Nonreciprocal synchronization in embryonic oscillator ensembles.

Synchronization of coupled oscillators is a universal phenomenon encountered across different scales and contexts, e.g., chemical wave patterns, superconductors, and the unison applause we witness in concert halls. The existence of common underlying coupling rules defines universality classes, revealing a fundamental sameness between seemingly distinct systems. Identifying rules of synchronization in any particular setting is hence of paramount relevance. Here, we address the coupling rules within an embryonic oscillator ensemble linked to vertebrate embryo body axis segmentation. In vertebrates, the periodic segmentation of the body axis involves synchronized signaling oscillations in cells within the presomitic mesoderm (PSM), from which somites, the prevertebrae, form. At the molecular level, it is known that intact Notch-signaling and cell-to-cell contact are required for synchronization between PSM cells. However, an understanding of the coupling rules is still lacking. To identify these, we develop an experimental assay that enables direct quantification of synchronization dynamics within mixtures of oscillating cell ensembles, for which the initial input frequency and phase distribution are known. Our results reveal a "winner-takes-it-all" synchronization outcome, i.e., the emerging collective rhythm matches one of the input rhythms. Using a combination of theory and experimental validation, we develop a coupling model, the "Rectified Kuramoto" (ReKu) model, characterized by a phase-dependent, nonreciprocal interaction in the coupling of oscillatory cells. Such nonreciprocal synchronization rules reveal fundamental similarities between embryonic oscillators and a class of collective behaviors seen in neurons and fireflies, where higher-level computations are performed and linked to nonreciprocal synchronization.