  Synchronization of mobile oscillators occurs in numerous contexts, including
physical, chemical, biological and engineered systems. In vertebrate embryonic
development, a segmental body structure is generated by a population of mobile
oscillators. Cells in this population produce autonomous gene expression
rhythms, and interact with their neighbors through local signaling. These cells
form an extended tissue where frequency and cell mobility gradients coexist.
Gene expression kinematic waves travel through this tissue and pattern the
segment boundaries. It has been shown that oscillator mobility promotes global
synchronization. However, in vertebrate segment formation, mobility may also
introduce local fluctuations in kinematic waves and impair segment boundaries.
Here we derive a general framework for mobile oscillators that relates local
mobility fluctuations to synchronization dynamics and pattern robustness. We
formulate a statistical description of mobile phase oscillators in terms of a
probability density. We obtain and solve diffusion equations for the average
phase and variance, revealing the relationship between local fluctuations and
global synchronization in a homogeneous population of oscillators. Analysis of
the probability density for large mobility identifies a mean-field transition,
where locally coupled oscillators start behaving as if each oscillator was
coupled with all the others. We extend the statistical description to
inhomogeneous systems to address the gradients present in the vertebrate
segmenting tissue. The theory relates pattern stability to mobility, coupling
and pattern wavelength. The general approach of the statistical description may
be applied to mobile oscillators in other contexts, as well as to other
patterning systems where mobility is present.
