The emergence of the deterministic Hodgkin–Huxley equations as a limit from the underlying stochastic ion-channel mechanism



The Annals of Applied Probability

The emergence of the deterministic Hodgkin–Huxley equations as a limit from the underlying stochastic ion-channel mechanism

Tim D. Austin

Source: Ann. Appl. Probab. Volume 18, Number 4 (2008), 1279-1325.

Abstract

In this paper we consider the classical differential equations of Hodgkin and Huxley and a natural refinement of them to include a layer of stochastic behavior, modeled by a large number of finite-state-space Markov processes coupled to a simple modification of the original Hodgkin–Huxley PDE. We first prove existence, uniqueness and some regularity for the stochastic process, and then show that in a suitable limit as the number of stochastic components of the stochastic model increases and their individual contributions decrease, the process that they determine converges to the trajectory predicted by the deterministic PDE, uniformly up to finite time horizons in probability. In a sense, this verifies the consistency of the deterministic and stochastic processes.

Primary Subjects: 60F17
Secondary Subjects: 60K99, 92C20
Keywords: Hodgkin–Huxley equations; stochastic Hodgkin–Huxley equations; action potential; convergence of Markov processes; nonlinear parabolic PDE

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Alternatively, the document is available for a cost of $15. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1216677123
Digital Object Identifier: doi:10.1214/07-AAP494
Mathematical Reviews number (MathSciNet): MR2434172

References

[1] Chow, C. C. and White, J. A. (1996). Spontaneous action potentials due to channel fluctuations. Biophys. J. 71 3013–3021.
[2] Cronin, J. (1987). Mathematical Aspects of Hodgkin–Huxley Neural Theory. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR909892
Zentralblatt MATH: 0627.92005
[3] DeFelice, L. J. and Isaac, A. (1992). Chaotic states in a random world. J. Stat. Phys. 70 339–352.
[4] Darling, R. W. R. Fluid limits of pure jump Markov processes: A practical guide. Available at arXiv:math.PR/0210109.
[5] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
Mathematical Reviews (MathSciNet): MR838085
Zentralblatt MATH: 0592.60049
[6] Evans, J. and Shenk, N. (1970). Solutions to axon equations. Biophys. J. 10 1090–1101.
[7] Evans, L. C. (1998). Partial Differential Equations. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR1625845
Zentralblatt MATH: 0902.35002
[8] Faisal, A. A., White, J. A. and Laughlin, S. B. (2005). Ion-channel noise places limits on the miniaturization of the brain’s wiring. Current Biology 15 1143–1149.
[9] Fox, R. F. and Lu, Y. (1994). Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels. Physical Review E 49 3421–3431.
[10] Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical Systems, 2nd ed. Springer, New York.
Mathematical Reviews (MathSciNet): MR1652127
Zentralblatt MATH: 0922.60006
[11] Hille, B. (2001). Ion Channels of Excitable Membranes. Sinauer Associates, Sunderland.
[12] Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conductation and excitation in nerve. J. Physiol. 117 500–544.
[13] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR959133
Zentralblatt MATH: 0635.60021
[14] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
Mathematical Reviews (MathSciNet): MR1876169
Zentralblatt MATH: 0996.60001
[15] Kurtz, T. G. (1981). Approximation of Population Processes. SIAM, Philadelphia.
Mathematical Reviews (MathSciNet): MR610982
Zentralblatt MATH: 0465.60078
[16] Lamberti, L. (1986). Solutions to the Hodgkin–Huxley equations. Appl. Math. Comput. 18 43–70.
Mathematical Reviews (MathSciNet): MR815772
Digital Object Identifier: doi:10.1016/0096-3003(86)90027-5
[17] Steinmetz, P. N., Manwani, A. and Koch C. (2001). Variability and coding efficiency of noisy neural spike encoders. BioSystems 62 87–97.
[18] Tuckwell, H. C. (1989). Stochastic Processes in the Neurosciences. SIAM, Philadelphia.
Mathematical Reviews (MathSciNet): MR1002192
Zentralblatt MATH: 0675.92001

2008 © Institute of Mathematical Statistics