A continuous Markovian model for neuronal activity

A diffusion model for the description of neurons' membrane potential fluctuations is proposed. Though retaining the well known feature consisting of the spontaneous exponential decay of the membrane potential to its resting value, the model discussed differs substantially from the ones in the current literature. Moreover, the Fokker-Planck equation now describing the membrane potential fluctuations is singular. The neuron's firing times probability density function is calculated in closed form as in a first passage time problem, and its expectation value and variance are evaluated. A detailed study of the mode of the firing times probability density function as related to the noise's intensity is performed. Some other auxiliary results are also obtained.