In this paper we study damage spreading in a one-dimensional model under two dynamics introduced by Hinrichsen and Domany. In particular, we study the effects of synchronous and asynchronous updating on the spreading properties. We show that the damage does not spread when the second dynamic is implemented in a synchronous way. We find that the rules for updating the damage produced by this dynamics, as the temperature goes to infinity and a certain parameter lambda is zero, are equivalent to those of Grassberger's well-known model A cellular automaton.