An organizing center in a planar model of neuronal excitability

SIAM Journal on Applied Dynamical Systems

Volumn 11, Issue 4, 2012, Pages 1698-1722

  Franci, Alessio ^a   Drion, Guillaume ^a   Sepulchre, Rodolphe ^a  

^a UNIVERSITY OF LIÈGE   (Belgium)

Author keywords

Bifurcation unfolding; Competitive and cooperative variables; Neuronal excitability; Planar neuron model; Voltage gated calcium channels

Indexed keywords

CALCIUM CHANNELS; CALCIUM CURRENT; COMPETITIVE AND COOPERATIVE VARIABLES; FITZHUGH-NAGUMO MODEL; NEURON MODEL; NEURONAL EXCITABILITY; PITCHFORK BIFURCATIONS; PLANAR MODEL;

BIFURCATION (MATHEMATICS);

NEURONS;

EID: 84871373684     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/120875016     Document Type: Article

References (38)

1. J. R. Clay, D. Paydarfar, and D. B. Forger, A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons, J. Roy. Soc. Interface, 5 (2008), pp. 1421- 1428.
2. A. Destexhe, D. Contreras, M. Steriade, T. J. Sejnowski, and J. R. Huguenard, In vivo, in vitro, and computational analysis of dendritic calcium currents in thalamic reticular neurons, J. Neurosci., 16 (1996), pp. 169-185.
3. G. Drion, A. Franci, V. Seutin, and R. Sepulchre, A novel phase portrait for neuronal excitability, PLoS ONE, 7 (2012), e41806.
4. G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, Interdisciplinary Applied Mathematics 35, Springer, New York, 2010.
5. N. Fenichel, Geometric singular perturbation theory, J. Differential Equations, 31 (1979), pp. 53-98.
6. R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), pp. 445-466.
7. Y. Gai, B. Doiron, V. Kotak, and J. Rinzel, Noise-gated encoding of slow inputs by auditory brain stem neurons with a low-threshold K+ current, J. Neurophysiol., 102 (2009), pp. 3447-3460.
8. A. A. Grace and B. S. Bunney, The control of firing pattern in nigral dopamine neurons: Burst firing, J. Neurosci., 4 (1984), pp. 2877-2890.
9. C. M. Gray and D. A. McCormick, Chattering cells: Superficial pyramidal neurons contributing to the generation of synchronous oscillations in the visual cortex, Science, 274 (1996), pp. 109-113.
10. R. Haberman, Slowly varying jump and transition phenomena associated with algebraic bifurcation problems, SIAM J. Appl. Math., 37 (1979), pp. 69-106.
11. N. E. Hallworth, C. J. Wilson, and M. D. Bevan, Apamin-sensitive small conductance calciumactivated potassium channels, through their selective coupling to voltage-gated calcium channels, are critical determinants of the precision, pace, and pattern of action potential generation in rat subthalamic nucleus neurons in vitro, J. Neurosci., 23 (2003), pp. 7525-7542.
12. P. Heyward, M. Ennis, A. Keller, and M. T. Shipley, Membrane bistability in olfactory bulb mitral cells, J. Neurosci., 21 (2001), pp. 5311-5320.
13. A. L. Hodgkin, The local electric changes associated with repetitive action in a non-medullated axon, J. Physiol., 107 (1948), pp. 165-181.
14. J. R. Huguenard and D. A. Prince, A novel T-type current underlies prolonged Ca(2+)-dependent burst firing in GABAergic neurons of rat thalamic reticular nucleus, J. Neurosci., 12 (1992), pp. 3804- 3817.
15. E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press, Cambridge, MA, 2007.
16. S. W. Johnson, V. Seutin, and R. A. North, Burst firing in dopamine neurons induced by N-methyl- D-aspartate: Role of electrogenic sodium pump, Science, 258 (1992), pp. 665-667.
17. C. K. R. Jones, Geometric singular perturbation theory, in Dynamical Systems, Lecture Notes in Math. 1609, Springer, Berlin, 1995, pp. 44-120.
18. M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points- fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), pp. 286-314.
19. M. Krupa and P. Szmolyan, Extending slow manifolds near transcritical and pitchfork singularities, Nonlinearity, 14 (2001), pp. 1473-1491.
20. M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), pp. 312-368.
21. N. R. Lebovitz and R. J. Schaar, Exchange of stabilities in autonomous systems, Stud. Appl. Math., 54 (1975), pp. 229-260.
22. W. Liu, Exchange lemmas for singular perturbation problems with certain turning points, J. Differential Equations, 167 (2000), pp. 134-180.
23. D. A. McCormick and T. Bal, Sleep and arousal: Thalamocortical mechanisms, Ann. Rev. Neurosci., 20 (1997), pp. 185-215.
24. D. A. McCormick and J. R. Huguenard, A model of the electrophysiological properties of thalamocortical relay neurons, J. Neurophysiol., 68 (1992), pp. 1384-1400.
25. C. Messina and R. Cotrufo, Different excitability of type 1 and type 2 alpha-motoneurones: The recruitment curve of H- and M-responses in slow and fast muscles of rabbits, J. Neurological Sci., 28 (1976), pp. 57-63.
26. C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J., 35 (1981), pp. 193-213.
27. R. E. 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.
28. L. S. Pontryagin, Asymptotic behavior of solutions of systems of differential equations with a small parameter in the derivatives of highest order, Izv. Ross. Akad. Nauk Ser. Mat., 21 (1957), pp. 605- 626.
29. J. Rinzel, Excitation dynamics: Insights from simplified membrane models, in Fed. Proc., Vol. 44, 1985, pp. 2944-2946.
30. J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, in Methods in Neuronal Modeling, MIT Press, Cambridge, MA, 1989, pp. 135-169.
31. S. Sessley and R. J. Butera, Evidence for "type I" excitability in molluscan neurons, in Engineering in Medicine and Biology, Proceedings of the 24th Annual Conference and the Annual Fall Meeting of the Biomedical Engineering Society (EMBS/BMES Conference), Vol. 3, IEEE, Washington, DC, 2002, pp. 1966-1967.
32. R. Seydel, Practical Bifurcation and Stability Analysis, 3rd ed., Interdisciplinary Applied Mathematics 5, Springer-Verlag, New York, 2010.
33. T. Tateno, A. Harsch, and H. P. C. Robinson, Threshold firing frequency-current relationships of neurons in rat somatosensory cortex: Type 1 and type 2 dynamics, J. Neurophysiol., 92 (2004), pp. 2283-2294.
34. R. D. Traub, R. K. Wong, R. Miles, and H. Michelson, A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances, J. Neurophysiol., 66 (1991), pp. 635-650.
35. M. Wechselberger, Canards, Scholarpedia J., 2 (2007), p. 1356.
36. C. Wilson, Up and down states, Scholarpedia J., 3 (2008), p. 1410.
37. C. J. Wilson and P. M. Groves, Spontaneous firing patterns of identified spiny neurons in the rat neostriatum, Brain Res., 220 (1981), pp. 67-80.
38. X. J. Zhan, C. L. Cox, J. Rinzel, and S. M. Sherman, Current clamp and modeling studies of lowthreshold calcium spikes in cells of the cat's lateral geniculate nucleus, J. Neurophysiol., 81 (1999), pp. 2360-2373.