Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Application to a stochastic Hodgkin−Huxley model
ESAIM: Probability and Statistics, Tome 20 (2016), pp. 527-554.

We formulate simple criteria for positive Harris recurrence of strongly degenerate stochastic differential equations with smooth coefficients on a state space with certain boundary conditions. The drift depends on time and space and is periodic in the time argument. There is no time dependence in the diffusion coefficient. Control systems play a key role, and we prove a new localized version of the support theorem. Beyond existence of some Lyapunov function, we only need one attainable inner point of full weak Hoermander dimension. Our motivation comes from a stochastic Hodgkin−Huxley model for a spiking neuron including its dendritic input. This input carries some deterministic periodic signal coded in its drift coefficient and is the only source of noise for the whole system. We have a 5d SDE driven by 1d Brownian motion. As an application of the general results above, we can prove positive Harris recurrence. Here analyticity of the coefficients and Nummelin splitting allow to formulate a Glivenko−Cantelli type theorem for the interspike intervals.

DOI : 10.1051/ps/2016020
Classification : 60J60, 60J25, 60H07
Mots clés : Degenerate diffusion processes, time inhomogeneous diffusion processes, weak Hoermander condition, support theorem, periodic ergodicity, Hodgkin−Huxley, dendritic input, spike trains
Höpfner, Reinhard 1 ; Löcherbach, Eva 2 ; Thieullen, Michèle 3

1 Johannes-Gutenberg-Universität Mainz, Institut für Mathematik, Staudingerweg 9, 55099 Mainz, Germany.
2 Université de Cergy-Pontoise, Laboratoire AGM, UMR CNRS 8088, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France.
3 UniversitéPierre et Marie Curie, Laboratoire de Probabilités et Modèles Aléatoires, UMR CNRS 7599, Case 188, 4 place Jussieu, 75252 Paris cedex 5, France.
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     title = {Ergodicity and limit theorems for degenerate diffusions with time periodic drift. {Application} to a stochastic {Hodgkin\ensuremath{-}Huxley} model},
     journal = {ESAIM: Probability and Statistics},
     pages = {527--554},
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Höpfner, Reinhard; Löcherbach, Eva; Thieullen, Michèle. Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Application to a stochastic Hodgkin−Huxley model. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 527-554. doi : 10.1051/ps/2016020. http://archive.numdam.org/articles/10.1051/ps/2016020/

L. Arnold and W. Kliemann, On unique ergodicity for degenerate diffusions. Stochastics 21 (1987) 41–61. | DOI | MR | Zbl

J. Azéma, M. Duflo and D. Revuz, Mesures invariantes des processus de Markov récurrents. Séminaire de Probabilités III. In vol. 88 of Lect. Notes Math. Springer (1969). | Numdam | MR | Zbl

R. Bass, Diffusions and elliptic operators. Springer (1998). | MR | Zbl

P. Brémaud, Introduction aux Probabilités. Springer (1984). | Zbl

Y. Chow and H. Teicher, Probability theory. 2nd Edition Springer (1988). | MR | Zbl

S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Feller neuronal model. Phys. Rev. E 73 (2006) 061910. | DOI | MR | Zbl

H. Doss and P. Priouret, Support d’un processus de reflexion. Z. Wahrscheinlichkeitstheorie verw. Geb. 61 (1982) 327–345. | DOI | MR | Zbl

K. Endler, Periodicities in the Hodgkin−Huxley model and versions of this model with stochastic input. Master Thesis, Institute of Mathematics, University of Mainz (2012). Available at https://publications.ub.uni-mainz.de/theses/frontdoor.php?source˙opus=3083&la=de (2016).

M. Hairer, On Malliavin’s proof of Hörmander’s theorem. Bull. Sci. Math. 135 (2011) 650–666. | DOI | MR | Zbl

T. Harris, The existence of stationary measures for certain Markov processes. Proc. of 3rd Berkeley Symp. II (1956) 113–124. | MR | Zbl

A. Hodgkin and A. Huxley, A quantitative description of ion currents and its applications to conduction and excitation in nerve embranes. J. Physiol. 117 (1951) 500–544. | DOI

R. Höpfner, On a set of data for the membrane potential in a neuron. Math. Biosci. 207 (2007) 275–301. | DOI | MR | Zbl

R. Höpfner, E. Löcherbach and M. Thieullen, Strongly degenerate time inhomogenous SDEs: densities and support properties. Application to Hodgkin−Huxley type systems. Preprint (2014). To appear in Bernoulli (2017). | arXiv | MR

R. Höpfner, E. Löcherbach and M. Thieullen, Ergodicity for a stochastic Hodgkin−Huxley model driven by Ornstein−Uhlenbeck type input. Ann. Inst. Henri Poincaré 52 (2016) 483–501. | DOI | MR | Zbl

R. Höpfner and Yu. Kutoyants, Estimating discontinuous periodic signals in a time inhomogeneous diffusion. Statist. Inference Stoch. Proc. 13 (2010) 193–230. | DOI | MR | Zbl

K. Ichihara and H. Kunita, A Classification of the Second Order Degenerate Elliptic Operators and its Probabilistic Characterization. Z. Wahrscheinlichkeitsth. verw. Geb. 30 (1974) 235–254. | DOI | MR | Zbl

E. Izhikevich, Dynamical systems in neuroscience: the geometry of excitability and bursting. MIT Press (2007). | MR

H. Kunita, Stochastic flows and stochastic differential equations. Cambridge Univ. Press (1990). | MR | Zbl

P. Lansky, L. Sacerdote and F. Tomassetti, On the comparison of Feller and Ornstein−Uhlenbeck models for neuronal activity. Biol. Cybern. 73 (1995) 457–465. | DOI | Zbl

J. Mattingly, A. Stuart and D. Highham, Ergodicity of SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Proc. Appl. 101 (2002) 185–232. | DOI | MR | Zbl

S. Meyn and R. Tweedie, Stability of Markovian processes I: criteria for discrete-time chains. Adv. Appl. Probab. 24 (1992) 542–574. | DOI | MR | Zbl

S. Meyn and R. Tweedie, Stochastic stablity of Markov chains. Springer (1993). | MR

A. Millet and M. Sanz-Solé, A simple proof of the support theorem for diffusion processes. Séminaire de Probabilités, Strasbourg, tome 28 (1994) 26–48. | Numdam | MR | Zbl

E. Nummelin, A splitting technique for Harris recurrent Markov chains. Zeitschr. Wahrscheinlichkeitstheorie Verw. Geb. 43 (1978) 309–318. | DOI | Zbl

E. Nummelin, General irreducible Markov chains and non-negative operators. Cambridge University Press (1985). | MR | Zbl

F. Petit, Théorème de support pour les diffusions réfléchis de type Ventcell. Ann. Inst. Henri Poincaré 32 (1996) 135–210. | Numdam | MR | Zbl

D. Revuz, Markov chains. Rev. Edition. Springer (1984). | MR

J. Rinzel and R. Miller, Numerical calculation of stable and unstable periodic solutions to the Hodgkin−Huxley equations. Math. Biosci. 49 (1980) 27–59. | DOI | MR | Zbl

D. Strook and S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle. Proc. of 6th Berkeley Symp. Math. Stat. Prob. III (1972) 333–359. | MR | Zbl

H. Sussmann, Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180 (1973) 171–188. | DOI | MR | Zbl

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