Quantifying self-organization with optimal predictors

Despite broad interest in self-organizing systems, there are few quantitative, experimentally applicable criteria for self-organization. The existing criteria all give counter-intuitive results for important cases. In this Letter, we propose a new criterion, namely, an internally generated increase in the statistical complexity, the amount of information required for optimal prediction of the system's dynamics. We precisely define this complexity for spatially extended dynamical systems, using the probabilistic ideas of mutual information and minimal sufficient statistics. This leads to a general method for predicting such systems and a simple algorithm for estimating statistical complexity. The results of applying this algorithm to a class of models of excitable media (cyclic cellular automata) strongly support our proposal.