Advertisement

Journal of Computational Neuroscience

, Volume 28, Issue 2, pp 267–284 | Cite as

Multiplicatively interacting point processes and applications to neural modeling

  • Stefano CardanobileEmail author
  • Stefan Rotter
Article

Abstract

We introduce a nonlinear modification of the classical Hawkes process allowing inhibitory couplings between units without restrictions. The resulting system of interacting point processes provides a useful mathematical model for recurrent networks of spiking neurons described as Wiener cascades with exponential transfer function. The expected rates of all neurons in the network are approximated by a first-order differential system. We study the stability of the solutions of this equation, and use the new formalism to implement a winner-takes-all network that operates robustly for a wide range of parameters. Finally, we discuss relations with the generalised linear model that is widely used for the analysis of spike trains.

Keywords

Poisson process Spiking neuron Interacting random process Recurrent network dynamics Generalised linear model Winner-takes-all 

Notes

Acknowledgements

We wish to thank an anonymous reviewer of the paper for discovering a flaw in our derivation of the master equation, and for suggesting the appropriate correction.

References

  1. Amit, D. J., & Brunel, N. (1997). Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cerebral Cortex, 7, 237–252.CrossRefPubMedGoogle Scholar
  2. Benci, V., & Di Nasso, M. (2003). Alpha-theory: An elementary axiomatics for nonstandard analysis. Expositiones Mathematicae, 21(4), 355–386.CrossRefGoogle Scholar
  3. Benci, V., Galatolo, S., & Ghimenti, M. (2008). An elementary approach to stochastic differential equations using the infinitesimals. In V. Bergelson, A. Blass, M. D. Nasso, & R. Jin (Eds.), Ultrafilters across mathematics, Cont. Math. AMS. http://arxiv.org/abs/0807.3477.
  4. Blake, R. (1989). A neural theory of binocular rivalry. Psychological Review, 96, 145–167.CrossRefPubMedGoogle Scholar
  5. Carandini, M. (2004). Amplification of trial-to-trial response variability by neurons in visual cortex. PLoS Biology,2, 1483–1493.CrossRefGoogle Scholar
  6. Destexhe, A., Rudolph, M., & Pare, D. (2003). The high-conductance state of neocortical neurons in vivo. Nature Reviews. Neuroscience, 4(9), 739–751 (2003). doi: 10.1038/nrn1198.CrossRefPubMedGoogle Scholar
  7. Fahle, M., & Palm, G. (1991). Perceptual rivalry between illusory and real contours. Biological Cybernetics, 66(1), 1–8.CrossRefPubMedGoogle Scholar
  8. Fukai, T., & Tanaka, S. (1997). A simple neural network exhibiting selective activation of neuronal ensembles: From winner-take-all to winners-share-all. Neural Computation, 9(1), 77–97.CrossRefPubMedGoogle Scholar
  9. Gerstner, W., & Kistler, W. M. (2002). Spiking neuron models. Single neurons, populations, plasticity. Cambridge: Cambridge University Press.Google Scholar
  10. Hawkes, A. G. (1971a). Point spectra of some mutually exciting point processes. Journal of the Royal Statistical Society. Series B, 33, 438–443.Google Scholar
  11. Hawkes, A. G. (1971b). Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58, 83–90.CrossRefGoogle Scholar
  12. Ilyashenko, Y. (2002). Centennial history of Hilbert’s 16th problem. Bulletin of the American Mathematical Society (New Series), 39(3), 301–354 (electronic).CrossRefGoogle Scholar
  13. Johnson, D. H. (1996). Point process models of single-neuron discharges. Journal of Computational Neuroscience, 3, 275–299.CrossRefPubMedGoogle Scholar
  14. Jolivet, R., Rauch, A., Lüscher, H., & Gerstner, W. (2006). Predicting spike timing of neocortical pyramidal neurons by simple threshold models. Journal of Computational Neuroscience, 21(1), 35–49. doi: 10.1007/s10827-006-7074-5.CrossRefPubMedGoogle Scholar
  15. Kriener, B., Tetzlaff, T., Aertsen, A., Diesmann, M., & Rotter, S. (2008). Correlations and population dynamics in cortical networks. Neural Computation, 20(9), 2185–2226.CrossRefPubMedGoogle Scholar
  16. Muller, E., Buesing, L., Schemmel, J., & Meier, K. (2007). Spike-frequency adapting neural ensembles: Beyond mean adaptation and renewal theories. Neural Computation, 19(11), 2958–3010. doi: 10.1162/neco.2007.19.11.2958. PMID: 17883347.CrossRefPubMedGoogle Scholar
  17. Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.CrossRefGoogle Scholar
  18. Nelson, E. (1987). Radically elementary probability theory. In Annals of mathematics studies (Vol. 117). Princeton: Princeton University Press.Google Scholar
  19. Ogata, Y. (1999). Seismicity analysis through point-process modeling: A review. Pure and Applied Geophysics, 155, 471–507.CrossRefGoogle Scholar
  20. Paninski, L. (2004). Maximum likelihood estimation of cascade point-process neural encoding models. Network: Comput. Neural Syst., 15, 243–262.CrossRefGoogle Scholar
  21. Pillow, J. W., Shlens, J., Paninski, L., Sher, A., Litke, A. M., Chichilnisky, E. J., et al. (2008). Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature, 454(7207), 995–999. doi: 10.1038/nature07140.CrossRefPubMedGoogle Scholar
  22. Rotter, S. (1994). Wechselwirkende stochastische Punktprozesse als Modell für neuronale Aktivität im Neocortex der Säugetiere. In Reihe physik (Vol. 21). Harri Deutsch Verlag.Google Scholar
  23. Rotter, S., Heck, D., & Aertsen, A. (1996). Spatio-temporal patterns of activity in cortical networks. In J. Bower (Ed.), Computational neuroscience: Trends in research 1995 (p. 261). London: Academic Press.Google Scholar
  24. Toyoizumi, T., Rad, K. R., & Paninski, L. (2009). Mean-field approximations for coupled populations of generalized linear model spiking neurons with Markov refractoriness. Neural Computation, 21, 1–41.CrossRefGoogle Scholar
  25. Truccolo, W., Eden, U. T., Fellows, M. R., Donoghue, J. P., & Brown, E. D. (2005). A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. Journal of Neurophysiology, 93, 1074–1089.CrossRefPubMedGoogle Scholar
  26. van den Boogaard, H. (1988). System identification based on point processes and correlation densities. II: The refractory neuron model. Mathematical Biosciences, 91(1), 35–65.CrossRefGoogle Scholar
  27. van den Boogaard, H., Hesselmans, G., & Johannesma, P. (1986). System identification based on point processes and correlation densities. I: The nonrefractory neuron model. Mathematical Biosciences, 80(2), 143–171.CrossRefGoogle Scholar
  28. Wiener, N. (1958). Nonlinear problems in random theory. Cambridge: MIT.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.BCCN & Faculty of BiologyAlbert-Ludwig University FreiburgFreiburgGermany

Personalised recommendations