Incoherent Input as a Key Strategy for Robust Time-Dependent Biphasic Dynamics.

Biological systems maintain functional order through precise responses, with time-dependent biphasic dynamics (TBD) serving as a key regulatory mechanism. However, the network topologies supporting TBD and the principles underlying their robustness remain poorly understood. Here, we systematically investigate all possible one-, two-, and three-node network motifs using extensive random parameter sampling to evaluate their capacity to generate TBDs. Statistical analysis of the resulting dynamics reveals that incoherent feedforward loops and negative feedback loops constitute core topological modules underlying TBD generation. We identify 13 minimal core three-node motifs and construct a comprehensive topological landscape linking network structure to functional robustness. Extending this framework, we propose a motif-to-hypermotif design paradigm and demonstrate that coupling motifs through node sharing or edge connectivity to form feedback-enriched hypermotifs significantly enhances TBD robustness. Our findings reveal that incoherent input targeting the output node, particularly via inhibitory regulation, is a key design strategy for robust TBDs, providing new insights into the modular principles governing biological network architecture.