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).
Many mathematical models of biological and life sciences applications have the form of a coupled collection or network of differential equations. We have explored how the general theory of networks of differential equations differs from that of a large system of differential equations and we have applied some of these results to specific neuroscience applications. In a network of differential equations it makes sense to compare the output from two or more nodes. We call the output synchronous if the output in two nodes is identical. If the dynamics is periodic and two nodes have the same output except for a phase lag, then we say that the solution has phase shift synchrony. A simple example of phase shift synchrony occurs when a two-legged animal walks. In this gait (or repetivie periodic motion) the motion of the left leg is one-half period out of phase with the motion of the right leg. We proved that this kind of (rigid) phase shift synchrony can only occur if the network has a symmetry. We also showed that the simplest solutions associated with certain small networks are surprising in a mathematical sense. In particular, coupliing three identical systems A -> B -> C leads naturally to an excellent band with filter. Binocular rivaly is the result of a psychophysics experiment where a subject is shown two different images (one to the left eye and one to the right eye) and asked to descibe what he or she perceives. These experiments explore how the brain reacts to contradictory information. Typically, subjects perceive alternation between the two images. Sometimes, however, more complicated percepts are reported. Hugh Wilson has proposed that rivalry in the brain can be modeled as competition between patterns and that patterns can be understood as choices of levels of a certain set of attributes. The associated mathematical model is a network of differential equations and rivalry alternation is related to phase shift synchrony of periodic solutions. We used our theory of coupled systems to show how the surprising percepts found in a number of psychophysics experiments can be understood using Wilson networks. In addition, with support of this grant, we studied the detailed dynamics of sytems of differential equations near neutrally stable equilibria when the system is periodically forced. We also studied the simplest ways that strategies in adaptive game theory change as parameters are varied. The research of four PhD students were supported in part by this grant; three postdoctoral fellows and two visiting faculty also contributed to this research. This supported research has led to eight published refereed journal articles, to one submitted article, and to two articles that are in preparation.
