First passage time distribution for spiking neuron with delayed excitatory feedback

A class of spiking neuronal models with threshold 2 is considered. It is defined by a set of conditions typical for basic threshold-type models, such as the leaky integrate-and-fire (LIF) or the binding neuron model and also for some artificial neurons. A neuron is stimulated with a Poisson stream of excitatory impulses. Each output impulse is conveyed through the feedback line to the neuron input after finite delay $\Delta$. This impulse is identical to those delivered from the input stream. We have obtained a general relation allowing calculating exactly the probability density function (PDF) $p(t)$ for distribution of the first passage time of crossing the threshold, which is the distribution of output interspike intervals (ISI) values for this neuron. The calculation is based on known PDF $p^0(t)$ for that same neuron without feedback, intensity of the input stream $\lambda$ and properties of the feedback line. Also, we derive exact relation for calculating the moments of $p(t)$ based on known moments of $p^0(t)$. The obtained general expression for $p(t)$ is checked numerically using Monte Carlo simulation for the case of LIF model. The course of $p(t)$ has a $\delta$-function type peculiarity. This fact contributes to the discussion about the possibility to model neuronal activity with Poisson process, supporting the "no" answer.