Homeostasis Patterns

TL;DR

This paper develops a mathematical framework using graph theory and matrix analysis to identify and classify homeostasis points and patterns in input-output networks, extending previous models to include sets of nodes with simultaneous homeostasis.

Contribution

It introduces the concept of homeostasis patterns and the homeostasis pattern network, providing a systematic way to analyze multiple nodes' simultaneous homeostasis in complex systems.

Findings

Defined homeostasis patterns as sets of nodes with infinitesimal homeostasis.

Proved each homeostasis type corresponds to a unique pattern.

Characterized all patterns via the homeostasis pattern network.

Abstract

Homeostasis is a regulatory mechanism that keeps a specific variable close to a set value as other variables fluctuate. The notion of homeostasis can be rigorously formulated when the model of interest is represented as an input-output network, with distinguished input and output nodes, and the dynamics of the network determines the corresponding input-output function of the system. In this context, homeostasis can be defined as an 'infinitesimal' notion, namely, the derivative of the input-output function is zero at an isolated point. Combining this approach with graph-theoretic ideas from combinatorial matrix theory provides a systematic framework for calculating homeostasis points in models and classifying the different homeostasis types in input-output networks. In this paper we extend this theory by introducing the notion of a homeostasis pattern, defined as a set of nodes, in addition to the output node, that are simultaneously infinitesimally homeostatic. We prove that each homeostasis type leads to a distinct homeostasis pattern. Moreover, we describe all homeostasis patterns supported by a given input-output network in terms of a combinatorial structure associated to the input-output network. We call this structure the homeostasis pattern network.