Applications of Coupled Cell Systems

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A system of differential equations is called a cell. A coupled cell network is a collection of cells that are coupled together and the architecture of that network is a graph that indicates which cells are identical and which cells are coupled to which. The primary question addressed by this proposal is: What part of the dynamics of coupled systems is due to network architecture? The theory of these systems was developed with recent NSF support: we will continue its development by focusing on bifurcations and forcing of feedforward networks, stability of solutions obtained by bifurcation, and bursting in coupled systems. We will also focus more directly on applications including the vestibular system (in particular, the network of connections between neurons of the six semicircular canals in the ears and eight neck muscle groups), the development of a frequency filter/amplifier associated to synchrony breaking Hopf bifurcation in a simple feedforward network, and `cortical songs' obtained from different dynamical patterns in coupled phase oscillators.The biologist J.B.S. Haldane, when asked what we can learn about the Creator by examining the world, replied that God seemed to have an inordinate fondness for beetles. Today's biologists could be forgiven for pointing to the deity's inordinate fondness for networks. Networks are ubiquitous in biology: examples include gene expression, neural circuitry, ecological food webs, and disease transmission. Networks are also common in many other branches of science, and there has been a recent explosion of interest in the topic. The research literature, including applications, now extends to many thousands of papers. We have been developing a theory of network dynamics, where the nodes in the network are systems of differential equations. In this proposal we focus on some generalizations of the theory and on specific applications including the dynamics associated to a network in the vestibular system called the canal-neck projection.