Continuation-based Computation of Global Isochrons

Isochrons are foliations of phase space that extend the notion of phase of a stable periodic orbit to the basin of attraction of this periodic orbit. Each point in the basin of attraction lies on only one isochron, and two points on the same isochron converge to the periodic orbit with the same phase. Global isochrons, that is, isochrons extended into the full basin of attraction rather than just a neighborhood of the periodic orbit, can form remarkable foliations. For example, accumulations of all isochrons can occur in arbitrarily small regions of phase space; the limit of such an accumulation is called the phaseless set, which lies on the boundary of the basin of attraction of the periodic orbit. Since global isochrons must typically be approximated numerically, such complicated geometries are often difficult to realize for actual examples. Indeed, the computation of global isochrons can be challenging, particularly for systems with multiple time scales. We present a novel method for computing isochron...

[1]  Hinke M. Osinga,et al.  Numerical study of manifold computations , 2005 .

[2]  A. Winfree.  Patterns of phase compromise in biological cycles , 1974 .

[3]  John Guckenheimer,et al.  A Survey of Methods for Computing (un)Stable Manifolds of Vector Fields , 2005, Int. J. Bifurc. Chaos.

[4]  Bernd Krauskopf,et al.  Growing 1D and Quasi-2D Unstable Manifolds of Maps , 1998 .

[5]  Bard Ermentrout,et al.  Simulating, analyzing, and animating dynamical systems - a guide to XPPAUT for researchers and students , 2002, Software, environments, tools.

[6]  Yuri A. Kuznetsov,et al.  Continuation of Connecting orbits in 3D-ODES (II) : Cycle-to-Cycle Connections , 2008, Int. J. Bifurc. Chaos.

[7]  Jacques Demongeot,et al.  The Isochronal Fibration: Characterization and Implication in Biology , 2010, Acta biotheoretica.

[8]  A. J. Grono,et al.  Setting and testing automatic generator synchronizers , 1999 .

[9]  Freddy Dumortier,et al.  Canard Cycles and Center Manifolds , 1996 .

[10]  J. Guckenheimer,et al.  Isochrons and phaseless sets , 1975, Journal of mathematical biology.

[11]  Willy Govaerts,et al.  Convergence analysis of a numerical method to solve the adjoint linearized periodic orbit equations , 2010 .

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

[13]  Rustam Singh,et al.  Simulating , 2012 .

[14]  Gemma Huguet,et al.  A Computational and Geometric Approach to Phase Resetting Curves and Surfaces , 2009, SIAM J. Appl. Dyn. Syst..

[15]  G. Ermentrout,et al.  Phase-response curves give the responses of neurons to transient inputs. , 2005, Journal of neurophysiology.

[16]  João Pedro Hespanha,et al.  Event-based minimum-time control of oscillatory neuron models , 2009, Biological Cybernetics.

[17]  D. Takeshita,et al.  Higher order approximation of isochrons , 2009, 0910.3890.

[18]  From Clocks to Chaos: The Rhythms of Life , 1988 .

[19]  Michael E. Henderson,et al.  Computing Invariant Manifolds by Integrating Fat Trajectories , 2005, SIAM J. Appl. Dyn. Syst..

[20]  Bernd Krauskopf,et al.  Computing One-Dimensional Global Manifolds of Poincaré Maps by Continuation , 2005, SIAM J. Appl. Dyn. Syst..

[21]  C. D. Boor,et al.  Collocation at Gaussian Points , 1973 .

[22]  James P. Keener,et al.  Mathematical physiology , 1998 .

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

[24]  G. Ermentrout,et al.  Multiple pulse interactions and averaging in systems of coupled neural oscillators , 1991 .

[25]  Bernd Krauskopf,et al.  A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits , 2008 .

[26]  Yuri A. Kuznetsov,et al.  Continuation of Connecting orbits in 3D-ODES (I): Point-to-Cycle Connections , 2007, Int. J. Bifurc. Chaos.

[27]  Theodosios Pavlidis,et al.  Biological Oscillators: Their Mathematical Analysis , 1973 .

[28]  A. Prinz,et al.  Phase resetting and phase locking in hybrid circuits of one model and one biological neuron. , 2004, Biophysical journal.

[29]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[30]  Eusebius J. Doedel,et al.  Lecture Notes on Numerical Analysis of Nonlinear Equations , 2007 .

[31]  D. W. Smaha,et al.  Generator synchronizing industry survey results , 1996 .

[32]  M. Wodzicki.  Lecture Notes in Math , 1984 .

[33]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1990 .

[34]  Michael Shub,et al.  The local theory of normally hyperbolic, invariant, compact manifolds , 1977 .

[35]  Eric Shea-Brown,et al.  On the Phase Reduction and Response Dynamics of Neural Oscillator Populations , 2004, Neural Computation.

[36]  Bernd Krauskopf,et al.  Computing Invariant Manifolds via the Continuation of Orbit Segments , 2007 .

[37]  J. Palis,et al.  Geometric theory of dynamical systems , 1982 .

[38]  H. A. Larrondo,et al.  Isochrones and the dynamics of kicked oscillators , 1989 .

[39]  Frank Schilder,et al.  Continuation of Quasi-periodic Invariant Tori , 2005, SIAM J. Appl. Dyn. Syst..

[40]  R. Llave,et al.  The parameterization method for invariant manifolds III: overview and applications , 2005 .

[41]  W. Govaerts,et al.  Computation of the Phase Response Curve: A Direct Numerical Approach , 2006, Neural Computation.

[42]  A. Winfree.  The geometry of biological time , 1991 .

[43]  M. Liserre,et al.  Synchronization methods for three phase distributed power generation systems - An overview and evaluation , 2005, 2005 IEEE 36th Power Electronics Specialists Conference.

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

[45]  John Guckenheimer,et al.  Dissecting the Phase Response of a Model Bursting Neuron , 2009, SIAM J. Appl. Dyn. Syst..

[46]  John Guckenheimer,et al.  Mixed-Mode Oscillations with Multiple Time Scales , 2012, SIAM Rev..

[47]  C. Simó.  On the analytical and numerical approximation of invariant manifolds. , 1990 .