Stability Analysis of Complex Networks with Linear-Threshold Rate Dynamics

Network models with linear-threshold rate dynamics have been widely used to explain the behavior of biological neural networks and replicate it using artificial neural networks. A full characterization of the stability properties of these networks, nevertheless, has remained elusive. This paper addresses the study of the existence and uniqueness of equilibria and asymptotic stability, leading to a thorough understanding of the conditions on the network structure that determine these properties. Given the stringency of these conditions for large-scale complex networks, we then study the stabilizability of linear-threshold network dynamics and show that, using either feedback or feedforward control, stabilization of the entire network is solely determined by the subnetwork of nodes that are not directly controlled. Illustrative examples demonstrate our results.