Firing Rate Analysis for an Integrate-and-Fire Neuronal Model

We investigate a stochastic linear integrate-and-fire (IF) neuronal model and use the corresponding Fokker-Planck equation (FPE) to study the mean firing rate of a population of IF neurons. The firing rate(or emission rate) function, is given in terms of an eigenfunction expansion solution of the FPE. We consider two parameter regimes of current input and prove the existence of infinitely many branches of eigenvalues and derive their asymptotic properties. We use the eigenfunction expansion solution to prove asymptotic properties of the firing rate function. We also perform a numerical experiment of 10,000 IF neurons and show that our simulation is in agreement with our theoretical results. Finally, we state several open problems for future research.