<p>Understanding how cells attenuate responses initiated by external signals and display adaptive dynamics is a fundamental question in biology. This study explores two well-known control mechanisms to gain insights into how cellular signal adaptation occurs in each. The first mechanism involves an incoherent feedforward loop(IFFL), and the second involves a negative feedback loop(NFBL). After deriving the sufficient conditions for the adaptive dynamics of each mechanism, we determine the parameter regime in which adaptation occurs. Mathematical analysis shows that the parameter regime for adaptive dynamics in the IFFL model is broader than that in the NFBL model. Additionally, the relationship between the signal level and the steady state of the intermediate protein is linear in the IFFL model, whereas this relationship is saturating in the NFBL model. Our simulation results show that for the constant prolonged input signals, the IFFL model is faster to produce a response and quicker to return to its pre-steady state compared to the NFBL model when the signal protein has a moderate to high affinity to the response protein. The NFBL model, with low affinity between the signal protein and the response protein, returns to its pre-steady state more quickly, but lacks the capacity to generate a strong response. However, the NFBL model with a high affinity of the signal protein to the response protein is capable of producing a higher maximal response compared to the IFFL model for weaker signals, but this dynamic is not robust against repetitive pulse-type signals.</p>