Unstable dynamics, nonequilibrium phases, and criticality in networked excitable media

Excitable systems are of great theoretical and practical interest in mathematics, physics, chemistry, and biology. Here, we numerically study models of excitable media, namely, networks whose nodes may occasionally be dormant and the connection weights are allowed to vary with the system activity on a short-time scale, which is a convenient and realistic representation. The resulting global activity is quite sensitive to stimuli and eventually becomes unstable also in the absence of any stimuli. Outstanding consequences of such unstable dynamics are the spontaneous occurrence of various nonequilibrium phases--including associative-memory phases and one in which the global activity wanders irregularly, e.g., chaotically among all or part of the dynamic attractors--and 1/f noise as the system is driven into the phase region corresponding to the most irregular behavior. A net result is resilience which results in an efficient search in the model attractor space that can explain the origin of some observed behavior in neural, genetic, and ill-condensed matter systems. By extensive computer simulation we also address a previously conjectured relation between observed power-law distributions and the possible occurrence of a "critical state" during functionality of, e.g., cortical networks, and describe the precise nature of such criticality in the model which may serve to guide future experiments.