Coupled Systems and Applications

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The principal investigator and his students and collaborators study the dynamics of networks of systems of differential equations and their applications. Previous work identified when network architecture forces synchrony to be present in such systems (generalizing network symmetry to a combinatorial notion of a balanced coloring). Balanced colorings also lead to quotient networks and previous work also showed that robust phase shift synchrony can be associated to symmetry in a quotient network. These theories were motivated by studies of locomotor central pattern generators of animal gaits. The studies have progressed and several general questions are now studied like: Are all robust phase shifts present in periodic solutions of network equations the result of symmetry (using quotient networks), what are the patterns of oscillations associated with network architecture, and do these patterns always appear through bifurcation? (Surprisingly the answer to the last question is no; additionally, network architecture can change Hopf bifurcation in ways that transcend symmetry.) How network dynamics affects Takens reconstruction is studied (is it possible to reconstruct network dynamics from the output of one node) and the dynamics of small networks (can general network studies lead to a better understanding of motifs in Systems Biology) is studied.The study of networks of differential equations is central to much of modern biology (from biochemical networks through protein interaction networks and gene transcription networks to ecology through food webs). Recent work by many research groups (usually based on chemical kinetics equations) has shown that network architecture does affect the kind of solutions that appear in coupled systems. Moreover, these solutions lead to function and to conjectures about how large biological networks work. General mathematical studies of network dynamics are providing both the tools needed to investigate more specific networks and an understanding of the kinds of phenomena that network models can produce. The project involves studying general theories for network dynamics, specific examples of networks that promise to yield new (mathematical) phenomena, and specific application areas (such as the ways in which sound waves are processed in the cochlear region of the inner ear, which in part is based on a network model of inner hair bundles).