Stochastic Neural Activity

The activity of a neuron subjected to an input of many small excitatory and inhibitory pulses is analysed. The theoretical model consists of a linear or nonlinear first order system followed by a threshold. Diffusion equations for the transition probability density of the somatic potential are derived. Analytical results for the stationary activity include expressions for the stationary distribution of the somatic potential and for the moments of the distribution of intervals between action potentials. Applications and numerical data are given for three specific models: an integrator, an imperfect integrator and an equivalent circuit for the membrane, each combined with a threshold. Linear regions of the input-output characteristics are indicated. The dynamical aspects are discussed in more general terms. A relation between the stochastic switch-response and the expectation density is derived. For small variations of the input a dynamical description is given on the base of a region-dependent stochastic transfer matrix. A general form of stationary and a linearised version of dynamical interaction equations for an ensemble of neurons are proposed.