Topology and Dynamics of Boolean Networks with Strong Inhibition

A major challenge in systems biology is to understand interactions within biological systems. Such a system often consists of units with various levels of activities that evolve over time, mathematically represented by the dynamics of the system. The interaction between units is mathematically represented by the topology of the system. We carry out some mathematical analysis on the connections between topology and dynamics of such networks. We focus on a specific Boolean network model the Strong Inhibition Model. This model defines a natural map from the space of all possible topologies on the network to the space of all possible dynamics on the same network. We prove this map is neither surjective nor injective. We introduce the notions of “redundant edges” and “dormant vertices” which capture the non-injectiveness of the map. Using these, we determine exactly when two different topologies yield the same dynamics and we provide an algorithm that determines all possible network solutions given a dynamics.