D. Anderson, G. B. Ermentrout, and P. Thomas, Stochastic representations of ion channel kinetics and exact stochastic simulation of neuronal dynamics, Journal of Computational Neuroscience, vol.23, issue.7, pp.67-82, 2015.
DOI : 10.1016/S0166-2236(99)01521-0

M. Assaf and B. Meerson, Spectral Theory of Metastability and Extinction in Birth-Death Systems, Physical Review Letters, vol.37, issue.20, 2006.
DOI : 10.1119/1.1972842

P. C. Bressloff and J. Newby, Metastability in a Stochastic Neural Network Modeled as a Velocity Jump Markov Process, SIAM Journal on Applied Dynamical Systems, vol.12, issue.3, pp.1394-1435, 2013.
DOI : 10.1137/120898978

P. C. Bressloff and J. Newby, Path integrals and large deviations in stochastic hybrid systems, Physical Review E, vol.16, issue.4, p.42701, 2014.
DOI : 10.1002/cpa.20238

P. C. Bressloff and J. Newby, Stochastic hybrid model of spontaneous dendritic NMDA spikes, Physical Biology, vol.11, issue.1, p.16006, 2014.
DOI : 10.1088/1478-3975/11/1/016006

P. Bressloff, Path-Integral Methods for Analyzing the Effects of Fluctuations in Stochastic Hybrid Neural Networks, The Journal of Mathematical Neuroscience, vol.37, issue.944, 2015.
DOI : 10.1103/PhysRevA.37.4865

H. A. Brooks and P. Bressloff, Quasicycles in the stochastic hybrid Morris-Lecar neural model, Physical Review E, vol.80, issue.1, p.12704, 2015.
DOI : 10.3389/fncom.2014.00139

E. Buckwar and M. Riedler, An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution, Journal of Mathematical Biology, vol.2, issue.3, pp.1051-1093, 2011.
DOI : 10.1049/iet-syb:20070045

C. Chow and J. White, Spontaneous action potentials due to channel fluctuations, Biophysical Journal, vol.71, issue.6, pp.3013-3021, 1996.
DOI : 10.1016/S0006-3495(96)79494-8
URL : http://doi.org/10.1016/s0006-3495(96)79494-8

M. Davis, Piecewise-deterministic Markov processes, Journal of the Royal Society, Series B (Methodological), vol.46, pp.353-388, 1984.
DOI : 10.1007/978-1-4899-4483-2_2

M. I. Dykman, E. Mori, R. J. Hunt, and P. , Large fluctuations and optimal paths in chemical kinetics, The Journal of Chemical Physics, vol.104, issue.8, pp.5735-5750, 1994.
DOI : 10.1063/1.462446

A. Faggionato, D. Gabrielli, and M. Crivellari, Non-equilibrium Thermodynamics of Piecewise Deterministic Markov Processes, Journal of Statistical Physics, vol.45, issue.1/2, pp.259-304, 2009.
DOI : 10.1007/978-3-662-03752-2

A. Faggionato, D. Gabrielli, and M. Crivellari, Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes and applications to molecular motors, Markov Processes and Related Fields, pp.497-548, 2010.

J. Feng and T. Kurtz, Large Deviations for Stochastic Processes, 2006.
DOI : 10.1090/surv/131
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.7544

R. F. Fox and Y. Lu, Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels, Physical Review E, vol.66, issue.4, pp.3421-3431, 1994.
DOI : 10.1002/jcp.1030660518

M. I. Freidlin and A. Wentzell, Random perturbations of dynamical systems, 1998.

C. Gardiner, Handbook of stochastic methods, 2009.

J. H. Goldwyn and E. Shea-brown, The What and Where of Adding Channel Noise to the Hodgkin-Huxley Equations, PLoS Computational Biology, vol.24, issue.11, p.1002247, 2011.
DOI : 10.1371/journal.pcbi.1002247.t001

G. R. Grimmett and D. Stirzaker, Probability and random processes 3rd ed, 2001.

P. Hanggi, H. Grabert, P. Talkner, and H. Thomas, Bistable systems: Master equation versus Fokker-Planck modeling, Physical Review A, vol.27, issue.1, pp.371-378, 1984.
DOI : 10.1007/BF01320526

J. P. Keener and J. Newby, Perturbation analysis of spontaneous action potential initiation by stochastic ion channels, Physical Review E, vol.94, issue.1, p.11918, 2011.
DOI : 10.1007/s10439-009-9635-z

T. B. Kepler and T. Elston, Stochasticity in Transcriptional Regulation: Origins, Consequences, and Mathematical Representations, Biophysical Journal, vol.81, issue.6, pp.3116-3136, 2001.
DOI : 10.1016/S0006-3495(01)75949-8
URL : http://doi.org/10.1016/s0006-3495(01)75949-8

Y. Kifer, Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging, Memoirs of the American Mathematical Society, vol.201, issue.944, 2009.
DOI : 10.1090/memo/0944

I. A. Khovanov, A. V. Polovinkin, D. G. Luchinsky, and P. Mcclintock, Noise-induced escape in an excitable system, Physical Review E, vol.50, issue.3, p.32116, 2013.
DOI : 10.1016/j.spa.2005.11.006
URL : http://eprints.lancs.ac.uk/63261/1/PRE2013NoiseInducedExcapeInExcitable.pdf

C. Knessl, B. J. Matkowsky, Z. Schuss, and C. Tier, An Asymptotic Theory of Large Deviations for Markov Jump Processes, SIAM Journal on Applied Mathematics, vol.45, issue.6, pp.1006-1028, 1985.
DOI : 10.1137/0145062

B. J. Matkowsky and Z. Schuss, The Exit Problem for Randomly Perturbed Dynamical Systems, SIAM Journal on Applied Mathematics, vol.33, issue.2, pp.365-382, 1977.
DOI : 10.1137/0133024

T. Naeh, M. M. Klosek, B. J. Matkowsky, and Z. Schuss, A Direct Approach to the Exit Problem, SIAM Journal on Applied Mathematics, vol.50, issue.2, pp.595-627, 1990.
DOI : 10.1137/0150036

C. Morris, H. Lecar, and H. , 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

J. Newby, Isolating intrinsic noise sources in a stochastic genetic switch, Physical Biology, vol.9, issue.2, p.26002, 2012.
DOI : 10.1088/1478-3975/9/2/026002

J. Newby and P. Bressloff, Local synaptic signaling enhances the stochastic transport of motor-driven cargo in neurons, Physical Biology, vol.7, issue.3, p.36004, 2010.
DOI : 10.1088/1478-3975/7/3/036004

J. Newby and J. Keener, An Asymptotic Analysis of the Spatially Inhomogeneous Velocity-Jump Process, Multiscale Modeling & Simulation, vol.9, issue.2, pp.735-765, 2011.
DOI : 10.1137/10080676X

J. M. Newby, P. Bressloff, and J. Keener, Breakdown of Fast-Slow Analysis in an Excitable System with Channel Noise, Physical Review Letters, vol.111, issue.12, p.128101, 2013.
DOI : 10.1103/PhysRevA.40.4065

J. M. Newby and S. Chapman, Metastable behavior in Markov processes with internal states, Journal of Mathematical Biology, vol.102, issue.52, pp.941-976, 2013.
DOI : 10.1073/pnas.0509547102
URL : http://arxiv.org/pdf/1304.6957

J. Newby, Spontaneous Excitability in the Morris--Lecar Model with Ion Channel Noise, SIAM Journal on Applied Dynamical Systems, vol.13, issue.4, pp.1756-1791, 2014.
DOI : 10.1137/140971385

J. Newby, Bistable switching asymptotics for the self regulating gene, Journal of Physics A: Mathematical and Theoretical, vol.48, issue.18, p.185001, 2015.
DOI : 10.1088/1751-8113/48/18/185001
URL : http://arxiv.org/pdf/1407.4344

K. Pakdaman, M. Thieullen, and G. Wainrib, Fluid limit theorems for stochastic hybrid systems with application to neuron models, Advances in Applied Probability, vol.46, issue.03, pp.761-794, 2010.
DOI : 10.1073/pnas.0236032100
URL : https://hal.archives-ouvertes.fr/hal-00447808

R. S. Maier and D. Stein, Limiting Exit Location Distributions in the Stochastic Exit Problem, SIAM Journal on Applied Mathematics, vol.57, issue.3, pp.752-790, 1997.
DOI : 10.1137/S0036139994271753

H. Touchette, The large deviation approach to statistical mechanics, Physics Reports, vol.478, issue.1-3, pp.1-69, 2009.
DOI : 10.1016/j.physrep.2009.05.002
URL : http://arxiv.org/pdf/0804.0327

M. Vellela and H. Qian, Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlogl model revisited, Journal of The Royal Society Interface, vol.4, issue.5, pp.925-940, 2009.
DOI : 10.1063/1.459735

G. Wainrib, M. Thieullen, and K. Pakdaman, Reduction of stochastic conductance-based neuron models with time-scales separation, Journal of Computational Neuroscience, vol.23, issue.3, pp.327-373, 2012.
DOI : 10.1007/978-1-4612-0627-9
URL : https://hal.archives-ouvertes.fr/hal-00640029