Structure and control of self-sustained target waves in excitable small-world networks

Abstract

Small-world networks describe many important practical systems among which neural networks consisting of excitable nodes are the most typical ones. In this paper we study self-sustained oscillations of target waves in excitable small-world networks. A dominant phase-advanced driving (DPAD) method, which is generally applicable for analyzing all oscillatory complex networks consisting of nonoscillatory nodes, is proposed to reveal the self-organized structures supporting this type of oscillations. The DPAD method explicitly explores the oscillation sources and wave propagation paths of the systems, which are otherwise deeply hidden in the complicated patterns of randomly distributed target groups. Based on the understanding of the self-organized structure, the oscillatory patterns can be controlled with an extremely high efficiency.

Topological control of synchronous patterns in systems of networked chaotic oscillators

“…4(a), this network owns two reflection symmetries: S 1 and S 2 . A checking of their eigenvalues shows that only S 1 satisfies the eigenvalue condition λ = 0.5, to make the pattern stable, the coupling strength should be chosen within the range ε ∈ (12,20). By ε = 12.8, we plot in Fig.…”

“…For instance, it is found that by intentionally removing a few of the network links at the beginning of the cascading, the network damage can be largely reduced [6]. Besides cascading, recently the idea of topological control has been also employed in many other problems in network science, e.g., epidemic propagation [7], global synchronization [8], oscillatory patterns [9], and control optimization [10], etc.…”

“…This is partially due to the fact that the synchronous patterns, if existing, are highly fragmented and scattered, making them difficult to be figured out from the complex network [18,20]; and also because that the patterns are highly dynamic and unstable, making them difficult to be manipulated [21,22]. Regarding these difficulties, to investigate the control of synchronous patterns in complex systems, a plausible and meaningful approach would be adopting the simplified network structures that capture some essential features of the general complex networks, e.g., regular networks with a few of random shortcut links.…”

Sustained dynamics of a weakly excitable system with nonlocal interactions

“…Network connections of nonoscillatory elements, for instance, give rise to such dynamics and have been studied in various contexts, including gene networks [1], epidemic spreading dynamics [2][3][4][5], and generic excitable units [6][7][8][9][10], Excitable units undergo oscillations in many ways: Simple two excitable systems can exhibit sustained dynamics when delay-coupled [11]; spatially extended excitable media can produce sustained spiral waves by introducing a perturbation leading to the formation of a spiral core [12]; one can even consider interactions at a distance through nonlocal links embedded in spatially extended systems [13][14][15][16], which eventually forms network structures composed of wave propagation and nonlocal interactions.…”

Oscillation sources and wave propagation paths in complex networks consisting of excitable nodes

“…In our previous papers [16,17], we proposed a dominant phase-advanced driving (DPAD) method to reveal the underlying dynamic structure of self-sustained target waves in small-world networks of excitable nodes. Based on the information embedded in the DPAD structures, we successfully revealed the oscillation source.…”