[-1,1]$. In this paper we shall use a uniform connectivity $k_i=K\,\,\forall\, i$. The magnetization $m$ is defined as $$m=\dfrac{1}{N}\sum_is_i,$$ with $m\in [-1,1]$.

In the following we consider multi-spin (plaquette) interactions. We moreover consider only completely asymmetric interactions [@Suzuki; @Sherrington], arranged to give a preferred direction that corresponds to time in the standard cellular automata language [@bagnoli2013; @SS-bif].

Considering up to 4-spin interactions, the Hamiltonian is $$\begin{aligned}
\label{Hamiltonian}
    \mathcal{H} (@s) =& -\sum_i s_i'\Bigl(\tilde H+ \tilde J\sum_j a_{ij} s_j +%
        \tilde Z \sum_{jk} a_{ij} a_{ik}s_j s_k + \nonumber\\
        &\tilde W \sum_{jkl} a_{ij} a_{ik} a_{il} s_j s_k s_l\Bigr),\end{aligned}$$ where $\tilde H$ is the external field, and $\tilde J$, $\tilde Z$, $\tilde W$ the two-spin, three-spin and four-spin interaction constants respectively as shown in Fig. \[plaquette\].

It is possible to recast the interaction constants in terms of the local field $h_i$. The terms containing $s'_i$ at “time” $t+1$ are, $$\begin{split}
&\text{2-spin: }\; s_i'\sum_j a_{ij} s_j = s_i' \tilde h_i,\\  
&\text{3-spin: }\; s_i'\sum_{jk} a_{ij} a_{jk}s_j s_k=s_i'Q^{(2)}_i,\\
&\text{4-spin: }\; s_i'\sum_{jkl} a_{ij} a_{ik} a_{il} s_j s_k s_l  = s_i'Q^{(3)}_i.\\
\end{split}$$ These expressions define $ Q^{(2)}$ and $ Q^{(3)}$. Since $$\begin{split}
&\tilde{h_i}^2 = K + 2Q_i^{(2)},\\
&\tilde{h_i}^3 =  (3K-2)\tilde h_i + 6 Q_i^{(3)},\\
\end{split}$$ the Hamiltonian can be 