Workshop 1: Dynamics in Networks with Special Properties

(January 25,2016 - January 29,2016 )


Peter Ashwin
College of Engineering, Mathematics and Physical Sciences, University of Exeter
Gheorghe Craciun
Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison
Thomas Gedeon
Department of Mathematical Sciences, Montana State University

Network structures underlie models for the dynamical descriptions and bifurcations in a wide range of biological phenomena. These include models for subcellular genomic and signaling processes, neural models at single-cell or multiple-cell level, high-level cognitive models, and many forms of chemically reacting systems. Naturally arising networks often have properties that make them especially pliant mathematically:

  1. Networks with special overall architectures (having, for example, special symmetries or special node-coupling rules) enable one to pose and answer questions about network dynamics by techniques that exploit those architectures.
  2. Networks often have natural and precise restrictions on the behavior of nodes or edges (for example, mass action kinetics, piecewise linear dynamics, or Boolean dynamics) that can lead to fairly deep and general theorems about network behavior.
  3. Small networks, quotient networks, factor networks, or network motifs can be used as building blocks to understand the dynamics of larger networks, allowing one to think of network structures in a constructive and synthetic way.

This workshop will explore the current state of affairs and ways of unifying emerging mathematical techniques by focusing on a variety of special biologically motivated structures. A few examples of applications with such structures include:

a. Chemical reaction network theory: In recent years separate theories have been developed for the dynamics of chemical reaction networks and for general networks with symmetry. Both theories have yielded interesting and nontrivial results. What has not been explored, however, are networks of identical and interconnected chemical cells. Although the dynamics within the cells themselves might exhibit a high degree of stability, there remain questions about the dynamics of the full multi-cell assembly that can exhibit important behavior through symmetry-breaking pattern formation.

b. Neuroscience: Classic central pattern generators and recent Wilson networks describing generalized rivalry have symmetries that dictate preferred kinds of patterned oscillation.

c. Gene expression networks: New technologies are providing massive data concerning the connectivity and functional control of these networks. Theoretical models based on these data typically incorporate combinatorial logical control of gene expression in dynamical models.

d. Epidemiology: Heterosexual contact networks are best explored by (quasi-) bipartite graphs, usually with fairly strong restrictions on degree distributions within each partition. Exploring sexually transmitted infection dynamics relies strongly on both this architecture (especially as the global description is preserved over time while local descriptions shift), but also on the time ordering of edge presence.

Development and analysis of models like these provide a rich source of new mathematical problems involving classification of dynamic states, bifurcations, and reverse engineering of complex networks based on observed dynamics.

Monday, January 25, 2016
Time Session
Tuesday, January 26, 2016
Time Session
Wednesday, January 27, 2016
Time Session
Thursday, January 28, 2016
Time Session
Friday, January 29, 2016
Time Session
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