Pitchfork Bifurcation in a Coupled Cell System

Various biological phenomena, like cell differentiation and pattern formation in multicellular organisms, are explained using bifurcation theory. Molecular network motifs such as positive feedback and mutual repression exhibit bifurcations and are responsible for the emergence of diverse cell types. Mathematical investigations of such problems usually focus on bifurcations in a molecular network within individual cells. However, in a multicellular organism, cells interact, and intercellular interactions affect individual cell dynamics. Therefore, bifurcations in an ensemble of cells could differ from those for a single cell. This work considers a ring of identical cells. When independent, each cell exhibits a supercritical pitchfork bifurcation. Using analytical and numerical tools, we investigate bifurcations in this ensemble when cells interact through positive or negative coupling. We show that within a specific parameter zone, an ensemble of positively coupled cells behaves like a single cell that can undergo a supercritical pitchfork bifurcation. In this regime, the system is homogeneous, with all cells at the same steady state. However, this unique behaviour is lost when cells interact through negative coupling. Apart from the homogeneous states, cell-cell coupling can lead to heterogeneous steady states with unique patterns. We also investigate the distribution of such heterogeneous states under positive and negative coupling.