibility is certa...

We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scalefree and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants generally exhibit poor synchronization. On the other hand, connection delays can actually be conducive to synchronization, so that it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior which the individual units are incapable of producing in isolation. (http://link.aps.org/abstract/PRL/v92/e144101) Recent years have witnessed a growing interest in the dynamics of interacting units. Particularly, a large number of studies have been devoted to synchronization in a variety of systems (see [1] and the references therein), including the coupled map lattices introduced by Kaneko [2]. Usually, such systems have been investigated under the assumption of a certain regularity in the connection topology, where units are coupled to their nearest neighbors or to all other units. Lately, more general networks with random, small-world, scale-free, and hierarchical architectures have been emphasized as appropriate models of interaction [3,4,5,6,7]. On the other hand, realistic modeling of many large networks with non-local interaction inevitably requires connection delays to be taken into account, since they naturally arise as a consequence of finite information transmission and processing speeds among the units. Some numerical studies have regarded synchronization under delays for special cases such as globally coupled logistic maps [8] or carefully chosen delays [9]. In this Letter we consider synchronization of coupled chaotic maps for general network architectures and connection delays. Because of the presence of the delays, the constituent units are unaware of the present state of the others; so it is not evident a priori that such a collection of chaotic units can operate in unison, i. e. synchronize. Based on analytical calculations, we show that this is indeed possible, and in fact may be facilitated by the presence of delays. Moreover, while the connection topology is important for synchronization, the delays have a crucial role in determining the resulting collective dynamics. As a result, the synchronized system can exhibit a plethora of new behavior in the presence of delays. We illustrate the results by numerical simulation of large networks of logistic maps.

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Stimulation with rhythmic light flicker (photic driving) plays an important role in the diagnosis of schizophrenia, mood disorder, migraine, and epilepsy. In particular, the adjustment of spontaneous brain rhythms to the stimulus frequency (entrainment) is used to assess the functional flexibility of the brain. We aim to gain deeper understanding of the mechanisms underlying this technique and to predict the effects of stimulus frequency and intensity. For this purpose, a modified Jansen and Rit neural mass model (NMM) of a cortical circuit is used. This mean field model has been designed to strike a balance between mathematical simplicity and biological plausibility. We reproduced the entrainment phenomenon observed in EEG during a photic driving experiment. More generally, we demonstrate that such a single area model can already yield very complex dynamics, including chaos, for biologically plausible parameter ranges. We chart the entire parameter space by means of characteristic Lyapunov spectra and Kaplan-Yorke dimension as well as time series and power spectra. Rhythmic and chaotic brain states were found virtually next to each other, such that small parameter changes can give rise to switching from one to another. Strikingly, this characteristic pattern of unpredictability generated by the model was matched to the experimental data with reasonable accuracy. These findings confirm that the NMM is a useful model of brain dynamics during photic driving. In this context, it can be used to study the mechanisms of, for example, perception and epileptic seizure generation. In particular, it enabled us to make predictions regarding the stimulus amplitude in further experiments for improving the entrainment effect.

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