The Annals of Applied Statistics

Estimation in the partially observed stochastic Morris–Lecar neuronal model with particle filter and stochastic approximation methods

Susanne Ditlevsen and Adeline Samson

Full-text: Open access

Enhanced PDF (745 KB)

Abstract

Parameter estimation in multidimensional diffusion models with only one coordinate observed is highly relevant in many biological applications, but a statistically difficult problem. In neuroscience, the membrane potential evolution in single neurons can be measured at high frequency, but biophysical realistic models have to include the unobserved dynamics of ion channels. One such model is the stochastic Morris–Lecar model, defined by a nonlinear two-dimensional stochastic differential equation. The coordinates are coupled, that is, the unobserved coordinate is nonautonomous, the model exhibits oscillations to mimic the spiking behavior, which means it is not of gradient-type, and the measurement noise from intracellular recordings is typically negligible. Therefore, the hidden Markov model framework is degenerate, and available methods break down. The main contributions of this paper are an approach to estimate in this ill-posed situation and nonasymptotic convergence results for the method. Specifically, we propose a sequential Monte Carlo particle filter algorithm to impute the unobserved coordinate, and then estimate parameters maximizing a pseudo-likelihood through a stochastic version of the Expectation–Maximization algorithm. It turns out that even the rate scaling parameter governing the opening and closing of ion channels of the unobserved coordinate can be reasonably estimated. An experimental data set of intracellular recordings of the membrane potential of a spinal motoneuron of a red-eared turtle is analyzed, and the performance is further evaluated in a simulation study.

Article information

Source
Ann. Appl. Stat., Volume 8, Number 2 (2014), 674-702.

Dates
First available in Project Euclid: 1 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1404229510

Digital Object Identifier
doi:10.1214/14-AOAS729

Mathematical Reviews number (MathSciNet)
MR3262530

Zentralblatt MATH identifier
06333772

Keywords
Sequential Monte Carlo diffusions pseudo likelihood Stochastic Approximation Expectation Maximization motoneurons conductance-based neuron models membrane potential

Citation

Ditlevsen, Susanne; Samson, Adeline. Estimation in the partially observed stochastic Morris–Lecar neuronal model with particle filter and stochastic approximation methods. Ann. Appl. Stat. 8 (2014), no. 2, 674--702. doi:10.1214/14-AOAS729. https://projecteuclid.org/euclid.aoas/1404229510


Export citation

References

  • Aït-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 223–262.
    Mathematical Reviews (MathSciNet): MR1926260
    Digital Object Identifier: doi:10.1111/1468-0262.00274
  • Andrieu, C., Doucet, A. and Punskaya, E. (2001). Sequential Monte Carlo methods for optimal filtering. In Sequential Monte Carlo Methods in Practice 79–95. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1847787
    Digital Object Identifier: doi:10.1007/978-1-4757-3437-9_4
  • Berg, R. W., Alaburda, A. and Hounsgaard, J. (2007). Balanced inhibition and excitation drive spike activity in spinal half-centers. Science 315 390–393.
  • Berg, R. W. and Ditlevsen, S. (2013). Synaptic inhibition and excitation estimated via the time constant of membrane potential fluctuations. J. Neurophys. 110 1021–1034.
  • Berg, R. W., Ditlevsen, S. and Hounsgaard, J. (2008). Intense synaptic activity enhances temporal resolution in spinal motoneurons. PLoS ONE 3 e3218.
  • Beskos, A., Papaspiliopoulos, O., Roberts, G. O. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 333–382.
    Mathematical Reviews (MathSciNet): MR2278331
    Digital Object Identifier: doi:10.1111/j.1467-9868.2006.00552.x
  • Borg-Graham, L. J., Monier, C. and Frégnac, Y. (1998). Visual input evokes transient and strong shunting inhibition in visual cortical neurons. Nature 393 369–373.
  • Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2159833
  • Delyon, B., Lavielle, M. and Moulines, E. (1999). Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27 94–128.
    Mathematical Reviews (MathSciNet): MR1701103
    Digital Object Identifier: doi:10.1214/aos/1018031103
    Project Euclid: euclid.aos/1018031103
  • Del Moral, P., Jacod, J. and Protter, P. (2001). The Monte-Carlo method for filtering with discrete-time observations. Probab. Theory Related Fields 120 346–368.
    Mathematical Reviews (MathSciNet): MR1843179
    Digital Object Identifier: doi:10.1007/PL00008786
  • Del Moral, P. and Miclo, L. (2000). Branching and interacting particle systems approximations of Feynman–Kac formulae with applications to non-linear filtering. In Séminaire de Probabilités, XXXIV. Lecture Notes in Math. 1729 1–145. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1768060
    Digital Object Identifier: doi:10.1007/BFb0103798
  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1–38.
    Mathematical Reviews (MathSciNet): MR501537
  • Ditlevsen, S. and Greenwood, P. (2013). The Morris–Lecar neuron model embeds a leaky integrate-and-fire model. J. Math. Biol. 67 239–259.
    Mathematical Reviews (MathSciNet): MR3071157
    Digital Object Identifier: doi:10.1007/s00285-012-0552-7
  • Douc, R., Garivier, A., Moulines, E. and Olsson, J. (2011). Sequential Monte Carlo smoothing for general state space hidden Markov models. Ann. Appl. Probab. 21 2109–2145.
    Mathematical Reviews (MathSciNet): MR2895411
    Digital Object Identifier: doi:10.1214/10-AAP735
    Project Euclid: euclid.aoap/1322057317
  • Doucet, A., de Freitas, N. and Gordon, N. (2001). An introduction to sequential Monte Carlo methods. In Sequential Monte Carlo Methods in Practice 3–14. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1847784
    Digital Object Identifier: doi:10.1007/978-1-4757-3437-9_1
  • Doucet, A., Godsill, S. and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filterin. Stat. Comput. 10 197–208.
  • Durham, G. B. and Gallant, A. R. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J. Bus. Econom. Statist. 20 297–338. With comments and a reply by the authors.
    Mathematical Reviews (MathSciNet): MR1939904
  • Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959–993.
    Mathematical Reviews (MathSciNet): MR1839375
    Digital Object Identifier: doi:10.1111/1468-0262.00226
  • Eraker, B. (2001). MCMC analysis of diffusion models with application to finance. J. Bus. Econom. Statist. 19 177–191.
    Mathematical Reviews (MathSciNet): MR1939708
  • Fearnhead, P., Papaspiliopoulos, O. and Roberts, G. O. (2008). Particle filters for partially observed diffusions. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 755–777.
    Mathematical Reviews (MathSciNet): MR2523903
    Digital Object Identifier: doi:10.1111/j.1467-9868.2008.00661.x
  • Gerstner, W. and Kistler, W. M. (2002). Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1923120
  • Gobet, E. and Labart, C. (2008). Sharp estimates for the convergence of the density of the Euler scheme in small time. Electron. Commun. Probab. 13 352–363.
    Mathematical Reviews (MathSciNet): MR2415143
    Digital Object Identifier: doi:10.1214/ECP.v13-1393
  • Golightly, A. and Wilkinson, D. J. (2006). Bayesian sequential inference for nonlinear multivariate diffusions. Stat. Comput. 16 323–338.
    Mathematical Reviews (MathSciNet): MR2297534
    Digital Object Identifier: doi:10.1007/s11222-006-9392-x
  • Golightly, A. and Wilkinson, D. J. (2008). Bayesian inference for nonlinear multivariate diffusion models observed with error. Comput. Statist. Data Anal. 52 1674–1693.
    Mathematical Reviews (MathSciNet): MR2422763
  • Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings-F 140 107–113.
  • Huys, Q. J. M., Ahrens, M. and Paninski, L. (2006). Efficient estimation of detailed single-neuron models. J. Neurophysiol. 96 872–890.
  • Huys, Q. J. M. and Paninski, L. (2009). Smoothing of, and parameter estimation from, noisy biophysical recordings. PLoS Comput. Biol. 5 e1000379, 16.
    Mathematical Reviews (MathSciNet): MR2516090
    Digital Object Identifier: doi:10.1371/journal.pcbi.1000379
  • Ionides, E. L., Bhadra, A., Atchadé, Y. and King, A. (2011). Iterated filtering. Ann. Statist. 39 1776–1802.
    Mathematical Reviews (MathSciNet): MR2850220
    Digital Object Identifier: doi:10.1214/11-AOS886
    Project Euclid: euclid.aos/1311600283
  • Jahn, P., Berg, R. W., Hounsgaard, J. and Ditlevsen, S. (2011). Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process. J. Comput. Neurosci. 31 563–579.
    Mathematical Reviews (MathSciNet): MR2864731
    Digital Object Identifier: doi:10.1007/s10827-011-0326-z
  • Jensen, A. C., Ditlevsen, S., Kessler, M. and Papaspiliopoulos, O. (2012). Markov chain Monte Carlo approach to parameter estimation in the FitzHugh–Nagumo model. Physical Review E 86 041114.
  • Kalogeropoulos, K. (2007). Likelihood-based inference for a class of multivariate diffusions with unobserved paths. J. Statist. Plann. Inference 137 3092–3102.
    Mathematical Reviews (MathSciNet): MR2364153
    Digital Object Identifier: doi:10.1016/j.jspi.2006.05.017
  • Kloeden, P. and Neuenkirch, A. (2012). Convergence of numerical methods for stochastic differential equations in mathematical finance. Available at arXiv:1204.6620v1 [math.NA].
    arXiv: 1204.6620v1
  • Künsch, H. R. (2005). Recursive Monte Carlo filters: Algorithms and theoretical analysis. Ann. Statist. 33 1983–2021.
    Mathematical Reviews (MathSciNet): MR2211077
    Digital Object Identifier: doi:10.1214/009053605000000426
    Project Euclid: euclid.aos/1132936554
  • Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032–1044.
    Mathematical Reviews (MathSciNet): MR1649198
    Digital Object Identifier: doi:10.1080/01621459.1998.10473765
  • Longtin, A. (2013). Neuronal noise. Scholarpedia 8 1618.
  • Monier, C., Fournier, J. and Frégnac, Y. (2008). In vitro and in vivo measures of evoked excitatory and inhibitory conductance dynamics in sensory cortices. J. Neurosci. Methods 169 323–365.
  • Morris, C. and Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35 193–213.
  • Pedersen, A. R. (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Stat. 22 55–71.
    Mathematical Reviews (MathSciNet): MR1334067
  • Pokern, Y., Stuart, A. M. and Wiberg, P. (2009). Parameter estimation for partially observed hypoelliptic diffusions. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 49–73.
    Mathematical Reviews (MathSciNet): MR2655523
    Digital Object Identifier: doi:10.1111/j.1467-9868.2008.00689.x
  • Pospischil, M., Piwkowska, Z., Bal, T. and Destexhe, A. (2009). Extracting synaptic conductances from single membrane potential traces. Neurosci. 158 545–552.
  • Rinzel, J. and Ermentrout, G. B. (1989). Methods in Neural Modeling, Chapter Analysis of Neural Excitability and Oscillations. MIT Press, Cambridge, MA.
  • Roberts, G. O. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm. Biometrika 88 603–621.
    Mathematical Reviews (MathSciNet): MR1859397
    Digital Object Identifier: doi:10.1093/biomet/88.3.603
  • Rudolph, M., Piwkowska, Z., Badoual, M., Bal, T. and Destexhe, A. (2004). A method to estimate synaptic conductances from membrane potential fluctuations. J. Neurophysiol. 91 2884–2896.
  • Samson, A. and Thieullen, M. (2012). A contrast estimator for completely or partially observed hypoelliptic diffusion. Stochastic Process. Appl. 122 2521–2552.
    Mathematical Reviews (MathSciNet): MR2926166
    Digital Object Identifier: doi:10.1016/j.spa.2012.04.006
  • Sørensen, H. (2004). Parametric inference for diffusion processes observed at discrete points in time: A survey. Int. Stat. Rev. 72 337–354.
  • Sørensen, M. (2012). Estimating functions for diffusion-type processes. In Statistical Methods for Stochastic Differential Equations. Monogr. Statist. Appl. Probab. 124 1–107. CRC Press, Boca Raton, FL.
    Mathematical Reviews (MathSciNet): MR2976982
    Digital Object Identifier: doi:10.1201/b12126-2
  • Tateno, T. and Pakdaman, K. (2004). Random dynamics of the Morris–Lecar neural model. Chaos 14 511–530.
    Mathematical Reviews (MathSciNet): MR2089477
    Digital Object Identifier: doi:10.1063/1.1756118

The Institute of Mathematical Statistics

The Annals of Applied Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more