Extended self organised criticality in asynchronously tuned cellular automata

Systems at a critical point in phase transitions can be regarded as being relevant to biological complex behaviour. Such a perspective can only result, in a mathematical consistent manner, from a recursive structure. We implement a recursive structure based on updating by asynchronously tuned elementary cellular automata (AT ECA), and show that a large class of elementary cellular automata (ECA) can reveal critical behavior due to the asynchronous updating and tuning. We show that the obtained criticality coincides with the criticality in phase transitions of asynchronous ECA with respect to density decay, and that multiple distributed ECAs, synchronously updated, can emulate critical behavior in AT_ECA. Our approach draws on concepts and tools from category and set theory, in particular on "adjunction dualities" of pairs of adjoint functors.