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M. Golubitsky, I. Stewart, F. Antoneli, Z. Huang and YY. Wang

Input-output networks, singularity theory, and homeostasis

In: *Proceedings on the Workshop of Dynamics, Optimization, and Computation.*
(O. Junge, S. Ober-Blobaum, K. Padburg-Gehle,G. Froyland, and O. Schütze, eds.)
Springer Nature Switzerland AG
To appear.

Homeostasis is a regulatory mechanism that keeps some specific variable close to a set value as other variables fluctuate, and is of particular interest in biochemical networks. We review and investigate a reformulation of homeostasis in which the system is represented as an input-output network, with two distinguished nodes `input' and `output', and the dynamics of the network determines the corresponding input-output function of the system. Interpreting homeostasis as an infinitesimal notion --- namely, the derivative of the input-output function is zero at an isolated point --- we apply methods from singularity theory to characterize homeostasis points in the input-output function. This approach, coupled to graph-theoretic ideas from combinatorial matrix theory, provides a sßystematic framework for calculating homeostasis points in models, classifying different types of homeostasis in input-output networks, and describing all small perturbations of the input-output function near a homeostasis point.