Homeostasis, singularities, and networks

Abstract

Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter $\lambda$ 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 $\lambda ^3 + a\lambda$ 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.