We demonstrate that chimera behavior can be observed in nonlocally coupled networks of excitable systems in the presence of noise. This phenomenon is distinct from classical chimeras, which occur in deterministic oscillatory systems, and it combines temporal features of coherence resonance, i.e., the constructive role of noise, and spatial properties of chimera states, i.e., coexistence of spatially coherent and incoherent domains in a network of identical elements. Coherence-resonance chimeras are associated with alternating switching of the location of coherent and incoherent domains, which might be relevant in neuronal networks. Chimera states are intriguing spatio-temporal patterns made up of spatially separated domains of synchronized (spatially coherent) and desynchronized (spatially incoherent) behavior, arising in networks of identical units. Originally discovered in a network of phase oscillators with a simple symmetric non-local coupling scheme [1,2], this sparked a tremendous activity of theoretical investigations . The first experimental evidence on chimera states was presented only one decade after their theoretical discovery [28][29][30][31][32][33][34][35][36][37][38]. In realworld systems chimera states might play a role, e.g., in power grids [39], in social systems [40], in the unihemispheric sleep of birds and dolphins [41], or in epileptic seizures [42]. In the context of the latter two applications it is especially relevant to explore chimera states in neuronal networks under conditions of excitability. However, while chimera states have previously been reported for neuronal networks in the oscillatory regime, e.g., in the FitzHugh-Nagumo system [17], they have not been detected in the excitable regime even for specially prepared initial conditions [17]. Therefore, the existence of chimera states for excitable elements remains unresolved.One of the challenging issues concerning chimera states is their behavior in the presence of random fluctuations, which are unavoidable in real-world systems. The robustness of chimeras with respect to external noise has been studied only very recently [43]. An even more intriguing question is whether the constructive role of noise in nonlinear systems, manifested for example in the counterintuitive increase of temporal coherence due to noise in coherence resonance [44][45][46][47], can be combined with the chimera behavior in spatially extended systems and networks. Coherence resonance, originally discovered for excitable systems like the FitzHugh-Nagumo model, implies that noise-induced oscillations become more regular for an optimum intermediate value of noise intensity. A question naturally arising in this context is whether noise can * corresponding author: anna.zakharova@tu-berlin.de also have a beneficial effect on chimera states. No evidence for the constructive role of noise for chimeras has been previously provided. Therefore, an important issue we aim to address here is to establish a connection between two intriguing counter-intuitive phenomena wh...

-We analyze nonlocally coupled networks of identical chaotic oscillators with either time-discrete or time-continuous dynamics (Henon map, Lozi map, Lorenz system). We hypothesize that chimera states, in which spatial domains of coherent (synchronous) and incoherent (desynchronized) dynamics coexist, can be obtained only in networks of nonhyperbolic chaotic systems and cannot be found in networks of hyperbolic systems. This hypothesis is supported by analytical results and numerical simulations for hyperbolic and nonhyperbolic cases.Introduction. -Chimera states in dynamical networks of nonlocally coupled chaotic oscillators have recently attracted much attention [1]. They represent an intriguing phenomenon where an ensemble of identical elements with symmetric coupling spontaneously splits into spatially separated coexisting domains of coherent (synchronized) and incoherent (desynchronized) dynamics. Since their first discovery in systems of coupled phase oscillators [2, 3] they have been found in a broad range of diverse models [1,4]

Using the model of a FitzHugh-Nagumo system in the excitable regime, we study the impact of multiplexing on coherence resonance in a two-layer network. We show that multiplexing allows for the control of the noise-induced dynamics. In particular, we find that multiplexing induces coherence resonance in networks that do not demonstrate this phenomenon in isolation. Examples are provided by deterministic networks and networks where the strength of interaction between the elements is not optimal for coherence resonance. In both cases, we show that the control strategy based on multiplexing can be successfully applied even for weak coupling between the layers. Moreover, for the case of deterministic networks, we obtain a counter-intuitive result: the multiplex-induced coherence resonance in the layer which is deterministic in isolation manifests itself even more strongly than that in the noisy layer.

We study numerically the dynamics of a network made of two coupled one-dimensional ensembles of discrete-time systems. The first ensemble is represented by a ring of nonlocally coupled Henon maps, and the second one -by a ring of nonlocally coupled Lozi maps. We find that the network of coupled ensembles can realize all the spatio-temporal structures which are observed both in the Henon map ensemble and in the Lozi map ensemble when uncoupled. Moreover, we reveal a new type of spatiotemporal structure, a solitary state chimera, in the considered network. We also establish and describe the effect of mutual synchronization of various complex spatiotemporal patterns in the system of two coupled ensembles of Henon and Lozi maps.PACS numbers: 05.45.-a, 02.60.-x Keywords: ensemble of nonlocally coupled oscillators, multilayer systems, chimera states, synchronization of spatiotemporal structures, synchronization region, inertial and dissipative couplingRecently studying the formation and evolution of various spatiotemporal patterns in ensembles or networks of coupled oscillators has become one of the most rapidly developing and highly attractive research topics in the nonlinear science and its applications. This exclusive interest is especially related to the discovery of a novel type of spatiotemporal structure -a chimera state. A lot of attention is paid to the dynamics of coupled ensembles of identical elements with various coupling topologies, but of particular interest are coupled ensembles with different types of network elements. In the latter case, the enrichment of regimes as well as the synchronization of spatiotemporal patterns is expected to be observed.In the present paper we analyze the spatiotemporal dynamics of a network made of two coupled rings of Henon and Lozi maps with nonlocal coupling. Our numerical studies have shown that this network can demonstrate both the spatiotemporal regimes, which are observed in separate rings, and a new type of chimera structure, called a solitary state chimera. We have also established the possibility of realizing the mutual synchronization of various complex spatiotemporal structures in the network of two coupled rings. The identity of synchronous patterns is confirmed by calculating the cross-correlation coefficient. The existence of a finite region of synchronization in the coupling parameters plane of the considered system is shown for an exemplary synchronous structure.

We describe numerical results for the dynamics of networks of nonlocally coupled chaotic maps. Switchings in time between amplitude and phase chimera states have been first established and studied. It has been shown that in autonomous ensembles, a nonstationary regime of switchings has a finite lifetime and represents a transient process towards a stationary regime of phase chimera. The lifetime of the nonstationary switching regime can be increased to infinity by applying short-term noise perturbations.

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We study and analyze the fundamental aspects of noise propagation in recurrent as well as deep, multi-layer networks. The main focus of our study are neural networks in analogue hardware, yet the methodology provides insight for networks in general. The system under study consists of noisy linear nodes, and we investigate the signal-to-noise ratio at the network's outputs which is the upper limit to such a system's computing accuracy. We consider additive and multiplicative noise which can be purely local as well as correlated across populations of neurons. This covers the chief internal-perturbations of hardware networks and noise amplitudes were obtained from a physically implemented recurrent neural network and therefore correspond to a real-world system. Analytic solutions agree exceptionally well with numerical data, enabling clear identification of the most critical components and aspects for noise management. Focusing on linear nodes isolates the impact of network connections and allows us to derive strategies for mitigating noise. Our work is the starting point in addressing this aspect of analogue neural networks, and our results identify notoriously sensitive points while simultaneously highlighting the robustness of such computational systems.

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