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In this work, we apply the spatial recurrence quantification analysis (RQA) to identify chaotic burst phase synchronisation in networks. We consider one neural network with small-world topology and another one composed of small-world subnetworks. The neuron dynamics is described by the Rulkov map, which is a two-dimensional map that has been used to model chaotic bursting neurons. We show that with the use of spatial RQA, it is possible to identify groups of synchronised neurons and determine their size. For the single network, we obtain an analytical expression for the spatial recurrence rate using a Gaussian approximation. In clustered networks, the spatial RQA allows the identification of phase synchronisation among neurons within and between the subnetworks. Our results imply that RQA can serve as a useful tool for studying phase synchronisation even in networks of networks.

The field of dynamical systems with time delay is an active research area that connects practically all scientific disciplines including mathematics, physics, engineering, biology, neuroscience, physiology, economics, and many others. This Focus Issue brings together contributions from both experimental and theoretical groups and emphasizes a large variety of applications. In particular, lasers and optoelectronic oscillators subject to time-delayed feedbacks have been explored by several authors for their specific dynamical output, but also because they are ideal test-beds for experimental studies of delay induced phenomena. Topics include the control of cavity solitons, as light spots in spatially extended systems, new devices for chaos communication or random number generation, higher order locking phenomena between delay and laser oscillation period, and systematic bifurcation studies of mode-locked laser systems. Moreover, two original theoretical approaches are explored for the so-called Low Frequency Fluctuations, a particular chaotical regime in laser output which has attracted a lot of interest for more than 30 years. Current hot problems such as the synchronization properties of networks of delay-coupled units, novel stabilization techniques, and the large delay limit of a delay differential equation are also addressed in this special issue. In addition, analytical and numerical tools for bifurcation problems with or without noise and two reviews on concrete questions are proposed. The first review deals with the rich dynamics of simple delay climate models for El Nino Southern Oscillations, and the second review concentrates on neuromorphic photonic circuits where optical elements are used to emulate spiking neurons. Finally, two interesting biological problems are considered in this Focus Issue, namely, multi-strain epidemic models and the interaction of glucose and insulin for more effective treatment., SCOPUS: ar.j, info:eu-repo/semantics/published

We consider the recurrent pulse-coupled networks of excitable elements with delayed connections, which are inspired by the biological neural networks. If the delays are tuned appropriately, the network can either stay in the steady resting state, or alternatively, exhibit a desired spiking pattern. It is shown that such a network can be used as a pattern-recognition system. More specifically, the application of the correct pattern as an external input to the network leads to a self-sustained reverberation of the encoded pattern. In terms of the coupling structure, the tolerance and the refractory time of the individual systems, we determine the conditions for the uniqueness of the sustained activity, i.e., for the functionality of the network as an unambiguous pattern detector. We point out the relation of the considered systems with cyclic polychronous groups and show how the assumed delay configurations may arise in a self-organized manner when a spike-time dependent plasticity of the connection delays is assumed. As excitable elements, we employ the simplistic coincidence detector models as well as the Hodgkin-Huxley neuron models. Moreover, the system is implemented experimentally on a Field-Programmable Gate Array.

We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures, and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators.