"Critical" has a precise meaning in these kinds of systems; it essentially means when correlation lengths diverge (or, with a finite brain, become the size of the whole). In physical systems this happens at 2nd order phase transitions. Unfortunately most familiar phase transitions are first-order (boiling and freezing, for example) but the development of macroscopic magnetism as iron cools is an example.
Away from the critical point the dynamics become either
1) too strong, meaning that information has a hard time getting from one spot to another because it's in too much traffic
2) too weak, meaning that information has trouble being processed, because the traffic is so light that cars don't spend enough time together to come to equilibrium
In these cases the correlations will be short-distance, in contrast to critical.
Away from the critical point the dynamics become either 1) too strong, meaning that information has a hard time getting from one spot to another because it's in too much traffic 2) too weak, meaning that information has trouble being processed, because the traffic is so light that cars don't spend enough time together to come to equilibrium
In these cases the correlations will be short-distance, in contrast to critical.