Coincidence of Homeostasis and Bifurcation in Feedforward Networks

Abstract

Homeostasis is an important and common biological phenomenon wherein an output variable does not change very much as an input parameter is varied over an interval. It can be studied by restricting attention to homeostasis points — points where the output variable has a vanishing derivative with respect to the input parameter. In a feedforward network, if a node has a homeostasis point then downstream nodes will inherit it. This is the case except when the downstream node has a bifurcation point coinciding with the homeostasis point. We apply singularity theory to study the behavior of the downstream node near these homeostasis-bifurcation points. The unfoldings of low codimension homeostasis-bifurcation points are found. In the case of steady-state bifurcation, the behavior includes multiple homeostatic plateaus separated by hysteretic switches. In the case of Hopf bifurcation, the downstream node may have limit cycles with a wide range of near-constant amplitudes and periods. Homeostasis-bifurcation is therefore a mechanism by which binary, switch-like responses or stable clock rhythms could arise in biological systems.