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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.