M. Golubitsky and Y. Wang

Homeostasis in three-node networks

preprint. Submitted.

Homeostasis occurs in a system where an output variable is approximately constant on an interval on variation of an input variable I. Homeostasis is known to play an important role in the regulation of biological systems. See Nijhout et al. As a way of finding homeostasis, Golubitsky and Stewart introduced the notion of infinitesimal homeostasis -- points where the derivative of the output variable with respect to I is zero. Reed et al. give two examples of infinitesimal homeostasis in three-node chemical reaction systems: feedforward excitation and substrate inhibition. In this paper we show that there are 13 different three-node networks leading to many possible input-output system configurations. The different configurations are based on which node is the input node, which node is the output node, and whether individually the couplings are excitatory, inhibitory, or neutral. We show nonetheless that there are only two basic mechanisms that lead to infinitesimal homeostasis and they are illustrated by feedforward excitation and substrate inhibition. Feedforward excitation occurs only when the network has a feedforward loop as a subnetwork; that is, when there are two distinct simple paths connecting the input node to the output node. Moreover, one of the paths must be excitatory and one inhibitory to support infinitesimal homeostasis. Substrate inhibition occurs when there is a single simple path from the input node to the output node and then only when one of the couplings along this path is neutral. The paper ends with an analysis of substrate inhibition infinitesimal chair singularities.