Collective canard explosions of globally-coupled rotators with adaptive coupling

Marzena Ciszak, Simona Olmi, 3 Giacomo Innocenti, Alessandro Torcini, 2 and Francesco Marino 3 CNR Consiglio Nazionale delle Ricerche Istituto Nazionale di Ottica, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy CNR Consiglio Nazionale delle Ricerche Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy INFN, Sezione di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy Department of Industrial Engineering, University of Florence, via di Santa Marta 3, 50139 Florence, Italy Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, CNRS, UMR 8089, 95302 Cergy-Pontoise cedex, France (Dated: October 22, 2021)

[1]  S. Strogatz,et al.  Phase diagram for the collective behavior of limit-cycle oscillators. , 1990, Physical review letters.

[2]  S. Olmi,et al.  Emergent excitability in adaptive networks of non-excitable units. , 2020, 2010.06249.

[3]  R. Spigler,et al.  The Kuramoto model: A simple paradigm for synchronization phenomena , 2005 .

[4]  Yoshiki Kuramoto,et al.  Self-entrainment of a population of coupled non-linear oscillators , 1975 .

[5]  Xiao-Jing Wang,et al.  Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle , 1993 .

[6]  E. Ott,et al.  Exact results for the Kuramoto model with a bimodal frequency distribution. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Ernest Montbri'o,et al.  Macroscopic description for networks of spiking neurons , 2015, 1506.06581.

[8]  F T Arecchi,et al.  Mixed-mode oscillations via canard explosions in light-emitting diodes with optoelectronic feedback. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Francesco Marin,et al.  Chaotically spiking canards in an excitable system with 2D inertial fast manifolds. , 2007, Physical review letters.

[10]  R. Veltz,et al.  Effective low-dimensional dynamics of a mean-field coupled network of slow-fast spiking lasers. , 2019, Physical review. E.

[11]  Martin Krupa,et al.  Mixed Mode Oscillations due to the Generalized Canard Phenomenon , 2006 .

[12]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[13]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[14]  M Desroches,et al.  Inflection, canards and excitability threshold in neuronal models , 2012, Journal of Mathematical Biology.

[15]  Jonathan Touboul,et al.  Noise-induced Canard and Mixed-Mode Oscillations in Large-Scale Stochastic Networks , 2015, SIAM J. Appl. Math..

[16]  Serafim Rodrigues,et al.  Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems , 2016, SIAM Rev..

[17]  M Desroches,et al.  Canards of mixed type in a neural burster. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  J. Hindmarsh,et al.  A model of neuronal bursting using three coupled first order differential equations , 1984, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[19]  R. Hilborn.  Lyapunov exponents: a tool to explore complex dynamics , 2017 .

[20]  Diego Pazó,et al.  Existence of hysteresis in the Kuramoto model with bimodal frequency distributions. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  J. M. Gonzalez-Miranda,et al.  Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model. , 2003, Chaos.

[22]  F. Tito Arecchi,et al.  Chaotic spiking and incomplete homoclinic scenarios in semiconductor lasers with optoelectronic feedback , 2009 .

[23]  S. F. Abdalah,et al.  Excitability of periodic and chaotic attractors in semiconductor lasers with optoelectronic feedback , 2010 .

[24]  Andrey Shilnikov,et al.  When weak inhibition synchronizes strongly desynchronizing networks of bursting neurons. , 2008, Physical review letters.

[25]  Michelle M. McCarthy,et al.  Excitable Neurons, Firing Threshold Manifolds and Canards , 2013, Journal of mathematical neuroscience.

[26]  Jürgen Kurths,et al.  Network mechanism for burst generation. , 2007, Physical review letters.

[27]  Nancy J Kopell,et al.  New dynamics in cerebellar Purkinje cells: torus canards. , 2008, Physical review letters.

[28]  N. Kopell,et al.  Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron. , 2008, Chaos.

[29]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[30]  Ernest Barreto,et al.  Generating macroscopic chaos in a network of globally coupled phase oscillators. , 2011, Chaos.

[31]  E. Ott,et al.  Low dimensional behavior of large systems of globally coupled oscillators. , 2008, Chaos.

[32]  J. Callot,et al.  Chasse au canard , 1977 .

[33]  Simona Olmi,et al.  Hysteretic transitions in the Kuramoto model with inertia. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Hakim,et al.  Dynamics of the globally coupled complex Ginzburg-Landau equation. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[35]  Eugene M. Izhikevich,et al.  Neural excitability, Spiking and bursting , 2000, Int. J. Bifurc. Chaos.

[36]  John Guckenheimer,et al.  A Geometric Model for Mixed-Mode Oscillations in a Chemical System , 2011, SIAM J. Appl. Dyn. Syst..

[37]  A. Lichtenberg,et al.  A First Order Phase Transition Resulting from Finite Inertia in Coupled Oscillator Systems , 1996 .

[38]  R. Veltz,et al.  Canard resonance: on noise-induced ordering of trajectories in heterogeneous networks of slow-fast systems , 2020, Journal of Physics: Photonics.

[39]  John Guckenheimer,et al.  Singular Hopf Bifurcation in Systems with Two Slow Variables , 2008, SIAM J. Appl. Dyn. Syst..

[40]  Alessandro Torcini,et al.  Dynamical phases of the Hindmarsh-Rose neuronal model: studies of the transition from bursting to spiking chaos. , 2007, Chaos.

[41]  F. Arecchi,et al.  Control of transient synchronization with external stimuli. , 2009, Chaos.

[42]  37 , 2018, In Pursuit.

[43]  J. Restrepo,et al.  Complex macroscopic behavior in systems of phase oscillators with adaptive coupling , 2012, 1207.3102.

[44]  Jeff Moehlis,et al.  Canards for a reduction of the Hodgkin-Huxley equations , 2006, Journal of mathematical biology.

[45]  Naoko Nakagawa,et al.  Collective Chaos in a Population of Globally Coupled Oscillators , 1993 .