Citation: |
[1] |
W. B. Adams and J. A. Benson, The generation and modulation of endogenous rhythmicity in the Aplysia bursting pacemaker neurone R15, Prog. Biophys. Molec. Biol., 46 (1985), 1-49.doi: 10.1016/0079-6107(85)90011-2. |
[2] |
R. Bertram, M. J. Butte, T. Kiemel and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), 413-439. |
[3] |
J. Best, A. Borisyuk, J. Rubin, D. Terman and M. Wechselberger, The dynamic range of bursting in a model respiratory pacemaker network, SIAM J. Appl. Dyn. Syst., 4 (2005), 1107-1139.doi: 10.1137/050625540. |
[4] |
T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic $\beta$ cell, Biophys. J., 42 (1983), 181-190.doi: 10.1016/S0006-3495(83)84384-7. |
[5] |
L. Duan, Q. Lu and Q. Wang, Two-parameter bifurcation analysis of firing activities in the Chay neuronal model, Neurocomputing, 72 (2008), 341-351.doi: 10.1016/j.neucom.2008.01.019. |
[6] |
F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts, Springer Lect. Notes Math., 1480 (1991), 1489-1500. |
[7] |
M. Golubitsky, K. Josić and T. J. Kaper, An unfolding theory approach to bursting in fast-slow systems, in "Global Analysis of Dynamical Systems'' (eds. H. W. Broer, B. Krauskopf and G. Vegter), Institute of Physics Publishing, Bristol, (2001), 277-308. |
[8] |
F. van Goor, Y.-X. Li and S. S. Stojilkovic, Paradoxical role of large-conductance calcium-activated K$^+$ (BK) channels in controlling action potential-driven $Ca^{2+}$ entry in anterior pituitary cells, J. Neurosci., 16 (2001), 5902-5915. |
[9] |
F. van Goor, D. Zivadinovic, A. Martinez-Fuentes and S. Stojilkovic, Dependence of pituitary hormone secretion on the pattern of spontaneous voltage-gated calcium influx. Cell type-specific action potential secretion coupling, J. Biol. Chem., 276 (2001), 33840-33846.doi: 10.1074/jbc.M105386200. |
[10] |
J. Hindmarsh and M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. London B, 221 (1984), 87-102.doi: 10.1098/rspb.1984.0024. |
[11] |
A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London), 117 (1952), 205-249. |
[12] |
F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'' Applied Mathematical Sciences, 126, Springer-Verlag, New York, 1997. |
[13] |
E. M. Izhikevich, Neural excitability, spiking and bursting, Intl. J. Bifurc. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266.doi: 10.1142/S0218127400000840. |
[14] |
J. Keener and J. Sneyd, "Mathematical Physiology," 2^{nd} edition, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 2009. |
[15] |
A. I. Khibnik, B. Krauskopf and C. Rousseau, Global study of a family of cubic Liénard equations, Nonlinearity, 11 (1998), 1505-1519.doi: 10.1088/0951-7715/11/6/005. |
[16] |
A. P. LeBeau, A. B. Rabson, A. E. McKinnon and J. Sneyd, Analysis of a reduced model of corticotroph action potentials, J. Theoretical Biol., 192 (1998), 319-339.doi: 10.1006/jtbi.1998.0656. |
[17] |
M. Pernarowski, Fast subsystem bifurcations in a slowly varying Liénard system exhibiting bursting, SIAM J. Appl. Math., 54 (1994), 814-832.doi: 10.1137/S003613999223449X. |
[18] |
J. Rinzel, Bursting oscillations in an excitable membrane model, in "Ordinary and Partial Differential Equations'' (Dundee, 1984) (eds. B. D. Sleeman and R. D. Jarvis), Lect. Notes Math., 1151, Springer, Berlin, (1985), 304-316. |
[19] |
J. Rinzel, A formal classification of bursting mechanisms in excitable systems, in "Proc. Intl. Cong. Math., Vol. 1, 2" (Berkeley, Calif., 1986) (ed. A. M. Gleason), American Mathematical Society, Providence, RI, (1987), 1578-1593. |
[20] |
J. Rinzel and B. Ermentrout, Analysis of neural excitability and oscillations, in "Methods in Neuronal Modeling" (eds. C. Koch and I. Segev), The MIT Press, (1998), 251-291. |
[21] |
J. Rinzel and Y. S. Lee, Dissection of a model for neuronal parabolic bursting, J. Math. Biol., 25 (1987), 653-675.doi: 10.1007/BF00275501. |
[22] |
A. Shilnikov, R. L. Calabrese and G. Cymbalyuk, Mechanism of bistability: Tonic spiking and bursting in a neuron model, Phys. Rev. E (3), 71 (2005), 056214, 9 pp. |
[23] |
J. V. Stern, H. M. Osinga, A. LeBeau and A. Sherman, Resetting behavior in a model of burting in secretory pituitary cells: Distinguishing plateaus from pseudo-plateaus, Bull. Math. Biol., 70 (2008), 68-88.doi: 10.1007/s11538-007-9241-x. |
[24] |
J. Tabak, N. Toporikova, M. E. Freeman and R. Bertram, Low dose of dopamine may stimulate prolactin secretion by increasing fast potassium currents, J. Comput. Neurosci., 22 (2007), 211-222.doi: 10.1007/s10827-006-0008-4. |
[25] |
W. Teka, K. Tsaneva-Atanasova, R. Bertram and J. Tabak, From plateau to pseudo-plateau bursting: Making the transition, Bull. Math. Biol., 73 (2011), 1292-1311.doi: 10.1007/s11538-010-9559-7. |
[26] |
N. Toporikova, J. Tabak, M. E. Freeman and R. Bertram, A-type K$^+$ current can act as a trigger for bursting in the absence of a slow variable, Neural Comput., 20 (2008), 436-451.doi: 10.1162/neco.2007.08-06-310. |
[27] |
K. Tsaneva-Atanasova, H. M. Osinga, T. Rieß and A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theoretical Biol., 264 (2010), 1133-1146.doi: 10.1016/j.jtbi.2010.03.030. |
[28] |
K. Tsaneva-Atanasova, A. Sherman, F. van Goor and S. S. Stojilkovic, Mechanism of spontaneous and receptor-controlled electrical activity in pituitary somatotrophs: Experiments and theory, J. Neurophysiology, 98 (2007), 131-144.doi: 10.1152/jn.00872.2006. |
[29] |
T. Vo, R. Bertram, J. Tabak and M. Wechselberger, Mixed mode oscillations as a mechanism for pseudo-plateau bursting, J. Comput. Neurosci., 28 (2010), 443-458.doi: 10.1007/s10827-010-0226-7. |
[30] |
G. de Vries, Multiple bifurcations in a polynomial model of bursting oscillations, J. Nonlinear Sci., 8 (1998), 281-316.doi: 10.1007/s003329900053. |