References and models cited by this paper

References and models that cite this paper

Austin TD (2008) The emergence of the deterministic Hodgkin–Huxley equations as a limit from the underlying stochastic ion-channel mechanism Ann Appl Probab 18:1279-1325

Bayer C, Szepessy A, Tempone R (2010) Adaptive weak approximation of reflected and stopped diffusions Monte Carlo Methods And Applications 16:1-67

Bezanilla F (2000) The voltage sensor in voltage-dependent ion channels. Physiol Rev 80:555-92 [Journal] [PubMed]

Bhattacharya RN, Waymire EC (2009) Stochastic Processes with Applications

Bossy M, Gobet E, Talay D (2004) A symmetrized Euler scheme for an efficient approximation of reflected diffusions J Appl Probab 41(3):877-889

Brillinger DR (2003) Simulating constrained animal motion using stochastic differential equations Lecture Notes in Statistics-monograph Series 41:35-48

Bruce IC (2007) Implementation issues in approximate methods for stochastic Hodgkin-Huxley models. Ann Biomed Eng 35:315-8; author reply 319 [Journal] [PubMed]

Bruce IC (2009) Evaluation of stochastic differential equation approximation of ion channel gating models. Ann Biomed Eng 37:824-38 [Journal] [PubMed]

Chen Y, Ye X (2011) Projection Onto A Simplex http:--arxiv.org-abs-1101.6081

Chow CC, White JA (1996) Spontaneous action potentials due to channel fluctuations. Biophys J 71:3013-21 [Journal] [PubMed]

Clay JR, DeFelice LJ (1983) Relationship between membrane excitability and single channel open-close kinetics. Biophys J 42:151-7 [Journal] [PubMed]

Combettes PL, Wajs VR (2005) Signal recovery by proximal forward-backward splitting Multiscale Modeling Simulation 4(4):1168-1200

Cox JC, Ingersoll JE, Ross SA (1985) A Theory of the Term Structure of Interest Rates Econometrica 53:385-407

Dangerfield CE, Kay D, Burrage K (2010) Stochastic models and simulation of ion channel dynamics Procedia Computer Science 1(1):1581-1590

Dangerfield CE, Kay D, Macnamara S, Burrage K (2012) A boundary preserving numerical algorithm for the Wright--Fisher model with mutation 52(2):283-304

De Vries G, Sherman A (2000) Channel sharing in pancreatic beta -cells revisited: enhancement of emergent bursting by noise. J Theor Biol 207:513-30 [Journal] [PubMed]

Ding D, Zhang YY (2008) A splitting-step algorithm for reflected stochastic differential equations in R^1_+ Comput Math Appl 55(11):2413-2425

Feller W (1951) Two singular diffusion problems The Annals Of Mathematics 54(1):173-182

Fox RF (1997) Stochastic versions of the Hodgkin-Huxley equations. Biophys J 72:2068-74 [Journal] [PubMed]

Fox RF, Lu Yn (1994) Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 49:3421-3431 [PubMed]

Gihman II, Skorokhod AV (1972) Stochastic Differential Equations

Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. Journal Of Physical Chemistry 81:2340-2361

Gillespie DT (1992) A rigorous derivation of the chemical master equation Physica A 188:404-425

Gillespie DT (2000) The chemical Langevin equation J Chem Phys 113:297-306

Gillespie DT (2002) The Chemical Langevin and Fokker−Planck Equations for the Reversible Isomerization Reaction J Phys Chem A 106(20):5063-5071

Gillespie DT (2007) Stochastic simulation of chemical kinetics. Annu Rev Phys Chem 58:35-55 [Journal] [PubMed]

Goldwyn JH, Imennov NS, Famulare M, Shea-Brown E (2011) Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons. Phys Rev E Stat Nonlin Soft Matter Phys 83:041908 [Journal] [PubMed]

Goldwyn JH, Shea-Brown E (2011) The what and where of adding channel noise to the Hodgkin-Huxley equations. PLoS Comput Biol 7:e1002247 [Journal] [PubMed]

Groff J, Deremigio H, Smith G (2010) Stochastic Methods in Neuroscience, Laing C:Lord G, ed. pp.29

Higham D (2011) Stochastic ordinary differential equations in applied and computational mathematics Ima J Appl Math 76(3):449-474

Hille B (1991) Ionic Channels of Excitable Membranes, 2nd Ed.

HODGKIN AL, HUXLEY AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500-44 [Journal] [PubMed]

Jahnke T, Huisinga W (2007) Solving the chemical master equation for monomolecular reaction systems analytically. J Math Biol 54:1-26 [Journal] [PubMed]

Kawamura T, Saisho Y (2006) Stochastic models describing human metabolism processes using stochastic differential equations Stochastic Models 22:273-287

Kloeden PE, Platen E (2011) Numerical Solution of Stochastic Differential Equations

Lemay M, de Lange E, Kucera JP (2011) Effects of stochastic channel gating and distribution on the cardiac action potential. J Theor Biol 281:84-96 [Journal] [PubMed]

Lepingle D (1995) Euler scheme for reflected stochastic differential equations Mathematics And Computers In Simulation 38:119-126

Lions PL, Sznitman AS (1984) Stochastic differential equations with reflecting boundary conditions Commun Pure Appl Math 37(4):511-537

Liu Y (1995) Discretization of a class of reflected diffusion processes Mathematics And Computers In Simulation 38(1-3):103-108

Lord R, Koekkoek R, Dijk DV (2010) A comparison of biased simulation schemes for stochastic volatility models Quantitative Finance 10(2):177-194

Macnamara S, Burrage K (2009) Krylov and steady-state techniques for the solution of the chemical master equation for the mitogen-activated protein kinase cascade Numerical Algorithms 51(3):281-307

Martins ML, Ferreira SC, Vilela MJ (2010) Multiscale models for biological systems Current Opinion In Colloid Interface Science 15(1):18-23

Mélykúti B, Burrage K, Zygalakis KC (2010) Fast stochastic simulation of biochemical reaction systems by alternative formulations of the chemical Langevin equation. J Chem Phys 132:164109 [Journal] [PubMed]

Mino H, Rubinstein JT, White JA (2002) Comparison of algorithms for the simulation of action potentials with stochastic sodium channels. Ann Biomed Eng 30:578-87 [PubMed]

Oosterhoff P, Oros A, Vos MA (2007) Beat-to-beat variability of repolarization: a new parameter to determine arrhythmic risk of an individual or identify proarrhythmic drugs. Anadolu Kardiyol Derg 7 Suppl 1:73-8 [PubMed]

Pakdaman K, Thieullen M, Wainrib G (2010) Fluid limit theorems for stochastic hybrid systems with application to neuron models Adv Appl Probab 42(3):761-794

Pettersson R (1995) Approximations for stochastic differential equations with reflecting convex boundaries Stochastic Processes And Their Applications 59(2):295-308

Pettersson R (1997) Penalization schemes for reflecting stochastic differential equations Bernoulli 3(4):403-414

Pueyo E, Corrias A, Virág L, Jost N, Szél T, Varró A, Szentandrássy N, Nánási PP, Burrage K, Rodríguez B (2011) A multiscale investigation of repolarization variability and its role in cardiac arrhythmogenesis. Biophys J 101:2892-902 [Journal] [PubMed]

Riley D, Koutsoukos X, Riley K (2008) Simulation of stochastic hybrid systems with switching and reflective boundaries Winter Simulation Conference :804-812

Rubinstein JT (1995) Threshold fluctuations in an N sodium channel model of the node of Ranvier. Biophys J 68:779-85 [Journal] [PubMed]

Rudy Y, Silva JR (2006) Computational biology in the study of cardiac ion channels and cell electrophysiology. Q Rev Biophys 39:57-116 [Journal] [PubMed]

Sakmann B, Neher E (1995) Single-channel Recording 2:487

Schneidman E, Freedman B, Segev I (1998) Ion channel stochasticity may be critical in determining the reliability and precision of spike timing. Neural Comput 10:1679-703 [PubMed]

Sengupta B, Laughlin SB, Niven JE (2010) Comparison of Langevin and Markov channel noise models for neuronal signal generation. Phys Rev E Stat Nonlin Soft Matter Phys 81:011918 [Journal] [PubMed]

Skorokhod AV (1961) Stochastic Equations for Diffusion Processes in a Bounded Region Theory Probab Appl 6-3:264-274

Skorokhod AV (1962) Stochastic Equations for Diffusion Processes in a Bounded Region. II Theory Probab Appl 7(1):3-23

Slominski L (1994) On approximation of solutions of multidimensional SDE's with reflecting boundary conditions Stochastic Processes And Their Applications 50(2):197-219

Sun XJ, Lei JZ, Perc M, Lu QS, Lv SJ (2011) Effects of channel noise on firing coherence of small-world Hodgkin-Huxley neuronal networks Eur Phys J B 79(1):61-66

Tanaka H (1979) Stochastic differential equations with reflecting boundary conditions in convex regions Hiroshima Math J 9:163-177

Van_kampen NG (2007) Stochastic Processes in Physics and Chemistry

Verveen AA, Derksen HE (1968) Fluctuation phenomena in nerve membrane Proc IEEE 56:906-916

Wantanebe S (1971) On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions J Math Kyoto Univ 11(1):169-180

White JA, Rubinstein JT, Kay AR (2000) Channel noise in neurons. Trends Neurosci 23:131-7 [PubMed]

Schmerl BA, McDonnell MD (2013) Channel noise induced stochastic facilitation in an auditory brainstem neuron model Physical Review E 88:052722 [Journal] [PubMed]