DFG-RSF: Complex dynamical networks: effects of heterogeneity, adaptivity, and delays
Final Report Abstract
Within the DFG-RSF project, both groups from Germany and the group from Russia collaborated successfully. In a series of publications, we unveil the self-organized emergence of hierarchical frequency-clusters in networks of adaptively coupled oscillators. This means that the dynamical network splits into groups or clusters of different sizes, each of which exhibits a common collective frequency of oscillation, but for different clusters the frequencies are different. Along with the characterization of the new phenomenon, an exhaustive analytical study was carried out, and novel methodological approaches for adaptive networks were developed. This study investigates in depth the cluster states, their existence and stability, and offers new perspectives on the mechanisms underlying the formation of clusters. From the viewpoint of application, already two major conclusions for neural networks with synaptic plasticity can be drawn. Firstly, the emergence of frequency-clusters in adaptive networks of spiking neurons can be qualitatively described by simple phase oscillator models. This statement was verified for a more sophisticated and realistic model of Hodgkin-Huxley neurons with Hebbian-like plasticity rule. Secondly, the effect of time scale separation on the dynamics of adaptive networks was analyzed. We found that there is an essential need for slow adaptation in a dynamical system in order to possess (stable) frequencyclusters. Thus, these dynamical states are of major importance when the oscillatory (or spiking) dynamics of the dynamical nodes is much faster than the adaptation of the coupling weights. Our novel results on adaptive dynamical networks, already, provide necessary insights in order to understand the pattern formation due to long-term synaptic plasticity as well as shed light on the dynamical mechanisms behind the hierarchical modular structure of the human brain. With this, we have started to close a major gap in the theory of neuromorphic network models. From the perspective of co-evolutionary networks, i.e., coupled dynamics of nodes and weights, our work on the emergence of hierarchical modular network structures uncovers a link between the structural properties of modular networks and functional activation processes on them. Our theoretical analysis further reveals the formation of non-local ring structures in the presence of causal adaptation rules such as spike timing-dependent plasticity. In addition to networks with adaptation, we made several contributions to the basic theory of phase-reduction and networks with time-delays. In particular, we extended the framework of phase response curves (PRC) to the case of strong or frequent pulses. Such an approach extends the applicability of the PRC modeling technique. Our results on networks with timedelays, e.g. a new destabilization scenarios, extend our knowledge in such a challenging subject. Furthermore, we investigated effects of the coupling topology upon relay synchronization in multilayer neuronal networks. In summary, the project contributes towards the understanding of the working principles of realistic dynamical networks, such as neuronal networks with plasticity, time-delays, and noise. The results establish important relations between the structural and functional properties of such networks.
Publications
- Embedding the dynamics of a single delay system into a feed-forward ring, Phys. Rev. E 96, 042217 (2017)
V. Klinshov, D. Shchapin, S. Yanchuk, M. Wolfrum, O. D’Huys, and V. Nekorkin
(See online at https://doi.org/10.1103/PhysRevE.96.042217) - Phase response function for oscillators with strong forcing or coupling, EPL (Europhysics Lett.) 118, 50006 (2017)
V. Klinshov, S. Yanchuk, A. Stephan, and V. Nekorkin
(See online at https://doi.org/10.1209/0295-5075/118/50006) - Self-organized emergence of multilayer structure and chimera states in dynamical networks with adaptive couplings, Phys. Rev. E 96, 062211 (2017)
D. V. Kasatkin, S. Yanchuk, E. Schöll, and V. I. Nekorkin
(See online at https://doi.org/10.1103/PhysRevE.96.062211) - Desynchronization by phase slip patterns in networks of pulse-coupled oscillators with delays, Eur. Phys. J. Spec. Top. 227, 1117 (2018)
V. Klinshov, L. Lücken, and S. Yanchuk
(See online at https://doi.org/10.1140/epjst/e2018-800073-7) - Frequency cluster formation and slow oscillations in neural populations with plasticity , PLoS ONE 14, e0225094 (2019)
V. Röhr, R. Berner, E. L. Lameu, O. V. Popovych, and S. Yanchuk
(See online at https://doi.org/10.1371/journal.pone.0225094) - Hierarchical frequency clusters in adaptive networks of phase oscillators, Chaos 29, 103134 (2019)
R. Berner, J. Fialkowski, D. V. Kasatkin, V. I. Nekorkin, S. Yanchuk, and E. Schöll
(See online at https://doi.org/10.1063/1.5097835) - Multiclusters in networks of adaptively coupled phase oscillators, SIAM J. Appl. Dyn. Syst. 18, 2227 (2019)
R. Berner, E. Schöll, and S. Yanchuk
(See online at https://doi.org/10.1137/18M1210150) - Birth and stabilization of phase clusters by multiplexing of adaptive networks, Phys. Rev. Lett. 124, 088301 (2020)
R. Berner, J. Sawicki, and E. Schöll
(See online at https://doi.org/10.1103/PhysRevLett.124.088301) - Solitary states in adaptive nonlocal oscillator networks, Eur. Phys. J. Spec. Top. (2020)
R. Berner, A. Polanska, E. Schöll, and S. Yanchuk
(See online at https://doi.org/10.1140/epjst/e2020-900253-0)
