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Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input. (English. French summary) Zbl 1335.60091
Summary: We consider a model describing a neuron and the input it receives from its dendritic tree when this input is a random perturbation of a periodic deterministic signal, driven by an Ornstein-Uhlenbeck process. The neuron itself is modeled by a variant of the classical Hodgkin-Huxley model. Using the existence of an accessible point where the weak Hörmander condition holds and the fact that the coefficients of the system are analytic, we show that the system is non-degenerate. The existence of a Lyapunov function allows to deduce the existence of (at most a finite number of) extremal invariant measures for the process. As a consequence, the complexity of the system is drastically reduced in comparison with the deterministic system.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J25 Continuous-time Markov processes on general state spaces
60H07 Stochastic calculus of variations and the Malliavin calculus
60H30 Applications of stochastic analysis (to PDEs, etc.)
92C20 Neural biology
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[1] K. Aihara, G. Matsumoto and Y. Ikegaya. Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator. J. Theoret. Biol. 109 (1984) 249-269.
[2] L. Arnold and W. Kliemann. On unique ergodicity for degenerate diffusions. Stochastics 21 (1987) 41-61. · Zbl 0617.60076
[3] M. Benaïm, S. Le Borgne, F. Malrieu and P. A. Zitt. Qualitative properties of certain piecewise deterministic Markov processes. Preprint. Available on . · Zbl 1325.60123
[4] G. Ben Arous, M. Gradinaru and M. Ledoux. Hölder norms and the support theorem for diffusions. Ann. Inst. H. Poincaré, Probab. Statist. 30 (1994) 415-436. · Zbl 0814.60075
[5] N. Berglund and B. Gentz. Stochastic dynamic bifurcations and excitability. In Stochastic Methods in Neuroscience , L. Carlo and G. J. Lord (Eds). Oxford Univ. Press, Oxford, 2010. · Zbl 1352.92023
[6] N. Berglund and D. Landon. Mixed-mode oscillations and interspike interval statistics in the stochastic Fitzhugh-Nagumo model. Nonlinearity 25 (8) (2012) 2303-2335. · Zbl 1248.60059
[7] F. Colonius and W. Kliemann. Some aspects of control systems as dynamical systems. J. Dynam. Differential Equations 5 (3) (1993) 469-494. · Zbl 0784.34050
[8] M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger. Mixed-mode oscillations with multiple time scales. SIAM Rev. 54 (2) (2012) 211-288. · Zbl 1250.34001
[9] A. Destexhe. Conductance-based integrate and fire models. Neural Comput. 9 (1997) 503-514.
[10] K. Endler. Periodicities in the Hodgkin-Huxley model and versions of this model with stochastic input. Master thesis, Institute of Mathematics, Univ. Mainz, 2012. Available at .
[11] J. Guckenheimer and R. A. Oliva. Chaos in the Hodgkin-Huxley model. SIAM J. Appl. Dyn. Syst. 1 (1) (2002) 105-114. · Zbl 1002.92005
[12] A. Hodgkin and A. Huxley. A quantitative description of ion currents and its applications to conduction and excitation in nerve. J. Physiol. 117 (1952) 500-544.
[13] R. Höpfner, E. Löcherbach and M. Thieullen. Transition densities for strongly degenerate time inhomogeneous random models, 2013. Available at . arXiv:1310.7373 · Zbl 1388.60096
[14] R. Höpfner. On a set of data for the membrane potential in a neuron. Math. Biosci. 207 (2) (2007) 275-301. · Zbl 1255.60132
[15] R. Höpfner and Y. Kutoyants. Estimating discontinuous periodic signals in a time inhomogeneous diffusion. Stat. Inference Stoch. Process. 13 (3) (2010) 193-230. · Zbl 1209.62195
[16] E. M. Izhikevich. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Computational Neuroscience . MIT Press, Cambridge, MA, 2007.
[17] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 32 (1985) 1-76. · Zbl 0568.60059
[18] S. P. Meyn and R. L. Tweedie. Stability of Markovian processes. I: Criteria for discrete-time chains. Adv. in Appl. Probab. 24 (3) (1992) 542-574. · Zbl 0757.60061
[19] A. Millet and M. Sanz-Solé. A simple proof of the support theorem for diffusion processes. In Séminaire de Probabilités XXVIII . J. Azéma, P. A. Meyer and M. Yor (Eds). Lecture Notes in Mathematics 1583 . Springer, Berlin, 1994. · Zbl 0807.60073
[20] T. Nagano. Linear differential systems with singularities and an application to transitive Lie algebras. J. Math. Soc. Japan 18 (1966) 398-404. · Zbl 0147.23502
[21] D. Nualart. The Malliavin Calculus and Related Topics. Probability and Its Applications . Springer, New York, 1995. · Zbl 0837.60050
[22] E. Nummelin. A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 (1978) 309-318. · Zbl 0364.60104
[23] J. Rinzel and R. N. Miller. Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations. Math. Biosci. 49 (1980) 27-59. · Zbl 0429.92014
[24] J. Rubin and M. Wechselberger. Giant squid-hidden canard: The 3d geometry of the Hodgkin-Huxley model. Biol. Cybernet. 97 (1) (2007) 5-32. · Zbl 1125.92015
[25] D. W. Stroock and S. R. S. Varadhan. On the support of diffusion processes with applications to the strong maximum principle. In Proc. 6th Berkeley Sympos. Math. Statist. Probab., Univ. Calif. 1970 3 333-359. Univ. California Press, Berkeley, CA, 1972. · Zbl 0255.60056
[26] H. J. Sussmann. Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180 (1973) 171-188. · Zbl 0274.58002
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