Entrainment as a means of controlling phase waves in populations of coupled oscillators

We explore waves and entrainment in a model of coupled oscillators, inspired from the cellular oscillators in the presomitic mesoderm (PSM) of mice. The internal clock in each cell is based on a negative feedback loop which couples to the clocks of neighboring cells through a Notch mechanism. We investigate how a morphogen gradient in the mesoderm, which affects the period of oscillating cells, gives rise to phase waves traveling from the posterior to the anterior part of the PSM. We show that the phase waves can be entrained by an external periodic variation in this morphogen and also observe that multiple oscillatory solutions can coexist in the cell population. Together, these provide a way to potentially control phase waves and thereby manipulate somite patterning in embryos, based on entrainment properties of coupled nonlinear oscillators.