W. Duncan and M. Golubitsky
Coincidence of homeostasis and bifurcation in feedforward networks
Intern. J. Bif. Chaos.
(2019) 1930037-1-29; DOI:10.1142/S0218127419300374
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.