Period variability of coupled noisy oscillators

Period variability, quantified by the standard deviation (SD) of the cycle-to-cycle period, is investigated for noisy phase oscillators. We define the checkpoint phase as the beginning or end point of one oscillation cycle and derive an expression for the SD as a function of this phase. We find that the SD is dependent on the checkpoint phase only when oscillators are coupled. The applicability of our theory is verified using a realistic model. Our work clarifies the relationship between period variability and synchronization from which valuable information regarding coupling can be inferred.

Figure 1

(a) An example of the time series of an oscillation. Periods are observed at two checkpoints, $\alpha$ and $\beta$. (b) The corresponding checkpoint phases in the phase description.

Figure 2

The quantity $\sigma \left({\theta }_{\text{cp}}\right)/\tau$ for cases (A) and (B) is shown in (a) and (b), respectively, where the vertical scale is expressed as a percentage. The distance $d\left({\theta }_{\text{cp}}\right)$ from in-phase synchronization for cases (A) and (B) is shown in (c) and (d), respectively. The points and lines are the numerical results of the simulation and analytical predictions given by Eqs. (16) and (13), respectively.

Figure 3

Validation of Eq. (16) in the FitzHugh-Nagumo model. Open symbols are the numerically obtained $\sigma$ values. Filled symbols are the $\sigma$ values evaluated from Eq. (16) with the numerically obtained $d$ values and fitting parameters $C_1$ and $C_2$. The triangles and circles represent the $V$ and $W$ coupling cases. The plus symbols are the numerically obtained $\sigma$ values for an uncoupled oscillator.