Journal of Computational Neuroscience

, Volume 24, Issue 1, pp 69–79

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


DOI: 10.1007/s10827-007-0042-x

Cite this article as:
Paninski, L., Haith, A. & Szirtes, G. J Comput Neurosci (2008) 24: 69. doi:10.1007/s10827-007-0042-x


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 

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

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