  The $\beta$-functions of O(N) and U(N) invariant Grosse-Wulkenhaar models are
computed at one loop using the matrix basis. In particular, for ``parallel
interactions", the model is proved asymptotically free in the UV limit for $N
>1$, and has a triviality problem or Landau ghost for $N<1$. The vanishing
$\beta$-function is recovered solely at N=1. We discuss various possible
consequences of these results.
