Gauss-Markov Processes for Neuronal Models Including Reversal Potentials

Gauss-Markov processes, restricted from below by a reflecting boundary, are here used to construct inhomogeneous leaky integrate-and-fire (LIF) stochastic models for single neuron's activity in the presence of a reversal hyperpolarization potential and different input signals. Under suitable assumptions, we are able to obtain the transition probability density function with a view to determine numeric, simulated and asymptotic solutions for the firing densities when the input signal is constant, decays exponentially or is a periodic function. The our results suggest the importance of the position of the lower boundary as well as that of the firing threshold when one studies the statistical properties of LIF neuron models.