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Persistent or reentrant activity (PA) in systems of coupled neurons, cardiac myocytes, etc is a common feature that can be desired or pathological, depending on the context. This activity arises in networks of excitable elements when there are reentrant paths. Simultaneous activation (synchronization) of the units leads to silencing of the persistent activity. Thus, PA and quiescence represent two stable states of the network (bistability). A natural question is how does the topology of the connectivity between the elements in the network affect the existence of reentrant or persistent activity. In this talk, I will first relate reentrant activity in excitable units, to nonsynchronous locked patterns in networks of coupled oscillators. With this convenient homotopy, I will turn my attention to systems of equations of the form:
xi' = sum_j g_{ij} sin(xj xi+alpha)
where g_{ij} is the connection graph of 0's and 1's. I will focus almost entirely on regular undirected graphs where each node has exactly k edges. With a brief introduction to k=2, I will present some recent results on k=3 and k>3 where the graphs either have some symmetry or are random. I will construct stable nonsynchronous solutions (reentrant) and also explore the dynamics on random regular graphs. Finally, I will also briefly describe some approaches that use algebraic geometry to exhaustively search for patterns.
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