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

J.
Computational Neuroscience 24, 2008: 69-79.

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.

Reprint | Related work
on integrate-and-fire neurons | Liam Paninski's research page