Homeostasis, singularities, and networks.

Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter [Formula: see text] varies over some interval. We discuss two main aspects of homeostasis, both related to the effect of coordinate changes on the input-output map. The first is a reformulation of homeostasis in the context of singularity theory, achieved by replacing 'approximately constant over an interval' by 'zero derivative of the output with respect to the input at a point'. Unfolding theory then classifies all small perturbations of the input-output function. In particular, the 'chair' singularity, which is especially important in applications, is discussed in detail. Its normal form and universal unfolding [Formula: see text] is derived and the region of approximate homeostasis is deduced. The results are motivated by data on thermoregulation in two species of opossum and the spiny rat. We give a formula for finding chair points in mathematical models by implicit differentiation and apply it to a model of lateral inhibition. The second asks when homeostasis is invariant under appropriate coordinate changes. This is false in general, but for network dynamics there is a natural class of coordinate changes: those that preserve the network structure. We characterize those nodes of a given network for which homeostasis is invariant under such changes. This characterization is determined combinatorially by the network topology.