Biological Cybernetics

, 99:253 | Cite as

A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models

Review

Abstract

Parameters in diffusion neuronal models are divided into two groups; intrinsic and input parameters. Intrinsic parameters are related to the properties of the neuronal membrane and are assumed to be known throughout the paper. Input parameters characterize processes generated outside the neuron and methods for their estimation are reviewed here. Two examples of the diffusion neuronal model, which are based on the integrate-and-fire concept, are investigated—the Ornstein–Uhlenbeck model as the most common one and the Feller model as an illustration of state-dependent behavior in modeling the neuronal input. Two types of experimental data are assumed—intracellular describing the membrane trajectories and extracellular resulting in knowledge of the interspike intervals. The literature on estimation from the trajectories of the diffusion process is extensive and thus the stress in this review is set on the inference made from the interspike intervals.

Keywords

Ornstein–Uhlenbeck Statistical inference Feller process First-passage times Maximum likelihood Moment method Laplace transform Fortets integral equation Interspike intervals 

References

  1. Aalen O, Gjessing H (2004) Survival models based on the Ornstein-Uhlenbeck process. Lifetime Data Anal 10: 407–423CrossRefPubMedGoogle Scholar
  2. Alili L, Patie P, Pedersen J (2005) Representations of the first hitting time density of an Ornstein-Uhlenbeck process. Stochastic Models 21: 967–980CrossRefGoogle Scholar
  3. Bibby B, Sørensen M (1996) On estimation for discretely observed diffusions: A review. Theory Stochastic Process 2: 49–56Google Scholar
  4. Borodin A, Salminen P (2002) Handbook of Brownian motion—Facts and Formulae. Probability and its applications. Birkhauser Verlag, BaselGoogle Scholar
  5. Brillinger D (1988) Maximum likelihood analysis of spike trains of interacting nerve cells. Biol Cybern 59: 189–200CrossRefPubMedGoogle Scholar
  6. Brunel N, van Rossum M (2008) Lapicque’s 1907 paper: from frogs to integrate-and-fire. Biol Cybern 97:337–339CrossRefGoogle Scholar
  7. Bulsara A, Elston T, Doering C, Lowen S, Lindberg K (1996) Cooperative behavior in periodically driven noisy integrate-and-fire models of neuronal dynamics. Phys Rev E 53: 3958–3969CrossRefGoogle Scholar
  8. Burkitt A (2006) A review of the integrate-and-fire neuron model: I. homogeneous synaptic input. Biol Cybern 95: 1–19CrossRefPubMedGoogle Scholar
  9. Clopath C, Jolivet R, Rauch A, Luscher HR, Gerstner W (2007) Predicting neuronal activity with simple models of the threshold type: Adaptive exponential integrate-and-fire model with two compartments. Neurocomput 70: 1668–1673CrossRefGoogle Scholar
  10. Cox J, Ingersoll J, Ross S (1985) A theory of the term structure of interest rates. Econometrica 53: 385–407CrossRefGoogle Scholar
  11. Ditlevsen S (2007) A result on the first-passage time of an Ornstein-Uhlenbeck process. Stat Probab Lett 77: 1744–1749CrossRefGoogle Scholar
  12. Ditlevsen S, Ditlevsen O (2008) Parameter estimation from observations of first-passage times of the Ornstein-Uhlenbeck process and the Feller process. Prob Eng Mech 23: 170–179CrossRefGoogle Scholar
  13. Ditlevsen S, Lansky P (2005) Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model. Phys Rev E 71:Art. No. 011,907Google Scholar
  14. Ditlevsen S, Lansky P (2006) Estimation of the input parameters in the Feller neuronal model. Phys Rev E 73:Art. No. 061,910Google Scholar
  15. Ditlevsen S, Lansky P (2007) Parameters of stochastic diffusion processes estimated from observations of first hitting-times: application to the leaky integrate-and-fire neuronal model. Phys Rev E 76:Art. No. 041,906Google Scholar
  16. Ditlevsen S, Lansky P (2008) Comparison of statistical methods for estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model from first-passage times data. American Institute of Physics Proceedings Series (to appear)Google Scholar
  17. Durbin J (1971) Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J Appl Prob 8: 431–453CrossRefGoogle Scholar
  18. Feller W (1951) Diffusion processes in genetics. In: Neyman J (eds) Proceedings of the second Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp 227–246Google Scholar
  19. Fortet R (1943) Les fonctions aléatories du type de markoff associées à certaines équations lineáires aux dérivées partiell es du type parabolique. J Math pures Appl 22: 177–243Google Scholar
  20. Gerstein G, Mandelbrot B (1964) Random walk models for the spike activity of a single neuron. Biophys J 4: 41–68CrossRefPubMedGoogle Scholar
  21. Giorno V, Lansky P, Nobile A, Ricciardi L (1988) Diffusion approximation and first-passage-time problem for a model neuron. Biol Cybern 58: 387–404CrossRefPubMedGoogle Scholar
  22. Hanson F, Tuckwell H (1983) Diffusion approximations for neuronal activity including synaptic reversal potentials. J Theor Neurobiol 2: 127–153Google Scholar
  23. Inoue J, Sato S, Ricciardi L (1995) On the parameter estimation for diffusion models of single neurons activity. Biol Cybern 73: 209–221CrossRefPubMedGoogle Scholar
  24. Johannesma P (1968) Diffusion models for the stochastic activity of neurons. In: Caianiello E (eds) Proceedings of the school on neural networks june 1967 in Ravello. Springer, Berlin, pp 116–144Google Scholar
  25. Jolivet R, Rauch A, Luscher HR, Gerstner W (2006) Predicting spike timing of neocortical pyramidal neurons by simple threshold models. J Comput Neurosci 21: 35–49CrossRefPubMedGoogle Scholar
  26. Jolivet R, Kobayashi R, Rauch A, Naud R, Shinomoto S, Gerstner W (2008) A benchmark test for a quantitative assessment of simple neuron models. J Neurosci Methods 169: 417–424CrossRefPubMedGoogle Scholar
  27. Karlin S, Taylor H (1981) A second course in stochastic processes. Academic Press, San DiegoGoogle Scholar
  28. Kostal L, Lansky P, Rospars JP (2007a) Neuronal coding and spiking randomness. Eur J Neurosci 26: 2693–2701CrossRefPubMedGoogle Scholar
  29. Kostal L, Lansky P, Zucca C (2007b) Randomness and variability of the neuronal activity described by the Ornstein-Uhlenbeck model. Netw Comput Neural Syst 18: 63–75CrossRefGoogle Scholar
  30. Kutoyants Y (2003) Statistical inference for ergodic diffusion processes. Springer Series in Statistics, New YorkGoogle Scholar
  31. Lansky P, Lanska V (1987) Diffusion approximations of the neuronal model with synaptic reversal potentials. Biol Cybern 56: 19–26CrossRefPubMedGoogle Scholar
  32. Lansky P, Sacerdote L, Tomasetti F (1995) On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity. Biol Cybern 73: 457–465CrossRefPubMedGoogle Scholar
  33. Lansky P, Sanda P, He J (2006) The parameters of the stochastic leaky integrate-and-fire neuronal model. J Comput Neurosci 21: 211–223CrossRefPubMedGoogle Scholar
  34. Lebedev N (1972) Special functions and their applications. Dover, New YorkGoogle Scholar
  35. Mullowney P, Iyengar S (2008) Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data. J Comput Neurosci 24: 179–194CrossRefPubMedGoogle Scholar
  36. Nobile A, Ricciardi L, Sacerdote L (1985) Exponential trends of Ornstein-Uhlenbeck 1st-passage-time densities. J Appl Prob 22: 360–369CrossRefGoogle Scholar
  37. Paninski L, Pillow J, Simoncelli E (2004) Maximum likelihood estimation of a stochastic integrate-and-fire neural encoding model. Neural Comput 16: 2533–2561CrossRefPubMedGoogle Scholar
  38. Pawlas Z, Klebanov L, Prokop M, Lansky P (2008) Parameters of spike trains observed in a short time window. Neural Comput (in press)Google Scholar
  39. Picchini U, Lansky P, De Gaetano A, Ditlevsen S (2008) Parameters of the diffusion leaky integrate-and fire neuronal model for a slowly fluctuating signal. Neural Comput (to appear)Google Scholar
  40. Prakasa Rao B (1999) Statistical inference for diffusion type processes. ArnoldGoogle Scholar
  41. Rauch A, G L, Luscher HR, Senn W, Fusi S (2003) Neocortical pyramidal cells respond as integrate-and fire neurons in vivo-like input currents. J Neurophysiol 90: 1598–1612CrossRefPubMedGoogle Scholar
  42. Ricciardi L (1977) Diffusion processes and related topics in biology. Springer, BerlinGoogle Scholar
  43. Ricciardi L, Sacerdote L (1979) The Ornstein-Uhlenbeck process as a model of neuronal activity. Biol Cybern 35: 1–9CrossRefPubMedGoogle Scholar
  44. Ricciardi L, Sato S (1988) First-passage-time density and moments of the Ornstein-Uhlenbeck process. J Appl Prob 25: 43–57CrossRefGoogle Scholar
  45. Ricciardi L, Di Crescenzo A, Giorno V, Nobile A (1999) An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling. Math Japonica 50(2): 247–322Google Scholar
  46. Shinomoto S, Sakai Y, Funahashi S (1999) The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex. Neural Comput 11: 935–951CrossRefPubMedGoogle Scholar
  47. Siegert A (1951) On the first passage time probability problem. Phys Rev 81: 617–623CrossRefGoogle Scholar
  48. Tuckwell H (1988) Introduction to theoretical neurobiology, vol. 2: Nonlinear and stochastic theories. Cambridge University Press, CambridgeGoogle Scholar
  49. Tuckwell H, Richter W (1978) Neuronal interspike time distributions and the estimation of neurophysiological and neuroanatomical parameters. J Theor Biol 71: 167–180CrossRefPubMedGoogle Scholar
  50. Wan F, Tuckwell H (1982) Neuronal firing and input variability. J Theoret Neurobiol 1: 197–218Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institute of PhysiologyAcademy of Sciences of the Czech RepublicPragueCzech Republic
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

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