  In this paper a random graph model $G_{\mathbb{Z}^2_N,p_d}$ is introduced,
which is a combination of fixed torus grid edges in $(\mathbb{Z}/N
\mathbb{Z})^2$ and some additional random ones. The random edges are called
long, and the probability of having a long edge between vertices
$u,v\in(\mathbb{Z}/N \mathbb{Z})^2$ with graph distance $d$ on the torus grid
is $p_d=c/Nd$, where $c$ is some constant. We show that, {\em whp}, the
diameter $D(G_{\mathbb{Z}^2_N,p_d})=\Theta (\log N)$. Moreover, we consider
non-monotonous bootstrap percolation on $G_{\mathbb{Z}^2_N,p_d}$. We prove the
presence of phase transitions in mean-field approximation and provide fairly
sharp bounds on the error of the critical parameters. Our model addresses
interesting mathematical questions of non-monotonous bootstrap percolation, and
it is motivated by recent results of brain research.
