Homeostasis, Catastrophes, and Networks
Networks consisting of nodes and unidirectional arrows encode systems of differential equations. The arrows indicate which nodes are coupled to which. The nodes and arrows can be annotated to indicate which nodes are identical and which kinds of coupling are identical. Our main point is that these types of annotated networks should be thought of as modeling assumptions. What distinguishes a coupled network of differential equations from a large system of differential equations is the desire to keep track of the output from each node individually. It is then possible to compare signals from different nodes (synchrony) and to keep track of singularities in individual nodes. The talk will focus on illustrating these ideas with two biological applications: binocular rivalry (based on Wilson networks) and homeostasis (in biochemical and gene regulatory networks).
Martin Golubitsky is Distinguished Professor of Natural and Mathematical Sciences at the Ohio State University. He received his PhD in Mathematics from M.I.T. in 1970 and has been Professor of Mathematics at Arizona State University (1979-83) and Cullen Distinguished Professor of Mathematics at the University of Houston (1983-2008). At Ohio State Golubitsky served as Director of the Mathematical Biosciences Institute (2008-16). Dr. Golubitsky works in the fields of nonlinear dynamics and bifurcation theory studying the role of symmetry in the formation of patterns in physical systems and the role of network architecture in the dynamics of coupled systems. His recent research focuses on some mathematical aspects of biological applications: animal gaits, the visual cortex, homeostasis, and coupled systems. He has co-authored four graduate texts, one undergraduate text, and two nontechnical trade books, (Fearful Symmetry: Is God a Geometer with Ian Stewart and Symmetry in Chaos with Michael Field) and over 100 research papers. Dr. Golubitsky is a Fellow of the American Academy of Arts and Sciences, the American Association for the Advancement of Science (AAAS), the American Mathematical Society (AMS), and the Society for Industrial and Applied Mathematics (SIAM). He is also the 1997 recipient of the University of Houston Esther Farfel Award, the 2001 corecipient of the Ferran Sunyer i Balaguer Prize (for The Symmetry Perspective) and the recipient of the 2009 Moser Lecture Prize of the SIAM Dynamical Systems Activity Group. He has been elected to the Councils of SIAM, AAAS, and AMS. Dr. Golubitsky was the founding Editor-in-Chief of the SIAM Journal on Applied Dynamical Systems and has served as President of SIAM (2005-06). |