A Computational Tool for the Reduction of Nonlinear ODE Systems Possessing Multiple Scales

Near an orbit of interest in a dynamical system, it is typical to ask which variables dominate its structure at what times. What are its principal local degrees of freedom? What local bifurcation structure is most appropriate? We introduce a combined numerical and analytical technique that aids the identification of structure in a class of systems of nonlinear ordinary differential equations (ODEs) that are commonly applied in dynamical models of physical processes. This "dominant scale" technique prioritizes consideration of the influence that distinguished "inputs" to an ODE have on its dynamics. On this basis a sequence of reduced models is derived, where each model is valid for a duration that is determined self-consistently as the system’s state variables evolve. The characteristic time scales of all sufficiently dominant variables are also taken into account to further reduce the model. The result is a hybrid dynamical system of reduced differential-algebraic models that are switched at discrete eve...

[1]  I. Jolliffe.  Principal Component Analysis , 2002 .

[2]  G. Ermentrout,et al.  Symmetry and phaselocking in chains of weakly coupled oscillators , 1986 .

[3]  John Guckenheimer,et al.  Computing Periodic Orbits and their Bifurcations with Automatic Differentiation , 2000, SIAM J. Sci. Comput..

[4]  R. Traub,et al.  Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation , 1995, Nature.

[5]  M. Krupa,et al.  Relaxation Oscillation and Canard Explosion , 2001 .

[6]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[7]  Alexander N. Gorban,et al.  Reduced description in the reaction kinetics , 2000 .

[8]  Danielle C. Tarraf,et al.  On the nature and stability of differential-algebraic systems , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[9]  G. Buzsáki,et al.  Gamma Oscillation by Synaptic Inhibition in a Hippocampal Interneuronal Network Model , 1996, The Journal of Neuroscience.

[10]  Ulrich Maas,et al.  Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space , 1992 .

[11]  Carson C. Chow,et al.  Synchronization and Oscillatory Dynamics in Heterogeneous, Mutually Inhibited Neurons , 1998, Journal of Computational Neuroscience.

[12]  S. L. Campbell,et al.  Intelligent DAE solvers and user friendly design, simulation, and analysis packages , 1998, SMC'98 Conference Proceedings. 1998 IEEE International Conference on Systems, Man, and Cybernetics (Cat. No.98CH36218).

[13]  J. Grasman.  Asymptotic Methods for Relaxation Oscillations and Applications , 1987 .

[14]  Ioannis G. Kevrekidis,et al.  Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum , 2002, SIAM J. Sci. Comput..

[15]  Muruhan Rathinam,et al.  Dynamic Iteration Using Reduced Order Models: A Method for Simulation of Large Scale Modular Systems , 2002, SIAM J. Numer. Anal..

[16]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[17]  C. Morris,et al.  Voltage oscillations in the barnacle giant muscle fiber. , 1981, Biophysical journal.

[18]  Eve Marder,et al.  Reduction of conductance-based neuron models , 1992, Biological Cybernetics.

[19]  S. Pilyugin.  Shadowing in dynamical systems , 1999 .

[20]  D. Sorensen,et al.  Approximation of large-scale dynamical systems: an overview , 2004 .

[21]  Nancy Kopell,et al.  Alpha-Frequency Rhythms Desynchronize over Long Cortical Distances: A Modeling Study , 2000, Journal of Computational Neuroscience.

[22]  M. Branicky.  Analyzing continuous switching systems: theory and examples , 1994, Proceedings of 1994 American Control Conference - ACC '94.

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

[24]  H. B. Keller,et al.  NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (II): BIFURCATION IN INFINITE DIMENSIONS , 1991 .

[25]  John Guckenheimer,et al.  Bifurcation and degenerate decomposition in multiple time scale dynamical systems , 2002 .

[26]  Vadim N. Biktashev,et al.  The asymptotic Structure of the Hodgkin-huxley Equations , 2003, Int. J. Bifurc. Chaos.

[27]  L. Abbott,et al.  Model neurons: From Hodgkin-Huxley to hopfield , 1990 .

[28]  Bruno Sportisse,et al.  Partitioning techniques and lumping computation for reducing chemical kinetics: APLA: an automatic partitioning and lumping algorithm , 2002 .

[29]  Ioannis G. Kevrekidis,et al.  International Journal of Bifurcation and Chaos in Applied Sciences and Engineering: Editorial , 2005 .

[30]  Ian A. Hiskens,et al.  Stability of limit cycles in hybrid systems , 2001, Proceedings of the 34th Annual Hawaii International Conference on System Sciences.

[31]  Ian A. Hiskens,et al.  Sensitivity analysis of differential-algebraic systems using the GMRES method-application to power systems , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

[32]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[33]  Iliya V. Karlin,et al.  Method of invariant manifold for chemical kinetics , 2003 .

[34]  Vadim N Biktashev,et al.  Comparison of asymptotics of heart and nerve excitability. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  J. Rinzel,et al.  INTEGRATE-AND-FIRE MODELS OF NERVE MEMBRANE RESPONSE TO OSCILLATORY INPUT. , 1981 .

[36]  J. Cowan,et al.  A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue , 1973, Kybernetik.

[37]  Muruhan Rathinam,et al.  A New Look at Proper Orthogonal Decomposition , 2003, SIAM J. Numer. Anal..

[38]  G. Ermentrout,et al.  Coupled oscillators and the design of central pattern generators , 1988 .

[39]  Wulfram Gerstner,et al.  Reduction of the Hodgkin-Huxley Equations to a Single-Variable Threshold Model , 1997, Neural Computation.

[40]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[41]  Carson C. Chow,et al.  Frequency Control in Synchronized Networks of Inhibitory Neurons , 1998, Journal of Computational Neuroscience.

[42]  N. K. Rozov,et al.  Differential Equations with Small Parameters and Relaxation Oscillations , 1980 .

[43]  J.H. Taylor,et al.  Modeling and simulation of hybrid systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[44]  Y. Kuramoto.  Collective synchronization of pulse-coupled oscillators and excitable units , 1991 .

[45]  Nancy Kopell,et al.  Synchronization of Strongly Coupled Excitatory Neurons: Relating Network Behavior to Biophysics , 2003, Journal of Computational Neuroscience.

[46]  A. Alonso,et al.  Oscillatory Activity in Entorhinal Neurons and Circuits: Mechanisms and Function , 2000, Annals of the New York Academy of Sciences.

[47]  R. Traub,et al.  Inhibition-based rhythms: experimental and mathematical observations on network dynamics. , 2000, International journal of psychophysiology : official journal of the International Organization of Psychophysiology.

[48]  Germán Mato,et al.  Synchrony in Excitatory Neural Networks , 1995, Neural Computation.

[49]  A. N. Gorbana,et al.  Reduced description in the reaction kinetics , 1999 .

[50]  W. Eckhaus.  Asymptotic Analysis of Singular Perturbations , 1979 .

[51]  N. Kopell,et al.  Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[52]  H. McKean.  Nagumo's equation , 1970 .

[53]  B. Van Der Pol,et al.  Forced Oscillations in a Circuit with Nonlinear Resistance , 1924 .

[54]  Michal Rewienski,et al.  A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems , 2003 .

[55]  Peter Deuflhard,et al.  Dynamic Dimension Reduction in ODE Models , 1996 .

[56]  S. Wiggins.  Normally Hyperbolic Invariant Manifolds in Dynamical Systems , 1994 .

[57]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[58]  James H. Taylor,et al.  MODELING AND SIMULATION OF HYBRID SYSTEMS IN MATLAB , 1996 .

[59]  Kenneth R. Jackson,et al.  Rigorous Shadowing of Numerical Solutions of Ordinary Differential Equations by Containment , 2003, SIAM J. Numer. Anal..

[60]  RewieÅ ski,et al.  A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems , 2003 .

[61]  Arjan van der Schaft,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[62]  Clarence W. Rowley,et al.  Model Reduction for fluids, Using Balanced Proper Orthogonal Decomposition , 2005, Int. J. Bifurc. Chaos.

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

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

[65]  Henry D. I. Abarbanel,et al.  Analysis of Observed Chaotic Data , 1995 .

[66]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[67]  John Guckenheimer,et al.  The Forced van der Pol Equation I: The Slow Flow and Its Bifurcations , 2003, SIAM J. Appl. Dyn. Syst..

[68]  A. Hodgkin,et al.  Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo , 1952, The Journal of physiology.

[69]  Bard Ermentrout,et al.  Type I Membranes, Phase Resetting Curves, and Synchrony , 1996, Neural Computation.

[70]  Nancy Kopell,et al.  Slow and fast inhibition and an H-current interact to create a theta rhythm in a model of CA1 interneuron network. , 2005, Journal of neurophysiology.

[71]  Christof Koch,et al.  Do neurons have a voltage or a current threshold for action potential initiation? , 1995, Journal of Computational Neuroscience.

[72]  Robert Clewley,et al.  Chapter 1 Dominant-scale analysis for the automatic reduction of high-dimensional ODE systems , 2005 .

[73]  G B Ermentrout,et al.  Fine structure of neural spiking and synchronization in the presence of conduction delays. , 1998, Proceedings of the National Academy of Sciences of the United States of America.