Stable Irregular Dynamics in Complex Neural Networks

For infinitely large sparse networks of spiking neurons mean field theory shows that a balanced state of highly irregular activity arises under various conditions. Here we analytically investigate the microscopic irregular dynamics in finite networks of arbitrary connectivity, keeping track of all individual spike times. For delayed, purely inhibitory interactions we demonstrate that the irregular dynamics is not chaotic but rather stable and convergent towards periodic orbits. Moreover, every generic periodic orbit of these dynamical systems is stable. These results highlight that chaotic and stable dynamics are equally capable of generating irregular activity.Comment: 10 pages, 2 figure

Iterative solutions to the steady state density matrix for optomechanical systems

We present a sparse matrix permutation from graph theory that gives stable incomplete Lower-Upper (LU) preconditioners necessary for iterative solutions to the steady state density matrix for quantum optomechanical systems. This reordering is efficient, adding little overhead to the computation, and results in a marked reduction in both memory and runtime requirements compared to other solution methods, with performance gains increasing with system size. Either of these benchmarks can be tuned via the preconditioner accuracy and solution tolerance. This reordering optimizes the condition number of the approximate inverse, and is the only method found to be stable at large Hilbert space dimensions. This allows for steady state solutions to otherwise intractable quantum optomechanical systems.Comment: 10 pages, 5 figure

Distributed LQR-based Suboptimal Control for Coupled Linear Systems

A well-established distributed LQR method for decoupled systems is extended to the dynamically coupled case where the open-loop systems are dynamically dependent. First, a fully centralized controller is designed which is subsequently substituted by a distributed state-feedback gain with sparse structure. The control scheme is obtained by optimizing an LQR performance index with a tuning parameter utilized to emphasize/de-emphasize relative state difference between interconnected systems. Overall stability is guaranteed via a simple test applied to a convex combination of Hurwitz matrices, the validity of which leads to stable global operation for a class of interconnection schemes. It is also shown that the suboptimality of the method can be assessed by measuring a certain distance between two positive definite matrices which can be obtained by solving two Lyapunov equations