Collective Oscillations in Coupled-Cell Systems

Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation

ABSTRACT

A fascinating phenomenon is the self-organization of coupled systems to a whole. This phenomenon is studied for a particular class of coupled oscillatory systems exhibiting so-called simultaneously diagonalizable matrices. For three exemplary systems, namely, an electric circuit, a coupled system of oscillatory neurons, and a system of coupled oscillatory gene regulatory pathways, eigenvectors and amplitude equations are derived. It is shown that for all three systems, only the unstable eigenvectors and their amplitudes matter for the dynamics of the systems on their respective limit cycle attractors. A general class of coupled second-order dynamical oscillators is presented in which stable limit cycles emerging via Hopf bifurcations are solely specified by appropriately defined unstable eigenvectors and their amplitudes. While the eigenvectors determine the orientation of limit cycles in state spaces, the amplitudes determine the evolution of states along those limit cycles. In doing so, it is shown that the unstable eigenvectors define reduced amplitude spaces in which the relevant long-term dynamics of the systems under consideration takes place. Several generalizations are discussed. First, if stable and unstable system parts exhibit a slow-fast dynamics, the fast variables may be eliminated and approximative descriptions of the emerging limit cycle dynamics in reduced amplitude spaces may be again obtained. Second, the principle of reduced amplitude spaces holds not only for coupled second-order oscillators, but can be applied to coupled third-order and higher order oscillators. Third, the possibility to apply the approach to multifrequency limit cycle attractors and other types of attractors is discussed.

Microfeature Segmentation Algorithm for Biological Images Using Improved Density Peak Clustering

To address the problem of low precision in feature segmentation of biological images with large noise, a microfeature segmentation algorithm for biological images using improved density peak clustering was proposed. First, the center pixel and edge information of a biological image were obtained to remove some redundant information. The three-dimensional space of the image is constructed, and the coordinate system is used to describe every superpixel of the biological image. Second, the image symmetry and reversibility are used to obtain the stopping position of pixels, other adjacent points are used to obtain the current color and shape information, and more vectors are used to express the density to complete the image pretreatment. Finally, the improved density peak clustering method is used to cluster the image, and the pixels completed by clustering and the remaining pixels are evenly distributed into the space to segment the image so as to complete the microfeature segmentation of the biological image based on the improved density peak clustering method. The results show that the proposed algorithm improves the segmentation efficiency, segmentation integrity rate, and segmentation accuracy. The time consumed by the proposed biological image microfeature segmentation algorithm is always less than 2 minutes, and the segmentation integrity rate can reach more than 90%. Furthermore, the proposed algorithm can reduce the missing condition and the noise of the segmented image and improve the image feature segmentation effect.