A Fisher-gradient complexity in systems with spatio-temporal dynamics

•Statistical complexity measures (SCM) for spatio-temporal systems studied.•We define a benchmark of complexity.•We focus in particular on the Collective Motion model.•We analyse LMC’s, Autocorrelation, and Fisher-gradient complexities.

We define a benchmark for definitions of complexity in systems with spatio-temporal dynamics and employ it in the study of Collective Motion. We show that LMC’s complexity displays interesting properties in such systems, while a statistical complexity model (SCM) based on autocorrelation reasonably meets our perception of complexity. However this SCM is not as general as desirable, as it does not merely depend on the system’s Probability Distribution Function. Inspired by the notion of Fisher information, we develop a SCM candidate, which we call the Fisher-gradient complexity, which exhibits nice properties from the viewpoint of our benchmark.