Fluid limit theorems for stochastic hybrid systems with application to neuron models

Advances in Applied Probability

Volumn 42, Issue 3, 2010, Pages 761-794

  Pakdaman, K ^a   Thieullen, M ^b   Wainrib, G ^c  

^a UNIVERSITÉ PARIS DESCARTES   (France)

^b SORBONNE UNIVERSITÉ   (France)

^c ECOLE POLYTECHNIQUE   (France)

Author keywords

Fluid limit; Hodgkin Huxley; Kinetic model; Langevin approximation; Morris Lecar; Neuron model; Piecewise deterministic Markov process; Stochastic hybrid system; Stochastic ion channels

Indexed keywords

FLUID LIMITS; HODGKIN-HUXLEY; KINETIC MODEL; LANGEVIN; MORRIS-LECAR; NEURON MODEL; PIECEWISE DETERMINISTIC MARKOV PROCESS; STOCHASTIC HYBRID SYSTEMS; STOCHASTIC ION CHANNELS;

ELECTRIC GROUNDING; FLUIDS; HYBRID COMPUTERS; HYBRID SENSORS; HYBRID SYSTEMS; IMAGE CODING; IONS; KINETIC THEORY; MARKOV PROCESSES; STOCHASTIC SYSTEMS;

STOCHASTIC MODELS;

EID: 78650962578     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1239/aap/1282924062     Document Type: Article

References (41)

1. Austin, T. D. (2008). The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism. Ann. Appl. Prob. 18, 1279-1325.
2. Blom, H. A. P. and Lygeros, J. (eds) (2006). Stochastic Hybrid Systems (Lecture Notes Control Inf. Sci. 337). Springer.
3. Chicone, C. (1999). Ordinary Differential Equations with Applications. Springer, NewYork.
4. Cronin, J. (1987). Mathematical Aspects of Hodgkin-Huxley Neural Theory. Cambridge University Press.
5. Darling, R. W. R. and Norris, J. R. (2005). Structure of large random hypergraphs. Ann. Appl. Prob. 15, 125-152.
6. Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353-388.
7. DeFelice, L. J. and Isaac, A. (1993). Chaotic states in a random world: Relationship between the nonlinear differential equations of excitability and the stochastic properties of ion channels. J. Statist. Phys. 70, 339-354.
8. Destexhe, A., Mainen, Z. F. and Sejnowski, T. J. (1994). Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism. J. Comput. Neuroscience 1, 195-230.
9. Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. JohnWiley, NewYork.
10. Faisal, A. A., Selen, L. P. J. andWolpert, D. M. (2008). Noise in the nervous system. Nature Rev. Neurosci. 9, 292-303.
11. Fox, R. F. and Lu, Y.-N. (1994). Emergent collective behaviour in large numbers of globally coupled independently stochastic ion channels. Phys. Rev. E 49, 3421-3431.
12. Gollisch, T. and Meister, M. (2008). Rapid neural coding in the retina with relative spike latencies. Science 319, 1108-1111.
13. Guckenheimer, J. and Oliva, R. A. (2002). Chaos in the Hodgkin-Huxley Model. SIAM J. Appl. Dynam. Systems 1, 105-114.
14. Hespanha, J. P. (2005). A model for stochastic hybrid systems with application to communication networks. Nonlinear Anal. 62, 1353-1383.
15. Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiology 117, 500-544.
16. Izhikevich, E. M. (2006). Dynamical Systems in Neuroscience. MIT Press, Cambridge, MA.
17. Josić, K. and Rosenbaum, R. (2008). Unstable solutions of nonautonomous linear differential equations. SIAM Rev. 50, 570-584.
18. Kallenberg, O. (1997). Foundations of Modern Probability. Springer, NewYork.
19. Keener, J. P. (2009). Invariant manifold reductions for markovian ion channel dynamics. J. Math. Biol. 58, 447-57.
20. Kouretas, P., Koutroumpas, K., Lygeros, J. and Lygerou, Z. (2006). Stochastic hybrid modeling of biochemical processes. In Stochastic Hybrid Systems, CRC Press, Boca Raton, FL, pp. 221-248.
21. Krausz, H. I. and Friesen,W.O. (1977). The analysis of nonlinear synaptic transmission. J. General Physiology 70, 243-265.
22. Kurtz, T. G. (1971). Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl Prob. 8, 344-356.
23. Lygeros, J. et al. (2008). Stochastic hybrid modeling of DNA replication across a complete genome. Proc. Nat. Acad. Sci. USA 105, 12295-12300.
24. Mainen, Z. F., Joerges, J., Huguenard, J. R. and Sejnowski, T. J. (1995). A model of spike initiation in neocortical pyramidal neurons. Neuron 15, 1427-1439.
25. Morris, C. and Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193-213.
26. Pakdaman, K., Thieullen, M. and Wainrib, G. (2010). Diffusion approximation of birth-death processes: Comparison in terms of large deviations and exit points. Statist. Prob. Lett. 80, 1121-1127.
27. Pankratova, E. V., Polovinkin, A. V. P. and Mosekilde, E. (2005). Resonant activation in a stochastic Hodgkin-Huxley model: interplay between noise and suprathreshold driving effects. Europ. Phys. J. B 45, 391-397.
28. Protter, P. F. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.
29. Rinzel, J. and Miller, R. N. (1980). Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations. Math. Biosci. 49, 27-59.
30. Rowat, P. (2007). Interspike interval statistics in the stochastic Hodgkin-Huxley model: Coexistence of gamma frequency bursts and highly irregular firing. Neural Computation 19, 1215-1250.
31. Segundo, J. P. et al. (1994). Noise and the neurosciences: A long history, a recent revival and some theory. In Origins: Brain and Self Organization, Lawrence Erlbaum, Hillsdale, pp. 299-331.
32. Shuai, J.W. and Jung, P. (2003). Optimal ion channel clustering for intracellular calcium signaling. Proc. Nat. Acad. Sci. USA 100, 506-512.
33. Skaugen, E. and Walloe, L. (1979). Firing behaviour in a stochastic nerve membrane model based upon the Hodgkin-Huxley equations. Acta Physiology Scand. 107, 343-363.
34. Steinmetz, P. N. et al. (2000). Subthreshold voltage noise due to channel fluctuations in active neuronal membranes. J. Comput. Neurosci. 9, 133-148.
35. Touboul, J. and Faugeras, O. (2008). A characterization of the first hitting time of double integral processes to curved boundaries. Adv. Appl. Prob. 40, 501-528.
36. Correction: 40 (2009), 309.
37. Tuckwell, H. C. (1987). Diffusion approximations to channel noise. J. Theoret. Biol. 127, 427-438.
38. Vandenberg, C. A. and 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.
39. Van Kampen, N. G. (1981). Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam.
40. VanRullen, R., Guyonneau, R. and Thorpe, S. J. (2005). Spike times make sense. Trends Neurosci. 28, 1-4.
41. White, J. A., Rubinstein, J. T. and Kay, A. R. (2000). Channel noise in neurons. Trends Neurosci. 23, 131-137.