Phase sensitivity for coherence resonance oscillators

Abstract

The phase reduction approach has manifested its power in analyzing the rhythmic behaviors for limit cycle oscillators. For coherent oscillation purely induced by noise, e.g., the coherence resonance oscillator, the stochastic dynamics exhibit almost deterministic limit cycle phenomenon which inspires the application of the phase reduction approach to this kind of systems. In this paper, the FitzHugh–Nagumo system in coherence resonance is modeled as a jump process. The phase sensitivity is obtained by applying the phase reduction approach for the hybrid system. A modified direct method is proposed to compare the theoretical results with those of Monte Carlo simulation for the stochastic system, which shows a relatively good agreement for the perturbation not being too small. The phase reduction results of the coherent oscillators are applied to two coupled FitzHugh–Nagumo neurons. It is interesting that the phase difference of the coupled coherent oscillators does not converge as the deterministic oscillators, but forms a distribution where the peaks and valleys correspond to the stable and unstable synchronizations predicted by the phase coupling functions. The idea in this paper could be applied to other coherent oscillators where the stochastic dynamics follow almost deterministic paths.

Appendices

Appendix A The direct method of phase sensitivity function for the CR jump system

The direct method for the CR jump system is calculated as the following:

$$\begin{aligned} \begin{aligned} \mathbf{Z}_i( \theta )&\approx \frac{\left[ \theta ( {\mathbf{X}}_0( t ) + \delta \overrightarrow{{\varvec{e}}}_i ) - \theta ( {\mathbf{X}}_0( t ) ) \right] }{\delta }\\&\approx \frac{\left[ \theta ( {\mathbf{X}}_0( kT ) + \delta \overrightarrow{{\varvec{e}}}_i ) - \theta _0 \right] }{\delta }, \end{aligned} \end{aligned}$$

(20)

where the subscript i represents the ith component of the phase sensitivity function $\mathbf{Z}(\theta )$ and the vector $\overrightarrow{{\varvec{e}}}_i )$ is the unite vector along the ith direction with the perturbation strength $\delta$. The phase $\theta _0=\theta ( {\mathbf{X}}_0(0) )$ and $k \ge 1$ is an integer to assure the convergence of the perturbed oscillation. Figure 11 illustrates the initial points (red square dots) and perturbed ones (green circles) where the points in the phaseless area (gray patch) are discarded. Vectorization can be applied for faster computation. The perturbation strength should be small so that the linear approximation satisfies.

Fig. 11

Schematic diagram of the direction method for the CR jump system. The red square dots (equal phase distance) are the initial points on the LC, and the green circles are the corresponding perturbed initial points with perturbation strength $\delta$. Any point that is kicked into the in-between area of the jump surfaces (gray patch in the figure) will be excluded from the initial points

Appendix B Monte Carlo simulation of the modified direct method of phase sensitivity for the CR stochastic system

For the direct method applied in the stochastic system, one should keep in mind that because of the noise in the system, the phase sensitivity is no longer a definite value but a distribution with a mean value. Since the phase value could be sensitive to different noise histories, we will adopt the same noise series for the initial points and the perturbed ones. The problem in this kind of setting is that common noise-induced synchronization [3, 32] will make us underestimate the phase change. However, two reasons will help us reduce this influence. First, the noise strength is small for oscillators in coherence resonance. Thus, the synchronization effect of common noise will be weak for relatively large perturbation (but not too large so as to meet the requirement of the linear approximation). Second, benefiting from the fast convergence of the limit cycle, we could choose less than one cycle as the total integral time. To avoid the final position to be in the phaseless area as in Fig. 11, we stop evolving immediately after the state crosses the $y=0$ from below to above for the initial points starting from the left-half limit cycle, or after the state crosses the $y=0$ from above to below for the initial points starting from the right-half limit cycle.

If one applies independent noise series for the modified direct method, different noise histories would significantly impact the phase change. As a comparison with Fig. 8, we apply independent noises in the modified direct method to compute the phase response curve. The results are in Fig. 12. Although increasing the perturbation strength could also reduce the large dispersion as the common noise case, the magnitude of the dispersion is much larger than those in Fig. 8. Besides, the results are overestimated which is opposite with the common noise case. This could be explained by independent additive noise-induced desynchronization [33].

As a result, we apply common noise in the Monte Carlo simulation for computing the phase response curve.

Fig. 12

Phase response curves (PRC) for the CR jump system (6) and its MC simulation for the CR stochastic system (2) for different perturbation strengths. The only difference from Fig. 8 is that independent noise series are used for the MC simulation. Note the axis range for ${\mathrm{PRC}}_x/\delta$ which is much larger than those in Fig. 8