The main goal of the “top-down” work package was to combine tools from nonlinear dynamics and statistical physics to describe complex oscillatory systems by virtue the dimension-reduction techniques. A substantial progress has been achieved in this direction, resulted in about 18 publications in high-quality journals by ESRs. A very hot topic was study of chimera states, which combine properties of synchronous and asynchronous regimes. Here, ESR10 established properties of phase and generalized synchronization between chimera states, while ESR1 discovered and described a novel state of blinking chimera. In studies of neural field networks, ESR7 developed a diffusion approximation for a stochastic model of neural activity, and ESR12 juxtaposed properties of scale-freeness and of partial synchrony for neural mass networks. Effects of noise on the dynamics of complex oscillatory networks were in focus of several projects. ESR7 studied desynchronization and pattern formation in a noisy feed-forward oscillatory network. He also proposed and studied a simple stochastic model of neuronal excitatory and inhibitory interactions on a directed lattice. ESR2 discovered a nontrivial mechanism of noise-induced stabilization of collective dynamics in oscillator populations. In a more general setup, studies by ESRs 5 and 3 showed that nonautonomous systems can have properties quite different from that of standard autonomous ones. An important theoretical advance in the analysis of complex oscillatory networks was made by ESR2, who created a novel dimension reduction technique that includes not only the phases, but also variations of the amplitudes. ESRs 12 and 13 established existence of first-order phase transitions in the Kuramoto model with compact bimodal frequency distributions. Two other studies dealt with effect of delay. ESR11 described the dynamics of a large system of spiking neurons with synaptic delay. ESR3 found evidence of a critical phase transition in a purely temporal dynamics with a long-delayed feedback.
In the frame of “bottom-up” work package, methods of inference have been developed and applied to synthetic and real data. One overlapping topic was characterization of spike trains. In a series of papers, ESR17 developed methods of reconstructing networks of pulse- coupling oscillators, without any preliminary knowledge about their properties. Work of ESR8 was concentrated on the characterization of similarities and diversities in spike trains. Furthermore, he developed measures of spike train synchrony for data with multiple time scales, and quantified consistency in spatio-temporal propagation patterns. ESR10 characterized robustness and versatility of a nonlinear interdependence method for directional coupling detection from spike trains. Another overlapping topic was network reconstruction. Work of ESR4 concentrated on the analysis of degree distributions of reconstructed networks in dependence on the statistics of false positive and false negative
conclusions about the links. ESR14 and 10 collaboratively studied, how a rank-based connectivity measure can be used to infer the network connections. Furthermore, ESR14 developed a method of evolutionary optimization of network reconstruction from derivative-variable correlations. A problem of cardio-respiratory interactions was attacked by ESR16. He developed a method of disentangling respiratory and non-respiratory components in the heart phase dynamics, and applied this to real physiological measurements. Significance is very important in the practical data analysis. Here ESR6 contributed by comparing different surrogate data techniques for hypothesis testing. In the frame of integration and comparison of different approaches, the consortium delivered 9 publicly available software packages summarizing created methods and techniques.
The ITN-EJD has contributed to train specialists in complex oscillatory systems, able to cope with different challenging setups from physics, engineering and biology, to apply multidisciplinary approaches, and to communicate top-level science to general public. Throughout academic and industrial secondments, the ESRs gained a broad palette of research methods, with an additional focus on transferrable skills such as entrepreneurship, presentation and communication. They are well-prepared to cope with complex data appearing in different fields of science, technology, and society. Through close interaction between beneficiaries and industrial partners, the ESRs developed excellent networking potential and good chances of further career development.
