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

Yu Qian, 1, 2 Xiaodong Huang, 1 Gang Hu, 1, ∗ and Xuhong Liao 1, † 

$^{1}$Department of Physics, Beijing Normal University, Beijing 100875, China 

$^{2}$Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 721007, China 

(Dated: November 5, 2018) 

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 novel 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 extremely high e ffi ciency. 

PACS numbers: : 82.40.Ck, 05.65. + b, 87.18.Bb, 89.75.Hc 

## I. INTRODUCTION 

Pattern dynamics in excitable media has attracted great attention in wide fields due to its relevance to various important systems, such as cardiac tissues and neural systems for typical examples [1–3]. Though single excitable cell is not oscillatory, organized oscillatory patterns, however, are extremely important issues in the media of coupled excitable cells [4–6]. Spiralwaves and target waves are two typical patterns in excitable tissues [1, 7]. The former can self-sustain in autonomous systems with the spiral tips serving as the oscillation sources while the latter can exist only by external pacing and can never exist in autonomous regular excitable media. So far most of researches studying pattern dynamics of excitable tissues have focused on locally and regularly coupled media, and little has been known on the influences of long-range or random links on the system dynamics. In many realistic systems of great importance, of which neural networks are typical examples, these long-range couplings do exist and they play crucial roles in deciding the functions of the systems [8–15]. Recently, A. Roxin et al. [16], S. Sinha et al. [17], A. J. Steele et al. [18] and Gray et al. [19] studied excitable systems in small-world networks and found some new types of self-sustained oscillations, including target waves in autonomous systems[18,19]. However, the 

