During development, cells sequentially lose their ability to differentiate into other cell types and become committed to different cellular states. This process can be described as a landscape in which the valleys are canalized one by one. This process of canalization is understood in terms of dynamical systems of interacting cells. In fact, as cells with oscillating gene expression proliferate and interact with each other, they differentiate into other expression states. Cells with oscillatory gene expression have pluripotency, either to replicate the same state or to differentiate into other cellular states, whereas cells that differentiate and lose their oscillations of expression simply replicate themselves, that is, they are committed. The proportion of each cell type is robust to changes in initial conditions and noise perturbations. Differentiation by protein expression dynamics is further stabilized by a feedback process of epigenetic modifications, such as DNA modification. The irreversibly differentiated cell state can be initialized to a pluripotent state by restoring an oscillatory state by forcing the expression of multiple genes from the outside, known experimentally as reprogramming.

In this chapter, we discuss dynamical system approaches for cellular differentiation. We explain how intracellular reaction dynamics can give rise to various attractors using a simple discrete-time and discrete-state reaction model known as a Boolean network. Subsequently, we outline the behavior of a simple stochastic differentiation model of stem cells, where the scaling law discovered therein aligns well with that observed in the distribution of clonal cell populations generated by epidermal stem cells. To integate both approaches, we introduce a theory wherein cell–cell interactions induce transitions between attractors, and stability at the cell-population level emerges through the regulation of these dynamic transitions. Such a circular relationship satisfies the consistency between the cell and the cell population. We expound on three types of differentiation processes, that by Turing instability, transition from an oscillatory state (limit-cycle) to a fixed point, and retaining oscillatory expression dynamics. Additionally, we analyze stability at the cell population level through the regulation of differentiation ratios and the differentiation dynamics of stem cells. Finally, we engage in a discussion of unresolved issues in the field.