Inferring noise intensity and phase response from noisy synchronous oscillators

Numerous biological and microscale systems exhibit synchronization in noisy environments. The theory of such noisy oscillators and their synchronization has been developed and experimentally demonstrated, but inferring the noise intensity and phase response is not always straightforward. In this paper, we propose a useful formula that enables us to infer the noise intensity and phase response of a noisy oscillator synchronized with periodic external forcing. Through asymptotic approximations for small noise, we show that noisy synchronous oscillators satisfy a simple relationship among the noise intensity and measurable quantities, i.e., the stationary distribution of the oscillation phase and stationary probability current obtained as the average phase velocity, which is verified through systematic numerical analysis. The proposed formula facilitates a unified analysis and design of synchronous oscillators in weakly noisy environments.