When the leak is weak – how the first-passage statistics of a biased random walk can approximate the ISI statistics of an adapting neuron

Sequences of first-passage times can describe the interspike intervals (ISI) between subsequent action potentials of sensory neurons. Here, we consider the ISI statistics of a stochastic neuron model, a leaky integrate-and-fire neuron, which is driven by a strong mean input current, white Gaussian current noise, and a spike-frequency adaptation current. In previous studies, it has been shown that without a leak current, i.e. for a so-called perfect integrate-and-fire (PIF) neuron, the ISI density can be well approximated by an inverse Gaussian corresponding to the first-passage-time density of a biased random walk. Furthermore, the serial correlations between ISIs, which are induced by the adaptation current, can be described by a geometric series. By means of stochastic simulations, we inspect whether these results hold true in the presence of a modest leak current. Specifically, we measure mean and variance of the ISI in the full model with leak and use the analytical results for the perfect IF model to relate these cumulants of the ISI to effective values of the mean input and noise intensity of an equivalent perfect IF model. This renormalization procedure yields semi-analytical approximations for the ISI density and the ISI serial correlation coeffcient in the full model with leak. We find that both in the absence and the presence of an adaptation current, the ISI density can be well approximated in this way if the leak current constitutes only a weak modification of the dynamics. Moreover, also the serial correlations of the model with leak are well reproduced by the expressions for a PIF model with renormalized parameters. Our results explain, why expressions derived for the rather special perfect integrate-and-fire model can nevertheless be often well fit to experimental data.