# Infinitestimal Homeostasis in Biological Networks

## Talk by Dr. William Duncan (Dept. of Mathematics, MSU, Duke University)

### 9/3/2020 3:10-4:00pm Webex Meeting

#### Abstract:

Homeostasis is the phenomenon in which a biological quantity (such as body temperature), is approximately constant with respect to an external parameter (such as external temperature). Homeostasis can be studied by restricting one’s attention to infinitesimal homeostasis points – points at which a component of a dynamical system has a vanishing derivative with respect to a parameter. If additional consecutive derivatives vanish, then the homeostasis point is more singular and the homeostatic plateau is expected to be larger. In biology, the dynamical system often arises from a network with a distinguished input parameter, I, and output node, o. Recently it was shown that in these input-output networks, there is a homeostasis matrix H such that the input-output function x_o(I) has infinitesimal homeostasis when det(H) = 0. H can be decomposed into irreducible blocks and these blocks can be associated to disjoint subnetworks so that homeostasis is induced by a particular subnetwork. After discussing these results, I will present my ongoing work which answers the following questions. 1) If a subnetwork is homeostasis inducing, which nodes are homeostatic? 2) If two subnetworks are homeostasis inducing, how many derivatives of x_o(I) vanish? Surprisingly, the answer to (2) is that either the first through third derivatives (as opposed to just the first and second) vanish or x_o(I) is not well defined. This is joint work with Marty Golubitsky.