The Annals of Applied Probability

Hypoelliptic stochastic FitzHugh–Nagumo neuronal model: Mixing, up-crossing and estimation of the spike rate

José R. León and Adeline Samson

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


The FitzHugh–Nagumo is a well-known neuronal model that describes the generation of spikes at the intracellular level. We study a stochastic version of the model from a probabilistic point of view. The hypoellipticity is proved, as well as the existence and uniqueness of the stationary distribution. The bi-dimensional stochastic process is $\beta$-mixing. The stationary density can be estimated with an adaptive nonparametric estimator. Then we focus on the distribution of the length between successive spikes. Spikes are difficult to define directly from the continuous stochastic process. We study the distribution of the number of up-crossings. We link it to the stationary distribution and propose an estimator of its expectation. We finally prove mathematically that the mean length of inter-up-crossings interval is equal to its up-crossings rate. We illustrate the proposed estimators on a simulation study. Different regimes are explored, with no, few or high generation of spikes.

Article information

Ann. Appl. Probab., Volume 28, Number 4 (2018), 2243-2274.

Received: March 2017
Revised: September 2017
First available in Project Euclid: 9 August 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]
Secondary: 62M05: Markov processes: estimation 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65] 62P10: Applications to biology and medical sciences 35Q62: PDEs in connection with statistics 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]

Hypoelliptic diffusion FitzHugh–Nagumo model invariant density nonparametric estimation up-crossings pulse rate spike rate estimation


León, José R.; Samson, Adeline. Hypoelliptic stochastic FitzHugh–Nagumo neuronal model: Mixing, up-crossing and estimation of the spike rate. Ann. Appl. Probab. 28 (2018), no. 4, 2243--2274. doi:10.1214/17-AAP1355.

Export citation


  • [1] Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. Wiley, Hoboken, NJ.
  • [2] Berglund, N. and Gentz, B. (2006). Noise-Induced Phenomena in Slow–Fast Dynamical Systems: A Sample-Paths Approach. Springer, London.
  • [3] Berglund, N. and Landon, D. (2012). Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model. Nonlinearity 25.
  • [4] Cattiaux, P., Leon, J., Pineda Centeno, A. and Prieur, C. (2017). An overlook on statistical inference issues for stochastic damping Hamiltonian systems under the fluctuation-dissipation condition. Statistics 51 11–29.
  • [5] Comte, F., Prieur, C. and Samson, A. (2017). Adaptive estimation for stochastic damping Hamiltonian systems under partial observation. Stochastic Process. Appl. 127 3689–3718.
  • [6] Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. Wiley, New York.
  • [7] Ditlevsen, S. and Samson, A. (2017). Hypoelliptic diffusion: Discretization, filtering and inference from complete and partial observations. Submitted.
  • [8] Di Bernardino, E., León, J. and Tchumatchenko, T. (2014). Cross-correlations and joint Gaussianity in multivariate level crossing models. J. Math. Neurosci. 4 Art. ID 22.
  • [9] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. Springer, New York.
  • [10] Gerstner, W. and Kistler, W. (2002). Spiking Neuron Models. Cambridge Univ. Press, Cambridge.
  • [11] Goldenshluger, A. and Lepski, O. (2011). Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. Ann. Statist. 39 1608–1632.
  • [12] Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes. J. Physiol. 117 500–544.
  • [13] Konakov, V., Menozzi, S. and Molchanov, S. (2010). Explicit parametrix and local limit theorems for some degenerate diffusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 46 908–923.
  • [14] Kumar, P. and Narayanan, S. (2006). Solution of Fokker–Planck equation by finite element and finite difference methods for nonlinear systems. Sādhanā 31 445–461.
  • [15] Lacour, C., Massart, P. and Rivoirard, V. (2017). Estimator selection: A new method with applications to kernel density estimation. Sankhya A 72 1–38.
  • [16] Lindner, B., Garcia-Ojalvo, J., Neiman, A. and Schimansky-Geier, L. (2004). Effects of noise in excitable systems. Phys. Rep. 392 321–424.
  • [17] Lindner, B. and Schimansky-Geier, L. (1999). Analytical approach to the stochastic FitzHugh–Nagumo system and coherence resonance. Phys. Rev. E 60 7270–7276.
  • [18] Longtin, A. (2000). Effect of noise on the tuning properties of excitable systems. Chaos Solitons Fractals 11 1835–1848.
  • [19] Mattingly, J. C., Stuart, A. M. and Higham, D. J. (2002). Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101 185–232.
  • [20] Morris, C. and Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35 193–213.
  • [21] Pakdaman, K., Thieullen, M. and Wainrib, G. (2010). Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv. in Appl. Probab. 42 761–794.
  • [22] Reynaud-Bouret, P., Rivoirard, V., Grammont, F. and Tuleau-Malot, C. (2014). Goodness-of-fit tests and nonparametric adaptive estimation for spike train analysis. J. Math. Neurosci. 4 Art. ID 3.
  • [23] Reynaud-Bouret, P., Rivoirard, V. and Tuleau-Malot, C. (2013). Inference of functional connectivity in neurosciences via Hawkes processes. In 1st IEEE Global Conference on Signal and Information Processing, Austin, TX.
  • [24] Samson, A. and Thieullen, M. (2012). Contrast estimator for completely or partially observed hypoelliptic diffusion. Stochastic Process. Appl. 122 2521–2552.
  • [25] Tuckwell, H. C. and Rodriguez, R. (1998). Analytical and simulation results for stochastic Fitzhugh–Nagumo neurons and neural networks. J. Comput. Neurosci. 5 91–113.
  • [26] Wu, L. (2001). Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stochastic Process. Appl. 91 205–238.