Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T14:40:52.316Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Limit theorems Physiological, cellular and medical topics Markov processes

Published online by Cambridge University Press:  01 July 2016

K. Pakdaman ,
M. Thieullen  and
G. Wainrib
K. Pakdaman*
Affiliation:
Université Paris VII
M. Thieullen*
Affiliation:
Université Paris VI
G. Wainrib*
Affiliation:
Ecole Polytechnique, Université Paris VII and Université Paris VI
*
∗ Postal address: Institut Jacques Monod UMR7592, Bâtiment Buffon, 15 rue Hélène Brion, 75205 Paris cedex 13, France.
∗∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires UMR7599, Boîte 188, 175 rue du Chevaleret, 75013 Paris, France.
∗∗∗ Email address: wainrib.gilles@ijm.univ-paris-diderot.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.

Keywords

Stochastic hybrid systempiecewise-deterministic Markov processfluid limitLangevin approximationkinetic modelneuron modelHodgkin-HuxleyMorris-Lecarstochastic ion channels

MSC classification

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60J75: Jump processes
Secondary: 92C20: Neural biology 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 761 - 794
Copyright
Copyright © Applied Probability Trust 2010 

References

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.CrossRefGoogle Scholar
Blom, H. A. P. and Lygeros, J. (eds) (2006). Stochastic Hybrid Systems (Lecture Notes Control Inf. Sci. 337). Springer.CrossRefGoogle Scholar
Chicone, C. (1999). Ordinary Differential Equations with Applications. Springer, New York.Google Scholar
Cronin, J. (1987). Mathematical Aspects of Hodgkin–Huxley Neural Theory. Cambridge University Press.CrossRefGoogle Scholar
Darling, R. W. R. and Norris, J. R. (2005). Structure of large random hypergraphs. Ann. Appl. Prob. 15, 125–152.CrossRefGoogle Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.CrossRefGoogle Scholar
Faisal, A. A., Selen, L. P. J. and Wolpert, D. M. (2008). Noise in the nervous system. Nature Rev. Neurosci. 9, 292–303.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Gollisch, T. and Meister, M. (2008). Rapid neural coding in the retina with relative spike latencies. Science 319, 1108–1111.CrossRefGoogle ScholarPubMed
Guckenheimer, J. and Oliva, R. A. (2002). Chaos in the Hodgkin–Huxley Model. SIAM J. Appl. Dynam. Systems 1, 105–114.CrossRefGoogle Scholar
Hespanha, J. P. (2005). A model for stochastic hybrid systems with application to communication networks. Nonlinear Anal. 62, 1353–1383.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Izhikevich, E. M. (2006). Dynamical Systems in Neuroscience. MIT Press, Cambridge, MA.CrossRefGoogle Scholar
Josić, K. and Rosenbaum, R. (2008). Unstable solutions of nonautonomous linear differential equations. SIAM Rev. 50, 570–584.CrossRefGoogle Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Keener, J. P. (2009). Invariant manifold reductions for markovian ion channel dynamics. J. Math. Biol. 58, 447–57.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Krausz, H. I. and Friesen, W. O. (1977). The analysis of nonlinear synaptic transmission. J. General Physiology 70, 243–265.CrossRefGoogle ScholarPubMed
Kurtz, T. G. (1971). Limit theorems for sequences of Jump Markov processes approximating ordinary differential processes. J. Appl Prob. 8, 344–356.CrossRefGoogle Scholar
Lygeros, J. et al. (2008). Stochastic hybrid modeling of DNA replication across a complete genome. Proc. Nat. Acad. Sci. USA 105, 12295–12300.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Morris, C. and Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–213.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Protter, P. F. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
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.Google Scholar
Shuai, J. W. and Jung, P. (2003). Optimal ion channel clustering for intracellular calcium signaling. Proc. Nat. Acad. Sci. USA 100, 506–512.CrossRefGoogle ScholarPubMed
Skaugen, E. and Wallœ, L. (1979). Firing behaviour in a stochastic nerve membrane model based upon the Hodgkin–Huxley equations. Acta Physiology Scand. 107, 343–363.CrossRefGoogle Scholar
Steinmetz, P. N. et al. (2000). Subthreshold voltage noise due to channel fluctuations in active neuronal membranes. J. Comput. Neurosci. 9, 133–148.CrossRefGoogle ScholarPubMed
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. (Correction: 40 (2009), 309.)CrossRefGoogle Scholar
Tuckwell, H. C. (1987). Diffusion approximations to channel noise. J. Theoret. Biol. 127, 427–438.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Van Kampen, N. G. (1981). Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam.Google Scholar
VanRullen, R., Guyonneau, R. and Thorpe, S. J. (2005). Spike times make sense. Trends Neurosci. 28, 1–4.CrossRefGoogle ScholarPubMed
White, J. A., Rubinstein, J. T. and Kay, A. R. (2000). Channel noise in neurons. Trends Neurosci. 23, 131–137.CrossRefGoogle ScholarPubMed