The what and where of adding channel noise to the Hodgkin-Huxley equations

PLoS Computational Biology

Volumn 7, Issue 11, 2011, Pages

  Goldwyn, Joshua H ^a   Shea Brown, Eric ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENTIAL EQUATIONS; ELECTROPHYSIOLOGY; INTEGRAL EQUATIONS; KINETIC ENERGY; KINETIC THEORY; KINETICS; NONLINEAR EQUATIONS; PHYSICAL CHEMISTRY; STOCHASTIC SYSTEMS;

ACTION-POTENTIALS; CELL VOLTAGES; CHANNEL NOISE; COMPUTATIONAL BIOLOGY; ELECTRICALLY ACTIVES; HODGKIN-HUXLEY EQUATIONS; IONIC CURRENT; KINETIC EQUATIONS; MATHEMATICAL FRAMEWORKS; NON-LINEAR IMPACT;

MATLAB;

ION CHANNEL; POTASSIUM CHANNEL; SODIUM CHANNEL;

ACTION POTENTIAL; CALCIUM TRANSPORT; CELL KINETICS; CHANNEL GATING; COMPUTER SIMULATION; ELECTRIC CAPACITANCE; ELECTROPHYSIOLOGY; GIANT AXON; HODGKIN HUXLEY EQUATION; ION CONDUCTANCE; MATHEMATICAL COMPUTING; MEMBRANE POTENTIAL; MOLECULAR DYNAMICS; NOISE; NONHUMAN; PROBABILITY; PROTEIN FUNCTION; REGULATORY MECHANISM; REVIEW; SIMULATION; STOCHASTIC MODEL; VOLTAGE CLAMP; ANIMAL; BIOLOGICAL MODEL; CEPHALOPOD; ELECTRIC CONDUCTIVITY; NERVE FIBER; PHYSIOLOGY; STATISTICS;

ACTION POTENTIALS; ANIMALS; AXONS; DECAPODIFORMES; ELECTRIC CONDUCTIVITY; MARKOV CHAINS; MODELS, NEUROLOGICAL; POTASSIUM CHANNELS; SODIUM CHANNELS; STOCHASTIC PROCESSES;

EID: 81355164214     PISSN: 1553734X     EISSN: 15537358     Source Type: Journal    
DOI: 10.1371/journal.pcbi.1002247     Document Type: Review

References (66)

1. Faisal AA, Selen LPJ, Wolpert DM, (2008) Noise in the nervous system. Nat Neurosci 9: 292-303.
2. Laing C, Lord GJ, eds. (2010) Stochastic methods in neuroscience New York Oxford University Press.
3. Rolls ET, Deco G, (2010) The noisy brain: stochastic dynamics as a principle of brain function New York Oxford University Press.
4. Sakmann B, Neher E, (1995) Single-channel recording New York Plenum Press.
5. White JA, Rubinstein JT, Kay AR, (2000) Channel noise in neurons. Trends Neurosci 23: 131-137.
6. Hille B, (2001) Ion channels of excitable membranes. 3rd edition Sunderland (MA) Sinauer Associates.
7. Imennov NS, Rubinstein JT, (2009) Stochastic population model for electrical stimulation of the auditory nerve. IEEE Trans Biomed Eng 10: 2493-2501.
8. Woo J, Miller CA, Abbas PJ, (2010) The dependence of auditory nerve rate adaptation on electric stimulus parameters, electrode position, and fiber diameter: a computer model study. J Assoc Res Otolaryngol 11: 283-296.
9. White JA, Klink R, Alonso A, Kay AR, (1998) Noise from voltage-gated ion channels may influence neuronal dynamics in the entorhinal cortex. J Neurophysiol 80: 262-269.
10. Saarinen A, Linne ML, Yli-Harja O, (2008) Stochastic differential equation model for cerebellar granule cell excitability. PLoS Compu Biol 4: e1000004 doi:10.1371/journal.pcbi.1000004.
11. Cannon RC, O'Donnell C, Nolan MF, (2010) Stochastic ion channel gating in dendritic neurons: morphology dependence and probabilistic synaptic activation of dendritic spikes. PLoS Comput Biol 6: e1000886 doi:10.1371/journal.pcbi.1000886.
12. 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.
13. 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: 1679-1703.
14. Schmid G, Goychuk I, Hänggi P, (2001) Stochastic resonance as a collective property of ion channel assemblies. Europhys Lett 56: 22.
15. Rowat P, (2007) Interspike interval statistics in the stochastic Hodgkin-Huxley model: Coexistence of gamma frequency bursts and highly irregular firing. Neural Comput 19: 1215-1250.
16. Faisal AA, Laughlin SB, (2007) Stochastic simulations on the reliability of action potential propagation in thin axons. PLoS Comput Biol 3: e79 doi:10.1371/journal.pcbi.0030079.
17. Finke C, Vollmer J, Postnova S, Braun HA, (2008) Propagation effects of current and conductance noise in a model neuron with subthreshold oscillations. Math Biosci 214: 109-121.
18. Keleshian AM, Edeson RO, Liu GJ, Madsen BW, (2000) Evidence for cooperativity between nicotinic acetylcholine receptors in patch clamp records. Biophys J 78: 1-12.
19. Shuai JW, Jung P, (2002) Optimal intracellular calcium signalling. Phys Rev Lett 88: 068102-1-068102-4.
20. Hodgkin AL, Huxley AF, (1952) A quantitative description of membrane current, its application to conduction, excitation in nerve. J Physiol 117: 500-544.
21. Groff JR, DeRemigio H, Smith GD, (2010) Markov chain models of ion channels and calcium release sites. In: Laing C, Lord G, editors. Stochastic methods in neuroscience New York Oxford University Press pp. 29-64.
22. Fox RF, Lu YN, (1994) Emergent collective behavior in large numbers of globally coupled independent stochastic ion channels. Phys Rev E Stat Nonlin Soft Matter Phys 49: 3421-3431.
23. Dayan P, Abbott LF, (2001) Theoretical neuroscience: computational and mathematical modeling of neural systems. Computational Neuroscience London MIT Press.
24. Austin TD, (2008) The emergence of the deterministic Hodgkin-Huxley equations as a limit theorem from the underlying stochastic ion-channel mechanism. Ann Appl Probab 18: 1279-1235.
25. Keener JP, (2009) Invariant manifold reductions for Markovian ion channel dynamics. J Math Biol 58: 447-457.
26. Pakdaman K, Thieullen M, Wainrib G, (2010) Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv Appl Probab 42: 761-794.
27. 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.
28. Gillespie DT, (1977) Exact stochastic simulation of coupled chemical-reactions. J Phys Chem 81: 2340-2361.
29. Skaugen E, Walløe L, (1979) Firing behaviour in a stochastic nerve model based upon the Hodgkin-Huxley equations. Acta Physiol Scand 107: 343-363.
30. Chow CC, White JA, (1996) Spontaneous action potentials due to channel fluctuations. Biophys J 71: 3013-3021.
31. Izhikevich EM, (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. Computational neuroscience London MIT Press.
32. Rinzel J, Ermentrout B, (1998) Analysis of neural excitability and oscillations. In: Koch C, Segev I, editors. Methods in neural modeling. 2nd edition Cambridge (MA) MIT Press pp. 251-292.
33. Fox RF, (1997) Stochastic versions of the Hodgkin-Huxley equations. Biophys J 72: 2068-2074.
34. Mino H, Rubinstein JT, White JA, (2002) Comparison of algorithms for the simulation of action potentials with stochastic sodium channels. Ann Biomed Eng 30: 578-587.
35. Zeng S, Jung P, (2004) Mechanism for neuronal spike generation by small and large ion channel clusters. Phys Rev E Stat Nonlin Soft Matter Phys 70: 011903.
36. Bruce IC, (2007) Implementation issues in approximate methods for stochastic Hodgkin-Huxley models. Ann Biomed Eng 35: 315-318.
37. Bruce IC, (2009) Evaluation of stochastic differential equation approximation of ion channel gating models. Ann Biomed Eng 37: 824-838.
38. Linaro D, Storace M, Giugliano M, (2011) Accurate and fast simulation of channel noise in conductance-based model neurons by diffusion approximation. PLoS Comput Biol 7: e1001102 doi:10.1371/journal.pcbi.1001102.
39. Orio P, (2011) Diffusion approximation algorithm for stochastic ion channel simulations with multiple states [abstract]. Computational and Systems Neuroscience 2011 meeting (Cosyne 2011); 24-27 February 2011; Salt Lake City, Utah, United States.
40. Vandenberg CA, Bezanilla F, (1991) A sodium channel gating model based on single channel, macroscopic ionic, and gating currents in the squid giant axon. Biophys J 60: 1511-1533.
41. Clay JR, (1998) Excitability of the squid giant axon revisited. J Neurophysiol 80: 903-913.
42. Li YX, Rinzel J, (1994) Equations for InsP3 receptor-mediated [Ca2+] oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. J Theor Biol 166: 461-473.
43. Gerstein GL, Mandelbrot B, (1964) Random walk models for the spike activity of a single neuron. Biophys J 81: 41-68.
44. Tuckwell HC, (1988) Introduction to theoretical neurobiology: nonlinear and stochastic theories. Volume 2 New York Cambridge University Press.
45. Brunel N, Chance FS, Fourcaud N, Abbott LF, (2001) Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys Rev Lett 86: 2186-2189.
46. Johnston D, Wu SM, (1995) Foundations of cellular neurophysiology Cambridge (MA) MIT Press.
47. Vries GD, Sherman A, (2000) Channel sharing in pancreatic β-cells revisited: Enhancement of emergent bursting by noise. J Theor Biol 207: 513-530.
48. Casado JM, (2003) Synchronization of two Hodgkin-Huxley neurons due to internal noise. Phys Lett A 310: 400-406.
49. Rowat P, Elson R, (2004) State-dependent effects of Na channel noise on neuronal burst generation. J Comput Neurosci 16: 87-112.
50. Wang M, Hou Z, Xin H, (2004) Double-system-size resonance for spiking activity of coupled Hodgkin-Huxley neurons. Chemphyschem 5: 1602-1605.
51. Jo J, Kang H, Choo MY, Koh DS, (2005) How noise and coupling induce bursting action potentials in Pancreatic β-cells. Biophys J 89: 1534-1542.
52. Ozer M, Ekmecki N, (2005) Effect of channel noise on the time-course of recovery from inactivation of sodium channels. Phys Lett A 338: 150-154.
53. Cudmore RH, Fronzaroli-Molinieres L, Giraud P, Debanne D, (2010) Spike-time precision and network synchrony are controlled by the homeostatic regulation of the D-type Potassium current. J Neurosci 30: 12885-12895.
54. Sato D, Xie LH, Nguyen TP, Weiss JN, Qu Z, (2010) Irregularly appearing early afterdepolarizations in cardiac myocytes: Random fluctuations or dynamical chaos? Biophys J 99: 765-773.
55. Faisal AA, (2010) Stochastic simulations of neurons, axons, and action potentials. In: Laing C, Lord G, editors. Stochastic methods in neuroscience New York Oxford University Press pp. 297-343.
56. Hida T, Hitsuda M, (1993) Gaussian processes. Translations of mathematical monographs Providence American Mathematical Society.
57. Gardiner CW, (2004) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer series in synergetics. 3rd edition New York Springer.
58. Higham DJ, (2001) An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev Soc Ind Appl Math 43: 525-546.
59. Woloshyn R, (1999) Mersenne Twister implemented in Fortran. Available: http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/VERSIONS/FORTRAN/mtfort90.f. Accessed 3 August 2010.
60. Press WH, Flannery BP, Teukolsky SA, Vetterling WT, (1988) Numerical Recipes: The Art of Scientific Computing New York Cambridge University Press.
61. Hines ML, Morse T, Migliore M, Carnevale NT, Shepherd GM, (2004) ModelDB: a database to support computational neuroscience. J Comput Neurosci 17: 7-11.
62. Clay JR, (2005) Axonal excitability revisited. Prog Biophys Mol Biol 88: 59-90.
63. Jones SW, (2006) Are rate constants constant? J Physiol 571: 502.
64. Liebovitch LS, Scheurle D, Rusek M, Zochowski M, (2001) Fractal methods to analyze ion channel kinetics. Methods 24: 359-375.
65. Freidlin M, Wentzell A, (1998) Random perturbations of dynamical systems New York Springer.
66. Alzubaidi H, Gilsing H, Shardlow T, (2010) Numerical simulations of SDEs and SPDEs from neural systems using SDELab. In: Laing C, Lord G, editors. Stochastic methods in neuroscience New York Oxford University Press pp. 344-366.