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, Journal of The Royal Society Interface, vol.414, issue.6859, pp.1421-1428, 2008.
DOI : 10.1038/35102009

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, vol.16, pp.169-185, 1996.

G. Drion, A. Franci, V. Seutin, and R. Sepulchre, A Novel Phase Portrait for Neuronal Excitability, PLoS ONE, vol.7, issue.8, p.41806, 2012.
DOI : 10.1371/journal.pone.0041806.s005

URL : https://hal.archives-ouvertes.fr/hal-00825278

R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane,B i o p h y s, J, vol.1, pp.445-466, 1961.

Y. Gai, B. Doiron, V. Kotak, and J. , Noise-Gated Encoding of Slow Inputs by Auditory Brain Stem Neurons With a Low-Threshold K+ Current, Journal of Neurophysiology, vol.102, issue.6, pp.3447-3460, 2009.
DOI : 10.1152/jn.00538.2009

A. A. Grace and B. S. Bunney, The control of firing pattern in nigral dopamine neurons: Burst firing, J. Neurosci, vol.4, pp.2877-2890, 1984.

C. M. Gray and D. A. Mccormick, Chattering Cells: Superficial Pyramidal Neurons Contributing to the Generation of Synchronous Oscillations in the Visual Cortex, Science, vol.274, issue.5284, pp.274-109, 1996.
DOI : 10.1126/science.274.5284.109

R. Haberman, Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems, SIAM Journal on Applied Mathematics, vol.37, issue.1, pp.69-106, 1979.
DOI : 10.1137/0137006

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, vol.23, pp.7525-7542, 2003.

P. Heyward, M. Ennis, A. Keller, and M. T. Shipley, Membrane bistability in olfactory bulb mitral cel ls, J. Neurosci, vol.21, pp.5311-5320, 2001.

A. L. Hodgkin, The local electric changes associated with repetitive action in a non-medullated axon, The Journal of Physiology, vol.107, issue.2, pp.165-181, 1948.
DOI : 10.1113/jphysiol.1948.sp004260

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, vol.12, pp.3804-3817, 1992.

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, 2007.

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, vol.258, issue.5082, pp.258-665, 1992.
DOI : 10.1126/science.1329209

C. K. Jones, Geometric singular perturbation theory, Dynamical Systems, Lecture Notes in Math. 1609, pp.44-120, 1995.
DOI : 10.1007/978-1-4612-4312-0

M. Krupa and P. Szmolyan, Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points---Fold and Canard Points in Two Dimensions, SIAM Journal on Mathematical Analysis, vol.33, issue.2, pp.286-314, 2001.
DOI : 10.1137/S0036141099360919

M. Krupa and P. Szmolyan, Extending slow manifolds near transcritical and pitchfork singularities, Nonlinearity, vol.14, issue.6, pp.14-1473, 2001.
DOI : 10.1088/0951-7715/14/6/304

M. Krupa and P. Szmolyan, Relaxation Oscillation and Canard Explosion, Journal of Differential Equations, vol.174, issue.2, pp.312-368, 2001.
DOI : 10.1006/jdeq.2000.3929

URL : http://doi.org/10.1006/jdeq.2000.3929

N. R. Lebovitz and R. J. Schaar, Exchange of Stabilities in Autonomous Systems, Studies in Applied Mathematics, vol.21, issue.31, pp.229-260, 1975.
DOI : 10.1002/sapm1975543229

W. Liu, Exchange lemmas for singular perturbation problems with certain turning points,J .D i ff e r e n t i a l Equations, pp.134-180, 2000.

D. A. Mccormick and T. Bal, SLEEP AND AROUSAL: Thalamocortical Mechanisms, Annual Review of Neuroscience, vol.20, issue.1, pp.185-215, 1997.
DOI : 10.1146/annurev.neuro.20.1.185

D. A. Mccormick and J. R. Huguenard, A model of the electrophysiological properties of thalamocortical relay neurons, J. Neurophysiol, vol.68, pp.1384-1400, 1992.

C. Messina and R. Cotrufo, Different excitability of type 1 and type 2 alpha-motoneurones, Journal of the Neurological Sciences, vol.28, issue.1, pp.8-57, 1976.
DOI : 10.1016/0022-510X(76)90047-2

C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophysical Journal, vol.35, issue.1, pp.193-213, 1981.
DOI : 10.1016/S0006-3495(81)84782-0

R. E. Plant and M. Kim, Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations, Biophysical Journal, vol.16, issue.3, pp.227-244, 1976.
DOI : 10.1016/S0006-3495(76)85683-4

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, vol.21, pp.605-626, 1957.

J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, Methods in Neuronal Modeling, pp.135-169, 1989.

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), pp.1966-1967, 2002.

R. Seydel, Practical Bifurcation and Stability Analysis, Interdisciplinary Applied Mathematics, vol.5, 2010.
DOI : 10.1007/978-1-4419-1740-9

T. Tateno, A. Harsch, and H. P. Robinson, Threshold Firing Frequency-Current Relationships of Neurons in Rat Somatosensory Cortex: Type 1 and Type 2 Dynamics, Journal of Neurophysiology, vol.92, issue.4, pp.2283-2294, 2004.
DOI : 10.1152/jn.00109.2004

R. D. Traub, R. K. Wong, R. Miles, and H. Michelson, Am o d e lo faC A 3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances, J. Neurophysiol, vol.66, pp.635-650, 1991.

C. Wilson, Up and down states, Scholarpedia, vol.3, issue.6, p.1410, 2008.
DOI : 10.4249/scholarpedia.1410

C. J. Wilson and P. M. Groves, Spontaneous firing patterns of identified spiny neurons in the rat neostriatum, Brain Research, vol.220, issue.1, pp.67-80, 1981.
DOI : 10.1016/0006-8993(81)90211-0

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, vol.81, pp.2360-2373, 1999.