In the previous subsection, we calculated the probability of firing in a short time step from the continuous-time firing intensity . Here we ask a similar question, but not on the level of a single spike, but on that of a full spike train.
Suppose we know that a spike train has been generated by an escape noise process
as in Eq. (9.2), where the membrane potential arises from the dynamics of one of the generalized integrate-and-fire models such as the SRM.
The likelihood that spikes occur at the times is (69)
where is the observation interval. The product on the right-hand side contains the momentary firing intensity at the firing times . The exponential factor takes into account that the neuron needs to ‘survive’ without firing in the intervals between the spikes.
Intuitively speaking, the likelihood of finding spikes at times depends on the instantaneous rate at the time of the spikes and the probability to survive the intervals in between without firing; cf. the survivor function introduced in Chapter 7, Eq. (7.26).
Instead of the likelihood, it is sometimes more convenient to work with the logarithm of the likelihood, called the log-likelihood
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