M. Golubitsky and I. Stewart
Homeostasis and Network Invariants
J. Mathematical Biology. Submitted.
Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter varies over some region. We reformulate homeostasis in the context of singularity theory by replacing `approximately constant over an interval' by `zero derivative with respect to the input at a point'. General coordinate changes need not leave this condition invariant, but in network dynamics there is a natural class of right network-preserving coordinate changes. We show that these coordinate changes preserve homeostasis in a cell if the cell forms a block in the right core of the network, a combinatorial condition that often occurs. The induced coordinate changes on the singularity are a version of contact equivalence. These results lead to new types of network invariant related to homeostasis. In particular, the `chair' singularity, which is especially important in applications, is discussed in detail. Its normal form and universal unfolding lambda^3 + a lambda is derived and the region of approximate homeostasis is deduced. The results are related to data on thermoregulation in two species of opossum and the spiny rat. We give a formula for finding chair points by implicit differentiation and discuss chair points in a simple network ODE. Finally we sketch generalizations of the theory to multiparameter systems.