Information Integration in Elementary Cellular Automata

We study the emergence of information integration in cellular automata (CA) with respect to states in the long run. Information integration is in this case quantified by applying the information-theoretic measure known as total correlation to the long-run distribution of CA states. Total correlation is the amount by which the total uncertainty associated with cell states surpasses the uncertainty of the CA state taken as a whole. It is an emergent property, in the sense that it can only be ascribed to how the cells interact with one another, and has been linked to the rise of consciousness in the brain. We investigate total correlation in the evolution of elementary CA for all update rules that are unique with respect to negation or reflection. For each rule we consider the usual, deterministic CA behavior, assuming that the initial state is chosen uniformly at random, and also the probabilistic variant in which every cell, at all time steps and independently of all others, disobeys the rule's prescription with a fixed probability. We have found rules that generate as much total correlation as possible, or nearly so, particularly in Wolfram classes 2 and 3. We conjecture that some of these rules can be used as CA models of information integration.