Abstract
In this chapter we review some mathematical aspects of FitzHugh–Nagumo systems of ordinary differential equations or partial differential equations. Our treatment is probabilistic. We focus on small noise asymptotics for these systems and their stochastic perturbations. The noise is either an external perturbation or already present when the system involves spatial propagation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aliev, R.R., Panfilov, A.V.: A simple two-variable model of cardiac excitation. Chaos Solitons Fractals 7, 293–301 (1995)
Doss, C., Thieullen, M.: Oscillations and Random Perturbations of a FitzHugh-Nagumo System. Preprint, arXiv:0906.2671v1, July (2009)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
Freidlin, M.I., Wentzell, A.D.: Random Perturbation of Dynamical Systems. Springer, New York (1984)
Gaertner, J.: Bistable Reaction-diffusion equations and excitable media. Math. Nach. 112, 125–152 (1983)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Lindner, B., Garcia-Ojalvo, J., Neiman, A., Schimansky-Geier, L.: Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004)
Nagumo, J.S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2071 (1962)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol. 293, 3rd edn. Springer, New York (1991)
Soravia, P., Souganidis, P.E.: Phase-field theory for FitzHugh-Nagumo-type systems. SIAM J. Math. Anal. 5, 1341–1359 (1996)
Tuckwell, H.C.: Introduction to Theoretical Neurobiology, vol. 1 and 2. Cambridge University Press, Cambridge (1988)
Tuckwell, H.C.: Nonlinear effects in white-noise driven spatial diffusion: general analytical results and probabilities of exceeding threshold. Phys. A 387, 1455–1463 (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Thieullen, M. (2013). Deterministic and Stochastic FitzHugh–Nagumo Systems. In: Bachar, M., Batzel, J., Ditlevsen, S. (eds) Stochastic Biomathematical Models. Lecture Notes in Mathematics(), vol 2058. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32157-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-32157-3_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32156-6
Online ISBN: 978-3-642-32157-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)