Time-dependent barriers and first-passage times--different types of neuronal models

To study the phenomenon of spike discharge in a single neuron, first-passage time density of the membrane potential to cross or reach the threshold value in different situations is analyzed. The neuron is taken as a leaky integrator and the inputs are due to Poisson events occurring independently. Closed solutions using the compensation method and imbedding techniques are arrived at. Using Keilson's limiting procedure, diffusion processes with boundaries are solved. To simulate the refractory period of the neuron, thresholds moving linearly or exponentially with time are considered. Very interesting Wald identities are obtained in the case of jump processes which go over to the corresponding identities for diffusion processes obtained by a limiting procedure.