Exploiting the nonlinear dynamics in the negative feedback loop, we propose a statistical signal-response model to describe the different oscillatory behaviour in a biological network motif. By choosing the delay as a bifurcation parameter, we discuss the existence of Hopf bifurcation and the stability of the periodic solutions of model equations with the centre manifold theorem and the normal form theory. It is shown that a periodic solution is bo in a Hopf bifurcation beyond a critical time delay, and thus the bifurcation phenomenon may be important to elucidate the mechanism of oscillatory activities in regulatory biological networks.