Spin models of neural networks and genetic networks are considered elegant as they are accessible to statistical mechanics tools for spin glasses and magnetic systems. However, the conventional choice of variables in spin systems may cause problems in some models when parameter choices are unrealistic from a biological perspective. Obviously, this may limit the role of a model as a template model for biological systems. Perhaps less obviously, also ensembles of random networks are affected and may exhibit different critical properties. We consider here a prototypical network model that is biologically plausible in its local mechanisms. We study a discrete dynamical network with two characteristic properties: Nodes with binary states 0 and 1, and a modified threshold function with Θ(0)(0)=0. We explore the critical properties of random networks of such nodes and find a critical connectivity K(c)=2.0 with activity vanishing at the critical point. Finally, we observe that the present model allows a more natural implementation of recent models of budding yeast and fission yeast cell-cycle control networks.