Asymptotic Phase for Stochastic Oscillators

Oscillations and noise are ubiquitous in physical and biological systems. When oscillations arise from a deterministic limit cycle, entrainment and synchronization may be analyzed in terms of the asymptotic phase function. In the presence of noise, the asymptotic phase is no longer well defined. We introduce a new definition of asymptotic phase in terms of the slowest decaying modes of the Kolmogorov backward operator. Our stochastic asymptotic phase is well defined for noisy oscillators, even when the oscillations are noise dependent. It reduces to the classical asymptotic phase in the limit of vanishing noise. The phase can be obtained either by solving an eigenvalue problem, or by empirical observation of an oscillating density’s approach to its steady state.

Figure 1

Trajectory of the heteroclinic oscillator and the histogram method to estimate the asymptotic phase. Trajectories in the $(Y_1,Y_2)$ plane like the one shown in (a) that all end up in the neighborhood of the reference point $(Y_1,Y_2)$ (red box) are used to estimate the time-dependent probability in the past in other points $(X_1,X_2)$ in the plane (blue boxes). This probability displays asymptotically damped oscillations [(c),(d)] characterized by the smallest nonvanishing eigenvalue and a space-dependent phase shift $\mathrm{\Delta }(x_1,x_2,y_1,y_2)=\psi (x_1,x_2)-\varphi (y_1,y_2)$, from which the asymptotic phase $\psi (x_1,x_2)$ can be extracted [the constant offset still depends on the reference point $(y_1,y_2)$]. Stochastic oscillations of the variables are shown in (b).

Figure 2

Asymptotic phase of the stochastic heteroclinic oscillator for two different noise levels. The complex phase of the backward eigenfunction (solid lines) is compared to the results of the histogram method [11] for $D=0.1$ (a) and $D=0.01125$ (b). Eigenfunctions used in (a) and (b) correspond to the slowest eigenvalues, marked by dashed boxes in (c). Isochrons at lower noise level [black in (d)] are more curled than for stronger noise [red in (d)]. Thick lines in (d) denote $2\pi$ jump in phase.

Figure 3

Trajectory, nullclines, eigenvalues of the backward operator, and asymptotic phase lines for the persistent-sodium–potassium model. (a) Sample trajectory (thin black line) for the $(V,N)$ process for $N_{\mathrm{tot}}=100$ channels, and nullclines for the deterministic $v$ (thick grey line) and $n$ (thick black line) dynamics. (b) Low-lying spectrum for ${\mathcal{L}}^{†}$ for two different channel numbers, $N_{\mathrm{tot}}=100$ (black dots) and $N_{\mathrm{tot}}=25$ (red crosses). Dashed boxes indicate the leading complex conjugate eigenvalue pairs. (c) Level curves (isochrons) of the asymptotic phase for $N_{\mathrm{tot}}=25$ (red), $N_{\mathrm{tot}}=100$ (black), and $N_{\mathrm{tot}}=\infty$ (blue; deterministic case). The thick lines indicate the locations of the phase jump by $2\pi$, which have been adjusted to coincide for the three cases. Isochrons are marked in equal increments of $2\pi /20$. Nullclines as in (a).