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Dangerfield CE, Kay D, Burrage K (2012) Modeling ion channel dynamics through reflected stochastic differential equations Phys Rev E 85(5):051907

   Reflected SDE Hodgkin-Huxley Model (Dangerfield et al. 2012)

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]

   Implementation issues in approximate methods for stochastic Hodgkin-Huxley models (Bruce 2007) [Model]

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

   Evaluation of stochastic diff. eq. approximation of ion channel gating models (Bruce 2009) [Model]

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]

   Spontaneous firing caused by stochastic channel gating (Chow, White 1996) [Model]

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]

   On stochastic diff. eq. models for ion channel noise in Hodgkin-Huxley neurons (Goldwyn et al. 2010) [Model]

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]

   Stochastic versions of the Hodgkin-Huxley equations (Goldwyn, Shea-Brown 2011) (pylab) [Model]
   Stochastic versions of the Hodgkin-Huxley equations (Goldwyn, Shea-Brown 2011) [Model]

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]

   Squid axon (Hodgkin, Huxley 1952) (LabAXON) [Model]
   Squid axon (Hodgkin, Huxley 1952) (NEURON) [Model]
   Squid axon (Hodgkin, Huxley 1952) (SNNAP) [Model]
   Squid axon (Hodgkin, Huxley 1952) used in (Chen et al 2010) (R language) [Model]
   Squid axon (Hodgkin, Huxley 1952) (SBML, XPP, other) [Model]

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]

   Simulating ion channel noise in an auditory brainstem neuron model (Schmerl & McDonnell 2013) [Model]

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