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.

Figure 1

Phases of the cyclic CA. Parameters are as described in the text, started from uniform random initial conditions. Color figures were prepared with 28. (a) Local oscillations $(T=1)$, in which the CA oscillates with period 4, each cell cycling through all colors; (b) spiral waves $(T=2)$; (c) the “turbulent” phase $(T=3)$; (d) fixation with solid color blocks $(T=4)$.

Figure 2

Complexity over time for CCA with different thresholds $T$, averaging 30 independent simulations at each value of $T$. The $T=2$ curve has the highest asymptote, followed by $T=3$, $T=4$, and $T=1$. Error bars: standard error in the complexity.