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Marty Golubitsky : Homeostasis and Network Invariants

We discuss the question: What properties of solutions to coupled cell network systems are invariant under changes of coordinates that preserve network structure? This question was motivated by trying to understand the biological phenomenon of homeostasis in a mathematically satisfactory way. In its simplest mathematical form homeostasis can be described as follows. Given a stable equilibrium $x(lambda)$ of a system that depends on an input parameter $lambda$: When is some coordinate (say $x_j(lambda)$) approximately constant? First, we translate approximately constant to derivative approximately 0. This allows us to search for regions of homeostasis in a model using bifurcation theory like formulas. Second, we claim that there is a sense in which homeostasis can be thought of as a network invariant. This is joint work with Ian Stewart.

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