메뉴 건너뛰기




Volumn 240, Issue 14-15, 2011, Pages 1164-1180

Voltage interval mappings for activity transitions in neuron models for elliptic bursters

Author keywords

Bifurcation; Bursting; Elliptic; Neuron model; Periodic orbit; Poincar mapping

Indexed keywords

BIFURCATION (MATHEMATICS); COMPUTER AIDED ANALYSIS; NEURONS; ONE DIMENSIONAL; OSCILLATORS (MECHANICAL);

EID: 79958006528     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2011.04.003     Document Type: Article
Times cited : (26)

References (67)
  • 1
    • 0002299089 scopus 로고
    • Toward a theory of modelling central pattern generators
    • N. Kopell Toward a theory of modelling central pattern generators A.H. Cohen, S. Rossingol, S. Grillner, Neural Control of Rhythmic Movements in Vertebrates 1988 Wiley New York
    • (1988) Neural Control of Rhythmic Movements in Vertebrates
    • Kopell, N.1
  • 2
    • 0029847273 scopus 로고    scopus 로고
    • Principles of rhythmic motor pattern generation
    • E. Marder, and R.L. Calabrese Principles of rhythmic motor pattern generation Physiol. Rev. 76 3 1996 687 717 (Pubitemid 26286076)
    • (1996) Physiological Reviews , vol.76 , Issue.3 , pp. 687-717
    • Marder, E.1    Calabrese, R.L.2
  • 3
    • 0002356724 scopus 로고
    • A formal classification of bursting mechanisms in excitable systems
    • AMS
    • J. Rinzel A formal classification of bursting mechanisms in excitable systems Proc. International Congress of Mathematics 1987 AMS 1578 1593
    • (1987) Proc. International Congress of Mathematics , pp. 1578-1593
    • Rinzel, J.1
  • 4
    • 0023461032 scopus 로고
    • Dissection of a model for neuronal parabolic bursting
    • J. Rinzel, and Y.S. Lee Dissection of a model for neuronal parabolic bursting J. Math. Biol. 25 6 1987 653 675
    • (1987) J. Math. Biol. , vol.25 , Issue.6 , pp. 653-675
    • Rinzel, J.1    Lee, Y.S.2
  • 5
    • 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 3 1995 413 439
    • (1995) Bull. Math. Biol. , vol.57 , Issue.3 , pp. 413-439
    • Bertram, R.1    Butte, M.J.2    Kiemel, T.3    Sherman, A.4
  • 10
    • 34250627892 scopus 로고
    • Geometric singular perturbation theory for ordinary differential equations
    • F. Fenichel Geometric singular perturbation theory for ordinary differential equations J. Differential Equations 31 1979 53 98
    • (1979) J. Differential Equations , vol.31 , pp. 53-98
    • Fenichel, F.1
  • 13
    • 18144427711 scopus 로고
    • Periodic solution of a system of ordinary differential equations with a small parameter in the terms containing derivatives
    • L.S. Pontryagin, and L.V. Rodygin Periodic solution of a system of ordinary differential equations with a small parameter in the terms containing derivatives Sov. Math. Dokl. 1 1960 611
    • (1960) Sov. Math. Dokl. , vol.1 , pp. 611
    • Pontryagin, L.S.1    Rodygin, L.V.2
  • 14
    • 0000559547 scopus 로고
    • On the dependence of solutions of differential equations from a small parameter
    • A.N. Tikhonov On the dependence of solutions of differential equations from a small parameter Mat. Sb. 22 64 1948 193 204
    • (1948) Mat. Sb. , vol.22 , Issue.64 , pp. 193-204
    • Tikhonov, A.N.1
  • 15
    • 0000881312 scopus 로고
    • On delayed stability loss under dynamical bifurcations i
    • A.I. Neishtadt On delayed stability loss under dynamical bifurcations I Differ. Equ. 23 1988 1385 1390
    • (1988) Differ. Equ. , vol.23 , pp. 1385-1390
    • Neishtadt, A.I.1
  • 16
    • 0000190869 scopus 로고
    • On delayed stability loss under dynamical bifurcations II
    • A.I. Neishtadt On delayed stability loss under dynamical bifurcations II Differ. Equ. 24 1989 171 176
    • (1989) Differ. Equ. , vol.24 , pp. 171-176
    • Neishtadt, A.I.1
  • 17
    • 0034311437 scopus 로고    scopus 로고
    • Homoclinic bifurcations leading to bursting oscillations in cell models
    • V.N. Belykh, I.V. Belykh, M. Colding-Jorgensen, and E. Mosekilde Homoclinic bifurcations leading to bursting oscillations in cell models Eur. Phys. J. E 3 3 2000 205 219
    • (2000) Eur. Phys. J. e , vol.3 , Issue.3 , pp. 205-219
    • Belykh, V.N.1    Belykh, I.V.2    Colding-Jorgensen, M.3    Mosekilde, E.4
  • 18
    • 34047181411 scopus 로고    scopus 로고
    • Origin of bursting through homoclinic spike adding in a neuron model
    • P. Channell, G. Cymbalyuk, and A. Shilnikov Origin of bursting through homoclinic spike adding in a neuron model Phys. Rev. Lett. 98 13 2007 134101
    • (2007) Phys. Rev. Lett. , vol.98 , Issue.13 , pp. 134101
    • Channell, P.1    Cymbalyuk, G.2    Shilnikov, A.3
  • 19
    • 18244408512 scopus 로고    scopus 로고
    • Coexistence of tonic spiking oscillations in a leech neuron model
    • DOI 10.1007/s10827-005-0354-7
    • G. Cymbalyuk, and A.L. Shilnikov Coexistence of tonic spiking oscillations in a leech neuron model J. Comput. Neurosci. 18 3 2005 255 263 (Pubitemid 40626655)
    • (2005) Journal of Computational Neuroscience , vol.18 , Issue.3 , pp. 255-263
    • Cymbalyuk, G.1    Shilnikov, A.2
  • 20
    • 0034147579 scopus 로고    scopus 로고
    • Homoclinic bifurcation in a hodgkin-huxley model of thermally sensitive neurons
    • U. Feudel, A. Neiman, X. Pei, W. Wojtenek, B. Braun, M. Huber, and F. Moss Homoclinic bifurcation in a hodgkin-huxley model of thermally sensitive neurons Chaos 10 1 2000 231 239
    • (2000) Chaos , vol.10 , Issue.1 , pp. 231-239
    • Feudel, U.1    Neiman, A.2    Pei, X.3    Wojtenek, W.4    Braun, B.5    Huber, M.6    Moss, F.7
  • 21
    • 0000732703 scopus 로고
    • From simple to simple bursting oscillatory behaviour via intermittent chaos in the RoseHindmarsh model for neuronal activity
    • A.V. Holden, and Y.S. Fan From simple to simple bursting oscillatory behaviour via intermittent chaos in the RoseHindmarsh model for neuronal activity Chaos Solitons Fractals 2 1992 349 369
    • (1992) Chaos Solitons Fractals , vol.2 , pp. 349-369
    • Holden, A.V.1    Fan, Y.S.2
  • 23
    • 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 2 1992 135 182
    • (1992) J. Nonlinear Sci. , vol.2 , Issue.2 , pp. 135-182
    • Terman, D.1
  • 24
    • 31144434673 scopus 로고    scopus 로고
    • Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems
    • M.A. Zaks, X. Sailer, L. Schimansky-Geier, and A.B. Neiman Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems Chaos 15 2 2005 26117
    • (2005) Chaos , vol.15 , Issue.2 , pp. 26117
    • Zaks, M.A.1    Sailer, X.2    Schimansky-Geier, L.3    Neiman, A.B.4
  • 26
    • 36549095130 scopus 로고
    • Mixed-mode oscillations in an electrochemical system. I. A farey sequence which does not occur on a torus
    • F.N. Albahadily, J. Ringland, and M. Schell Mixed-mode oscillations in an electrochemical system. I. A farey sequence which does not occur on a torus J. Chem. Phys. 90 2 1989 813 822
    • (1989) J. Chem. Phys. , vol.90 , Issue.2 , pp. 813-822
    • Albahadily, F.N.1    Ringland, J.2    Schell, M.3
  • 27
    • 0001527863 scopus 로고
    • Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium
    • P. Gaspard, and X.J. Wang Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium J. Stat. Phys. 48 12 1987 151 199
    • (1987) J. Stat. Phys. , vol.48 , Issue.12 , pp. 151-199
    • Gaspard, P.1    Wang, X.J.2
  • 28
    • 0002467474 scopus 로고
    • Bifurcations of mixed-mode oscillations in a three-variable autonomous van der pol-duffing model with a cross-shaped phase diagram
    • M. Koper Bifurcations of mixed-mode oscillations in a three-variable autonomous van der pol-duffing model with a cross-shaped phase diagram Phys. D 80 12 1995 72 94
    • (1995) Phys. D , vol.80 , Issue.12 , pp. 72-94
    • Koper, M.1
  • 29
    • 36749113474 scopus 로고
    • An experimental study of multiple peak periodic and nonperiodic oscillations in the BelousovZhabotinskii reaction
    • J.L. Hudson, and D. Marinko An experimental study of multiple peak periodic and nonperiodic oscillations in the BelousovZhabotinskii reaction J. Chem. Phys. 71 4 1979 1601 1606
    • (1979) J. Chem. Phys. , vol.71 , Issue.4 , pp. 1601-1606
    • Hudson, J.L.1    Marinko, D.2
  • 30
    • 33845516657 scopus 로고    scopus 로고
    • Return map characterizations for a model of bursting with two slow variables
    • DOI 10.1137/050635201
    • R.E. Griffiths, and M.C. Pernarowski Return map characterizations for a model of bursting with two slow variables SIAM J. Appl. Math. 66 6 2006 1917 1948 (Pubitemid 44925756)
    • (2006) SIAM Journal on Applied Mathematics , vol.66 , Issue.6 , pp. 1917-1948
    • Griffiths, R.E.1    Pernarowski, M.2
  • 31
    • 14944346854 scopus 로고    scopus 로고
    • Reduction of a model of an excitable cell to a one-dimensional map
    • G.M. Medvedev Reduction of a model of an excitable cell to a one-dimensional map Physica D 202 12 2005 87 106
    • (2005) Physica D , vol.202 , Issue.12 , pp. 87-106
    • Medvedev, G.M.1
  • 32
    • 33746613350 scopus 로고    scopus 로고
    • Transition to bursting via deterministic chaos
    • G.M. Medvedev Transition to bursting via deterministic chaos Phys. Rev. Lett. 97 2006 048102
    • (2006) Phys. Rev. Lett. , vol.97 , pp. 048102
    • Medvedev, G.M.1
  • 33
    • 54949130175 scopus 로고    scopus 로고
    • Methods of the qualitative theory for the HindmarshRose model: A case study. A tutorial
    • A.L. Shilnikov, and M.L. Kolomiets Methods of the qualitative theory for the HindmarshRose model: a case study. A tutorial Int. J. Bifurcation Chaos 18 7 2008 1 32
    • (2008) Int. J. Bifurcation Chaos , vol.18 , Issue.7 , pp. 1-32
    • Shilnikov, A.L.1    Kolomiets, M.L.2
  • 34
    • 40849105601 scopus 로고    scopus 로고
    • Blue sky catastrophe in singularly perturbed systems
    • A.L. Shilnikov, L.P. Shilnikov, and D.V. Turaev Blue sky catastrophe in singularly perturbed systems Mosc. Math. J. 5 1 2005 205 211
    • (2005) Mosc. Math. J. , vol.5 , Issue.1 , pp. 205-211
    • Shilnikov, A.L.1    Shilnikov, L.P.2    Turaev, D.V.3
  • 35
    • 34147183714 scopus 로고    scopus 로고
    • Chaotic synchronization in ensembles of coupled neurons modeled by the FitzHugh-Rinzel system
    • DOI 10.1007/s11141-006-0124-z
    • V.N. Belykh, and E.V. Pankratova Chaotic synchronization in ensembles of coupled neurons modeled by the FitzhhughRinzel system Radiophys. and Quantum Electronics 49 11 2006 910 921 (Pubitemid 46562843)
    • (2006) Radiophysics and Quantum Electronics , vol.49 , Issue.11 , pp. 910-921
    • Belykh, V.N.1    Pankratova, E.V.2
  • 36
    • 0035359894 scopus 로고    scopus 로고
    • Synchronization of elliptic bursters
    • PII S0036144500382064
    • E.M. Izhikevich Synchronization of elliptic bursters SIAM Rev. 43 2 2001 315 344 (Pubitemid 32697798)
    • (2001) SIAM Review , vol.43 , Issue.2 , pp. 315-344
    • Izhikevich, E.M.1
  • 37
    • 2942560675 scopus 로고    scopus 로고
    • Multimodal regimes in a compartmental model of the dopamine neuron
    • G. Medvedev, and J. Cisternas Multimodal regimes in a compartmental model of the dopamine neuron Physica D 194 2004 333 356
    • (2004) Physica D , vol.194 , pp. 333-356
    • Medvedev, G.1    Cisternas, J.2
  • 39
    • 0347931871 scopus 로고    scopus 로고
    • Effects of noise on elliptic bursters
    • DOI 10.1088/0951-7715/17/1/009, PII S0951771504583500
    • J. Su, J. Rubin, and D. Terman Effects of noise on elliptic bursters Nonlinearity 17 2004 133 157 (Pubitemid 38085739)
    • (2004) Nonlinearity , vol.17 , Issue.1 , pp. 133-157
    • Su, J.1    Rubin, J.2    Terman, D.3
  • 41
    • 33644526607 scopus 로고    scopus 로고
    • Generation of very slow neuronal rhythms and chaos near the hopf bifurcation in single neuron models
    • DOI 10.1007/s10827-005-2895-1
    • J. Doi, and S. Kumagai Generation of very slow neuronal rhythms and chaos near the hopf bifurcation in single neuron models J. Comput. Neurosci. 19 3 2005 325 356 (Pubitemid 43293059)
    • (2005) Journal of Computational Neuroscience , vol.19 , Issue.3 , pp. 325-356
    • Doi, S.1    Kumagai, S.2
  • 42
    • 55949135112 scopus 로고    scopus 로고
    • Mixed-mode oscillations in three time-scale systems: A prototypical example
    • M. Krupa, N. Popovich, and N.J. Kopell Mixed-mode oscillations in three time-scale systems: a prototypical example SIAM J. Appl. Dyn. Syst. 7 2 2008 361 420
    • (2008) SIAM J. Appl. Dyn. Syst. , vol.7 , Issue.2 , pp. 361-420
    • Krupa, M.1    Popovich, N.2    Kopell, N.J.3
  • 43
    • 41549101374 scopus 로고    scopus 로고
    • The selection of mixed-mode oscillations in a HodgkinHuxley model with multiple timescales
    • J. Rubin, and M. Wechselberger The selection of mixed-mode oscillations in a HodgkinHuxley model with multiple timescales Chaos 18 2008 015105 015117
    • (2008) Chaos , vol.18 , pp. 015105-015117
    • Rubin, J.1    Wechselberger, M.2
  • 45
    • 3142532617 scopus 로고    scopus 로고
    • Subthreshold oscillations in a map-based neuron model
    • A.L. Shilnikov, and N.F. Rulkov Subthreshold oscillations in a map-based neuron model Phys. Lett. A 328 23 2004 177 184
    • (2004) Phys. Lett. A , vol.328 , Issue.23 , pp. 177-184
    • Shilnikov, A.L.1    Rulkov, N.F.2
  • 46
    • 34247524521 scopus 로고    scopus 로고
    • Applications of the poincare mapping technique to analysis of neuronal dynamics
    • P. Channell, G. Cymbalyuk, and A.L. Shilnikov Applications of the poincare mapping technique to analysis of neuronal dynamics Neurocomputing 70 2007 10 12
    • (2007) Neurocomputing , vol.70 , pp. 10-12
    • Channell, P.1    Cymbalyuk, G.2    Shilnikov, A.L.3
  • 47
    • 70350573483 scopus 로고    scopus 로고
    • Variability of bursting patterns in a neuron model in the presence of noise
    • P. Channell, I. Fuwape, A.B. Neiman, and A.L. Shilnikov Variability of bursting patterns in a neuron model in the presence of noise J. Comput. Neurosci. 27 3 2009 527 542
    • (2009) J. Comput. Neurosci. , vol.27 , Issue.3 , pp. 527-542
    • Channell, P.1    Fuwape, I.2    Neiman, A.B.3    Shilnikov, A.L.4
  • 48
    • 0001910840 scopus 로고
    • Mixed-mode and chaotic oscillations in a simple model of electrochemical osillator
    • M. Koper, and P. Gaspard Mixed-mode and chaotic oscillations in a simple model of electrochemical osillator J. Phys. Chem. 95 1991 4945 4947
    • (1991) J. Phys. Chem. , vol.95 , pp. 4945-4947
    • Koper, M.1    Gaspard, P.2
  • 49
    • 33745452482 scopus 로고
    • Mixed-mode oscillations and incomplete homoclinic scenarios to a saddle-focus in the indium/thiocyanate electrochemical oscillators
    • M. Koper, and P. Gaspard Mixed-mode oscillations and incomplete homoclinic scenarios to a saddle-focus in the indium/thiocyanate electrochemical oscillators J. Chem. Phys. 97 11 1992 8250 8260
    • (1992) J. Chem. Phys. , vol.97 , Issue.11 , pp. 8250-8260
    • Koper, M.1    Gaspard, P.2
  • 50
    • 46549093797 scopus 로고
    • Chaos in a three-variable model of an excitable cell
    • T.R. Chay Chaos in a three-variable model of an excitable cell Physica D 16 2 1985 233 242
    • (1985) Physica D , vol.16 , Issue.2 , pp. 233-242
    • Chay, T.R.1
  • 51
    • 33745452482 scopus 로고
    • Mixed-mode oscillations and incomplete homoclinic scenarios to a saddle-focus in the indium/thiocyanate electrochemical oscillator
    • M. Koper, P. Gaspard, and J. Sluyters Mixed-mode oscillations and incomplete homoclinic scenarios to a saddle-focus in the indium/thiocyanate electrochemical oscillator J. Chem. Phys. 97 11 1992 8250 8260
    • (1992) J. Chem. Phys. , vol.97 , Issue.11 , pp. 8250-8260
    • Koper, M.1    Gaspard, P.2    Sluyters, J.3
  • 52
    • 0000764166 scopus 로고
    • On bifurcations of the Lorenz attractor in the ShimizuMorioka model
    • A.L. Shilnikov On bifurcations of the Lorenz attractor in the ShimizuMorioka model Physica D 62 14 1993 338 346
    • (1993) Physica D , vol.62 , Issue.14 , pp. 338-346
    • Shilnikov, A.L.1
  • 54
    • 49749152086 scopus 로고    scopus 로고
    • New dynamics in cerebellar purkinje cells: Torus canards
    • M.A. Kramer, R.D. Traub, and N.J. Kopell New dynamics in cerebellar purkinje cells: torus canards Phys. Rev. Lett. 101 6 2008 068103
    • (2008) Phys. Rev. Lett. , vol.101 , Issue.6 , pp. 068103
    • Kramer, M.A.1    Traub, R.D.2    Kopell, N.J.3
  • 56
    • 79955577320 scopus 로고
    • On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve
    • N.K. Gavrilov, and L.P. Shilnikov On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve Math. USSR Sb. 17 3 1972 467 485
    • (1972) Math. USSR Sb. , vol.17 , Issue.3 , pp. 467-485
    • Gavrilov, N.K.1    Shilnikov, L.P.2
  • 57
    • 79958000278 scopus 로고    scopus 로고
    • On chaotic subthreshold oscillations in a simple neuronal model
    • M. Zaks On chaotic subthreshold oscillations in a simple neuronal model Math. Model. Nat. Phenom. 6 1 2011 1 14
    • (2011) Math. Model. Nat. Phenom. , vol.6 , Issue.1 , pp. 1-14
    • Zaks, M.1
  • 58
    • 51649161951 scopus 로고
    • Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
    • A.B. Katok Lyapunov exponents, entropy and periodic orbits for diffeomorphisms Publ. Math. Inst. Hautes tudes Sci. 51 1980 137 173
    • (1980) Publ. Math. Inst. Hautes Tudes Sci. , vol.51 , pp. 137-173
    • Katok, A.B.1
  • 59
    • 0141519849 scopus 로고    scopus 로고
    • Zeros of the kneading invariant and topological entropy for Lorenz maps
    • P. Glendinning, and T. Hall Zeros of the kneading invariant and topological entropy for Lorenz maps Nonlinearity 9 1996 999 1014 (Pubitemid 126689428)
    • (1996) Nonlinearity , vol.9 , Issue.4 , pp. 999-1014
    • Glendinning, P.1    Hall, T.2
  • 61
    • 56449117107 scopus 로고    scopus 로고
    • Singular hopf bifurcation in systems with two slow variables
    • J. Guckenheimer Singular hopf bifurcation in systems with two slow variables SIAM J. Appl. Dyn. Syst. 7 4 2008 1355 1377
    • (2008) SIAM J. Appl. Dyn. Syst. , vol.7 , Issue.4 , pp. 1355-1377
    • Guckenheimer, J.1
  • 62
    • 46549102025 scopus 로고
    • Quasiperiodicity in chemistry: An experimental path in the neighborhood of a codimension-two bifurcation
    • F. Argoul, and J.C. Roux Quasiperiodicity in chemistry: an experimental path in the neighborhood of a codimension-two bifurcation Phys. Lett. A 108 8 1985 426 430
    • (1985) Phys. Lett. A , vol.108 , Issue.8 , pp. 426-430
    • Argoul, F.1    Roux, J.C.2
  • 65
    • 79957991754 scopus 로고    scopus 로고
    • An elementary model of torus canards
    • (submitted for publication)
    • N. Benes, A. Barry, T. Kaper, M. Kramer, J. Burke, An elementary model of torus canards. J. Chaos (2010) (submitted for publication).
    • (2010) J. Chaos
    • Benes, N.1    Barry, A.2    Kaper, T.3    Kramer, M.4    Burke, J.5
  • 66
    • 79958014085 scopus 로고    scopus 로고
    • Spontaneous voltage oscillations enhance sensitivity of saccular hair cells
    • (submitted for publication)
    • A. Neiman, A. Shilnikov, Spontaneous voltage oscillations enhance sensitivity of saccular hair cells. J Neurophysiol. (2011) (submitted for publication).
    • (2011) J Neurophysiol.
    • Neiman, A.1    Shilnikov, A.2
  • 67
    • 0036549260 scopus 로고    scopus 로고
    • Activity Patterns in a Model for the Subthalamopallidal Network of the Basal Ganglia
    • D. Terman, J.E. Rubin, A.C. Yew, and C.J. Wilson Activity patterns in a model for the subthalamopallidal network of the basal ganglia J. Neurosci. 22 7 2002 2963 2976 (Pubitemid 37465845)
    • (2002) Journal of Neuroscience , vol.22 , Issue.7 , pp. 2963-2976
    • Terman, D.1    Rubin, J.E.2    Yew, A.C.3    Wilson, C.J.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.