Find Topologies Exhibit Adaptation and Oscillation

Understanding the relationship between the topology and its function remain an important question in nonlinear physics. Previous works taught us that negative feedback is essential for the topology to show oscillation while positive feedback is essential for the appearance of bistable states. Many signaling systems show different behaviors when they are stimulated by different level of input signal. For example, when the system is stimulated by low input signal, it shows adaptation. When it is stimulated by high input signal, it shows oscillation. We found all simple transcriptional regulatory network topologies that could perform adaptation and oscillation, then we investigate how the system switches between two functions. We found that some topologies can transform two function by changing one parameter while others cannot. Finally, we find some biological network correspond to our findings.