Patterns of desynchronization and resynchronization in heteroclinic networks

We prove results that enable the efficient and natural realization of a large class of robust heteroclinic networks in coupled identical cell systems. We also propose some general conjectures that relate a natural and large class of robust heteroclinic networks that occur in networks modelled by equations of Lotka–Volterra type, and certain networks of symmetric systems, to robust heteroclinic networks in coupled cell networks.

[1]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[2]  Lambros Lambrou,et al.  Combinatorial Dynamics , 2004 .

[3]  Martin Krupa,et al.  Asymptotic stability of heteroclinic cycles in systems with symmetry. II , 2004, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[4]  M. Field Equivariant bifurcation theory and symmetry breaking , 1989 .

[5]  Michael Field,et al.  Dynamical equivalence of networks of coupled dynamical systems: II. General case , 2010 .

[6]  G. L. D. Reis,et al.  Structural stability of equivariant vector fields on two-manifolds , 1984 .

[7]  Ian Stewart,et al.  Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows , 2005, SIAM J. Appl. Dyn. Syst..

[8]  Peter Ashwin,et al.  On designing heteroclinic networks from graphs , 2013, 1302.0984.

[9]  Ian Melbourne,et al.  An example of a nonasymptotically stable attractor , 1991 .

[10]  E. Izhikevich,et al.  Weakly connected neural networks , 1997 .

[11]  P. Ashwin,et al.  Discrete computation using a perturbed heteroclinic network , 2005 .

[12]  Ramón Huerta,et al.  Transients versus attractors in Complex Networks , 2010, Int. J. Bifurc. Chaos.

[13]  M. Golubitsky,et al.  Interior symmetry and local bifurcation in coupled cell networks , 2004 .

[14]  Vivien Kirk,et al.  A competition between heteroclinic cycles , 1994 .

[15]  R. Huerta,et al.  Winnerless competition principle and prediction of the transient dynamics in a Lotka-Volterra model. , 2008, Chaos.

[16]  Michael Field,et al.  Dynamical equivalence of networks of coupled dynamical systems: I. Asymmetric inputs , 2010 .

[17]  Gábor Orosz,et al.  Dynamics on Networks of Cluster States for Globally Coupled Phase Oscillators , 2007, SIAM J. Appl. Dyn. Syst..

[18]  Alexandre A. P. Rodrigues,et al.  Attractors in complex networks. , 2017, Chaos.

[19]  Jeff Moehlis,et al.  Equivariant dynamical systems , 2007, Scholarpedia.

[20]  M. Peixoto,et al.  On an approximation theorem of Kupka and Smale , 1967 .

[21]  Michael Field,et al.  Stationary bifurcation to limit cycles and heteroclinic cycles , 1991 .

[22]  A. Konno,et al.  Dynamics , 2019, Humanoid Robots.

[23]  L. Chua,et al.  Methods of qualitative theory in nonlinear dynamics , 1998 .

[24]  M. Golubitsky,et al.  Nonlinear dynamics of networks: the groupoid formalism , 2006 .

[25]  M. Field Transversality in $G$-manifolds , 1977 .

[26]  Gilles Laurent,et al.  Transient Dynamics for Neural Processing , 2008, Science.

[27]  Claire M. Postlethwaite,et al.  Designing Heteroclinic and Excitable Networks in Phase Space Using Two Populations of Coupled Cells , 2015, J. Nonlinear Sci..

[28]  S. Strogatz,et al.  Synchronization of pulse-coupled biological oscillators , 1990 .

[29]  P. Holmes,et al.  Structurally stable heteroclinic cycles , 1988, Mathematical Proceedings of the Cambridge Philosophical Society.

[30]  M. Field HETEROCLINIC CYCLES IN SYMMETRICALLY COUPLED SYSTEMS , 1999 .

[31]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[32]  R. W. Richardson,et al.  Symmetry breaking and branching patterns in equivariant bifurcation theory II , 1992 .

[33]  M. J. Field,et al.  Heteroclinic Networks in Homogeneous and Heterogeneous Identical Cell Systems , 2015, Journal of Nonlinear Science.

[34]  D. Hansel,et al.  Phase Dynamics for Weakly Coupled Hodgkin-Huxley Neurons , 1993 .

[35]  Ian Stewart,et al.  Linear equivalence and ODE-equivalence for coupled cell networks , 2005 .

[36]  Manuela A. D. Aguiar,et al.  The Lattice of Synchrony Subspaces of a Coupled Cell Network: Characterization and Computation Algorithm , 2014, J. Nonlinear Sci..

[37]  P. Ashwin,et al.  Heteroclinic Networks in Coupled Cell Systems , 1999 .

[38]  Thomas Nowotny,et al.  Dynamical origin of independent spiking and bursting activity in neural microcircuits. , 2007, Physical review letters.

[39]  Melbourne,et al.  Asymptotic stability of heteroclinic cycles in systems with symmetry , 1995, Ergodic Theory and Dynamical Systems.

[40]  Werner Brannath,et al.  Heteroclinic networks on the tetrahedron , 1994 .

[41]  Peter Ashwin,et al.  Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation , 2011, J. Nonlinear Sci..

[42]  M. Aguiar,et al.  Heteroclinic network dynamics on joining coupled cell networks , 2017 .

[43]  Josef Hofbauer,et al.  The theory of evolution and dynamical systems , 1988 .

[44]  Michael Field,et al.  Dynamics and Symmetry , 2007 .

[45]  P. Ashwin,et al.  Multi-cluster dynamics in coupled phase oscillator networks , 2014, 1409.7527.

[46]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[47]  P. Ashwin,et al.  Designing the Dynamics of Globally Coupled Oscillators , 2009 .

[48]  Sofia B. S. D. Castro,et al.  Dynamics near a heteroclinic network , 2005 .

[49]  V. Zhigulin,et al.  On the origin of reproducible sequential activity in neural circuits. , 2004, Chaos.

[50]  Marcus Pivato,et al.  Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks , 2003, SIAM J. Appl. Dyn. Syst..

[51]  R. May,et al.  Nonlinear Aspects of Competition Between Three Species , 1975 .

[52]  Ian Stewart,et al.  The lattice of balanced equivalence relations of a coupled cell network , 2007, Mathematical Proceedings of the Cambridge Philosophical Society.

[53]  Josef Hofbauer,et al.  Heteroclinic cycles in ecological differential equations , 1994 .

[54]  M. Krupa Robust heteroclinic cycles , 1997 .

[55]  P. Ashwin,et al.  Encoding via conjugate symmetries of slow oscillations for globally coupled oscillators. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  Martin Golubitsky,et al.  Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.