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Although cellular automata (CAs) were conceptualized as utter discrete mathematical models in which the states of all their spatial entities are updated simultaneously at every consecutive time step, i.e.

synchronously, various CA-based models that rely on so-called asynchronous update methods have been constructed in order to overcome the limitations that are tied up with the classical way of evolving CAs.

► Substantial discrepancies arise between synchronously and asynchronously updated CA.

► Discrepancies between different asynchronous update schemes are less pronounced.

The state of every cell in the model is updated together, before any of the new states influence other cells.We use Kauffman's approach to model GRNs with random Boolean networks (RBNs).We start this paper by proving that, if we fix the values of the source nodes (nodes with in-degree 0), the expected number of fixed points of any RBN is one (independently of the topology we choose).► The employed update method has repercussions on the stability of CAs.► A decision on the employed update method should be taken with care.

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