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Volumn 158, Issue 1, 1999, Pages 48-78

Uniqueness and stability of periodic bursting solutions

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EID: 21944438553     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-0396(99)80018-7     Document Type: Article
Times cited : (28)

References (40)
  • 1
    • 0025941210 scopus 로고
    • On the dynamics of bursting systems
    • J. C. Alexander and D. Cai, On the dynamics of bursting systems, J. Math. Biol. 29 (1991), 405-423.
    • (1991) J. Math. Biol. , vol.29 , pp. 405-423
    • Alexander, J.C.1    Cai, D.2
  • 4
    • 0029303855 scopus 로고
    • Topological and phenomenological classification of bursting oscillations
    • R. Bertram, M. J. Butte, T. Kiemel, and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol. 57 (1995), 413-439.
    • (1995) Bull. Math. Biol. , vol.57 , pp. 413-439
    • Bertram, R.1    Butte, M.J.2    Kiemel, T.3    Sherman, A.4
  • 6
    • 0020957719 scopus 로고
    • Minimal model for membrane oscillations in the pancreatic beta-cell
    • T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophys. J. 42 (1983), 181-190.
    • (1983) Biophys. J. , vol.42 , pp. 181-190
    • Chay, T.R.1    Keizer, J.2
  • 7
    • 0022037388 scopus 로고
    • Bursting, beating, and chaos in an excitable membrane model
    • T. R. Chay and J. Rinzel, Bursting, beating, and chaos in an excitable membrane model, Biophys. J. 47 (1985), 357-366.
    • (1985) Biophys. J. , vol.47 , pp. 357-366
    • Chay, T.R.1    Rinzel, J.2
  • 8
    • 0001755071 scopus 로고
    • AUTO, a program for the automatic bifurcation and analysis of autonomous systems
    • E. J. Doedel, AUTO, a program for the automatic bifurcation and analysis of autonomous systems, Congr. Numer. 30 (1981), 265-284.
    • (1981) Congr. Numer. , vol.30 , pp. 265-284
    • Doedel, E.J.1
  • 9
    • 0022693272 scopus 로고
    • Parabolic bursting in an excitable system coupled with a slow oscillation
    • G. B. Ermentrout and N. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math. 46 (1986), 233-253.
    • (1986) SIAM J. Appl. Math. , vol.46 , pp. 233-253
    • Ermentrout, G.B.1    Kopell, N.2
  • 10
    • 0001356311 scopus 로고
    • Persistence and smoothness of invariant manifolds for flows
    • N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971), 193-226.
    • (1971) Indiana Univ. Math. J. , vol.21 , pp. 193-226
    • Fenichel, N.1
  • 11
    • 0016070051 scopus 로고
    • Asymptotic stability with rate conditions
    • N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23 (1974), 1109-1137.
    • (1974) Indiana Univ. Math. J. , vol.23 , pp. 1109-1137
    • Fenichel, N.1
  • 12
    • 0001957566 scopus 로고
    • Asymptotic stability with rate conditions, II
    • N. Fenichel, Asymptotic stability with rate conditions, II, Indiana Univ. Math. J. 26 (1977), 87-93.
    • (1977) Indiana Univ. Math. J. , vol.26 , pp. 87-93
    • Fenichel, N.1
  • 13
    • 34250627892 scopus 로고
    • Geometric singular perturbation theory for ordinary differential equation
    • N. Fenichel, Geometric singular perturbation theory for ordinary differential equation, J. Differential Equations 31 (1979), 53-98.
    • (1979) J. Differential Equations , vol.31 , pp. 53-98
    • Fenichel, N.1
  • 14
    • 53349102813 scopus 로고
    • Impulses and physiological states in models of nerve membrane
    • R. Fitzhugh, Impulses and physiological states in models of nerve membrane, Biophys. J. 1 (1961), 445-466.
    • (1961) Biophys. J. , vol.1 , pp. 445-466
    • Fitzhugh, R.1
  • 16
    • 0021768729 scopus 로고
    • A model of neuronal bursting using three coupled first order differential equation
    • J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equation, Proc. Roy. Soc. London B 221 (1984), 87-102.
    • (1984) Proc. Roy. Soc. London B , vol.221 , pp. 87-102
    • Hindmarsh, J.L.1    Rose, R.M.2
  • 17
    • 0001441740 scopus 로고
    • Invariant Manifolds
    • Springer-Verlag, New York/Heidelberg/Berlin
    • M. W. Hirsch and C. C. Pugh, "Invariant Manifolds," Springer Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, New York/Heidelberg/Berlin, 1977.
    • (1977) Springer Lecture Notes in Mathematics , vol.583
    • Hirsch, M.W.1    Pugh, C.C.2
  • 19
    • 35649001607 scopus 로고
    • A quantitative description of membrane current and its application to conduction and excitation in nerve
    • 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), 500-544.
    • (1952) J. Physiol. (London) , vol.117 , pp. 500-544
    • Hodgkin, A.L.1    Huxley, A.F.2
  • 21
    • 0019405994 scopus 로고
    • Voltage oscillations in the barnacle giant muscle fiber
    • C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J. 35 (1981), 193-213.
    • (1981) Biophys. J. , vol.35 , pp. 193-213
    • Morris, C.1    Lecar, H.2
  • 22
    • 6344238035 scopus 로고
    • An active pulse transmission line simulating nerve axon
    • J. Nagumo, S. Aritomo, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE (1964), 2061-2070.
    • (1964) Proc. IRE , pp. 2061-2070
    • Nagumo, J.1    Aritomo, S.2    Yoshizawa, S.3
  • 23
    • 0027043895 scopus 로고
    • Perturbation technique for models of bursting electrical activity in pancreatic β-cells
    • M. Pernarowski, R. M. Miura, and J. Kevorkian, Perturbation technique for models of bursting electrical activity in pancreatic β-cells, SIAM J. Appl. Math. 52 (1992), 1627-1650.
    • (1992) SIAM J. Appl. Math. , vol.52 , pp. 1627-1650
    • Pernarowski, M.1    Miura, R.M.2    Kevorkian, J.3
  • 24
    • 23044509493 scopus 로고
    • Approximate solution of a system of ordinary differential equations involving a small parameter in the derivatives
    • L. S. Pontryagin and L. V. Rodygin, Approximate solution of a system of ordinary differential equations involving a small parameter in the derivatives, Izdat. Akad. Nauk SSSR Moscow 3 (1956), 570.
    • (1956) Izdat. Akad. Nauk SSSR Moscow , vol.3 , pp. 570
    • Pontryagin, L.S.1    Rodygin, L.V.2
  • 26
    • 23044506065 scopus 로고
    • Periodic solution of a system of ordinary differential equations with a small parameter in the terms containing derivatives, I, II
    • Periodic solution of a system of ordinary differential equations with a small parameter in the terms containing derivatives, I, II, Izdat. Akad. Nauk SSSR, Moscow 3 (1956), 570.
    • (1956) Izdat. Akad. Nauk SSSR, Moscow , vol.3 , pp. 570
  • 27
    • 0000117122 scopus 로고
    • B. D. Sleeman and R. J. Jarvis, Eds., Springer-Verlag, New York
    • J. Rinzel, "Ordinary and Partial Differential Equations" (B. D. Sleeman and R. J. Jarvis, Eds.), pp. 304-316, Springer-Verlag, New York, 1985.
    • (1985) Ordinary and Partial Differential Equations , pp. 304-316
    • Rinzel, J.1
  • 28
    • 0002356724 scopus 로고
    • A. M. Gleason, Ed., Amer. Math. Soc., Providence
    • J. Rinzel, "Pro. Inter. Congress. of Math." (A. M. Gleason, Ed.), pp. 1578-1593, Amer. Math. Soc., Providence, 1987.
    • (1987) Pro. Inter. Congress. of Math. , pp. 1578-1593
    • Rinzel, J.1
  • 29
    • 0002453062 scopus 로고
    • Analysis of neural excitability and oscillations
    • C. Koch and I. Seger, Eds., MIT Press, Cambridge, MA
    • J. Rinzel and G. Ermentrout, Analysis of neural excitability and oscillations, in "Methods in Neural Modeling, from Synapses to Networks" (C. Koch and I. Seger, Eds.), pp. 135-169, MIT Press, Cambridge, MA, 1989.
    • (1989) Methods in Neural Modeling, from Synapses to Networks , pp. 135-169
    • Rinzel, J.1    Ermentrout, G.2
  • 30
    • 0003063371 scopus 로고
    • On different mechanism for membrane potential bursting
    • Springer-Verlag, New York/ Heidelberg/Berlin
    • J. Rinzel and Y. S. Lee, On different mechanism for membrane potential bursting, in "Lecture Notes in Biomathematics," Vol. 66, pp. 19-33, Springer-Verlag, New York/ Heidelberg/Berlin, 1986.
    • (1986) Lecture Notes in Biomathematics , vol.66 , pp. 19-33
    • Rinzel, J.1    Lee, Y.S.2
  • 31
    • 0022557304 scopus 로고
    • Calcium and delayed potassium currents in mouse pancreatic B-cells under voltage-clamp conditions
    • P. Rorsman and G. Trube, Calcium and delayed potassium currents in mouse pancreatic B-cells under voltage-clamp conditions, J. Physiol. 374 (1986), 531-550.
    • (1986) J. Physiol. , vol.374 , pp. 531-550
    • Rorsman, P.1    Trube, G.2
  • 32
    • 0029784059 scopus 로고    scopus 로고
    • Contributions of modeling to understanding stimulus-secretion coupling in pancreatic β-cells
    • A. Sherman, Contributions of modeling to understanding stimulus-secretion coupling in pancreatic β-cells, Amer. J. Physiol. 271 (1996), E362-E372.
    • (1996) Amer. J. Physiol. , vol.271
    • Sherman, A.1
  • 33
    • 0023791327 scopus 로고
    • Emergence of organized bursting in clusters of pancreatic β-cells by channel sharing
    • A. Sherman, J. Rinzel, and J. Keizer, Emergence of organized bursting in clusters of pancreatic β-cells by channel sharing, Biophys. J. 54 (1988), 411-425.
    • (1988) Biophys. J. , vol.54 , pp. 411-425
    • Sherman, A.1    Rinzel, J.2    Keizer, J.3
  • 35
    • 44949272344 scopus 로고
    • Transversal heteroclinic and homoclinic orbits in singular perturbation problems
    • P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations 92 (1991), 252-281.
    • (1991) J. Differential Equations , vol.92 , pp. 252-281
    • Szmolyan, P.1
  • 36
    • 0026239399 scopus 로고
    • Chaotic spikes arising from a model for bursting in excitable membranes
    • D. Terman, Chaotic spikes arising from a model for bursting in excitable membranes, SIAM J. Appl. Math, 51 (1991), 1418-1450.
    • (1991) SIAM J. Appl. Math , vol.51 , pp. 1418-1450
    • Terman, D.1
  • 37
    • 34249839599 scopus 로고
    • The transition from bursting to continuous spiking in excitable membrane models
    • D. Terman, The transition from bursting to continuous spiking in excitable membrane models, J. Nonlinear Sci. 2 (1992), 135-182.
    • (1992) J. Nonlinear Sci. , vol.2 , pp. 135-182
    • Terman, D.1
  • 38
    • 0000559545 scopus 로고
    • On the dependence of the solutions of the differential equations on a small parameter
    • A. N. Tibonov, On the dependence of the solutions of the differential equations on a small parameter, Mat. Sh. 31 (1948), 575-586.
    • (1948) Mat. Sh. , vol.31 , pp. 575-586
    • Tibonov, A.N.1
  • 39
    • 0001736010 scopus 로고
    • Oscillatory and bursting properties of neurons
    • M. A. Arbib, Ed., MIT Press, Cambridge, MA
    • X-J. Wang and J. Rinzel, Oscillatory and bursting properties of neurons, in "The Handbook of Brain Theory and Neural Networks" (M. A. Arbib, Ed.), pp. 686-691, MIT Press, Cambridge, MA, 1995.
    • (1995) The Handbook of Brain Theory and Neural Networks , pp. 686-691
    • Wang, X.-J.1    Rinzel, J.2


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