Aggregates of Monotonic Step Response systems: a structural classification

Complex dynamical networks can often be analysed as the interconnection of subsystems, to simplify the model and better understand the global behaviour. Some biological networks can be analysed as aggregates of monotone subsystems. Yet, monotonicity is a strong requirement and relies on the knowledge of an explicit state model. Systems with a Monotonic Step Response (MSR), which include input-output monotone systems, are a broader class and have interesting features. The property of having a monotonically increasing step response can be evinced from experimental data. We consider networks that can be decomposed as aggregates of MSR subsystems and we provide a structural (parameter-free) classification of oscillatory and multistationary behaviours. The classification is based on the exclusive or concurrent presence of negative and positive cycles in the system <formula><tex>$aggregate graph$</tex></formula>, whose nodes are the MSR subsystems. The result is analogous to our earlier classification for aggregates of monotone subsystems. Our classification is applied to models of biomolecular networks and helps build synthetic biomolecular circuits that, by design, are well suited to exhibit the desired dynamics.