Symmetry groupoids and patterns of synchrony\ud in coupled cell networks

Article English OPEN
Stewart, Ian ; Golubitsky, Martin ; Pivato, Marcus (2003)
  • Publisher: Society for Industrial and Applied Mathematics
  • Related identifiers: doi: 10.1137/S1111111103419896
  • Subject: QA
    arxiv: Quantitative Biology::Cell Behavior

A coupled cell system is a network of dynamical systems, or “cells,” coupled together. Such systems\ud can be represented schematically by a directed graph whose nodes correspond to cells and whose\ud edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that\ud preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized\ud cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only\ud mechanism that can create such states in a coupled cell system and show that it is not.\ud The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information\ud about the input sets of cells. (The input set of a cell consists of that cell and all cells\ud connected to that cell.) The admissible vector fields for a given graph—the dynamical systems with\ud the corresponding internal dynamics and couplings—are precisely those that are equivariant under\ud the symmetry groupoid. A pattern of synchrony is “robust” if it arises for all admissible vector\ud fields. The first main result shows that robust patterns of synchrony (invariance of “polydiagonal”\ud subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an\ud equivalence relation on cells is “balanced.” The second main result shows that admissible vector\ud fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled\ud cell network, the “quotient network.” The existence of quotient networks has surprising implications\ud for synchronous dynamics in coupled cell systems.
  • References (8)

    [1] H. Brandt, U¨ber eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1927), pp. 360-366.

    [2] S. Boccaletti, L. M. Pecora, and A. Pelaez, Unifying framework for synchronization of coupled dynamical systems, Phys. Rev. E (3), 63 (2001), 066219.

    [3] R. Brown, From groups to groupoids: A brief survey, Bull. London Math. Soc., 19 (1987), pp. 113-134.

    [4] P. L. Buono and M. Golubitsky, Models of central pattern generators for quadruped locomotion I. Primary gaits, J. Math. Biol., 42 (2001), pp. 291-326.

    [5] A. Dias and I. Stewart, Symmetry groupoids and admissible vector fields for coupled cell networks, submitted.

    [6] M. Golubitsky, M. Nicol, and I. Stewart, Some curious phenomena in coupled cell networks, submitted.

    [7] M. Golubitsky and I. Stewart, The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progr. Math. 200, Birkh¨auser Verlag, Basel, 2002.

    [8] M. Golubitsky and I. Stewart, Patterns of oscillation in coupled cell systems, in Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J. E. Marsden, P. Holmes, P. Newton, and A. Weinstein, eds., Springer-Verlag, New York, 2002, pp. 243-286.

  • Metrics
    views in OpenAIRE
    views in local repository
    downloads in local repository

    The information is available from the following content providers:

    From Number Of Views Number Of Downloads
    Warwick Research Archives Portal Repository - IRUS-UK 0 55
Share - Bookmark