Abstract
We give a constructive method for realising an arbitrary directed graph (with no one-cycles) as a heteroclinic or an excitable dynamic network in the phase space of a system of coupled cells of two types. In each case, the system is expressed as a system of first-order differential equations. One of the cell types (the p-cells) interacts by mutual inhibition and classifies which vertex (state) we are currently close to, while the other cell type (the y-cells) excites the p-cells selectively and becomes active only when there is a transition between vertices. We exhibit open sets of parameter values such that these dynamical networks exist and demonstrate via numerical simulation that they can be attractors for suitably chosen parameters.
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Notes
We refer to a “(weak) heteroclinic network (in phase space)” simply as a “heteroclinic network” for the remainder of the paper.
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Acknowledgments
We thank the following for stimulating conversations that contributed to the development of this paper: Mike Field, Marc Timme, John Terry, Ilze Ziedins. We also thank the London Mathematical Society for support of a visit of CMP to Exeter, the University of Auckland Research Council for supporting a visit of PA to Auckland during the development of this research and a referee for a number of questions that helped clarify the exposition. We are grateful to the Mathematics Departments at both Exeter and Auckland Universities for their hospitality during these visits.
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Communicated by Paul Newton.
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Ashwin, P., Postlethwaite, C. Designing Heteroclinic and Excitable Networks in Phase Space Using Two Populations of Coupled Cells. J Nonlinear Sci 26, 345–364 (2016). https://doi.org/10.1007/s00332-015-9277-2
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DOI: https://doi.org/10.1007/s00332-015-9277-2