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Biological Cybernetics

, Volume 107, Issue 1, pp 95–106 | Cite as

On dependency properties of the ISIs generated by a two-compartmental neuronal model

  • Elisa Benedetto
  • Laura Sacerdote
Original Paper

Abstract

One-dimensional leaky integrate and fire neuronal models describe interspike intervals (ISIs) of a neuron as a renewal process and disregarding the neuron geometry. Many multi-compartment models account for the geometrical features of the neuron but are too complex for their mathematical tractability. Leaky integrate and fire two-compartment models seem a good compromise between mathematical tractability and an improved realism. They indeed allow to relax the renewal hypothesis, typical of one-dimensional models, without introducing too strong mathematical difficulties. Here, we pursue the analysis of the two-compartment model studied by Lansky and Rodriguez (Phys D 132:267–286, 1999), aiming of introducing some specific mathematical results used together with simulation techniques. With the aid of these methods, we investigate dependency properties of ISIs for different values of the model parameters. We show that an increase of the input increases the strength of the dependence between successive ISIs.

Keywords

Two-compartment neural model  ISI dependency properties 

Notes

Acknowledgments

This work was supported in part by MIUR Project PRIN-Cofin 2008. The authors are grateful to Petr Lansky for useful suggestions and to the anonymous referees for their constructive comments.

References

  1. Benedetto E, Sacerdote L, Zucca C (2013) A first passage problem for a bivariate diffusion process: numerical solution with an application to neuroscience. J comput Appl Math 242:41–52Google Scholar
  2. Bressloff PC (1995) Dynamics of a compartmental integrate-and-fire neuron without dendritic potential reset. Phys D 80:399–412Google Scholar
  3. Burkitt AN (2006a) A review of the integrate and fire neuron model: I. Homogeneous synaptic input. Biol Cybern 95:1–19Google Scholar
  4. Burkitt AN (2006b) A review of the integrate and fire neuron model: II. Inhomogeneous synaptic input and network properties. Biol Cybern 95:97–112PubMedCrossRefGoogle Scholar
  5. Bush PC, Sejnowski TJ (1993) Reduced compartmental models of neocortical pyramidal cells. J Neurosci Method 46:159–166CrossRefGoogle Scholar
  6. De Schutter E, Bower JM (1994) An active membrane model of the cerebellar Purkinje cell. J Neurophysiol 71:375–400PubMedGoogle Scholar
  7. Ditlevsen S, Greenwood P (2012) The Morris–Lecar neuron model embeds a leaky integrate-and-fire model. J math Biol. doi: 10.1007/s00285-012-0552-7
  8. Ferguson KA, Campbell SA (2009) A two compartment model of a CA1 pyramidal neuron. Can Appl Math Q 17(2):293–307Google Scholar
  9. Fredricks GA, Nelsen RB (2007) On the relationship between Spearman’s rho and Kendall’s tau for pairs of continuous random variables. J Stat Plan Inference 137(7):2143–2150CrossRefGoogle Scholar
  10. Folland GB (1999) Real analysis: modern techniques and their applications. Wiley, New YorkGoogle Scholar
  11. Giraudo MT, Greenwood P, Sacerdote L (2011) How sample paths of leaky integrate-and-fir models are influenced by the presence of a firing threshold. Neural Comput 23(7):1743–1767PubMedCrossRefGoogle Scholar
  12. Godfrey K (1983) Compartmental models and their application. Academic Press, OrlandoGoogle Scholar
  13. Kendall MG (1938) A new measure of rank correlation. Biometrika 30(1/2):81–93CrossRefGoogle Scholar
  14. Kohn AF (1989) Dendritic transformations on random synaptic inputs as measured from a neuron’s spike train: modeling and simulation. IEEE Trans Biomed Eng 36:44–54PubMedCrossRefGoogle Scholar
  15. Lansky P, Rodriguez R (1999) Two-compartment stochastic model of a neuron. Phys D 132:267–286CrossRefGoogle Scholar
  16. Lansky P, Rospars JP (1993) Stochastic model neuron without resetting of dendritic potential. Application to the olfactory system. Biol Cybern 69:283–294PubMedCrossRefGoogle Scholar
  17. Lansky P, Rospars JP (1995) Ornstein–Uhlenbeck model neuron revisited. Biol Cybern 72:397–406CrossRefGoogle Scholar
  18. Lansky P, Ditlevsen S (2008) A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models. Biol Cybern 99:253–262PubMedCrossRefGoogle Scholar
  19. Mino H, Grill WM (2000) Modeling of mammalian myelinated nerve with stochastic sodium ionic channels. In: Engineering in medicine and biology society, Proceedings of the 22nd annual international conference of the IEEE, vol 2, pp 915–917.Google Scholar
  20. Nawrot MP (2010) Analysis and interpretation of interval and count variability in neural spike trains. In: Gruen S, Rotter S (eds) Analysis of parallel spike trains. Springer, New York, pp 37–58CrossRefGoogle Scholar
  21. Nelsen RB (1999) An introduction to copulas. Springer, New YorkCrossRefGoogle Scholar
  22. Ricciardi LM, Sacerdote L (1979) The Ornstein–Uhlenbeck process as a model for neuronal activity. Biol Cybern 35:1–9PubMedCrossRefGoogle Scholar
  23. Sacerdote L, Giraudo MT (2012) Leaky integrate and fire models: a review on mathematicals methods and their applications. Lecture Notes in Mathematics, vol. 2058. Springer, pp 95–142Google Scholar
  24. Shinomoto S, Shima K, Tanji J (2003) Differences in spiking patterns among cortical neurons. Neural Comput 15:2823–2842PubMedCrossRefGoogle Scholar
  25. Shinomoto S, Kim H, Shimokawa T, Matsuno N, Funahashi S, Shima K, Fujita I, Tamura H, Doi T, Kawano K, Inaba N, Fukushima K, Kurkin S, Kurata K, Taira M, Tsutsui K, Komatsu H, Ogawa T, Koida K, Tanji J, Toyama K (2009) Relating neuronal firing patterns to functional differentiation of cerebral cortex. PLoS Comput Biol 5:e1000433 Google Scholar
  26. Sklar A (1959) Functions de repartition a n dimensions et leurs marges, vol 8. Publications of the Institute of Statistics of the University of Paris, Paris, pp 229–231Google Scholar
  27. Traub RD, Wong RKS, Miles R, Michelson H (1973) A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. J Neurophysiol 66(2): 635–650 (1991). Kybernetika 9(6):449–460Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics “G. Peano”University of TorinoTorinoItaly

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