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## Desynchronization Transitions in Adaptive Networks

Rico Berner, Simon Vock, Eckehard Schöll and Serhiy Yanchuk

Adaptive Networks change their connectivity with time, depending on their dynamical state. While synchronization in structurally static networks has been studied extensively, this problem is much more challenging for adaptive networks. In this Letter, we develop the master stability appoach for a large class of adaptive networks. This approach allows for reducing the synchronization problem for adaptive networks to a low-dimensional system, by decoupling topological and dynamical properties. We show how the interplay between adaptivity and network structure gives rise to the formation of stability islands. Moreover, we report a desynchronization transition and the emergence of complex partial synchronization patterns induced by an increasing overall coupling strength. We illustrate our findings using adaptive networks of coupled phase oscillators and FithHugh-Nagumo neurons with synaptic plasticity.

Reference: PRL 126, 028301 (2021)

## FitzHugh–Nagumo oscillators on complex networks mimic epileptic-seizure-related synchronization phenomena

Moritz Gerster, Rico Berner, Jakub Sawicki, Anna Zakharova, Antonín Škoch, Jaroslav Hlinka, Klaus Lehnertz and Eckehard Schöll

We study patterns of partial synchronization in a network of FitzHugh–Nagumo oscillators with empirical structural connectivity measured in human subjects. We report the spontaneous occurrence of synchronization phenomena that closely resemble the ones seen during epileptic seizures in humans. In order to obtain deeper insights into the interplay between dynamics and network topology, we perform long-term simulations of oscillatory dynamics on different paradigmatic network structures: random networks, regular nonlocally coupled ring networks, ring networks with fractal connectivities, and small-world networks with various rewiring probability. Among these networks, a small-world network with intermediate rewiring probability best mimics the findings achieved with the simulations using the empirical structural connectivity. For the other network topologies, either no spontaneously occurring epileptic-seizure-related synchronization phenomena can be observed in the simulated dynamics, or the overall degree of synchronization remains high throughout the simulation. This indicates that a topology with some balance between regularity and randomness favors the self-initiation and self-termination of episodes of seizure-like strong synchronization.

Reference: Chaos 30, 123130 (2020)

Selected as Editor's Pick and highlighted on Journal Cover

## Invited Keynote Talk at the G-RISC International Student Conference Science and Progress 2020, St. Petersburg, Russia

11.11.2020

Eckehard Schöll: Physics and Control - interplay of nonlinear dynamics, complex network topology, and time delay

## Solitary states in adaptive nonlocal oscillator networks

Rico Berner, Alicja Polanska, Eckehard Schöll and Serhiy Yanchuk

In this article, we analyze a nonlocal ring network of adaptively coupled phase oscillators. We oberseve a variety of frequencysynchronized states such as phase-locked, multicluster and solitary states. For an important subclass of the phase-locked solutions, the rotating waves, we provide a rigorous stability analysis. This analysis shows a strong dependence of their stability on the coupling structure and the wavenumber which is a remarkable difference to an all-to-all coupled network. Despite the fact that solitary states have been observed in a plethora of dynamical systems, the mechanisms behind their emergence were largely unadressed in the literature. Here, we show how solitary states emerge due to the adaptive feature of the network and classify several bifurcation scenarios in which these states are created and stabilized.

Reference: Eur. Phys. J. Spec. Top. (2020)

## Invited Talk at the Institute for Basic Science, Research Center Daejeon, South Korea

21.07.2020

*"Partial synchronization patterns in brain networks - interplay of dynamics, delay, and network topology*"

## Invited Talk at the Annual Meeting of the National Academy of Sciences Leopoldina

Halle 2019

*"Chimeras in Physics and Biology: Synchronization and Desynchronization of Rhythms"*

## Birth and stabilization of phase clusters by multiplexing of adaptive networks

Rico Berner, Jakub Sawicki, and Eckehard Schöll

We propose a concept to generate and stabilize diverse partial synchronization patterns (phase clusters) in adaptive networks which are widespread in neuro- and social sciences, as well as biology, engineering, and other disciplines. We show by theoretical analysis and computer simulations that multiplexing in a multilayer network with symmetry can induce various stable phase cluster states in a situation where they are not stable or do not even exist in the single layer. Further, we develop a method for the analysis of Laplacian matrices of multiplex networks which allows for insight into the spectral structure of these networks enabling a reduction to the stability problem of single layers. We employ the multiplex decomposition to provide analytic results for the stability of the multilayer patterns. As local dynamics we use the paradigmatic Kuramoto phase oscillator, which is a simple generic model and has been successfully applied in the modeling of synchronization phenomena in a wide range of natural and technological systems.

Reference: Phys. Rev. Lett. (2020)

## Partial synchronization in empirical brain networks as a model for unihemispheric sleep

Lukas Ramlow, Jakub Sawicki, Anna Zakharova, Jaroslav Hlinka, Jens Christian Claussen and Eckehard Schöll

We analyze partial synchronization patterns in a network of FitzHugh-Nagumo os- cillators with empirical structural connectivity measured in healthy human subjects. We report a dynamical asymmetry between the hemispheres, induced by the natural structural asymmetry. We show that the dynamical asymmetry can be enhanced by introducing the inter-hemispheric cou- pling strength as a control parameter for partial synchronization patterns. We discuss a minimum model elucidating the modalities of unihemispheric sleep in human brain, where one hemisphere sleeps while the other remains awake. In fact, this state is common among migratory birds and mammals like aquatic species.

Reference: Europhys. Lett. (2019), Featured on Phys.org

## Coherence-Resonance Chimeras in a Network of Excitable Elements

Nadezhda Semenova, Anna Zakharova, Vadim Anishchenko, and Eckehard Schöll

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.

Reference: Phys. Rev. Lett. 117, 014102 (2016)

## Tweezers for Chimeras in Small Networks

Iryna Omelchenko, Oleh E. Omel'chenko, Anna Zakharova, Matthias Wolfrum, and Eckehard Schöll

We propose a control scheme which can stabilize and fix the position of chimera states in small networks. Chimeras consist of coexisting domains of spatially coherent and incoherent dynamics in systems of nonlocally coupled identical oscillators. Chimera states are generally difficult to observe in small networks due to their short lifetime and erratic drifting of the spatial position of the incoherent domain. The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized.

Reference: Phys. Rev. Lett. 116, 114101 (2016)

## Chimera Death: Symmetry Breaking in Dynamical Networks

Anna Zakharova, Marie Kapeller, and Eckehard Schöll

For a network of generic oscillators with nonlocal topology and symmetry breaking coupling we establish novel partially coherent inhomogeneous spatial patterns, which combine the features of chimera states (coexisting incongruous coherent and incoherent domains) and oscillation death (oscillation suppression), which we call “chimera death”. We show that due to the interplay of nonlocality and breaking of rotational symmetry by the coupling, two distinct scenarios from oscillatory behavior to a stationary state regime are possible: a transition from an amplitude chimera to chimera death via in-phase synchronized oscillations and a direct abrupt transition for larger coupling strength.

Reference: Phys. Rev. Lett. 112, 154101 (2014)

## When Nonlocal Coupling between Oscillators Becomes Stronger: Patched Synchrony or Multichimera States

Iryna Omelchenko, Oleh E. Omel’chenko, Philipp Hövel, and Eckehard Schöll

Systems of nonlocally coupled oscillators can exhibit complex spatiotemporal patterns, called chimera states, which consist of coexisting domains of spatially coherent (synchronized) and incoherent dynamics. We report on a novel form of these states, found in a widely used model of a limit-cycle oscillator if one goes beyond the limit of weak coupling typical for phase oscillators. Then patches of synchronized dynamics appear within the incoherent domain giving rise to a multi-chimera state. We find that, depending on the coupling strength and range, different multichimera states arise in a transition from classical chimera states. The additional spatial modulation is due to strong coupling interaction and thus cannot be observed in simple phase-oscillator models.

Reference: Phys. Rev. Lett. 110, 224101 (2013)

## Control of Synchronization Patterns in Neural-like Boolean Networks

David P. Rosin, Damien Rontani, Daniel J. Gauthier, and Eckehard Schöll

We study experimentally the synchronization patterns in time-delayed directed Boolean networks of excitable systems. We observe a transition in the network dynamics when the refractory time of the individual systems is adjusted. When the refractory time is on the same order of magnitude as the mean link time delays or the heterogeneities of the link time delays, cluster synchronization patterns change, or are suppressed entirely, respectively. We also show that these transitions occur when we change the properties of only a small number of driver nodes identified by their larger in degree; hence, the synchronization patterns can be controlled locally by these nodes. Our findings have implications for synchronization in biological neural networks.

Reference: Phys. Rev. Lett. 110, 104102 (2013)

## Experimental Observation of Chimeras in Coupled-Map Lattices

Aaron M. Hagerstrom, Thomas E. Murphy, Rajarshi Roy, Philipp Hövel, Iryna Omelchenko and Eckehard Schöll

Networks of nonlocally coupled phase oscillators can support chimera states in which identical oscillators evolve into distinct groups that exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Similar nonlocal coupling topologies implemented in networks of chaotic iterated maps also yield dynamical states displaying coexisting spatial domains of coherence and incoherence. In these discrete-time systems, the phase is not a continuous variable, so these states are generalized chimeras with respect to a broader notion of incoherence. Chimeras continue to be the subject of intense theoretical investigation, but have yet to be realized experimentally. Here we show that these chimeras can be realized in experiments using a liquid crystal spatial light modulator to achieve optical nonlinearity in a spatially extended iterated map system. We study the coherence-incoherence transition that gives rise to these chimera states through experiment, theory, and simulation.

Reference: Nature Physics 8, 658 (2012)

Highlighted in Physics Today (2012)

Also highlighted in Physics Today, Search and Discovery (2012)

## Dynamics, control and information in delay-coupled systems

The International Conference on Delayed Complex Systems held from 4 to 8 June 2012 at the Institute for Cross-Disciplinary Physics and Complex Systems (IFISC) in Palma de Mallorca, Spain, and co-organized with the Collaborative Research Center SFB 910 Control of Self-Organizing Nonlinear Systems—Theoretical Methods and Concepts of Application, Berlin, provided a forum for such topics. We took that opportunity to assemble a list of world leading experts, which now enables us to present perspectives of the state of the art in this field. This Theme Issue covers both applications and experiments, as well as mathematical foundations. The individual contributions summarize recent research results, but also address the broader context. Thus, the presentation is kept accessible for a large audience. The 14 articles cover various aspects of delay dynamics, control and information, ranging from fundamental mathematical aspects via delayed networks and time-delayed feedback control, to applications in neural science, optoelectronics and genetic control in cells. The articles are grouped into four parts.

http://rsta.royalsocietypublishing.org/content/371/1999.toc

## Loss of Coherence in Dynamical Networks: Spatial Chaos and Chimera States

Iryna Omelchenko, Yuri Maistrenko, Philipp Hövel, and Eckehard Schöll

We discuss the breakdown of spatial coherence in networks of coupled oscillators with nonlocal interaction. By systematically analyzing the dependence of the spatiotemporal dynamics on the range and strength of coupling, we uncover a dynamical bifurcation scenario for the coherence-incoherence transition which starts with the appearance of narrow layers of incoherence occupying eventually the whole space. Our findings for coupled chaotic and periodic maps as well as for time-continuous Rossler systems reveal that intermediate, partially coherent states represent characteristic spatiotemporal patterns at the transition from coherence to incoherence.

Reference: Phys. Rev. Lett. 106, 234102 (2011)