PHYSICAL REVIEW E 101, 052210 (2020) Spatial correlations in a finite-range Kuramoto model Sebastian Wüster 1, * and Rajasekaran Bhavna 2 1 Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462 023, India 2 Department of Biological Sciences, Tata Institute of Fundamental Research, 400005 Mumbai, India (Received 19 November 2019; accepted 21 April 2020; published 15 May 2020) 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. DOI: 10.1103/PhysRevE.101.052210 I. INTRODUCTION The Kuramoto model [1–3] is paradigmatic for the study of synchronization [4]. It has been applied in a diverse range of settings, such as neuronal activity [5], coupled magnetic spin torque oscillators [6], coupled Josephson junction arrays [7,8], atomic lasing [9], and flashing fireflies [10]. Also during vertebrate development, genetic oscillations in a mechanism called the segmentation clock are synchronized to generate a rhythmic pattern with a temporal periodicity that is converted into a striped spatial pattern of gene expression that makes up the embryonic segments across vertebrates [11–14]. Despite the conceptual simplicity of the basic Kuramoto model, containing only a set of phase oscillators coupled via a phase synchronizing interaction that is identical for all oscillator pairs, the presence of noise already gives rise to a host of additional mathematical features such as pattern formation, bistability, and bifurcations [3,15,16]. Extensions of the model additionally consider range-dependent couplings [17,18] or time-delayed coupling [19], further enriching the phenomenology. Here we study phase correlations in a specific variant of the model, with a finite-interaction range such that only oscil- lators that are spatially separated by less than R 0 interact with constant instantaneous coupling strength κ 0 , while subjected to noise [20,21]. Our objective is to contribute to experi- mental estimates of oscillator coupling strengths leading to synchronization in the segmentation clock [11–14,22–24] of the developing zebrafish embryos, but the results are more generally applicable. We demonstrate that in the case of homogenous frequen- cies, where all oscillators share the same natural frequency, * sebastian@iiserb.ac.in the essential parameters of our interaction model, the range R 0 and coupling strength κ 0 , can be directly inferred from nonlocal phase correlations in weakly synchronized regimes. They can still be constrained through fitting the model to data in more strongly synchronized regimes, or for heterogeneous frequencies. We follow up on earlier studies on parameter reconstruction in a similar model [25], while significantly extending these results for cases with unknown or mobile os- cillator positions, unknown network connectivity, and finite- range interactions. We finally explore correlations in the context of data from genetic oscillators of the segmentation clock within develop- ing zebrafish embryos. The oscillating quantity is the level of gene expression within a cell. These cells characteristically behave as autonomous cellular oscillators, while synchroniz- ing interactions are provided by the intercellular delta-notch coupling [26]. Parameters for models describing the coupled cell system, such as coupling strength, cell-autonomous pe- riod, and the coupling delays, have been inferred by disruption of delta-notch intercellular coupling under various genetic conditions [27,28] and the range of such interactions has been theoretically estimated in another study [29]. Recently, single-cell-based phase oscillator measurements provided a framework to constrain the interaction parameters [23]. Since, within a biological context, individual systems are typically insufficiently reproducible to allow a thorough en- semble average, an interesting question is to what extent the model used here is ergodic and hence allows the inference of ensemble averages by replacing them with a time average in a single system. We investigate this question numerically, and find that in some cases of interest to us, the model behaves ergodically. This article is organized as follows: In Sec. II we describe the version of Kuramoto (or Kuramoto-Sakaguchi) model that we study and its known steady-state solution [25], which 2470-0045/2020/101(5)/052210(10) 052210-1 ©2020 American Physical Society