Cyclops states in repulsive theta-neuron networks
- Authors: Bolotov M.I.1, Munyayev V.O.1, Smirnov L.A.1, Osipov G.V.1, Belykh I.2
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Affiliations:
- National Research Lobachevsky State University of Nizhny Novgorod
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University
- Issue: Vol 18, No 4 (2023)
- Pages: 844-846
- Section: Conference proceedings
- Submitted: 15.11.2023
- Accepted: 21.11.2023
- Published: 15.12.2023
- URL: https://genescells.ru/2313-1829/article/view/623432
- DOI: https://doi.org/10.17816/gc623432
- ID: 623432
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Abstract
Networks of phase oscillators have become a widely established paradigmatic model for studying emergent collective behavior across several real-world systems, including neuronal networks, populations of chemical oscillators, and power grids. The Kuramoto model, involving one-dimensional or two-dimensional phase oscillators, demonstrates the potential for networks to showcase exceptional collective dynamics. This encompasses various outcomes such as full, partial, explosive, and asymmetry-induced synchronization, clusters, chimeras, solitary states, and generalized splay states. Notably, increasing all-to-all coupling in the classical Kuramoto model induces full synchronization as the most probable outcome and dominant rhythm. Kuramoto networks with repulsive coupling usually display splay, generalized, and cluster splay states, but the conditions under which a certain rhythm can arise and prevail are not entirely understood.
Equally important for connecting Kuramoto networks to practical physical systems is understanding the function of higher-order coupling terms. These terms display a Fourier decomposition of a general 2π-periodic interaction function [1]. Previous studies have demonstrated that the inclusion of higher-order terms in the classical Kuramoto model of oscillators with all-to-all attractive coupling can lead to multiple synchronous states and switching between synchronization clusters. However, the impact of higher-order coupling modes on rhythm generation in repulsive networks remains unexplored.
In this work, we present significant progress in addressing the critical issue related to repulsive Kuramoto–Sakaguchi networks of phase oscillators with phase-lagged first-order and higher-order coupling. We demonstrate that weakly repulsive networks of even and odd numbers of oscillators with first-order coupling are dominated by two-cluster and three-cluster splay states, respectively. The three-cluster splay states consist of two distinct coherent clusters and one solitary oscillator. These tripod states can be considered a fusion of a two-body chimera and a solitary state. We have dubbed these patterns of three oscillators as “Cyclops states” in reference to the Greek mythological giant with a single eye. The solitary oscillator and synchronous clusters respectively represent the Cyclops’ eye and shoulders. We present a remarkable discovery that the inclusion of higher-order coupling modes leads to worldwide stability of cyclops states across almost the entire range of the phase-lag parameter controlling repulsion [2].
Beyond the Kuramoto oscillators, we demonstrate the robust presence of this effect in networks of canonical theta-neurons with adaptive coupling. Furthermore, our results provide insight into identifying dominant rhythms within repulsive physical and biological networks.
Full Text
Networks of phase oscillators have become a widely established paradigmatic model for studying emergent collective behavior across several real-world systems, including neuronal networks, populations of chemical oscillators, and power grids. The Kuramoto model, involving one-dimensional or two-dimensional phase oscillators, demonstrates the potential for networks to showcase exceptional collective dynamics. This encompasses various outcomes such as full, partial, explosive, and asymmetry-induced synchronization, clusters, chimeras, solitary states, and generalized splay states. Notably, increasing all-to-all coupling in the classical Kuramoto model induces full synchronization as the most probable outcome and dominant rhythm. Kuramoto networks with repulsive coupling usually display splay, generalized, and cluster splay states, but the conditions under which a certain rhythm can arise and prevail are not entirely understood.
Equally important for connecting Kuramoto networks to practical physical systems is understanding the function of higher-order coupling terms. These terms display a Fourier decomposition of a general 2π-periodic interaction function [1]. Previous studies have demonstrated that the inclusion of higher-order terms in the classical Kuramoto model of oscillators with all-to-all attractive coupling can lead to multiple synchronous states and switching between synchronization clusters. However, the impact of higher-order coupling modes on rhythm generation in repulsive networks remains unexplored.
In this work, we present significant progress in addressing the critical issue related to repulsive Kuramoto–Sakaguchi networks of phase oscillators with phase-lagged first-order and higher-order coupling. We demonstrate that weakly repulsive networks of even and odd numbers of oscillators with first-order coupling are dominated by two-cluster and three-cluster splay states, respectively. The three-cluster splay states consist of two distinct coherent clusters and one solitary oscillator. These tripod states can be considered a fusion of a two-body chimera and a solitary state. We have dubbed these patterns of three oscillators as “Cyclops states” in reference to the Greek mythological giant with a single eye. The solitary oscillator and synchronous clusters respectively represent the Cyclops’ eye and shoulders. We present a remarkable discovery that the inclusion of higher-order coupling modes leads to worldwide stability of cyclops states across almost the entire range of the phase-lag parameter controlling repulsion [2].
Beyond the Kuramoto oscillators, we demonstrate the robust presence of this effect in networks of canonical theta-neurons with adaptive coupling. Furthermore, our results provide insight into identifying dominant rhythms within repulsive physical and biological networks.
ADDITIONAL INFORMATION
Authors’ contribution. All authors made a substantial contribution to the conception of the work, acquisition, analysis, interpretation of data for the work, drafting and revising the work, final approval of the version to be published and agree to be accountable for all aspects of the work.
Funding sources. This study was supported by the Russian Science Foundation (grant No. 22-12-00348).
Competing interests. The authors declare that they have no competing interests.
About the authors
M. I. Bolotov
National Research Lobachevsky State University of Nizhny Novgorod
Author for correspondence.
Email: maxim.i.bolotov@gmail.com
Russian Federation, Nizhny Novgorod
V. O. Munyayev
National Research Lobachevsky State University of Nizhny Novgorod
Email: maxim.i.bolotov@gmail.com
Russian Federation, Nizhny Novgorod
L. A. Smirnov
National Research Lobachevsky State University of Nizhny Novgorod
Email: maxim.i.bolotov@gmail.com
Russian Federation, Nizhny Novgorod
G. V. Osipov
National Research Lobachevsky State University of Nizhny Novgorod
Email: maxim.i.bolotov@gmail.com
Russian Federation, Nizhny Novgorod
I. Belykh
Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University
Email: maxim.i.bolotov@gmail.com
United States, Atlanta
References
- Delabays R. Dynamical equivalence between Kuramoto models with first- and higher-order coupling. Chaos. 2019;29(11):113129. doi: 10.1063/1.5118941
- Munyayev VO, Bolotov MI, Smirnov LA, et al. Cyclops states in repulsive kuramoto networks: the role of higher-order coupling. Phys Rev Lett. 2023;130(10):107201. doi: 10.1103/PhysRevLett.130.107201
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