Phase Locking

Module 2: Synchronization and Collective Behaviour

A single firefly can lock its flashing to a periodic light. The model below, from Strogatz (2024, sec. 4.5), is based on (ermentrout1984beyond?). It is the smallest model of synchronization: one oscillator driven by one rhythm.

The model

Let $\Theta(t)$ be the phase of the stimulus and $\theta(t)$ the phase of the firefly. A flash happens when a phase passes through $0$. The stimulus runs at constant frequency $\Omega$. The firefly has natural frequency $\omega$ and is pulled toward the stimulus with strength $A > 0$:

\[ \dot\Theta = \Omega, \qquad \dot\theta = \omega + A\sin(\Theta - \theta). \tag{1}\]

If the stimulus is ahead ($0 < \Theta - \theta < \pi$), the firefly speeds up. If it is behind, the firefly slows down.

Phase difference

The quantity you care about is the phase difference $\phi = \Theta - \theta$. It obeys

\[ \dot\phi = \Omega - \omega - A\sin\phi. \tag{2}\]

With the dimensionless time $\tau = At$ and detuning $\mu = (\Omega - \omega)/A$, this becomes

\[ \phi' = \mu - \sin\phi, \tag{3}\]

which is the nonuniform oscillator from Flows on the Circle.

Locking range

If $|\mu| \le 1$, Equation 3 has a stable fixed point $\phi^* = \arcsin\mu$. The firefly runs at the stimulus frequency $\Omega$ with a constant phase lag. This is phase locking, or entrainment.
If $|\mu| > 1$, there are no fixed points and $\phi$ keeps increasing. The firefly drifts relative to the stimulus. This is phase drift.

In the original variables the firefly entrains when

\[ \omega - A \le \Omega \le \omega + A. \tag{4}\]

Outside this range, the time for $\phi$ to grow by $2\pi$ is

\[ T_{\text{drift}} = \frac{2\pi}{\sqrt{(\omega - \Omega)^2 - A^2}}. \tag{5}\]

Both predictions can be tested with a flashing light, because $\omega$ and the width of the locking range are measurable (Strogatz 2024, sec. 4.5). Real fireflies differ between species. Some can even shift their own natural frequency, which needs an extra adaptation variable (ermentrout1991adaptive?).

Figure 1: Phase difference $\phi = \Theta - \theta$ for a firefly with $\omega = 1$ and $A = 0.2$. Inside the locking range ($\Omega = 1.15$) the difference settles to $\arcsin\mu$. Outside it ($\Omega = 1.3$) the difference keeps growing, one turn every $T_{\text{drift}}$.

Predicted drift period for Omega = 1.3: 28.10

Exercise

Use Figure 1 and the printed value. Measure the drift period in the plot for $\Omega = 1.3$ and compare it with Equation 5.

Solution

With $\omega = 1$, $\Omega = 1.3$ and $A = 0.2$, Equation 5 gives $T_{\text{drift}} = 2\pi/\sqrt{0.09 - 0.04} = 2\pi/\sqrt{0.05} \approx 28.1$. In the plot $\phi$ climbs by about $2\pi \approx 6.28$ every 28 time units, with slow stretches where $\phi$ is near $\pi/2$, the bottleneck.

Exercise

A firefly has a natural period of $0.9$ s and entrains to stimulus periods between $0.78$ s and $1.06$ s. Estimate $A$.

Solution

Convert periods to angular frequencies with $\Omega = 2\pi/T$. The extremes of the locking range are $\Omega_{\max} = 2\pi/0.78 \approx 8.06$ and $\Omega_{\min} = 2\pi/1.06 \approx 5.93$ rad/s. By Equation 4, $\Omega_{\max} - \Omega_{\min} = 2A$, so $A \approx 1.06$ rad/s. The midpoint, $6.99$ rad/s, is close to $\omega = 2\pi/0.9 \approx 6.98$ rad/s, which is a consistency check. (These numbers are illustrative, chosen for the exercise.)

From one oscillator to many

Here the stimulus is fixed and does not respond to the firefly. In a swarm every firefly is both a stimulus and a responder. The Kuramoto model in Session 11 treats that case.

References

Strogatz, Steven H. 2024. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Chapman; Hall/CRC.

--- title: "Phase Locking" subtitle: "Module 2: Synchronization and Collective Behaviour" format: html --- A single firefly can lock its flashing to a periodic light. The model below, from @strogatz2024nonlinear [Section 4.5], is based on @ermentrout1984beyond. It is the smallest model of synchronization: one oscillator driven by one rhythm. ## The model Let $\Theta(t)$ be the phase of the stimulus and $\theta(t)$ the phase of the firefly. A flash happens when a phase passes through $0$. The stimulus runs at constant frequency $\Omega$. The firefly has natural frequency $\omega$ and is pulled toward the stimulus with strength $A > 0$: $$ \dot\Theta = \Omega, \qquad \dot\theta = \omega + A\sin(\Theta - \theta). $$ {#eq-firefly} If the stimulus is ahead ($0 < \Theta - \theta < \pi$), the firefly speeds up. If it is behind, the firefly slows down. ## Phase difference The quantity you care about is the phase difference $\phi = \Theta - \theta$. It obeys $$ \dot\phi = \Omega - \omega - A\sin\phi. $$ {#eq-phase-difference} With the dimensionless time $\tau = At$ and detuning $\mu = (\Omega - \omega)/A$, this becomes $$ \phi' = \mu - \sin\phi, $$ {#eq-firefly-nondim} which is the nonuniform oscillator from [Flows on the Circle](flows-on-the-circle.qmd). ## Locking range - If $|\mu| \le 1$, @eq-firefly-nondim has a stable fixed point $\phi^* = \arcsin\mu$. The firefly runs at the stimulus frequency $\Omega$ with a constant phase lag. This is **phase locking**, or entrainment. - If $|\mu| > 1$, there are no fixed points and $\phi$ keeps increasing. The firefly drifts relative to the stimulus. This is **phase drift**. In the original variables the firefly entrains when $$ \omega - A \le \Omega \le \omega + A. $$ {#eq-locking-range} Outside this range, the time for $\phi$ to grow by $2\pi$ is $$ T_{\text{drift}} = \frac{2\pi}{\sqrt{(\omega - \Omega)^2 - A^2}}. $$ {#eq-drift-period} Both predictions can be tested with a flashing light, because $\omega$ and the width of the locking range are measurable [@strogatz2024nonlinear, Section 4.5]. Real fireflies differ between species. Some can even shift their own natural frequency, which needs an extra adaptation variable [@ermentrout1991adaptive]. ```{python} #| label: fig-firefly #| fig-cap: "Phase difference $\\phi = \\Theta - \\theta$ for a firefly with $\\omega = 1$ and $A = 0.2$. Inside the locking range ($\\Omega = 1.15$) the difference settles to $\\arcsin\\mu$. Outside it ($\\Omega = 1.3$) the difference keeps growing, one turn every $T_{\\text{drift}}$." #| fig-width: 7 #| fig-height: 3.5 #| echo: false import numpy as np import matplotlib.pyplot as plt from amlab.synchronization.phase_models import ( firefly_drift_period, firefly_phase_difference, ) omega, amplitude = 1.0, 0.2 fig, ax = plt.subplots(figsize=(7, 3.5)) for big_omega in [1.15, 1.3]: t, phi = firefly_phase_difference(0.0, omega, big_omega, amplitude, t_end=120, dt=0.05) mu = (big_omega - omega) / amplitude ax.plot(t, phi, label=rf"$\Omega$ = {big_omega}, $\mu$ = {mu:.2f}") mu_locked = (1.15 - omega) / amplitude ax.axhline(np.arcsin(mu_locked), color="grey", linestyle="--", label=r"$\arcsin\mu$") ax.set_xlabel("t") ax.set_ylabel(r"$\phi$ (rad)") ax.legend() plt.tight_layout() plt.show() print(f"Predicted drift period for Omega = 1.3: {firefly_drift_period(omega, 1.3, amplitude):.2f}") ``` ::: {.callout-note} ## Exercise Use @fig-firefly and the printed value. Measure the drift period in the plot for $\Omega = 1.3$ and compare it with @eq-drift-period. ::: ::: {.callout-tip collapse="true"} ## Solution With $\omega = 1$, $\Omega = 1.3$ and $A = 0.2$, @eq-drift-period gives $T_{\text{drift}} = 2\pi/\sqrt{0.09 - 0.04} = 2\pi/\sqrt{0.05} \approx 28.1$. In the plot $\phi$ climbs by about $2\pi \approx 6.28$ every 28 time units, with slow stretches where $\phi$ is near $\pi/2$, the bottleneck. ::: ::: {.callout-note} ## Exercise A firefly has a natural period of $0.9$ s and entrains to stimulus periods between $0.78$ s and $1.06$ s. Estimate $A$. ::: ::: {.callout-tip collapse="true"} ## Solution Convert periods to angular frequencies with $\Omega = 2\pi/T$. The extremes of the locking range are $\Omega_{\max} = 2\pi/0.78 \approx 8.06$ and $\Omega_{\min} = 2\pi/1.06 \approx 5.93$ rad/s. By @eq-locking-range, $\Omega_{\max} - \Omega_{\min} = 2A$, so $A \approx 1.06$ rad/s. The midpoint, $6.99$ rad/s, is close to $\omega = 2\pi/0.9 \approx 6.98$ rad/s, which is a consistency check. (These numbers are illustrative, chosen for the exercise.) ::: ## From one oscillator to many Here the stimulus is fixed and does not respond to the firefly. In a swarm every firefly is both a stimulus and a responder. The Kuramoto model in Session 11 treats that case.