# Research

Current Research projects:

“The structure of Infinitesimal Homeostasis in Input-Output Networks”

• In a joint project with Prof. Martin Golubitsky at The Ohio State University, Dr. Yangyang Wang at Univesity of Iowa, and Dr. Fernando Antoneli at Universidade Federal de São Paulo, we studied homeostasis, a phenomenon whereby the output of a system is approximately constant on a variation of an input. Our project follows [Golubtisky and Stewart (2017)], and considers infinitesimal homeostasis, a mathematical concept. Specifically, we say that an input-output map Xo(I) has infinitesimal homeostasis at a point I0 if x'(I0) = 0. A consequence of infinitesimal homeostasis is that xo(I) is approximately constant in a neighborhood of I0. An input-output network is a network that has a designated input node i, a designated output node o, and a set of regulatory nodes. We assume that the system of network differential equations X’ = F(X, I) has a stable equilibrium at X0. The implicit function theorem implies that there exists a family of equilibria, where xo(I) is the network input-output map. We show that there is an (n + 1) × (n + 1) homeostasis matrix H(I) for which dxo/dI = 0 if and only if det(H) = 0. Using the combinatorial matrix theory, we are able to factor the polynomial det(H) and associate each factor to a unique type of infinitesimal homeostasis. Furthermore, we prove that there are exactly two combinatorically defined classes: structural and appendage. Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostatic subnetwork motif corresponding to each factor of det(H) without performing numerical simulations on model equations.

“A Classification of Homeostasis Types in Four-node Input-output Networks”

• This project follows [Wang et al. 2020] and studies infinitesimal homeostasis in Four-node input-output networks. While the classification of infinitesimal homeostasis in a general admissible system provides a global view of homeostasis characteristics, the study of four-node input-output networks draws a connection between biochemical networks and mathematical networks.

Presentation: