Bifurcations in Morris-Lecar neuron model

Neurocomputing

Volumn 69, Issue 4-6, 2006, Pages 293-316

  Tsumoto, Kunichika ^a,e   Kitajima, Hiroyuki ^b   Yoshinaga, Tetsuya ^c   Aihara, Kazuyuki ^d,e   Kawakami, Hiroshi ^c  

^a NATIONAL INSTITUTE OF TECHNOLOGY   (Japan)

^b KAGAWA UNIVERSITY   (Japan)

^c UNIVERSITY OF TOKUSHIMA   (Japan)

^d INSTITUTE OF INDUSTRIAL SCIENCE   (Japan)

^e JAPAN SCIENCE AND TECHNOLOGY AGENCY   (Japan)

Author keywords

Classes I and II neuron; Homoclinic bifurcation; Morris Lecar model; Saddle node bifurcation

Indexed keywords

BIFURCATION (MATHEMATICS); BIOLOGICAL MEMBRANES; DIFFERENTIAL EQUATIONS; MATHEMATICAL MODELS; SET THEORY;

HOMOCLINIC BIFURCATION; MORRIS-LECAR MODEL; NEURONS; SADDLE NODE BIFURCATION;

NEURAL NETWORKS;

CALCIUM ION; POTASSIUM; POTASSIUM ION;

ANALYTIC METHOD; ARTICLE; CALCIUM CURRENT; MATHEMATICAL MODEL; MATHEMATICAL PARAMETERS; MEMBRANE POTENTIAL; NERVE CELL; POTASSIUM CURRENT; PRIORITY JOURNAL; STEADY STATE;

EID: 29444442155     PISSN: 09252312     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.neucom.2005.03.006     Document Type: Article

References (43)

1. R.H. Adrian W.K. Chandler A.L. Hodgkin Voltage clamp experiments in striated muscle fibers J. Physiol. (London) 208 1970 607-644
2. R.I. Bogdanov Versal deformations of a singular point on the plane in the case of zero eigenvalues Funct. Anal. Appl. 9 1975 144-145
3. + current by the tyrosine phosphatase, SHP-1 J. Biol. Chem. 277 50 2002 48130-48138
4. S. Choe Potassium channel structures Nat. Rev. Neurosci. 3 2002 115-121
5. I. Chung L.C. Schlichter Regulation of native Kv1.3 channels by cAMP-dependent protein phosphorylation Am. J. Physiol. Cell. Physiol. 273 1997 622-633
6. J.A. Conner C.F. Stevens Prediction of repetitive firing behavior from voltage-clamped date on an isolated neuron soma J. Physiol. (London) 214 1971 31-53
7. J.A. Connor D. Walter R. Mckown Neural repetitive firing: Modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons Biophys. J. 18 1977 81-102
8. B. Ermentrout Type I membranes, Phase resetting curves, and synchrony Neural Comput. 8 1996 979-1001
9. R. FitzHugh Impulses and physiological state in theoretical models of nerve membrane Biophys J. 1 1961 445-467
10. C.M. Gray P. Konig A.K. Engel W. Singer Oscillatory responses in the cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties Nature 338 23 1989 334-337
11. B.S. Gutkin G.B. Ermentrout Dynamics of membrane excitability determine interspike interval variability: A link between spike generation mechanisms and cortical spike train statistics Neural Comput. 10 1998 1047-1065
12. D. Hansel G. Mato C. Meunier Synchrony in excitatory neural networks Neural Comput. 7 1995 307-335
13. B. Hille 3rd (Eds.), Ion Channels of Excitable Membranes, Sinauer Associates Inc., 2001, pp. 1-814.
14. J.L. Hindmarsh R.M. Rose A model of neuronal bursting using three coupled first order differential equations Proc. R. Soc. London B 221 1984 87-102
15. A.L. Hodgkin The local changes associated with repetitive action in a non-modulated axon J. Physiol. (London) 107 1948 165-181
16. A.L. Hodgkin A.F. Huxley A qualitative description of membrane current and its application to conduction and excitation in nerve J. Physiol. (London) 117 1952 500-544
17. E.M. Izhikevich Neural excitability, spiking and bursting Int. J. Bifurcation Chaos 10 6 2000 1171-1266
18. H. Kawakami Bifurcation of periodic responses in forced dynamic nonlinear circuits: Computation of bifurcation values of the system parameters IEEE Trans. Circuits Systems CAS-31 3 1984 248-260
19. H. Kawakami Y. Ozaki Analysis of a separatrix loop in n-dimensional autonomous system IEICE Trans. J67-A 11 1984 1098-1099 (in Japanese)
20. S. Kim H. Kook S.G. Lee M.H. Park Synchronization and clustering in a network of three globally coupled neural oscillators Int. J. Bifurcation Chaos 8 4 1998 731-739
21. S. Kim S.G. Lee H. Kook J.H. Shin Phase dynamics of two and three coupled Hodgkin-Huxley neurons under DC currents in: J.H. Oh C. Kwon S. Cho (Eds.), Neural Networks: The Statistical Mechanics Perspective 1995 World Scientific Singapore 141-155
22. C. Koch Phase space analysis of neuronal excitability in: C. Koch (Ed.), Biophysics of Computation: Information Processing in Single Neurons 1999 Oxford University press New York 172-192
23. Y.A. Kuznetsov, Two-parameter bifurcations of equilibria in continuous-time dynamical system, in: Y.A. Kuznetsov 2nd (Eds.), Elements of Applied Bifurcation Theory: Applied Mathematical Science, vol. 112, Springer, New York, 1998, pp. 293-392.
24. S.G. Lee S. Kim H. Kook Synchrony and clustering in two and three synaptically coupled Hodgkin-Huxley neurons with a time delay Int. J. Bifurcation Chaos 7 4 1997 889-895
25. C.M. Macica L.K. Kaczmarek Casein kinase 2 determines the voltage dependence of the Kv3.1 channel in auditory neurons and transfected cells J. Neurosci. 21 4 2001 1160-1168
26. C. Morris H. Lecar Voltage oscillations in the barnacle giant muscle fiber Biophys. J. 35 1981 193-213
27. J. Nagumo S. Arimoto S. Yoshizawa An active pulse transmission line simulating nerve axon Proc. IRE 50 1962 2061-2070
28. D. Noble The development of mathematical models of the heart Chaos Solitons Fractals 5 1995 321-333
29. M. Nomura T. Fukai T. Aoyagi Synchrony of fast-spiking interneurons interconnected by GABAergic and electrical synapses Neural Comput. 15 2003 2179-2198
30. L.S. Pontryagin (Ed.), Ordinary Differential Equations, Addison-Wesley, Reading, MA, 1962.
31. J. Rinzel in: W.E. Fitzgibbon III H.R. Walker (Eds.), Research Notes in Mathematics, Nonlinear Diffusion 1977 Pitman London
32. J. Rinzel, G.B. Ermentrout, Analysis of neuronal excitability and oscillations, in: C. Koch, I. Segev 2nd (Eds.), Methods in Neuronal Modeling: From Ions to Networks, MIT Press, London, 1999, pp. 251-292.
33. M.E. Rush J. Rinzel Analysis of bursting in a thalamic neuron model Biol. Cybernet. 71 1994 281-291
34. M.E. Rush J. Rinzel The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models Bull. Math. Biol. 57 1995 899-929
35. P.N. Steinmetz A. Roy P.J. Fitzgerald S.S. Hsiao K.O. Johnson E. Niebur Attention modulates synchronized neuronal firing in primate somatosensory cortex Nature 404 2000 187-190
36. K.J. Swartz Towards a structural view of gating in potassium channels Nat. Rev. Neurosci. 5 2004 905-916
37. F. Takens Singularities of vector fields Publ. Math. IHES 43 1974 47-100
38. Z. Tiran A. Peretz B. Attali A. Elson Phosphorylation-dependent regulation of Kv2.1 Channel activity at tyrosine 124 by Src and by protein-tyrosine phosphatase epsilon J. Biol. Chem. 278 19 2003 17509-17514
39. K. Tsumoto T. Yoshinaga K. Aihara H. Kawakami Bifurcations in synaptically coupled Hodgkin-Huxley neurons with a periodic α-function train Int. J. Bifurcation Chaos 11 4 2001 1053-1064
40. E. Vaddla I. Haalman M. Abeles H. Bergman Y. Prut H. Slovin A. Aertsen Dynamics of neuronal interactions in monkey cortex in relation to behavioral events Nature 373 9 1995 515-518
41. W.M. Yamada, C. Koch, P.R. Adams, Multiple channels and calcium dynamics, in: C. Koch, I. Segev 2nd (Eds.), Methods in Neuronal Modeling: From Ions to Networks, MIT Press, London, 1999, pp. 137-170.
42. T. Yoshinaga H. Kawakami Bifurcation and chaotic state in forced oscillatory circuits containing saturable inductors in: L. Pecora T. Carroll (Eds.), Nonlinear Dynamics in Circuits 1995 World Scientific Singapore 89-119
43. T. Yoshinaga Y. Sano H. Kawakami A method to calculate bifurcations in synaptically coupled Hodgkin-Huxley equations Int. J. Bifurcation Chaos 9 7 1999 1451-1458