Neuronal spike trains with exponential decay

Stochastic models for the spike discharge activity of neurons are analyzed. Model I treats only excitatory impulses occurring as Poisson events with exponential density of the jump magnitude, and, in the absence of these events, the subthreshold potential decays exponentially. A closed solution for the Laplace transform of the renewal density of the discharge is obtained by the imbedding method. Model III takes care of both excitatory and inhibitory impulses with exponential decay of the subthreshold potential between the jumps. Analytical closed solutions for the first passage problem are obtained in this care also. The mean interval time between the spikes and the stationary density of firing are obtained.