Winfree's Model of Synchronization

up:: 040 MOC Complex Systems

In 1967, Winfree posited a simple model to predict synchronization in biological systems.

Assumptions of Winfree’s model

1) Individuals are essentially oscillators in state space

The individuals in the system are essentially oscillators, i.e. they are “rotating around some Limit Cycle” in state space. Thus, they can be approximately describe by one variable (some angle $\theta$).

2) Symmetric frequency distribution (and unimodal)

Each oscillator has some natural frequency $\omega$, sampled from a symmetric distribution $g(\omega )$, reasonably narrow, and implicitly assumed unimodal.

3) Oscillators are equally coupled

All oscillators can send some signal of strength $P(\theta )$ and adjust their own frequency by an amount $R(\theta )$, with $P,R$ equal for all oscillators for simplicity.

4) All oscillators respond only to a mean field

Oscillators only respond to a mean response of all others. Thus, their equations of motion are

$${\dot{\theta }}_i={\omega }_i+\left(\frac{1}{N}\sum _{k=1}^{N}P({\theta }_k)\right)R({\theta }_i)$$

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References

• 2011 Simons Lectures - Steven Strogatz, Coupled Oscillators That Synchronize Themselves - YouTube
• WINFREE, Arthur T. Biological rhythms and the behavior of populations of coupled oscillators. Journal of theoretical biology, v. 16, n. 1, p. 15-42, 1967.