Kopell N, Rotstein HG, Clewley R. (2005). A Computational Tool for the Reduction of Nonlinear ODE Systems Possessing Multiple Scales Multiscale modeling & simulation. 4(3)

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References and models cited by this paper

Abarbanel HDI. (1996). Analysis of Observed Chaotic Data.

Abbott LF, Kepler TB. (1990). Model neurons: from Hodgkin Huxley to Hopfield. Statistical Mechanics of Neural Networks Lecture Notes in Physics.

Acker CD, Kopell N, White JA. (2003). Synchronization of strongly coupled excitatory neurons: relating network behavior to biophysics. Journal of computational neuroscience. 15 [PubMed]

Branicky MS. (1994). Analysis of continuous switching systems: Theory and examples Proceedings of the IEEE American Control Conference.

Campbell SL. (1998). Intelligent DAE solvers and user friendly design, simulation, and analysis packages Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics.

Chaniotis D, Pai MA, Hiskens IA. (2001). Sensitivity analysis of differential-algebraic systems using the GMRES method-application to power systems Proceedings of the IEEE International Symposium on Circuits and Systems, Sydney, Australia.

Chow CC, White JA, Ritt J, Kopell N. (1998). Frequency control in synchronized networks of inhibitory neurons. Journal of computational neuroscience. 5 [PubMed]

Clewley R. (2004). Dominant-scale analysis for the automatic reduction of high-dimensional ODE systems Proceedings of the International Conference on Computational Science, Krakow, Poland.

Deuflhard P, Heroth J. (1996). Dynamic dimension reduction in ODE models cientific Computing in Chemical Engineering.

Dickson CT, Magistretti J, Shalinsky M, Hamam B, Alonso A. (2000). Oscillatory activity in entorhinal neurons and circuits. Mechanisms and function. Annals of the New York Academy of Sciences. 911 [PubMed]

Djouad R, Sportisse B. (2002). Partitioning techniques and lumping computation for reducing chemical kinetics Appl Numer Math. 43

Eckhaus W. (1979). Asymptotic Analysis of Singular Perturbations.

Ermentrout B. (1996). Type I membranes, phase resetting curves, and synchrony. Neural computation. 8 [PubMed]

Ermentrout GB, Kopell N. (1998). Fine structure of neural spiking and synchronization in the presence of conduction delays. Proceedings of the National Academy of Sciences of the United States of America. 95 [PubMed]

Fenichel N. (1971). Persistence and smoothness of invariant manifolds for flows Ind Univ Math J. 21

Fitzhugh R. (1961). Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical journal. 1 [PubMed]

Gear CW, Kevrekidis IG. (2003). Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum Siam J Sci Comput. 24

Gorban AN, Karlin I. (2003). Method of invariant manifold for chemical kinetics Chemical Engineering Science. 58

Gorban AN, Karlin IV, Zmievskii VB, Dymova SV. (2000). Reduced description in reaction kinetics Phys A. 275

Grasman J. (1987). Asymptotic methods for relaxaton oscillations and applications applied Mathematical Sciences. 63

Griewank A. (2000). Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation Frontiers Appl. Math.. 19

Guckenheimer J. (2003). Bifurcation and degenerate decomposition in multiple time scale dynamical systems Nonlinear Dynamics and Chaos: Where Do We Go from Here?.

Guckenheimer J, Hoffman K, Weckesser W. (2003). The forced van der Pol equation I: The slow flow and its bifurcations Siam J Appl Dyn Syst. 2

Guckenheimer J, Holmes P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 2nd edition..

Guckenheimer J, Meloon B. (2000). Computing periodic orbits and their bifurcations with automatic differentiation Siam J Sci Comput. 22

HODGKIN AL, HUXLEY AF. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of physiology. 117 [PubMed]

Hansel D, Mato G, Meunier C. (1995). Synchrony in excitatory neural networks. Neural computation. 7 [PubMed]

Hayes WB, Jackson KR. (2003). Rigorous shadowing of numerical solutions of ordinary differential equations by containment Siam J Numer Anal. 41

Hiskens IA. (2001). Stability of limit cycles in hybrid systems Proceedings of the 34th Hawaii International Conference on System Sciences.

Izhikevich EM, Hoppensteadt FC. (1997). Weakly Connected Neural Networks.

Jolliffe IT. (1986). Principal component analysis.

Jones CKRT. (1995). Geometric singular perturbation theory Dynamical Systems Lecture Notes in Math.. 1609

Jones SR, Pinto DJ, Kaper TJ, Kopell N. (2000). Alpha-frequency rhythms desynchronize over long cortical distances: a modeling study. Journal of computational neuroscience. 9 [PubMed]

Keener J, Sneyd J. (1998). Mathematical Physiology Interdisciplinary Applied Mathematics. 8

Kepler TB, Abbott LF, Marder E. (1992). Reduction of conductance-based neuron models. Biological cybernetics. 66 [PubMed]

Kernevez JP, Doedel E, Keller HB. (1991). Numerical analysis and controlof bifurcation problems. (I) Bifurcation in finite dimensions Int J Bifn Chaos. 1

Kistler WM, van Hemmen JL, Gerstner W. (1997). Reduction of Hodgkin-Huxley equations to a single-variable threshold model. Neural Comput. 9

Koch C, Bernander O, Douglas RJ. (1995). Do neurons have a voltage or a current threshold for action potential initiation? Journal of computational neuroscience. 2 [PubMed]

Kopell N, Ermentrout GB. (1986). Symmetry and phaselocking in chains of weakly coupled oscillators. Comm Pure Appl Mat. 39

Kopell N, Ermentrout GB. (1988). Coupled oscillators and the design of central pattern generators Math Biosci. 90

Kopell N, LeMasson G. (1994). Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture. Proceedings of the National Academy of Sciences of the United States of America. 91 [PubMed]

Kopell N, Rotstein HG, Wechselberger M, Clewley R. (). Dynamics of an Entorhinal Cortex Stellate Cell Model manuscript.

Kuramoto Y. (1991). Collective synchronization of pulse-coupled oscillators and excitable units Physica D. 50

Lapique L. (1907). Researches quatatives sur l'excitation electrique des nerfs traitee comme une polarization. J Physiol Pathol Gen. 9

Mckean HP. (1970). Nagumo's equation Advances In Math. 4

Mishchenko EF, Rozov NK. (1980). Differential Equations with Small Parameters and Relaxation Oscillators.

Morris C, Lecar H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical journal. 35 [PubMed]

Murray JD. (1989). Mathematical Biology.

Petzold LR, Rathinam M. (2002). Dynamic iteration using reduced order models: A method for simulation of large scale modular systems Siam J Numer Anal. 40

Petzold LR, Rathinam M. (2003). A new look at proper orthogonal decomposition Siam J Numer Anal. 41

Pilyugin YUS. (1999). Shadowing in Dynamical Systems Lecture Notes in Math.. 1706

Pope SB, Maas U. (1992). Simplifying chemical kinetics: Intrinsic low dimensional manifolds in composition space Combust Flame. 88

Rewienski MJ, Thesis PHD. (1998). A Trajectory Piecewise-Linear Approach to Model Order Reduction of Nonlinear Dynamical Systems Ph.D. thesis.

Rinzel J, Hoppensteadt FC, Keener JP. (1981). Integrate and fire modelsof nerve membrane response to oscillatory input SIAM J Appl Math. 41

Rowley CW. (2005). Model reduction for fluids, using balanced proper orthogonal decomposition Internat J Bifur Chaos Appl Sci Engrg. 15

Sorensen DC, Antoulas AC. (2001). Approximation of large-scale dynamical systems: An overview Int J Appl Math Comput Sci. 11

Strogatz SH, Mirollo RE. (1990). Synchronization of pulse-coupled biological oscillators. Siam J Appl Math. 6

Suckley R, Biktashev VN. (2003). Comparison of asymptotics of heart and nerve excitability. Physical review. E, Statistical, nonlinear, and soft matter physics. 68 [PubMed]

Suckley R, Biktashev VN. (2003). The asymptotic structure of the Hodgkin-Huxley equations Internat J Bifur Chaos Appl Sci Engrg. 13

Szmolyan P, Krupa M. (2001). Relaxation oscillation and canard explosion J Diff Eq. 174

Tarraf DC, Asada HH. (2002). On the nature and stability of differential-algebraic systems Proceedings of the IEEE American Control Conference, Anchorage, AK.

Taylor JH, Kebede D. (1996). Modeling and simulation of hybrid systems in Matlab Proceedings of the IFAC World Congress, San Francisco, CA.

Van_der_pol B. (1964). Forced oscillations in a circuit with nonlinear resistance (receptance with reactive triode) Selected Papers on Mathematical Trends in Control Theory Reprint of article in the London, Edinburgh and Dublin Phil. Mag.. 3

Van_der_schaft A, Schumacher JM. (2000). An Introduction to Hybrid Dynamical Systems.

Wang XJ, Buzsáki G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. The Journal of neuroscience : the official journal of the Society for Neuroscience. 16 [PubMed]

White JA, Chow CC, Ritt J, Soto-Treviño C, Kopell N. (1998). Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons. Journal of computational neuroscience. 5 [PubMed]

White JA et al. (2003). Slow and fast inhibition and h-current interact to create a theta rhythm in CA1 Soc Neurosci Abstr.

Whittington MA, Traub RD, Jefferys JG. (1995). Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation. Nature. 373 [PubMed]

Whittington MA, Traub RD, Kopell N, Ermentrout B, Buhl EH. (2000). Inhibition-based rhythms: experimental and mathematical observations on network dynamics. International journal of psychophysiology : official journal of the International Organization of Psychophysiology. 38 [PubMed]

Wiggins S. (1994). Normally Hyperbolic Invariant Manifolds in Dynamical Systems.

Wilson HR, Cowan JD. (1973). A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik. 13 [PubMed]

Winfree AT. (1980). The geometry of biological time..

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