메뉴 건너뛰기




Volumn 66, Issue 6, 2006, Pages 1917-1948

Return map characterizations for a model of bursting with two slow variables

Author keywords

Bursting; Return map; Singular perturbation solutions

Indexed keywords

BURSTING; RETURN MAP; SINGULAR PERTURBATION SOLUTIONS;

EID: 33845516657     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/050635201     Document Type: Article
Times cited : (11)

References (56)
  • 1
    • 0014245305 scopus 로고
    • Spontaneous activity in isolated somata of Aplysia pacemaker neurons
    • B. ALVING, Spontaneous activity in isolated somata of Aplysia pacemaker neurons, J. Gen. Physiol., 51 (1968), pp. 29-45.
    • (1968) J. Gen. Physiol. , vol.51 , pp. 29-45
    • Alving, B.1
  • 3
    • 0029206121 scopus 로고
    • Analysis of an autonomous phase model for neuronal parabolic bursting
    • S. BAER, J. RINZEL, AND H. CARRILLO, Analysis of an autonomous phase model for neuronal parabolic bursting, J. Math. Biol., 33 (1995), pp. 309-333.
    • (1995) J. Math. Biol. , vol.33 , pp. 309-333
    • Baer, S.1    Rinzel, J.2    Carrillo, H.3
  • 4
    • 0029303855 scopus 로고
    • Topological and phenomenological classification of bursting oscillations
    • R. BERTRAM, M. BUTTE, T. KIEMEL, AND A. SHERMAN, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), pp. 413-439.
    • (1995) Bull. Math. Biol. , vol.57 , pp. 413-439
    • Bertram, R.1    Butte, M.2    Kiemel, T.3    Sherman, A.4
  • 6
    • 0029024670 scopus 로고
    • A role for calcium release-activated current (CRAC) in cholinergic modulation of electrical activity in pancreatic β-cells
    • R. BERTRAM, P. SMOLEN, A. SHERMAN, D. MEARS, I. ATWATER, F. MARTIN, AND B. SORIA, A role for calcium release-activated current (CRAC) in cholinergic modulation of electrical activity in pancreatic β-cells, Biophys. J., 68 (1995), pp. 2323-2332.
    • (1995) Biophys. J. , vol.68 , pp. 2323-2332
    • Bertram, R.1    Smolen, P.2    Sherman, A.3    Mears, D.4    Atwater, I.5    Martin, F.6    Soria, B.7
  • 8
    • 0346053690 scopus 로고    scopus 로고
    • Almost-synchronous solutions for mutually coupled excitatory neurons
    • A. BOSE, N. KOPELL, AND D. TERMAN, Almost-synchronous solutions for mutually coupled excitatory neurons, Phys. D, 140 (2000), pp. 69-94.
    • (2000) Phys. D , vol.140 , pp. 69-94
    • Bose, A.1    Kopell, N.2    Terman, D.3
  • 9
    • 0005981765 scopus 로고    scopus 로고
    • Multirhythmic bursting
    • R. BUTERA, Multirhythmic bursting, Chaos, 8 (1998), pp. 274-284.
    • (1998) Chaos , vol.8 , pp. 274-284
    • Butera, R.1
  • 10
    • 0027251798 scopus 로고
    • Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity
    • C. C. CANAVIER, D. A. BAXTER, J. W. CLARK, AND J. H. BYRNE, Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity, J. Neurophysiology, 69 (1993), pp. 2252-2257.
    • (1993) J. Neurophysiology , vol.69 , pp. 2252-2257
    • Canavier, C.C.1    Baxter, D.A.2    Clark, J.W.3    Byrne, J.H.4
  • 11
    • 0028106120 scopus 로고
    • Multiple modes of activity in a model neuron suggest a novel mechanism for the effects of neuromodulatros
    • C. C. CANAVIER, D. A. BAXTER, J. W. CLARK, AND J. H. BYRNE, Multiple modes of activity in a model neuron suggest a novel mechanism for the effects of neuromodulatros, J. Neurophysiology, 72 (1994), pp. 872-882.
    • (1994) J. Neurophysiology , vol.72 , pp. 872-882
    • Canavier, C.C.1    Baxter, D.A.2    Clark, J.W.3    Byrne, J.H.4
  • 12
    • 46549093797 scopus 로고
    • Chaos in a three-variable model of an excitable cell
    • T. CHAY, Chaos in a three-variable model of an excitable cell, Phys. D, 16 (1985), pp. 233-242.
    • (1985) Phys. D , vol.16 , pp. 233-242
    • Chay, T.1
  • 13
    • 0025422914 scopus 로고
    • 2+ ions on electrical bursting activity of pancreatic β-cells
    • 2+ ions on electrical bursting activity of pancreatic β-cells, Am. J. Physiol., 258 (1990), pp. C955-C965.
    • (1990) Am. J. Physiol. , vol.258
    • Chay, T.1
  • 14
    • 0023726535 scopus 로고
    • Endogenous bursting patterns in excitable cells
    • T. CHAY AND D. COOK, Endogenous bursting patterns in excitable cells, Math. Biosci., 90 (1988), pp. 139-153.
    • (1988) Math. Biosci. , vol.90 , pp. 139-153
    • Chay, T.1    Cook, D.2
  • 15
    • 0020957719 scopus 로고
    • Minimal model for membrane oscillations in the pancreatic β-cell
    • T. CHAY AND J. KEIZER, Minimal model for membrane oscillations in the pancreatic β-cell, Biophys. J., 42 (1983), pp. 181-190.
    • (1983) Biophys. J. , vol.42 , pp. 181-190
    • Chay, T.1    Keizer, J.2
  • 16
    • 0347802074 scopus 로고    scopus 로고
    • Multiple bifurcations in a polynomial model of bursting oscillations
    • G. DE VRIES, Multiple bifurcations in a polynomial model of bursting oscillations, J. Nonlinear Sci., 8 (1998), pp. 281-316.
    • (1998) J. Nonlinear Sci. , vol.8 , pp. 281-316
    • De Vries, G.1
  • 17
    • 0031796185 scopus 로고    scopus 로고
    • Diffusively coupled bursters: Effects of cell heterogeneity
    • G. DE VRIES, A. SHERMAN, AND H. ZHU, Diffusively coupled bursters: Effects of cell heterogeneity, Bull. Math. Biol., 60 (1998), pp. 1167-1200.
    • (1998) Bull. Math. Biol. , vol.60 , pp. 1167-1200
    • De Vries, G.1    Sherman, A.2    Zhu, H.3
  • 18
    • 0020053555 scopus 로고
    • Thalamic bursting mechanism: An inward slow current revealed by membrane hyperpolarization
    • M. DESCHÊNES, J. ROY, AND M. STERIADE, Thalamic bursting mechanism: An inward slow current revealed by membrane hyperpolarization, Brain Res., 239 (1982), pp. 289-293.
    • (1982) Brain Res. , vol.239 , pp. 289-293
    • Deschênes, M.1    Roy, J.2    Steriade, M.3
  • 19
    • 0001755071 scopus 로고
    • AUTO: A program for the automatic bifurcation analysis of autonomous systems
    • E. J. DOEDEL, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), pp. 265-284.
    • (1981) Congr. Numer. , vol.30 , pp. 265-284
    • Doedel, E.J.1
  • 23
    • 53349102813 scopus 로고
    • Impulses and physiological states in theoretical models of nerve membrane
    • R. FITZHUGH, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), pp. 445-466.
    • (1961) Biophys. J. , vol.1 , pp. 445-466
    • Fitzhugh, R.1
  • 26
    • 0023185823 scopus 로고
    • Multiple mechanism of bursting in a conditioned bursting neuron
    • R. HARRIS-WARRICK AND R. FLAMM, Multiple mechanism of bursting in a conditioned bursting neuron, J. Neurosci., 7 (1987), pp. 2113-2128.
    • (1987) J. Neurosci. , vol.7 , pp. 2113-2128
    • Harris-Warrick, R.1    Flamm, R.2
  • 27
    • 0023088280 scopus 로고
    • Theoretical studies on the electrical activity of pancreatic β-cells as a function of glucose
    • D. HIMMEL AND T. CHAY, Theoretical studies on the electrical activity of pancreatic β-cells as a function of glucose, Biophys. J., 51 (1987), pp. 89-107.
    • (1987) Biophys. J. , vol.51 , pp. 89-107
    • Himmel, D.1    Chay, T.2
  • 28
    • 0021768729 scopus 로고
    • A model of neuronal bursting using three coupled first order differential equations
    • J. HINDMARSH AND R. ROSE, A model of neuronal bursting using three coupled first order differential equations, Proc. Roy. Soc. Lond. Ser. B, 221 (1984), pp. 87-102.
    • (1984) Proc. Roy. Soc. Lond. Ser. B , vol.221 , pp. 87-102
    • Hindmarsh, J.1    Rose, R.2
  • 29
    • 35649001607 scopus 로고
    • A quantitative description of membrane current and its application to conduction and excitation in nerve
    • Lond.
    • A. HODGKIN AND A. HUXLEY, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (Lond.), 117 (1952), pp. 500-544.
    • (1952) J. Physiol. , vol.117 , pp. 500-544
    • Hodgkin, A.1    Huxley, A.2
  • 31
    • 0024323358 scopus 로고
    • Atp-sensitive potassium channel and bursting in the pancreatic β cell
    • J. KEIZER AND G. MAGNUS, Atp-sensitive potassium channel and bursting in the pancreatic β cell, Biophys. J., 56 (1989), pp. 229-242.
    • (1989) Biophys. J. , vol.56 , pp. 229-242
    • Keizer, J.1    Magnus, G.2
  • 33
    • 18544394928 scopus 로고    scopus 로고
    • Analysis of complex bursting in cortical pyramidal neuron models
    • A. KEPEOS AND X. WANG, Analysis of complex bursting in cortical pyramidal neuron models, Neurocomputing, 32/33 (2000), pp. 181-187.
    • (2000) Neurocomputing , vol.32-33 , pp. 181-187
    • Kepeos, A.1    Wang, X.2
  • 34
    • 14944346854 scopus 로고    scopus 로고
    • Reduction of a model of an excitable cell to a one-dimensional map
    • G. S. MEDVEDEV, Reduction of a model of an excitable cell to a one-dimensional map, Phys. D, 202 (2005), pp. 37-59.
    • (2005) Phys. D , vol.202 , pp. 37-59
    • Medvedev, G.S.1
  • 35
    • 1842857898 scopus 로고
    • A theory of degree of mapping based on infinitesimal analysis
    • M. NAGUMO, A theory of degree of mapping based on infinitesimal analysis, Amer. J. Math., 73 (1951), pp. 485-496.
    • (1951) Amer. J. Math. , vol.73 , pp. 485-496
    • Nagumo, M.1
  • 36
    • 0028444759 scopus 로고
    • Fast subsystem, bifurcations in a slowly varying Liénard system exhibiting bursting
    • M. PERNAROWSKI, Fast subsystem, bifurcations in a slowly varying Liénard system exhibiting bursting, SIAM J. Appl. Math., 54 (1994), pp. 814-832.
    • (1994) SIAM J. Appl. Math. , vol.54 , pp. 814-832
    • Pernarowski, M.1
  • 37
    • 0032186805 scopus 로고    scopus 로고
    • Fast and slow subsystems for a continuum model of bursting activity in the pancreatic islet
    • M. PERNAROWSKI, Fast and slow subsystems for a continuum model of bursting activity in the pancreatic islet, SIAM. J. Appl. Math., 58 (1998), pp. 1667-1687.
    • (1998) SIAM. J. Appl. Math. , vol.58 , pp. 1667-1687
    • Pernarowski, M.1
  • 38
    • 0033955202 scopus 로고    scopus 로고
    • Fast subsystem bifurcations in strongly coupled heterogeneous collections of excitable cells
    • M. PERNAROWSKI, Fast subsystem bifurcations in strongly coupled heterogeneous collections of excitable cells, Bull. Math. Biol., 62 (2000), pp. 101-120.
    • (2000) Bull. Math. Biol. , vol.62 , pp. 101-120
    • Pernarowski, M.1
  • 39
    • 0027043895 scopus 로고
    • Perturbation techniques for models of bursting electrical activity in the pancreatic β-cell
    • M. PERNAROWSKI, R. M. MIURA, AND J. KEVORKIAN, Perturbation techniques for models of bursting electrical activity in the pancreatic β-cell, SIAM J. Appl. Math., 52 (1992), pp. 1627-1650.
    • (1992) SIAM J. Appl. Math. , vol.52 , pp. 1627-1650
    • Pernarowski, M.1    Miura, R.M.2    Kevorkian, J.3
  • 40
    • 0017225621 scopus 로고
    • Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations
    • R. PLANT AND M. KIM, Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations, Biophys. J., 16 (1976), pp. 227-244.
    • (1976) Biophys. J. , vol.16 , pp. 227-244
    • Plant, R.1    Kim, M.2
  • 41
    • 0000357907 scopus 로고
    • A formal classification of bursting mechanisms in excitable systems
    • Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences, E. Teramato and M. Yamaguti, eds., Springer-Verlag, Berlin
    • J. RINZEL, A formal classification of bursting mechanisms in excitable systems, in Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences, E. Teramato and M. Yamaguti, eds., Lecture Notes in Biomath. 71, Springer-Verlag, Berlin, 1987, pp. 267-281.
    • (1987) Lecture Notes in Biomath. , vol.71 , pp. 267-281
    • Rinzel, J.1
  • 42
    • 0022557304 scopus 로고
    • Calcium and delayed potassium currents in mouse pancreatic β-cells under voltage-clamp conditions
    • Lond.
    • P. RORSMAN AND G. TRUBE, Calcium and delayed potassium currents in mouse pancreatic β-cells under voltage-clamp conditions, J. Physiol. (Lond.), 374 (1986), pp. 531-550.
    • (1986) J. Physiol. , vol.374 , pp. 531-550
    • Rorsman, P.1    Trube, G.2
  • 43
    • 0028500362 scopus 로고
    • Anti-phase, asymmetric, and aperiodic oscillations in excitable cells - I. Coupled bursters
    • A. SHERMAN, Anti-phase, asymmetric, and aperiodic oscillations in excitable cells - I. Coupled bursters, Bull. Math. Biol., 56 (1994), pp. 811-835.
    • (1994) Bull. Math. Biol. , vol.56 , pp. 811-835
    • Sherman, A.1
  • 45
    • 0025979886 scopus 로고
    • Model for synchronization of pancreatic β-cells by gap junctions
    • A. SHERMAN AND J. RINZEL, Model for synchronization of pancreatic β-cells by gap junctions, Biophys. J., 59 (1991). pp. 547-559.
    • (1991) Biophys. J. , vol.59 , pp. 547-559
    • Sherman, A.1    Rinzel, J.2
  • 46
    • 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), pp. 411-425.
    • (1988) Biophys. J. , vol.54 , pp. 411-425
    • Sherman, A.1    Rinzel, J.2    Keizer, J.3
  • 47
    • 24944590712 scopus 로고    scopus 로고
    • Mechanism of bistability: Tonic spiking and bursting in a neuron model
    • A. SHILNIKOV, R. CALABRESE, AND G. CYMBALYUK, Mechanism of bistability: Tonic spiking and bursting in a neuron model Phys. Rev. E, 71 (2005), p. 056214.
    • (2005) Phys. Rev. E , vol.71 , pp. 056214
    • Shilnikov, A.1    Calabrese, R.2    Cymbalyuk, G.3
  • 49
    • 0026587453 scopus 로고
    • 2+ currents and bursting mechanisms for the mouse pancreatic β-cell
    • 2+ currents and bursting mechanisms for the mouse pancreatic β-cell, J. Memb. Biol., 127 (1992), pp. 9-19.
    • (1992) J. Memb. Biol. , vol.127 , pp. 9-19
    • Smolen, P.1    Keizer, J.2
  • 50
    • 0027161335 scopus 로고
    • Why pancreatic islets burst but single β-cells do not: The heterogeneity hypothesis
    • P. SMOLEN, J. RINZEL, AND A. SHERMAN, Why pancreatic islets burst but single β-cells do not: The heterogeneity hypothesis, Biophys. J., 64 (1993), pp. 1668-1680.
    • (1993) Biophys. J. , vol.64 , pp. 1668-1680
    • Smolen, P.1    Rinzel, J.2    Sherman, A.3
  • 51
    • 0027608561 scopus 로고
    • Properties of a bursting model with two slow inhibitory variables
    • P. SMOLEN, D. TERMAN. AND J. RINZEL. Properties of a bursting model with two slow inhibitory variables, SIAM J. Appl. Math., 53 (1993), pp. 861-892.
    • (1993) SIAM J. Appl. Math. , vol.53 , pp. 861-892
    • Smolen, P.1    Terman, D.2    Rinzel, J.3
  • 53
    • 0026239399 scopus 로고
    • Chaotic spikes arising from a. model of bursting in excitable membranes
    • D. TERMAN, Chaotic spikes arising from a. model of bursting in excitable membranes, SIAM J. Appl. Math., 51 (1991), pp. 1418-1450.
    • (1991) SIAM J. Appl. Math. , vol.51 , pp. 1418-1450
    • Terman, D.1
  • 54
    • 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), pp. 135-182.
    • (1992) J. Nonlinear Sci. , vol.2 , pp. 135-182
    • Terman, D.1
  • 55
    • 0033103124 scopus 로고    scopus 로고
    • Fast burst firing and short-term synaptic plasticity: A model of neocortical chattering neurons
    • X. WANG, Fast burst firing and short-term synaptic plasticity: A model of neocortical chattering neurons, Neurosci., 89 (1999), pp. 347-362.
    • (1999) Neurosci. , vol.89 , pp. 347-362
    • Wang, X.1
  • 56
    • 0019505714 scopus 로고
    • Afterpotential generation in hippocampal pyramidal cells
    • R. WONG AND D. PRINCE, Afterpotential generation in hippocampal pyramidal cells, J. Neurophysiol., 45 (1981), pp. 86-97.
    • (1981) J. Neurophysiol. , vol.45 , pp. 86-97
    • Wong, R.1    Prince, D.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.