Mathematical Homeostasis Motivated by Nijhout, Reed, and Best

Marty Golubitsky (June 28, 2019)

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We say that an input-output map xo(I) has infinitesimal homeostasis at I0 if x′o(I0) = 0. A consequence of infinitesimal homeostasis is that xo(I) is approximately constant on a neighborhood of I0. An input-output network is a network that has a designated input node , a designated output node o, and a set of regulatory nodes  = (i, . . . , n). We assume that the system of network differential equations Ë™X = F(X, I) has a stable equilibrium at X0. The implicit function theorem implies that there exists a family of equilibria X(I) = (x(I), x(I), xo(I)), where xo(I) is the network input-output map. We use the network architecture of input-output networks to classify infinitesimal homeostasis into three types: structural homeostasis, Haldane homeostasis, and appendage homeostasis. The first two types generalize feedforward excitation and substrate inhibition. The third type appears to be a new form of homeostasis. This research is a joint project with Yangyang Wang, Ian Stewart, Joe Huang, and Fernando Antoneli.