  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 $\Theta_0(0)=0$. 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.
