Numéro spécial : Special Issue on Statistics and Neurosciences
Parameter estimation in neuronal stochastic differential equation models from intracellular recordings of membrane potentials in single neurons: a Review
Journal de la société française de statistique, Volume 157 (2016) no. 1, pp. 6-21.

Dynamics of the membrane potential in a single neuron can be studied by estimating biophysical parameters from intracellular recordings. Diffusion processes, given as continuous solutions to stochastic differential equations, are widely applied as models for the neuronal membrane potential evolution. One-dimensional models are the stochastic integrate-and-fire neuronal diffusion models. Biophysical neuronal models take into account the dynamics of ion channels or synaptic activity, leading to multidimensional diffusion models. Since only the membrane potential can be measured, this complicates the statistical inference and parameter estimation from these partially observed detailed models. This paper reviews parameter estimation techniques from intracellular recordings in these diffusion models.

On peut étudier la dynamique du potentiel de la membrane d’un neurone en estimant des paramètres biophysiques à partir d’enregistrement intracellulaire. Les processus de diffusion, définis comme solution à temps continu d’équations différentielles stochastiques ont été très utilisés pour modéliser l’évolution du potentiel membranaire. Parmi les processus de dimension un, les plus connus sont les modèles de diffusion intègre-et-tire. D’autres modèles neuronaux sont plus biophysiques et prennent en compte la dynamique des canaux ioniques ou de l’activité synaptique. Ce sont des processus de diffusion multidimensionnels. L’estimation des paramètres de ces modèles est difficile car seulement le potentiel membranaire peut être mesuré. Ce papier résume les techniques d’estimation qui ont été proposées pour ces modèles de diffusion de données intracellulaires.

Keywords: integrate-and-fire models, conductance based models, state space models, synaptic input estimation, maximum likelihood estimation, particle filter, estimating functions, MCMC methods, partial observations
Mot clés : modèles de diffusion intègre-et-tire, modèles de conductances, modèles à espace d’états, estimation synaptique, maximum de vraisemblance, filtre particulaire, fonctions estimantes, MCMC, observations partielles
@article{JSFS_2016__157_1_6_0,
     author = {Ditlevsen, Susanne and Samson, Adeline},
     title = {Parameter estimation in neuronal stochastic differential equation models from intracellular recordings of membrane potentials in single neurons: a {Review}},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {6--21},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {157},
     number = {1},
     year = {2016},
     mrnumber = {3491720},
     zbl = {1357.92010},
     language = {en},
     url = {http://www.numdam.org/item/JSFS_2016__157_1_6_0/}
}
TY  - JOUR
AU  - Ditlevsen, Susanne
AU  - Samson, Adeline
TI  - Parameter estimation in neuronal stochastic differential equation models from intracellular recordings of membrane potentials in single neurons: a Review
JO  - Journal de la société française de statistique
PY  - 2016
SP  - 6
EP  - 21
VL  - 157
IS  - 1
PB  - Société française de statistique
UR  - http://www.numdam.org/item/JSFS_2016__157_1_6_0/
LA  - en
ID  - JSFS_2016__157_1_6_0
ER  - 
%0 Journal Article
%A Ditlevsen, Susanne
%A Samson, Adeline
%T Parameter estimation in neuronal stochastic differential equation models from intracellular recordings of membrane potentials in single neurons: a Review
%J Journal de la société française de statistique
%D 2016
%P 6-21
%V 157
%N 1
%I Société française de statistique
%U http://www.numdam.org/item/JSFS_2016__157_1_6_0/
%G en
%F JSFS_2016__157_1_6_0
Ditlevsen, Susanne; Samson, Adeline. Parameter estimation in neuronal stochastic differential equation models from intracellular recordings of membrane potentials in single neurons: a Review. Journal de la société française de statistique, Volume 157 (2016) no. 1, pp. 6-21. http://www.numdam.org/item/JSFS_2016__157_1_6_0/

[Albert et al., 2015] Albert, M., Bouret, Y., Fromont, M., and Reynaud-Bouret, P. (2015). Bootstrap and permutation tests of independence for point processes. submitted. | MR

[Andrieu et al., 2010] Andrieu, C., Doucet, A., and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. J. R. Statist. Soc. B, 72(3):269–342. | MR

[Bachar et al., 2013] Bachar, M., Batzel, J., and Ditlevsen, S., editors (2013). Stochastic Biomathematical Models with Applications to Neuronal Modeling. Springer. | Zbl

[Berg et al., 2007] Berg, R., Alaburda, A., and Hounsgaard, J. (2007). Balanced inhibition and excitation drive spike activity in spinal half-centers. Science, 315:390–393.

[Berg and Ditlevsen, 2013] 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.

[Bibbona and Ditlevsen, 2013] Bibbona, E. and Ditlevsen, S. (2013). Estimation in discretely observed diffusions killed at a threshold. Scandinavian Journal of Statistics, 40(2):274–293. | MR

[Bibbona et al., 2010] Bibbona, E., Lansky, P., and Sirovich, R. (2010). Estimating input parameters from intracellular recordings in the Feller neuronal model. Phys Rev E, 81(3,1):031916.

[Burkitt, 2006] Burkitt, A. (2006). A review of the integrate-and-fire neuron model: I. homogeneous synaptic input. Biol. Cybern., 95:1–19. | MR | Zbl

[Cappé et al., 2005] Cappé, O., Moulines, E., and Ryden, T. (2005). Inference in Hidden Markov Models (Springer Series in Statistics). Springer-Verlag New York, USA. | MR | Zbl

[Destexhe et al., 2004] Destexhe, A., Badoual, M., Piwkowska, Z., Bal, T., and Rudolph, M. (2004). A novel method for characterizing synaptic noise in cortical neurons. Neurocomputing, 58-60:191–196.

[Destexhe et al., 2001] Destexhe, A., Rudolph, M., Fellous, J.-M., and Sejnowski, T. (2001). Fluctuating synaptic conductances recreate in-vivo-like activity in neocortical neurons. Neuroscience, 107:13–24.

[Ditlevsen and Samson, 2013] Ditlevsen, S. and Samson, A. (2013). Stochastic Biomathematical Models with Applications to Neuronal Modeling, chapter Introduction to Stochastic Models in Biology. Springer. | MR

[Ditlevsen and Samson, 2014] Ditlevsen, S. and Samson, A. (2014). Estimation in the partially observed stochastic Morris-Lecar neuronal model with particle filter and stochastic approximation methods. Annals of Applied Statistics, 8(2):674–702. | MR

[Forman and Sørensen, 2008] Forman, J. L. and Sørensen, M. (2008). The Pearson diffusions: A class of statistically tractable diffusion processes. Scand J Stat, 35(3):438–465. | MR | Zbl

[Gerstein and Mandelbrot, 1964] Gerstein, G. and Mandelbrot, B. (1964). Random walk models for the spike activity of a single neuron. Biophysical Journal, 4:41–68.

[Gerstner and Kistler, 2002] Gerstner, W. and Kistler, W. (2002). Spiking Neuron Models. Cambridge University Press. | MR

[Gerstner et al., 2014] Gerstner, W., Kistler, W. M., Naud, R., and Paninski, L. (2014). Neuronal Dynamics. From single neurons to networks and models of cognition. Cambridge University Press.

[Habib and Thavaneswaran, 1990] Habib, M. and Thavaneswaran, A. (1990). Inference for stochastic neuronal models. Applied Mathematics and Computation, 38(1):51–73. | MR | Zbl

[Hodgkin and Huxley, 1952] Hodgkin, A. and Huxley, A. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4):500–544.

[Hoepfner, 2007] Hoepfner, R. (2007). On a set of data for the membrane potential in a neuron. Mathematical Biosciences, 207(2):275–301. | MR | Zbl

[Huys et al., 2006] Huys, Q. J. M., Ahrens, M., and Paninski, L. (2006). Efficient estimation of detailed single-neuron models. Journal of Neurophysiology, 96(2):872–890.

[Huys and Paninski, 2009] Huys, Q. J. M. and Paninski, L. (2009). Smoothing of, and parameter estimation from, noisy biophysical recordings. PLOS Computational Biology, 5(5). | MR

[Iacus, 2008] Iacus, S. M. (2008). Simulation and Inference for Stochastic Differential Equations with R examples. Springer. | MR | Zbl

[Izhikevich, 2007] Izhikevich, E. M. (2007). Dynamical Systems in Neuroscience. The MIT Press, Cambridge, Massachesetts. | MR

[Jahn et al., 2011] Jahn, P., Berg, R. W., Hounsgaard, J., and Ditlevsen, S. (2011). Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process. Journal of Computational Neuroscience, 31:563–579. | MR

[Jensen et al., 2012] 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.

[Kloeden and Platen, 1992] Kloeden, P. and Platen, E. (1992). Numerical solution of stochastic differential equations. Springer. | MR | Zbl

[Kostuk et al., 2012] Kostuk, M., Toth, B. A., Meliza, C. D., Margoliash, D., and Abarbanel, H. D. I. (2012). Dynamical estimation of neuron and network properties II: path integral Monte Carlo methods. Biological Cybernetics, 106(3):155–167. | MR

[Laing and Lord, 2010] Laing, C. and Lord, G. J., editors (2010). Stochastic Methods in Neuroscience. Oxford University Press. | MR | Zbl

[Lanska and Lansky, 1998] Lanska, V. and Lansky, P. (1998). Input parameters in a one-dimensional neuronal model with reversal potentials. Biosystems, 48(1-3):123–129.

[Lansky, 1983] Lansky, P. (1983). Inference for the diffusion-models of neuronal-activity. Mathematical Biosciences, 67(2):247–260. | MR | Zbl

[Lansky et al., 2006] Lansky, P., Sanda, P., and He, J. (2006). The parameters of the stochastic leaky integrate-and-fire neuronal model. Journal of Computational Neuroscience, 21:211–223. | MR | Zbl

[Louis, 1982] Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society. Series B, 44(2):226–233. | MR | Zbl

[Morris and Lecar, 1981] Morris, C. and Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophys. J., 35:193–213.

[Øksendal, 2010] Øksendal, B. (2010). Stochastic Differential Equations: An Introduction with Applications. Springer. | MR | Zbl

[Paninski et al., 2010] Paninski, L., Ahmadian, Y., Ferreira, D. G., Koyama, S., Rad, K. R., Vidne, M., Vogelstein, J., and Wu, W. (2010). A new look at state-space models for neural data. Journal of Computational Neuroscience, 29(1-2, Sp. Iss. SI):107–126. | MR

[Paninski et al., 2005] Paninski, L., Pillow, J., and Simoncelli, E. (2005). Comparing integrate-and-fire models estimated using intracellular and extracellular data. Neurocomputing, 65:379–385.

[Paninski et al., 2012] Paninski, L., Vidne, M., DePasquale, B., and Ferreira, D. G. (2012). Inferring synaptic inputs given a noisy voltage trace via sequential Monte Carlo methods. Journal of Computational Neuroscience, 33(1):1–19. | MR

[Papaspiliopoulos et al., 2013] Papaspiliopoulos, O., Roberts, G., and Stramer, O. (2013). Data augmentation for diffusions. Journal of Computational and Graphical Statistics, 22:3:665D688. | MR

[Pedersen, 1994] Pedersen, A. R. (1994). Uniform residuals for discretely observed diffusion processes. Technical Report 292, Department of Theoretical Statistics, University of Aarhus.

[Picchini et al., 2008] Picchini, U., Ditlevsen, S., De Gaetano, A., and Lansky, P. (2008). Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal. Neural Computation, 20(11):2696–2714. | Zbl

[Pospischil et al., 2009a] Pospischil, M., Piwkowska, Z., Bal, T., and Destexhe, A. (2009a). Characterizing neuronal activity by describing the membrane potential as a stochastic process. Journal of Physiology-Paris, 103(1-2):98–106.

[Pospischil et al., 2009b] Pospischil, M., Piwkowska, Z., Bal, T., and Destexhe, A. (2009b). Extracting synaptic conductances from single membrane potential traces. Neuroscience, 158(2):545–52.

[Pospischil et al., 2007] Pospischil, M., Piwkowska, Z., Rudolph, M., Bal, T., and Destexhe, A. (2007). Calculating event-triggered average synaptic conductances from the membrane potential. Journal of Neurophysiology, 97(3):2544–2552.

[Prakasa Rao, 1999] Prakasa Rao, B. (1999). Statistical inference for diffusion type processes. Arnold. | MR | Zbl

[Roberts and Stramer, 2001] Roberts, G. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika, 88(3):603–621. | MR | Zbl

[Rudolph and Destexhe, 2003] Rudolph, M. and Destexhe, A. (2003). Characterization of subthreshold voltage fluctuations in neuronal membranes. Neural Computation, 15(11):2577–2618. | Zbl

[Rudolph et al., 2004a] Rudolph, M., Pelletier, J., Pare, D., and Destexhe, A. (2004a). Estimation of synaptic conductances and their variances from intracellular recordings of neocortical neurons in vivo. Neurocomputing, 58:387–392.

[Rudolph et al., 2004b] Rudolph, M., Piwkowska, Z., Badoual, M., Bal, T., and Destexhe, A. (2004b). A method to estimate synaptic conductances from membrane potential fluctuations. J Neurophysiol, 91(6):2884–96.

[Samson and Thieullen, 2012] Samson, A. and Thieullen, M. (2012). A contrast estimator for completely or partially observed hypoelliptic diffusion. Stochastic Processes and their Applications, 122(7):2521–2552. | MR | Zbl

[Sørensen, 2004] Sørensen, H. (2004). Parametric inference for diffusion processes observed at discrete points in time: a survey. International Statistical Review, 72(3):337–354.

[Sørensen, 2012] Sørensen, M. (2012). Statistical Methods for Stochastic Differential Equations, chapter Estimating functions for diffusion type processes. Chapman & Hall. | MR

[Tuckwell, 1988] Tuckwell, H. (1988). Introduction to theoretical neurobiology, Vol.2: Nonlinear and stochastic theories. Cambridge Univ. Press, Cambridge. | MR | Zbl