Phase reduction of a limit cycle oscillator perturbed by a strong amplitude-modulated high-frequency force

The phase reduction method for a limit cycle oscillator subjected to a strong amplitude-modulated high-frequency force is developed. An equation for the phase dynamics is derived by introducing a new, effective phase response curve. We show that if the effective phase response curve is everywhere positive (negative), then an entrainment of the oscillator to an envelope frequency is possible only when this frequency is higher (lower) than the natural frequency of the oscillator. Also, by using the Pontryagin maximum principle, we have derived an optimal waveform of the perturbation that ensures an entrainment of the oscillator with minimal power. The theoretical results are demonstrated with the Stuart-Landau oscillator and model neurons.

Figure 1

The Arnold tongue of the SL system (11) for $\omega /\mathrm{\Omega }=100$. Straight lines represent analytical Eq. (16), black circles and red crosses show the numerical results derived from the averaged Eq. (7) and original Eq. (11), respectively.

Figure 2

The effective phase response curve for the Morris-Lecar neuron model (17). The inset shows an enlarged segment of the effective PRC, where it has negative values.

Figure 3

The Arnold tongues for the Morris-Lecar neuron (17). The blue color represents the harmonic wave envelope $\psi \left(\mathrm{\Omega }t\right)=(1-cos(\mathrm{\Omega }t\left)\right)/2$, while the red color corresponds to the square wave envelope $\psi \left(\mathrm{\Omega }t\right)=H(sin(\mathrm{\Omega }t\left)\right)$. The strait lines show the theoretical values defined by Eqs. (18) and (19), circles show the numerical results obtained from averaged system (7) and the crosses represent the results of direct numerical simulation of the original system (17).

Figure 4

The threshold amplitude as the function of the carrier frequency for the Morris-Lecar neuron (17). The numerical computations are performed for the fixed frequency mismatch $\mathrm{\Delta }=0.01$ using the square wave envelope $\psi \left(\mathrm{\Omega }t\right)=H(sin(\mathrm{\Omega }t\left)\right)$ with the varying carrier frequency $\omega$. The solid line shows the theoretical value obtained from Eq. (19), while the dashed line is computed from the averaged system (7). The crosses represent the results of direct numerical simulation of the original system (17).

Figure 5

Waveform optimization for the FHN neuron (25): (a) the effective PRC, (b) an example of optimal envelope for $\mathrm{\Delta }=0.1$, (c) the functions $M^{\pm }\left(u\right)$ defined by Eq. (21), and (d) the functions $N^{\pm }\left(u\right)$ defined by Eq. (24).

Figure 6

The Arnold tongues of the FHN system (25). The red and blue colors show the results obtained with the optimal ${\psi }^{*}$ and “quarter” ${\psi }_{1/4}$ envelope, respectively. Solid curves are derived from the phase Eq. (9), circles represent the results of the averaged Eq. (7) and the crosses show the results obtained from the original system (25). When computing the solid curves for the optimal envelope, we fixed $I_0=70$, while for circles and crosses, at each given $\mathrm{\Delta }$, we used the same waveform as for the solid curve and varied slightly $I_0$ until the entrainment threshold was reached.