Pigment color patterns of molluscs as an autonomous process generated by asynchronous automata

Pigment color patterns of molluscs are studied from the viewpoint of autonomy. Brownian algebra developed by Spencer-Brown (1969) is extensively used for the expression of cellular-automaton rules. When asynchronous updating is introduced for the transition of cellular automata, various kinds of patterns such as traveling waves, kinks, oscillatory local patterns etc. are generated from the same transitional rule. The type of patterns depends more sensitively on the asynchronous updating relationship rather than the transitional rule itself. Therefore, pattern changes in ontogeny can be explained without any changes in transitional rules or reaction processes. It is proposed that asynchronousness is intrinsic to living systems and that recognition of the intrinsic time is essential in understanding living systems.