We provide an analytical expression for the mean frequency shift in phase oscillators as a function of the standard deviation, $\sigma$ and the autocorrelation time, $\tau$ of small random perturbations. We show that the frequency shift is negative and proportional to ${\sigma }^2$. Its absolute value increases monotonically with $\tau$, approaching an asymptote determined by the ${\text{L}}^2$-norm of the phase-response curve. We validate our theoretical predictions with computer simulations and discuss their implications for the design of electronic oscillators and for the encoding of information in biological neural networks.

• Received 23 June 2009

DOI:https://doi.org/10.1103/PhysRevE.80.036113