Growing a network via oscillations on nodes

Small cell networks, if not formed by cell aggregation, arise when a founder cell and its progeny divide with incomplete cytokinesis. These incomplete divisions result in intercellular bridges that allow the cells to share cytoplasm, enabling coordinated cellular behaviors like apoptosis and mitosis. Motivated by germline cysts in insect ovaries, we construct a minimal model for generating small cell networks: each node is an oscillator, and completion of a cycle causes a node to generate a daughter to which it remains connected. Importantly, oscillations on the nodes are transient -- under the control of a pulse -- and coupled via diffusion over the edges of the network. We, first, verify that our simple model can generate the diverse germline cyst structures observed in nature and identify where, in the space of model parameters, these naturally observed cyst structures appear. Inspired by cases like the lacewing C. perla, where a few different cyst structures are found in the same ovary, we test if a simple variation in model parameters can rationalize the observed variability. We conclude by proposing extensions of our model to the cases of germline cysts in mammals and annelid worms. Our simple framework, one of the first reported model for growing finite-sized networks based on oscillations, can be extended to study many naturally grown networks.