Entrainment and chaos in a pulse-driven hodgkin-huxley oscillator

SIAM Journal on Applied Dynamical Systems

Volumn 5, Issue 2, 2006, Pages 179-204

  Lin, Kevin K ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

Driven nonlinear oscillators.; Entrainment and phase locking; Hodgkin Huxley model; Strange attractors

Indexed keywords

ACTION POTENTIALS; DRIVEN NONLINEAR OSCILLATORS.; ENTRAINMENT AND PHASE-LOCKING; EXCITABLE CELLS; GENERAL FEATURES; HODGKIN-HUXLEY MODEL; INPUT PULSES; NEURON MODELS; STANDARD MODELS; STRANGE ATTRACTORS;

MILITARY OPERATIONS; OSCILLATORS (ELECTRONIC);

CHAOS THEORY;

EID: 45849094864     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/050631677     Document Type: Article

References (41)

1. K. AIHARA AND G. MATSUMOTO, Chaotic oscillations and bifurcations in squid giant axons, in Chaos, Nonlinear Science: Theory and Applications, Manchester University Press, Manchester, UK, 1986.
2. J. C. ALEXANDER, E. J. DOEDEL, AND H. G. OTHMER, On the resonance structure in a forced excitable system, SIAM J. Appl. Math., 50 (1990), pp. 1373-1418.
3. E. N. BEST, Null space in the Hodgkin-Huxley equations: A critical test, Biophys. J., 27 (1979), pp. 87-104.
4. E. BROWN, J. MOEHLIS, AND P. HOLMES, On the phase reduction and response dynamics of neural oscillator populations, Neural Comput., 16 (2004), pp. 673-715.
5. J. CRONIN, Mathematical Aspects of Hodgkin-Huxley Neural Theory, Cambridge University Press, Cambridge, UK, 1987.
6. S. DOI, A chaotic map with a flat segment can produce a noise-induced order, J. Statist. Phys., 55 (1989), pp. 941-964.
7. G. B. ERMENTROUT, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, Software Environ. Tools 14, SIAM, Philadelphia, 2002.
8. G. B. ERMENTROUT AND N. KOPELL, Multiple pulse interactions and averaging in systems of coupled neural oscillators, J. Math. Biol., 29 (1991), pp. 195-217.
9. G. B. ERMENTROUT, M. PASCAL, AND B. GUTKIN, The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators, Neural Comput., 13 (2001), pp. 1285-1310.
10. K. GEIST, U. PARLITZ, AND W. LAUTERBORN, Comparison of different methods for computing Lyapunov exponents, Progr. Theoret. Phys., 83 (1990), pp. 875-893.
11. L. GLASS AND M. C. MACKEY, Frorn Clocks to Chaos: The Rhythms of Life, Princeton University Press, Princeton, NJ, 1988.
12. J. GUCKENHEIMER, Isochrons and phaseless sets, J. Theoret. Biol., 1 (1974), pp. 259-273.
13. J. GUCKENHEIMER AND P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
14. R. GUTTMAN, L. FELDMAN, AND E. JAKOBSSON, Frequency entrainment of squid axon membrane, J. Membrane Biol., 56 (1980), pp. 9-18.
15. H. HAYASHI, S. ISHIZUKA, AND K. HLRAKAWA, Chaotic response of the pacemaker neuron, J. Phys. Soc. Japan, 54 (1985), pp. 2337-2346.
16. H. HAYASHI, S. ISHIZUKA, M. OHTA, AND K. HLRAKAWA, Chaotic behavior in the onchidium giant neuron under sinusoidal stimulation, Phys. Lett. A, 88 (1982). pp. 435-438.
17. H. HAYASHI, M. NAKAO, AND K. HIRAKAWA, Entrained, harmonic, quasiperiodic and chaotic responses of the self-sustained oscillation of nitella to sinusoidal stimulation, J. Phys. Soc. Japan, 52 (1983), pp. 344-351.
18. A. L. HODGKIN AND A. F. HUXLEY, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), pp. 500-544.
19. A. V. HOLDEN, The response of excitable membrane models to a cyclic input, Biology and Cybernetics, 21 (1975), pp. 1-7.
20. D. H. HUBEL, Eye, Brain, and Vision, Scientific American Library, W. H. Freeman, New York, 1988.
21. J. D. HUNTER, J. G. MILTON, P. J. THOMAS, AND J. D. COWAN, Resonance effect for neural spike time reliability, J. Neurophysiol., 80 (1998), pp. 1427-1438.
22. F. LEDRAPPIER AND L.-S. YOUNG, Entropy formula for random transformations, Probab. Theory Related Fields, 80 (1988), pp. 217-240.
23. Z. F. MAINEN AND T. J. SEJNOWSKI, Reliability of spike timing in neocortical neurons, Science, 268 (1995), pp. 1503-1506.
24. G. MATSUMOTO, K. AIHARA, Y. HANYU, N. TAKAHASHI, S. YOSHIZAWA, AND J. ICHI NAGUMO, Chaos and phase locking in normal squid axons, Phys. Lett. A, 123 (1987), pp. 162-166.
25. G. MATSUMOTO, N. TAKAHASHI, AND Y. HANYU, Chaos, phase locking and bifurcation in normal squid axons, in Chaos in Biological Systems, Plenum Press, New York, 1987, pp. 143-156.
26. H. NAKAO, K. SUKE ARAI K. NAGAI, Y. TSUBO, AND Y. KURAMOTO, Synchrony of limit-cycle oscillators induced by random external impulses, Phys. Rev. E (3), 72 (2005), paper 026220.
27. A. OKSASOGLU AND Q. WANG, Strange attractors in periodically-kicked Chua's circuit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2005), pp. 83-98.
28. S. A. OPRISAN AND C. C. CANAVIER, The influence of limit cycle topology on the phase resetting curve, Neural Comput., 14 (2002), pp. 1027-1057.
29. K. PAKDAMAN AND S. TANABE, Random dynamics of the Hodgkin-Huxley neuron model, Phys. Rev. E (3), 64 (2001), paper 050902.
30. W. H. PRESS, B. P. FLANNERY, S. A. TEUKOLSKY, AND W. T. VETTERLING, Numerical Recipes in C, Cambridge University Press, Cambridge, UK, 1992.
31. F. RIEKE, Spikes: Exploring the Neural Code, MIT Press, Cambridge, MA, 1997.
32. T. TAKABE, K. AIHARA, AND G. MATSUMOTO, Response characteristics of the Hodgkin-Huxley equations to pulse-train stimulation, Transactions of IEICE Japan (Japanese edition), J71-A (1988), pp. 744-750.
33. Q. WANG AND L.-S. YOUNG, Strange attractors with one direction of instability, Comm. Math. Phys., 218 (2001), pp. 1-97.
34. Q. WANG AND L.-S. YOUNG, From invariant curves to strange attractors, Comm. Math. Phys., 225 (2002), pp. 275-304.
35. Q. WANG AND L.-S. YOUNG, Strange attractors in periodically-kicked limit cycles and Hopf bifurcations, Comm. Math. Phys., 240 (2003), pp. 509-529.
36. Q. WANG AND L.-S. YOUNG, Toward a theory of rank-1 attractors, Ann. Math., to appear.
37. T. L. WILLIAMS AND G. BOWTELL, The calculation of frequency-shift functions for chains of coupled oscillators, with application to a network model of the lamprey locomotor pattern generator, J. Comput. Neurosci., 4 (1997), pp. 47-55.
38. A. WINFREE, The Geometry of Biological Time, 2nd ed., Springer-Verlag, New York, 2000.
39. L.-S. YOUNG, Ergodic theory of differentiable dynamical systems, in Real and Complex Dynamics, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 464, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995, pp. 293-336.
40. G. ZASLAVSKY, The simplest case of a strange attractor, Phys. Lett. A, 69 (1978), pp. 145-147.
41. C. ZHOU AND J. KURTHS, Noise-induced synchronization and, coherence resonance of a Hodgkin-Huxley model of thermally sensitive neurons. Chaos, 13 (2003), pp. 401-409.