An introduction to dynamical systems and neuronal dynamics

Lecture Notes in Mathematics

Volumn 1860, Issue , 2005, Pages 21-68

  Terman, David ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33748932017     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-540-31544-5_2     Document Type: Review

References (46)

1. R.L. Albin, A.B. Young, and J.B. Penney. The functional anatomy of basal ganglia disorders. Trends in Neurosci., 12:366-375, 1989.
2. M.D. Bevan and C.J. Wilson. Mechanisms underlying spontaneous oscillation and rhythmic firing in rat subthalamic neurons. J. Neurosci., 19:7617-7628, 1999.
3. R.J. Butera, J. Rinzel, and J.C. Smith. Models of respiratory rhythm generation in the pre-Botzinger complex: 1. Bursting pacemaker model. J. Neurophysiology, 82:382-397, 1999.
4. R.J. Butera, J. Rinzel, and J.C. Smith. Models of respiratory rhythm generation in the pre-Botzinger complex: II. Populations of coupled pacemaker neurons. J. Neurophysiology, 82:398-415, 1999.
5. T.R Chay and J. Keizer. Minimal model for membrane oscillations in the pancreatic β-cell. Biophys. J., 42:181-190, 1983.
6. T.R. Chay and J. Rinzel. Bursting, beating, and chaos in an excitable membrane model. Biophys. J., 47:357-366, 1985.
7. G. de Vries and R.M. Miura. Analysis of a class of models of bursting electrical activity in pancreatic β-cells. SIAM J. Appl. Math., 58:607-635, 1998.
8. M.R. DeLong. Activity of pallidal neurons during movement. J. Neurophysiol., 34:414-427, 1971.
9. W. Friesen. Reciprocal inhibition, a mechanism underlying oscillatory animal movements. Neurosci. Behavior, 18:547-553, 1994.
10. D. Golomb, X.-J. Wang, and J. Rinzel. Synchronization properties of spindle oscillations in a thalamic reticular nucleus model. J. Neurophysiol., 72:1109-1126, 1994.
11. J.L Hindmarsh and R.M. Rose. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond., Ser. B, 221:87-102, 1984.
12. A.L. Hodgkin and A.F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in a nerve. Journal of Physiology, 117:165-181, 1952.
13. F.C. Hoppensteadt and E.M. Izhikevich. Weakly Connected Neural Networks. Springer-Verlag, New York, Berlin, and Heidelberg, 1997.
14. J.M. Hurtado, C.M. Gray, L.B. Tamas, and K.A. Sigvardt. Dynamics of tremor-related oscillations in the human globus pallidus: A single case study. Proc. Natl. Acad. Sci. USA, 96:1674-1679, 1999.
15. E.M. Izhikevich. Neural excitability, spiking, and bursting. International Journal of Bifurcation and Chaos, 10, 2000.
16. D. Johnston and S. M-S. Wu. Foundations of Cellular Neurophysiology. MIT Press, Cambridge, Ma. and London, 1997.
17. C.K.R.T. Jones. Stability of the traveling wave solutions of the fitzhugh-nagumo system. Trans. Amer. Math. Soc., 286:431-469, 1984.
18. E.R. Kandel, J.H. Schwartz, and T.M. Jessell. Principles of Neural Science. Appleton & Lange, Norwalk, Conn., 1991.
19. J. Keener and J. Sneyd. Mathematical Physiology. Springer, New York, Berlin, Heidelberg, 1998.
20. N. Kopell and B. Ermentrout. Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators. In B. Fiedler, G. Iooss, and N. Kopell, editors, Handbook of Dynamical Systems, vol. 9: Towards Applications. Elsevier, 2003.
21. N. Kopell and D. Somers. Anti-phase solutions in relaxation oscillators coupled through excitatory synapses. J. Math. Biol., 33:261-280, 1995.
22. E. Lee and D. Terman. Uniqueness and stability of periodic bursting solutions. To appear in Journal of Differential Equations.
23. T. LoFaro and N. Kopell. Timing regulation in a network reduced from voltagegated equations to a one-dimensional map. J. Math. Biol. To appear.
24. C. Morris and H. Lecar. Voltage oscillations in the barnacle giant muscle fiber. Biophys. J., 35:193-213, 1981.
25. M. Pernarowski. Fast subsystem bifurcations in a slowly varying Lienard system exhibiting bursting. SIAM J. Appl. Math., 54:814-832, 1994.
26. P. Pinsky and J. Rinzel. Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons. J. Comput. Neurosci., 1:39-60, 1994.
27. J. Rinzel. A formal classification of bursting mechanisms in excitable systems. In A.M. Gleason, editor, Proceedings of the International Congress of Mathematicians, pages 1578-1594. American Mathematical Society, Providence, RI, 1987.
28. J. Rinzel and G.B. Ermentrout. Analysis of neural excitability and oscillations. In C.Koch and I. Segev, editors, Methods in Neuronal Modeling: From Ions to Networks, pages 251-291. The MIT Press, Cambridge, MA, second edition, 1998.
29. J. Rinzel, D. Terman, X.-J. Wang, and B. Ermentrout. Propagating activity patterns in large-scale inhibitory neuronal networks. Science, 279:1351-1355, 1998.
30. J. Rubin and D. Terman. Geometric analysis of population rhythms in synaptically coupled neuronal networks. Neural Comput., 12:597-645, 2000.
31. J. Rubin and D. Terman. Geometric singular perturbation analysis of neuronal dynamics. In B. Fiedler, G. Iooss, and N. Kopell, editors, Handbook of Dynamical Systems, vol. 3: Towards Applications. Elsevier, 2003.
32. A. Sherman. Contributions of modeling to understanding stimulus-secretion coupling in pancreatic β-cells. American. J. Phyciology, 271:547-559, 1996.
33. P. Smolen, D. Terman, and J. Rinzel. Properties of a bursting model with two slow inhibitory variables. SIAM J. Appl. Math., 53:861-892, 1993.
34. D. Somers and N. Kopell. Rapid synchronization through fast threshold modulation. Biol. Cybern., 68:393-407, 1993.
35. D. Somers and N. Kopell. Waves and synchrony in networks of oscillators of relaxation and non- relaxation type. Physica D, 89:169-183, 1995.
36. S.H. Strogatz. Nonlinear Dynamics and Chaos. Addison-Wesley Publishing Company, Reading, Ma., 1994.
37. J. Tabak, W. Senn, M.J. O'Donovan, and J. Rinzel. Comparison of two models for pattern generation based on synaptic depression. Neurocomputing, 26-27:551-556, 1999.
38. D. Terman. Chaotic spikes arising from a model for bursting in excitable membranes. SIAM J. Appl. Math., 51:1418-1450, 1991.
39. D. Terman. The transition from bursting to continuous spiking in an excitable membrane model. J. Nonlinear Sci., 2:133-182, 1992.
40. D. Terman, N. Kopell, and A. Bose. Dynamics of two mutually coupled inhibitory neurons. Physica D, 117:241-275, 1998.
41. D. Terman and E. Lee. Partial synchronization in a network of neural oscillators. SIAM J. Appl. Math., 57:252-293, 1997.
42. D. Terman, J. Rubin, A.C. Yew, and C.J. Wilson. Activity patterns in a model for the subthalamopallidal network of the basal ganglia. J. Neuroscience, 22:2963-2976, 2002.
43. D. Terman and D. L. Wang. Global competition and local cooperation in a network of neural oscillators. Physica D, 81:148-176, 1995.
44. D.H. Terman, G.B. Ermentrout, and A.C. Yew. Propagating activity patterns in thalamic neuronal networks. SIAM J. Appl. Math., 61:1578-1604, 2001.
45. X.-J. Wang and J. Rinzel. Spindle rhythmicity in the reticularis thalamic nucleus: synchronization among mutually inhibitory neurons. Neuroscience, 53:899-904, 1993.
46. X.-J. Wang and J. Rinzel. Oscillatory and bursting properties of neurons. In M.A. Arbib, editor, The Handbook of Brain Theory and Neural Networks, pages 686-691. The MIT Press, Cambridge, London, 1995.