Synchronization in Stochastic Biochemical Oscillators Subject to Common Multiplicative Extrinsic Noise

In this work we study the level of synchronization in stochastic biochemical reaction networks that support stable mean-field limit cycles and are subject to common external switching noise. Synchronization in stochastic limit cycle oscillators due to common noise is usually demonstrated by applying Ito's lemma to the logarithm of the phase difference. However, this argument cannot be straightforwardly extended to our case because of its discrete state space. Assuming the intrinsic and extrinsic noises operate at different time scales, we prove that the average level of synchronization is of the order of the rate of the intrinsic noise (inversely proportional to the system volume) times the square of the switching rate of the external noise. Moreover, we show in numerical experiments the approximate asymptotic value of the synchronization level by applying this result to classical oscillators found in the literature.