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Oscillations are a prominent feature of neuronal activity and are associated with a variety of phenomena in brain tissue, both healthy and unhealthy. Characterizing how oscillations spread through regions of the brain is of particular interest when studying countermeasures to pathological brain synchronizations. This paper models neuronal activity using networks of interconnected excitatoryinhibitory pairs with linear threshold dynamics, and presents strategies to design networks with desired robustness properties. In particular, we develop a dynamical description of the brain through a network where the state of each node models the firing rate of a region of neurons and where edges capture the structural connectivity between the regions. We characterize the presence of oscillations and study conditions on their spreading. We also discuss strategies to optimally design networks which are robust to oscillation spreading. We demonstrate our results with numerical simulations.

This paper considers the problem of designing a continuous time dynamical system to solve constrained nonlinear optimization problems such that the feasible set is forward invariant and asymptotically stable. The invariance of the feasible set makes the dynamics anytime, when viewed as an algorithm, meaning that it is guaranteed to return a feasible solution regardless of when it is terminated. The system is obtained by augmenting the gradient flow of the objective function with inputs, then designing a feedback controller to keep the state evolution within the feasible set using techniques from the theory of control barrier functions. The equilibria of the system correspond exactly to critical points of the optimization problem. Since the state of the system corresponds to the primal optimizer, and the steady-state input at equilibria corresponds to the dual optimizer, the method can be interpreted as a primal-dual approach. The resulting closed-loop system is locally Lipschitz continuous, so classical solutions to the system exist. We characterize conditions under which local minimizers are Lyapunov stable, drawing connections between various constraint qualification conditions and the stability of the local minimizer. The algorithm is compared to other continuous time methods for optimization.

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