Decomposing the dynamics of delayed networks: equilibria and rhythmic patterns in neural systems*

A method is presented that allows one to decompose the dynamics of networked dynamical systems with time-delayed coupling. The key idea is to “block-diagonalize” the system with the help of the eigenvectors of the coupling matrix. While large delayed networks are difficult to handle both analytically and numerically, the “small” blocks obtained by the decomposition can be investigated using standard methods. The proposed technique is applied to a neural network where the stability of the synchronized equilibria and periodic solutions are investigated. The methodology may also be applied for large engineered networks like electric power grids.