This Master's 2 + PhD focuses on the collective behavior of active agents interacting with their own trail, with applications to understanding stem cell differentiation in organoids. Preliminary data uncovers a feedback loop between differentiation and migration, as differentiating cells deposit a chemical trail that serves as a guiding track for subsequent differentiating cells. We will adapt a recently proposed theoretical framework to describe a phase separation process, by coupling several fields – the fraction of stem versus differentiated cells and associated fluxes, and the local nematic order in guiding filaments – in terms of the distinct time scales for cells to differentiate, migrate over the system size, and remodel the guiding matrix. We expect that the phase separation between stem and differentiated cells to be maintained by a difference in their motilities. Self-propelled objects exhibit a tendency to accumulate in areas where their motility is lower. Such decreased motility could be caused by a higher particle density, resulting in the celebrated motility-induced phase separation. In our model, matrix deposition could compete with density, resulting in counter-intuitive results. This PhD is an opportunity to gain joint expertise in active matter modeling and/or advanced image analysis methods. The modeling will guide new experiments in which the applicant might be involved according to their taste.