Journal of Computational Neuroscience

, Volume 24, Issue 1, pp 69–79 | Cite as

Integral equation methods for computing likelihoods and their derivatives in the stochastic integrate-and-fire model



We recently introduced likelihood-based methods for fitting stochastic integrate-and-fire models to spike train data. The key component of this method involves the likelihood that the model will emit a spike at a given time t. Computing this likelihood is equivalent to computing a Markov first passage time density (the probability that the model voltage crosses threshold for the first time at time t). Here we detail an improved method for computing this likelihood, based on solving a certain integral equation. This integral equation method has several advantages over the techniques discussed in our previous work: in particular, the new method has fewer free parameters and is easily differentiable (for gradient computations). The new method is also easily adaptable for the case in which the model conductance, not just the input current, is time-varying. Finally, we describe how to incorporate large deviations approximations to very small likelihoods.


Volterra integral equation Markov process Large deviations approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Boyd, S., & Vandenberghe, L. (2004). Convex optimization. New York: Oxford University Press.Google Scholar
  2. Buoncore, A., Nobile, A., & Ricciardi, L. (1987). A new integral equation for the evaluation of first-passage-time probability densities. Advances in Applied Probability, 19, 784–800.CrossRefGoogle Scholar
  3. Burkitt, A., & Clark, G. (1999). Analysis of integrate-and-fire neurons: Synchronization of synaptic input and spike output. Neural Computation, 11, 871–901.PubMedCrossRefGoogle Scholar
  4. Dembo, A., & Zeitouni, O. (1993). Large deviations techniques and applications. New York: Springer.Google Scholar
  5. DiNardo, E., Nobile, A., Pirozzi, E., & Ricciardi, L. (2001). A computational approach to first-passage-time problems for Gauss–Markov processes. Advances in Applied Probability, 33, 453–482.CrossRefGoogle Scholar
  6. Freidlin, M., & Wentzell, A. (1984). Random perturbations of dynamical systems. Berlin Heidelberg New York: Springer.Google Scholar
  7. Genz, A. (1992). Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics, 1, 141–149.CrossRefGoogle Scholar
  8. Haith, A. (2004). Estimating the parameters of a stochastic integrate-and-fire neural model. Master’s thesis, University of Edinburgh.Google Scholar
  9. Haskell, E., Nykamp, D., & Tranchina, D. (2001). Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics. Network: Computation in Neural Systems, 12, 141–174.CrossRefGoogle Scholar
  10. Iyengar, S., & Liao, Q. (1997). Modeling neural activity using the generalized inverse Gaussian distribution. Biological Cybernetics, 77, 289–295.PubMedCrossRefGoogle Scholar
  11. Jolivet, R., Lewis, T., & Gerstner, W. (2004). Generalized integrate-and-fire models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy. Journal of Neurophysiology, 92, 959–976.PubMedCrossRefGoogle Scholar
  12. Karatzas, I., & Shreve, S. (1997). Brownian motion and stochastic calculus. Berlin Heidelberg New York: Springer.Google Scholar
  13. Karlin, S., & Taylor, H. (1981). A second course in stochastic processes. New York: Academic.Google Scholar
  14. Knight, B., Omurtag, A., & Sirovich, L. (2000). The approach of a neuron population firing rate to a new equilibrium: an exact theoretical result. Neural Computation, 12, 1045–1055.PubMedCrossRefGoogle Scholar
  15. Lamm, P. (2000). A survey of regularization methods for first-kind Volterra equations. In Surveys on solution methods for inverse problems (pp. 53–82). Berlin Heidelberg New York: Springer.Google Scholar
  16. Mainen, Z., & Sejnowski, T. (1995). Reliability of spike timing in neocortical neurons. Science, 268, 1503–1506.PubMedCrossRefGoogle Scholar
  17. Paninski, L. (2006). The most likely voltage path and large deviations approximations for integrate-and-fire neurons. Journal of Computational Neuroscience, 21, 71–87.PubMedCrossRefGoogle Scholar
  18. Paninski, L., Pillow, J., & Simoncelli, E. (2004a). Comparing integrate-and-fire-like models estimated using intracellular and extracellular data. Neurocomputing, 65, 379–385.CrossRefGoogle Scholar
  19. Paninski, L., Pillow, J., & Simoncelli, E. (2004b). Maximum likelihood estimation of a stochastic integrate-and-fire neural model. Neural Computation, 16, 2533–2561.CrossRefGoogle Scholar
  20. Pillow, J., Paninski, L., Uzzell, V., Simoncelli, E., & Chichilnisky, E. (2005). Accounting for timing and variability of retinal ganglion cell light responses with a stochastic integrate-and-fire model. Journal of Neuroscience, 25, 11003–11013.PubMedCrossRefGoogle Scholar
  21. Plesser, H., & Gerstner, W. (2000). Noise in integrate-and-fire neurons: From stochastic input to escape rates. Neural Computation, 12, 367–384.PubMedCrossRefGoogle Scholar
  22. Plesser, H., & Tanaka, S. (1997). Stochastic resonance in a model neuron with reset. Physics Letters. A, 225, 228–234.CrossRefGoogle Scholar
  23. Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. (1992). Numerical recipes in C. Cambridge: Cambridge University Press.Google Scholar
  24. Ricciardi, L. (1977). Diffusion processes and related topics in biology. Berlin Heidelberg New York: Springer.Google Scholar
  25. Schervish, M. (1995). Theory of statistics. New York: Springer.Google Scholar
  26. Siegert, A. (1951). On the first passage time probability problem. Physical Review, 81, 617–623.CrossRefGoogle Scholar
  27. Stevens, C., & Zador, A. (1998). Novel integrate-and-fire-like model of repetitive firing in cortical neurons. In Proc. 5th Joint Symp. Neural Computation, UCSD.Google Scholar
  28. Tuckwell, H. (1989). Stochastic processes in the neurosciences. Philadelphia, PA: SIAM.Google Scholar
  29. van der Vaart, A. (1998). Asymptotic statistics. Cambridge: Cambridge University Press.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of StatisticsColumbia UniversityNew YorkUSA
  2. 2.Institute for Perception, Action and BehaviourUniversity of EdinburghEdinburghUK
  3. 3.Center for Theoretical NeuroscienceColumbia UniversityNew YorkUSA

Personalised recommendations