Winding Numbers and Average Frequencies in Phase Oscillator Networks |
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Authors: | M Golubitsky K Josic E Shea-Brown |
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Institution: | (1) Department of Mathematics, University of Houston, Houston TX 77204-3008, USA;(2) Courant Institute of Mathematical Sciences and Center for Neural Science, 251 Mercer St., New York University, New York, NY 10012, USA |
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Abstract: | We study networks of coupled phase oscillators and show that
network architecture can force relations between average
frequencies of the oscillators. The main tool of our analysis is
the coupled cell theory developed by Stewart, Golubitsky, Pivato,
and Torok, which provides precise relations between network
architecture and the corresponding class of ODEs in RM and
gives conditions for the flow-invariance of certain polydiagonal
subspaces for all coupled systems with a given network
architecture. The theory generalizes the notion of fixed-point
subspaces for subgroups of network symmetries and directly extends
to networks of coupled phase oscillators. For systems of coupled phase oscillators (but not generally for ODEs in RM, where M ≥ 2), invariant polydiagonal subsets of
codimension one arise naturally and strongly restrict the network
dynamics. We say that two oscillators i and j coevolve if the polydiagonal θi = θj is flow-invariant, and show that the average frequencies of these
oscillators must be equal. Given a network architecture, it is shown that coupled cell theory
provides a direct way of testing how coevolving oscillators form
collections with closely related dynamics. We give a
generalization of these results to synchronous clusters of phase
oscillators using quotient networks, and discuss implications for
networks of spiking cells and those connected through buffers that
implement coupling dynamics. |
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Keywords: | |
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