Bifurcations in the fast dynamics of neurons: Implications for bursting

Bursting: The Genesis of Rhythm in the Nervous System

Volumn , Issue , 2005, Pages 89-122

  Guckenheimer, John ^a   Tien, Joseph H ^a   Willms, Allan R ^b  

^a Department of Obstetrics Gynecology   (United States)

^b UNIVERSITY OF GUELPH   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77955400172     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/9789812703231_0004     Document Type: Chapter

References (29)

1. E. Benoit, J. L Callot, F. Diener, and M. Diener, Chasse au canards. Collect. Math. 31, 37-119, (1981).
2. N. M. Berezetskaya, V. N. Kharkyanen, and N. I. Kononenko, Mathematical model of pacemaker activity in bursting neurons of snail, helix pomatia. J. Theor. Biol. 183, 207-218, (1996).
3. R. Bertram, M. Butte, T. Kiemel, and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol. 57(3), 413-39, (1995).
4. R. J. Butera Jr, J. W. Clark, C. C. Canavier, D. A. Baxter, and J. H. Byrne, Analysis of the effects of modulatory agents on a modeled bursting neuron: Dynamic interactions between voltage and calcium dependent systems, J. Comp. Neurosci. 2, 19-44, (1995).
5. R. J. Butera Jr, J. Rinzel, and J. C. Smith, Models of respiratory rhythm generation in the pre-Botzinger complex. I. Bursting pacemaker neurons, J Neurophysiol. 82, 382-97, (1999).
6. G. S. Cymbalyuk and R. L. Calabrese, Oscillatory behaviors in pharmacologically isolated heart interneurons from the medicinal leech, Neurocomput. 32, 97-104, (2000).
7. + current in a window mode, Neurocomput. 38, 159-166, (2001).
8. G. S. Cymbalyuk, Q. Gaudry, M. A. Masino, and R. L. Calabrese Bursting in leech heart interneurons: Cell-autonomous and network based mechanisms, J. Neurosci. 22, 10580-10592, (2002).
9. F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc. 121, no. 577, (1996).
10. W. Eckhaus, Relaxation oscillations, including a standard chase on french ducks, Lecture Notes in Mathematics 985, 449-494, (1983).
11. R. Fitz Hugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445-466, (1961).
12. J. M. Gambaudo, P. Glendinning, and C. Tresser, The gluing bifurcation I: symbolic dynamics of the closed curves, Nonlinearity 1, 203-214, (1988).
13. J. Guckenheimer, S. Gueron, and R. M. Harris-Warrick, Mapping the dynamics of a bursting neuron, Phil. Trans. R. Soc. Lond. B 341, 345-359, (1993).
14. J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, (1983).
15. J. Guckenheimer and R. Oliva, Chaos in the Hodgkin-Huxley Model, SIAM J. Appl. Dyn. Systems 1, 105-114, (2002).
16. A. A. V. Hill, J. Lu, M. A. Masino, O. H. Olsen, and R. L. Calabrese, A model of a segmental oscillator in the leech heartbeat neuronal network, J. Comput. Neurosci. 10, 281-302, (2001).
17. B. Hille, Ionic Channels of Excitable Membranes, 3rd edition Sinauer, (2001).
18. A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its applications to conduction and excitation in nerve, J. Physiol. (Lond.) 116, 500-544, (1952).
19. E. M. Izhikevich, Neural excitability, spiking and bursting. Internat. J. Bifur. Chaos Appl. Sei. Engrg. 10, 1171-1266, (2000).
20. C. Jones, Geometric singular perturbation theory, Lecture Notes in Math., Springer, Berlin, 1609, 44-118, (1995).
21. R. H. Kramer and R. S. Zucker, Calcium-dependent inward current in aplysia bursting pace-maker neurones, J. Physiol. 362, 107-130, (1985).
22. F. Nadim, O. H. Olsen, E. DeSchutter, and R. L. Calabrese, Modeling the leech heartbeat elemental oscillator: I. Interactions of intrinsic and synaptic currents, J. Comput. Neurosci. 2, 215-235, (1995).
23. O. H. Olsen, F. Nadim, and R. L. Calabrese, Modeling the leech heartbeat elemental oscillator: II. Exploring the parameter space, J. Comput. Neurosci. 2, 237-257, (1995).
24. R. E. Plant, Bifurcation and resonance in a mathematical model for bursting nerve cells, J. Math. Biol. 11, 15-32, (1981).
25. J. Rinzel, A formal classification of bursting mechanisms in excitable systems, Mathematical topics in population biology, morphogenesis and neurosciences (Kyoto, 1985), Lecture Notes in Biomath. 71, 267-281, Springer, Berlin, (1987).
26. J. Rinzel and B. Ermentrout, Analysis of neural excitability and oscillations, in Methods of Neural Modeling: From Synapses to Networks, C. Koch and I. Segev, eds., MIT Press, 135-169, (1989).
27. J. Rinzel and Y. S. Lee, Dissection of a model for neuronal parabolic bursting, J. Math. Biol. 25, 653-675, (1987).
28. R. Roussarie, Bifurcation of planar vector fields and Hilbert’s sixteenth problem. Progress in Mathematics 164, Birkhuser Verlag, Basel, (1998).
29. X. J. Wang, Multiple dynamical modes of thalamic relay neurons: Rhythmic bursting and intermittent phase-locking, Neurosci. 59, 2131, (1994).