Maxmin-$ω$ is a new threshold model, where each node in a network waits for the arrival of states from a fraction $ω$ of neighborhood nodes before processing its own state, and subsequently transmitting it to downstream nodes. Repeating this sequence of events leads to periodic behavior. We show that maxmin-$ω$ reduces to a smaller, simpler, system represented by tropical mathematics, which forges a useful link between the nodal state update times and circuits in the network. Thus, the behavior of the system can be analyzed directly from the smaller network structure and is computationally faster. We further show that these reduced networks: (i) are not unique; they are dependent on the initialisation time, (ii) tend towards periodic orbits of networks -- "attractor networks." In light of these features, we vary the initial condition and $ω$ to obtain statistics on types of attractor networks. The results suggest the case $ω=0.5$ to give the most stable system. We propose that the most prominent attractor networks of nodes and edges may be seen as a 'backbone' of the original network. We subsequently provide an algorithm to construct attractor networks, the main result being that they must necessarily contain circuits that are deemed critical and that can be identified without running the maxmin-$ω$ system. Finally, we apply this work to the C. elegans neuron network, giving insight into its function and similar networks. This novel area of application for tropical mathematics adds to its theory, and provides a new way to identify important edges and circuits of a network.