Spatial correlations in a finite-range Kuramoto model.

We study spatial correlations between oscillator phases in the steady state of a Kuramoto model, in which phase oscillators that are randomly distributed in space interact with constant strength but within a limited range. Such a model could be relevant, for example, in the synchronization of gene expression oscillations in cells, where only oscillations of neighboring cells are coupled through cell-cell contacts. We analytically infer spatial phase-phase correlation functions from the known steady-state distribution of oscillators for the case of homogenous frequencies and show that these can contain information about the range and strength of interactions, provided the noise in the system can be estimated. We suggest a method for the latter, and also explore when correlations appear to be ergodic in this model, which would enable an experimental measurement of correlation functions to utilize temporal averages. Simulations show that our techniques also give qualitative results for the model with heterogenous frequencies. We illustrate our results by comparison with experimental data on genetic oscillations in the segmentation clock of zebrafish embryos.