Physics Colloquium 
Synchrony and Phase Relations in Network Dynamics
Ian Stewart 

Emeritus Professor of Mathematics
University of Warwick, England.
Abstract 

The talk will summarize some of the main ideas in a formal theory of networks of coupled dynamical systems, developed over the past 20 years or so. It is intended for non-specialists, so much of the material will be familiar to experts, but I will include some more recent results. I will also mention applications to neuroscience, synthetic gene circuits, and animal locomotion.

By 'network' I mean a directed graph whose nodes represent dynamical systems, and whose arrows represent coupling between those systems. Both nodes and arrows can be assigned 'types': roughly speaking, nodes of the same type have identical internal dynamics, and arrows of the same type represent identical coupling. Each network diagram defines a class of 'admissible' differential equations, which are compatible with the network topology (and types). Two nodes are synchronous, for a specific solution of this equation, if they have the same state at all times. For periodic states, nodes can also be related by a phase shift. 

Symmetries of the network can create synchrony and phase patterns; however, such patterns can also exist without symmetry. The key concept here is that of a 'balanced coloring' of the nodes: one in which nodes of the same color receive inputs whose colors and types match. I will discuss how balanced colorings  determine synchrony and phase patterns, summarise some basic conjectures in  the theory that have been partly proved, and describe recent results on networks with the topology of a lattice. 

Background

Ian Nicholas Stewart  FRS CMath FIMA (born 24 September 1945 in Folkestone, England)[3] is a British mathematician and a popular-science and science-fiction writer.[4] He is Emeritus Professor of Mathematics at the University of Warwick, England.