Boolean networks have been extensively studied, however I didn't find a reference to the following problem. May be you can provide one, or give hints to a possible solution.
Define K Boolean variables $x_i \in \{-1, 1 \}$ and a K-ary Boolean function of threshold type, $f(\{x_i\}) = {\rm sign} ({\sum_{i=1}^K w_i x_i})$ with real weights $w_i$.
The iteration over discrete timesteps $t$ of the Boolean variables follows the logic $$ x_1(t+1) = f(\{x_i(t)\})\\ x_i(t+1) =x_{i-1}(t) \quad \forall i \ge 2 $$
The latter conditions may be inserted to give the one-dimensional equation
$$ x_1(t) = {\rm sign} ({\sum_{i=1}^K w_i x_1(t-i)}) $$
and likewise for all other $x_j$.
Question: which cycles of lengths $N$ exist, i.e. $x_i(t+N) =x_{i}(t)$ ?
Clearly, a few observations can be made: the cycles depend on the choice of the weights $w_i$, there may be several cycles with different sets of $\{x_i \}$ and different $N$, and $1 \le N \le 2^K$.
The choice of the Boolean function as a threshold function may lead to an access to the problem, since it offers some special features:
The Boolean function is balanced, i.e. its output yields as many $1$s as $-1$s.
The choice $x_i^* = {\rm sign} (w_i )$ yields the output $1$, and this is the Boolean input vector "around" which all other Boolean input vectors are grouped which also yield the output $1$. This grouping is understood geometrically from the fact that the Boolean threshold function selects a halfspace in $K$-space, or in terms of Hamming distances, selects Boolean input vectors with "small" Hamming distances to $x_i^*$.