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.

Figure 1

Assembly of $N$ phase oscillators (small green circles) spread in 2D over a square of side-length $L$. Each oscillator interacts with equal strength ${\kappa }_0$ with all neighbours within a range $R_0$ only, indicated by large magenta circles around selected oscillators. This is described by a finite range Kuramoto model. Oscillators can still affect one another over distances $d$ larger than $R_0$, through chains of intermediary oscillators as sketched by the red lines without arrows.

Figure 2

Spatial phase-phase correlation functions $g\left(d\right)$ for $d>0$ (solid black line). The standard error of the mean, $\mathrm{\Delta }g$, is indicated by dotted lines at $g\pm \mathrm{\Delta }g$. We compare with the analytical calculation of $g\left(d\right)$ from Sec. 3a to leading order (red dashed) and second order (blue dot dashed). The synchronization $Z$ from Eq. (5) is indicated in each panel. We compare different representative sets of parameters: (a) $\overline{\omega }=0$, $\mathrm{\Delta }\omega =0$, ${\kappa }_0=0.03$, $R_0=0.1$, $\beta =1$. (b) As (a), but with less noise, $\beta =2$. (c) As (a), but with a distribution of frequencies $\mathrm{\Delta }\omega =0.5$. (d) As (a), but with longer-range interactions, $R_0=0.2$. Additionally, the resummed nonperturbative result (12) is shown as magenta ($\times$); see text. (e) Strongly synchronized scenario with $\overline{\omega }=0$, $\mathrm{\Delta }\omega =0.1$, ${\kappa }_0=0.2$, $R_0=0.07$, $\beta =5$. (f) Using $\overline{\omega }=0$, $\mathrm{\Delta }\omega =0.1$, ${\kappa }_0=0.05$, $R_0=0.1$, $\beta =10$.

Figure 3

Impact of frequency disorder on spatial phase correlation functions. (a) The same parameters as Fig. 2, except, from top to bottom, $\mathrm{\Delta }\omega =0$ (black solid line), $\mathrm{\Delta }\omega =0.5$ (red dashed line), $\mathrm{\Delta }\omega =1$ (blue dot dashed line), $\mathrm{\Delta }\omega =2$ (magenta solid line), and $\mathrm{\Delta }\omega =4$ (black dashed line). Dotted color matched lines indicate the stochastic sampling error as before. (b) The same parameters as in Fig. 2, with line styles as in (a).

Figure 4

Correlation function ergodicity. Black solid lines are correlation functions $g\left(d\right)$ obtained from the standard ensemble average, with black dashed lines indicating the sampling error. These are compared with $g\left(d\right)$ obtained from a time average within a single realization of the simulation, assuming ergodicity (red dashed). (a) Same parameters as Fig. 2. (b) Same parameters as Fig. 2.

Figure 5

Spatial phase correlation function from zebrafish embryo genetic oscillations compared with simulations of model (1). The experimental data described in the text are shown as a red dashed line, with statistical errors indicated by red dotted lines. We show only simulations where the parameters $\overline{\omega }=0.15{\min }^{-1}$, $\mathrm{\Delta }\omega =0.07{\min }^{-1}$ are matched to those extracted from Ref. [23]. In the following we list the varied parameters, where simulations are sorted in descending order of their correlation strength near $d\approx 0$: (a) ${\kappa }_0=0.55{\min }^{-1}$, $R_0=11\mu \mathrm{m}$, $\beta =0.84$ min; (b) the same, but $\beta =0.4$ min; (c) ${\kappa }_0=1.0{\min }^{-1}$, $R_0=11\mu \mathrm{m}$, $\beta =0.4$ min; (d) ${\kappa }_0=0.5{\min }^{-1}$, $R_0=13\mu \mathrm{m}$, $\beta =0.66$ min; and (e) ${\kappa }_0=0.18{\min }^{-1}$, $R_0=18\mu \mathrm{m}$, $\beta =0.4$ min. Dashed color-matched lines indicate the sampling error, as before. The legend indicates the respective overall level of synchronization $Z$ and the inset shows the initial spatial arrangement of cells in the experiment, for orientation.

Figure 6

Cancellation of disconnected elements of diagrams in the series expansion of correlations. The top row indicates some selected contributions to the numerator with target sites $ij$ marked as red dots. In addition to one identical connection via two intermediate oscillators, additional closed loops may appear. Only the closed loops also appear in the numerator. In a rigorous order-by-order expansion of both, the numerator thus cancels all disconnected elements.

Figure 7

Extraction of noise amplitude in model (1) through sampling of the time-increment histogram. Black ($•$): histogram of phase increments $\mathrm{\Delta }\mathrm{\Theta }$ for 240 000 time samples of length $\mathrm{\Delta }t=2.5$ for the same parameters as in Fig. 2. The red line is the theoretical expectation as described in the text.