  We consider the problem of studying the simulation capabilities of the
dynamics of arbitrary networks of finite states machines. In these models, each
node of the network takes two states 0 (passive) and 1 (active). The states of
the nodes are updated in parallel following a local totalistic rule, i.e.,
depending only on the sum of active states. Four families of totalistic rules
are considered: linear or matrix defined rules (a node takes state 1 if each of
its neighbours is in state 1), threshold rules (a node takes state 1 if the sum
of its neighbours exceed a threshold), isolated rules (a node takes state 1 if
the sum of its neighbours equals to some single number) and interval rule (a
node takes state 1 if the sum of its neighbours belong to some discrete
interval). We focus in studying the simulation capabilities of the dynamics of
each of the latter classes. In particular, we show that totalistic automata
networks governed by matrix defined rules can only implement constant functions
and other matrix defined functions. In addition, we show that t by threshold
rules can generate any monotone Boolean functions. Finally, we show that
networks driven by isolated and the interval rules exhibit a very rich spectrum
of boolean functions as they can, in fact, implement any arbitrary Boolean
functions. We complement this results by studying experimentally the set of
different Boolean functions generated by totalistic rules on random graphs.
