Journal of Computational Neuroscience

, Volume 21, Issue 2, pp 211–223 | Cite as

The parameters of the stochastic leaky integrate-and-fire neuronal model

Article

Abstract

Five parameters of one of the most common neuronal models, the diffusion leaky integrate-and-fire model, also known as the Ornstein-Uhlenbeck neuronal model, were estimated on the basis of intracellular recording. These parameters can be classified into two categories. Three of them (the membrane time constant, the resting potential and the firing threshold) characterize the neuron itself. The remaining two characterize the neuronal input. The intracellular data were collected during spontaneous firing, which in this case is characterized by a Poisson process of interspike intervals. Two methods for the estimation were applied, the regression method and the maximum-likelihood method. Both methods permit to estimate the input parameters and the membrane time constant in a short time window (a single interspike interval). We found that, at least in our example, the regression method gave more consistent results than the maximum-likelihood method. The estimates of the input parameters show the asymptotical normality, which can be further used for statistical testing, under the condition that the data are collected in different experimental situations. The model neuron, as deduced from the determined parameters, works in a subthreshold regimen. This result was confirmed by both applied methods. The subthreshold regimen for this model is characterized by the Poissonian firing. This is in a complete agreement with the observed interspike interval data.

Keywords

Leaky integrate-and-fire model Ornstein-Uhlenbeck neuronal model Parameters estimation Spontaneous firing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brillinger DR, Segundo JP (1979) Empirical examination of the threshold model of neuron firing. Biol. Cybern. 35: 213–220.Google Scholar
  2. Christodoulou C, Bugmann G (2000) Near poisson-type firing produced by concurrent excitation and inhibition. BioSystems 58: 41–48.Google Scholar
  3. Dayan P, Abbot LF (2001) Theoretical Neuroscience. MIT Press, Cambridge, MA.Google Scholar
  4. Ditlevsen S, Lansky P (2005) Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model. Phys. Rev. E 71: Art. No. 011907.Google Scholar
  5. Duchamp-Viret P, Kostal L, Chaput M, Lansky P, Rospars J-P (2005) Patterns of spontaneous activity in single rat olfactory receptor neurons are different in normally breathing and tracheotomized animals. J Neurobiol.Google Scholar
  6. Eggermont JJ, Smith GM, Bowman D (1993) Spontaneous burst firing in cat primary auditory-cortex—Age and depth dependency and its effect on neural interaction measures. J. Neurophysiol. 69: 1292–1313.Google Scholar
  7. Feigin P (1976) Maximum likelihood estimation for stochastic processes—A martingale approach. Adv. Appl. Probab. 8: 712–736.Google Scholar
  8. Gardiner CW (1983) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin.Google Scholar
  9. Gerstner W, Kistler W (2002) Spiking Neuron Models. Cambridge University Press, Cambridge.Google Scholar
  10. He J (2003) Slow oscillation in non-lemniscal auditory thalamus. J Neurosci. 23: 8281–8290.Google Scholar
  11. Inoue J, Sato S, Ricciardi LM (1995) On the parameter estimation for diffusion models of single neurons' activities. Biol. Cybern. 73: 209–221.Google Scholar
  12. Johnson DH (1996) Point process models of single-neuron discharges. J. Comput. Neurosci. 3: 275–300.Google Scholar
  13. Jolivet R, Rauch A, Lüscher H-R, Gerstner W (2006) Integrate-and-fire models with adaptation are good enough: Predicting spike times under random current injection. In: Y Weiss, B Schölkopf, J Platt, eds. Advances in Neural Information Processing Systems 18, MIT Press, Cambridge MA.Google Scholar
  14. Jones TA, Jones SM (2000) Spontaneous activity in the statoacoustic ganglion of the chicken embryo. J. Neurphysiol. 83: 1452–1468.Google Scholar
  15. Keat J, Reinagel P, Reid RC, Meister M (2001) Predicting every spike: A model for the responses of visual neurons. Neuron 30: 803–817.Google Scholar
  16. Kistler WM, Gerstner W, van Hemmen JL (1997) Reduction of the Hodgkin-Huxley equations to a single-variable threshold model. Neural Comput. 9: 1015–1045.Google Scholar
  17. Kloeden PE, Platen E (1992) Numerical Solution of Stochastic Differential Equations. Springer, Berlin.Google Scholar
  18. Koch C (1999) Biophysics of Computation: Information Processing in Single Neurons. Oxford University Press, Oxford.Google Scholar
  19. Koyama S, Shinomnoto S (2005) Empirical Bayes interpretation of random point events. J. Phys. A: Math. Gen. 38: L531–L537.Google Scholar
  20. La Camera G, Rauch A, Luscher HR, Senn W, Fusi S (2004) Minimal models of adapted neuronal response to in vivo-like input currents. Neural Comput. 16: 2101–2124.Google Scholar
  21. Lansky P (1983) Inference for the diffusion models of neuronal activity. Math. Biosci. 67: 247–260.Google Scholar
  22. Lansky P (1997) Sources of periodical force in noisy integrate-and-fire models of neuronal dynamics. Phys. Rev. E 55: 2040–2043.Google Scholar
  23. Lansky P, Lanska V (1987) Diffusion approximation of the neuronal model with synaptic reversal potentials. Biol. Cybern. 56:19–26.Google Scholar
  24. Lansky P, Giorno V, Nobile AG, Ricciardi LM (1998) A diffusion neuronal model and its parameters. In: LM Ricciardi, ed. Proceedings of International Workshop Biomathematics and related Computational Problems. Kluwer, Dordrecht.Google Scholar
  25. Lansky P, Lanska V (1994) First-passage-time problem for simulated stochastic diffusion processes. Comp. Biol. Med. 24: 91–101.Google Scholar
  26. Lansky P, Smith CE (1989) The effect of a random initial value in neural 1st-passage-time models. Math. Biosci. 93: 191–215.Google Scholar
  27. Laughlin SB (2001) Energy as a constraint on the coding and processing of sensory information. Curr. Opin. Neurobiol. 11: 475–480.Google Scholar
  28. Lin X, Chen SP (2000) Endogenously generated spontaneous spiking activities recorded from postnatal spiral ganglion neurons in vitro. Developmental Brain Res. 119: 297–305.Google Scholar
  29. Nobile AG, Ricciardi LM, Sacerdote L (1985) Exponential trends of Ornstein-Uhlenbeck 1st-passage-time densities. J. Appl. Prob. 22: 360–369.Google Scholar
  30. Paninski L, Pillow J, Simoncelli E (2005) Comparing integrate-and-fire models estimated using intracellular and extracellular data. Neurocomputing 65: 379–385.Google Scholar
  31. Paninski L, Pillow JW, Simoncelli EP (2004) Maximum likelihood estimation of a stochastic integrate-and-fire neural encoding model. Neural Comput. 16: 2533–2561.Google Scholar
  32. Pinsky PF, Rinzel J (1994) Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons 1: 39–60.Google Scholar
  33. Prakasa Rao BLS (1999) Statistical inference for diffusion type processes. Arnold, London.Google Scholar
  34. Rauch A, La Camera G, Luscher HR, Senn W, Fusi S (2003) Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents. J. Neurophysiol. 90: 1598–1612.Google Scholar
  35. Ricciardi LM, Lansky P (2003) Diffusion models of neuronal activity. In: MA Arbib, ed. The Handbook of the Brain Theory and Neural Networks, (2nd edn.) MIT Press, Cambridge, MA.Google Scholar
  36. Richardson MJE, Gerstner W (2005) Synaptic shot noise and conductance fluctuations affect the membrane voltage with equal significance Neural Comput. 17: 923–947.Google Scholar
  37. Rodriguez R, Lansky P (2000) Effect of spatial extension on noise enhanced phase-locking in a leaky integrate-and-fire model of a neuron. Phys. Rev. E 62: 8427–8437.Google Scholar
  38. Rospars J-P, Lansky P, Vaillant J, Duchamp-Viret P, Duchamp A (1994) Spontaneous activity of first- and second-order neurons in the olfactory system. Brain Res. 662: 31–44.Google Scholar
  39. Segev I (1992) Single neurone models: Oversimple, complex and reduced. TINS 15:414–421.Google Scholar
  40. Shinomoto S, Sakai Y, Funahashi S (1999) The ornstein-uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex. Neural Comp. 11: 935–951.Google Scholar
  41. Stein RB (1965) A theoretical analysis of neuronal variability. Biophys. J. 5: 173–195.Google Scholar
  42. Stevens CF, Zador AM (1998) Novel Integrate-and-fire-like Model of repetitive firing in cortical neurons. Proceedings of the 5th Joint Symposium on Neural Computation, UCSD, La Jolla CA.Google Scholar
  43. Stevens CF (1964) Letter to the editor. Biophys. J. 4: 417–419.Google Scholar
  44. Tateno T, Kawana A, Jimbo Y (2002) Analytical characterization of spontaneous firing in networks of developing rat cultured cortical neurons. Phys. Rev. E 65: Art. No. 051924.Google Scholar
  45. Tuckwell HC (1988) Introduction to Theoretical Neurobiology. Cambridge Univ. Press, Cambridge.Google Scholar
  46. Tuckwell HC, Lansky P (1997) On the simulation of biological diffusion processes. Comput. Biol. Med. 27: 1–7.Google Scholar
  47. Tuckwell HC, Richter W (1978) Neuronal interspike time distribution and the estimation of neurophysiological and neuroanatomical parameters. J. theor. Biol. 71: 167–183.Google Scholar
  48. Xiong Y, Yu YQ, Chan YS He J (2003) An in-vivo intracellular study of the auditory thalamic neurons. Thalamus Related Sys. 2: 253–260.Google Scholar
  49. Yu YQ, Xiong Y, Chan YS He JF (2004) Corticofugal gating of auditory information in the thalamus: An in vivo intracellular recording study. J. Neurosci. 24: 3060–3069.Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Institute of PhysiologyAcademy of Sciences of the Czech RepublicPragueCzech Republic
  2. 2.Department of Rehabilitation SciencesThe Hong Kong Polytechnic UniversityHung HomHong Kong

Personalised recommendations