Reduction of stochastic conductance-based neuron models with time-scales separation

Journal of Computational Neuroscience

Volumn 32, Issue 2, 2012, Pages 327-346

  Wainrib, Gilles ^a   Thieullen, Michèle ^b   Pakdaman, Khashayar ^c  

^a STANFORD UNIVERSITY   (United States)

^b SORBONNE UNIVERSITÉ   (France)

^c INSTITUT JACQUES MONOD   (France)

Author keywords

Averaging method; Hodgkin Huxley; Stochastic ion channels; Time scales separation

Indexed keywords

HOPF BIFURCATION; NEURONS; PERTURBATION TECHNIQUES; RELIABILITY ANALYSIS; SEPARATION; STOCHASTIC MODELS;

% REDUCTIONS; AVERAGING METHOD; DISCHARGE TIME; HODGKIN-HUXLEY; HODGKIN-HUXLEY MODELS; NEURON MODELING; REDUCED MODEL; STOCHASTIC ION CHANNELS; STOCHASTICS; TIME SCALE SEPARATION;

STOCHASTIC SYSTEMS;

POTASSIUM CHANNEL; SODIUM CHANNEL;

BIOLOGICAL MODEL; CHANNEL GATING; CHEMICAL REACTION KINETICS; HODGKIN HUXLEY EQUATION; NERVE CELL MEMBRANE CONDUCTANCE; NERVE CELL MEMBRANE POTENTIAL; NOISE; PRIORITY JOURNAL; PROBABILITY; RELIABILITY; REVIEW; SPIKE; STATISTICAL ANALYSIS; STATISTICAL CONCEPTS; STATISTICAL MODEL; TIME SCALES SEPARATION;

EID: 84861189013     PISSN: 09295313     EISSN: 15736873     Source Type: Journal    
DOI: 10.1007/s10827-011-0355-7     Document Type: Review

References (35)

1. Arnold, V. I. (1983). Geometric methods for ordinary differential equations. New York: Springer.
2. Cecchi, G. A., Sigman, M., Alonso, J. M., Martinez, L., Chialvo, D. R., & Magnasco, M. O. (2000). Noise in neurons is message dependent. Proceedings of the National Academy of Sciences, 97(10), 5557-5561. (Pubitemid 30313757)
3. Chow, C. C., & White, J. A. (1996). Spontaneous action potentials due to channel fluctuations. Biophysical Journal, 71(6), 3013-3021. (Pubitemid 26422344)
4. Faggionato, A., Gabrielli, D., & Crivellari M. R. (2010). Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes and applications to molecular motors. Markov Processes and Related Fields, 16(3), 497-548.
5. Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry, 81(25), 2340-2361.
6. Hille, B. (2001). Ion channels of excitable membranes. Massachusettes: Sinauer Sunderland.
7. Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117(4), 500-544.
8. Holden, A. V., Muhamad, M. A., & Schierwagen, A. K. (1985). Repolarizing currents and periodic activity in nerve membrane. Journal of Theoretical Neurobiology, 4, 61-71. (Pubitemid 16187502)
9. Izhikevich, E.M. (2007). Dynamical systems in neuroscience: The geometry of excitability and bursting. Cambridge: MIT Press.
10. Jung, P., & Shuai, J. W. (2001). Optimal sizes of ion channel clusters. Europhysics Letters, 56, 29-35.
11. Keener, J. (2009). Invariant manifold reductions for Markovian ion channel dynamics. Journal of Mathematical Biology, 58(3), 447-57.
12. Kepler, T. B., Abbott, L. F., & Marder, E. (1992). Reduction of conductance-based neuron models. Biological Cybernetics, 66, 381-387.
13. Mastny, E. A., Haseltine, E. L., & Rawlings, J. B. (2007). Two classes of quasi-steady-state model reductions for stochastic kinetics. The Journal of Chemical Physics, 127, 094106.
14. Pakdaman, K., Tanabe, S., & Shimokawa, T. (2001). Coherence resonance and discharge time reliability in neurons and neuronal models. Neural Networks, 14(6-7), 895-90.
15. Pakdaman, K., Thieullen, M., & Wainrib, G. (2010). Fluid limit theorems for stochastic hybrid systems with application to neuron models. Advances in Applied Probability, 42(3), 761-794.
16. Rinzel, J. (1985). Excitation dynamics: Insights from simplified membrane models. Feredation Proceedings, 44, 2944-2946.
17. Rowat, P. (2007). Interspike interval statistics in the stochastic Hodgkin-Huxley model: Coexistence of gamma frequency bursts and highly irregular firing. Neural Computation, 19(5), 1215.
18. Rubin, J., & Wechselberger, M. (2007). Giant squid-hidden canard: The 3d geometry of the Hodgkin Huxley model. Biological Cybernetics, 97, 5-32. (Pubitemid 47149652)
19. Schmid, G., Goychuk, I., & Hänggi, P. (2001). Stochastic resonance as a collective property of ion channel assemblies. Europhysics Letters, 56, 22-28.
20. Schmid, G., Goychuk, I., & Hänggi, P. (2003). Channel noise and synchronization in excitable membranes. Physica A: Statistical Mechanics and its Applications, 325(1-2), 165-175.
21. Schneidmann, E., Freedman, B., & Segev, I. (1998). Ion channel stochasticity may be critical in determining the reliability and precision of spike timing. Neural Computation, 10, 1679-1703.
22. Shuai, J. W.,&Jung, P. (2003).Optimal ion channel clustering for intracellular calcium signaling. Proceedings of the National Academy of Sciences, 100(2), 506-512.
23. Shuai, J. W.,&Jung, P. (2005). Entropically enhanced excitability in small systems. Physical Review Letters, 95(11), 114501.
24. Skaugen, E., & Walloe, L. (1979). Firing behavior in a stochastic nerve membrane model based upon the Hodgkin-Huxley equations. Acta Physiologica Scandinavica, 107(4), 343-63. (Pubitemid 10202647)
25. Steinmetz, P. N., Manwani, A., Koch, C., London, M., & Segev, I. (2000). Subthreshold voltage noise due to channel fluctuations in active neuronal membranes. Journal of Computational Neuroscience, 9(16), 133-148.
26. Suckley, R., & Biktashev, V. N. (2003). Comparison of asymptotics of heart and nerve excitability. Physical Review E, 68, 011902, 1-15.
27. Takahata, T., Tanabe, S., & Pakdaman, K. (2002). White noise stimulation of the Hodgkin-Huxley model. Biological Cybernetics, 86, 403-417. (Pubitemid 135689581)
28. Tanabe, S., & Pakdaman, K. (2001). Noise-induced transition in excitable neuron models. Biological Cybernetics 85, 269-280. (Pubitemid 33605221)
29. Tanabe, S., & Pakdaman, K. (2001). Noise-enhanced neuronal reliability. Physical Review E, 64, 041904.
30. Tanabe, S., & Pakdaman, K. (2001). Dynamics of moments of Fitz-Hugh-Nagumo neuronal models and stochastic bifurcations. Physical Review E, 63, 031911.
31. Tanabe, S., Sato, S., & Pakdaman, K. (1999). Response of an ensemble of noisy neuron models to a single input. Physical Review E, 60, 7235-7238. (Pubitemid 129651377)
32. Tateno, T., & Pakdaman, K. (2004). Random dynamics of the Morris-Lecar neural model. Chaos, 14, 511-530. (Pubitemid 39447420)
33. Wainrib, G., Thieullen, M., & Pakdaman, K. (2010). Intrinsic variability of latency to first spike. Biological Cybernetics, 103(1), 43-56.
34. White, J. A., Rubinstein, J. T., & Kay, A. R. (2000). Channel noise in neurons. Trends in Neurosciences, 23(3), 131-137. (Pubitemid 30110629)
35. Yin, G. G., & Zhang, Q. (1998). Continuous-time Markov chains and applications: A singular perturbation approach. New York: Springer.