An analytic solution of the reentrant Poisson master equation and its application in the simulation of large groups of spiking neurons

Population density techniques are statistical methods to describe large populations of spiking neurons. They describe the response of such a population to a stochastic input. These techniques are sometimes defined as the interaction of neuronal dynamics and a Poisson point process. In earlier work I showed that one can transform away neuronal dynamics, which leaves only the problem of solving the master equation for the Poisson point process. Previously, I used a numerical solution for the master equation. In this work, I will present an analytic solution, which is based on a formal solution by Sirovich (2003). I will show that using this solution for solving the population density equation results in a much faster and manifestly stable algorithm.