Spatial neuron model with two-parameter Ornstein–Uhlenbeck input current

We consider a new and extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation. The synaptic input current is also a function of space and time and satisfies a first order linear partial differential equation driven by a two-parameter random process. A natural choice for these random input processes is to make them two-parameter Poisson processes for both excitation and inhibition. For such inputs the mean subthreshold voltage is found in the case of an infinite cable and of finite cables. Assuming uniform amplitudes and rates exact expressions are obtained in the case of particular boundary conditions. We then consider a diffusion approximation and show that the membrane current is a two-parameter Ornstein–Uhlenbeck process, whose statistical properties are derived. Using representations for the voltage in terms of stochastic integrals in the plane we find, in the case of finite space intervals the mean, variance and covariance of the subthreshold voltage. For large times the voltage process is shown to be covariance stationary in time and the corresponding spectral density is found and compared with the result for a purely (two-parameter) white noise driven cable. The limiting white noise case is obtained from the extended model as the decay parameter becomes infinite. Finally, we develop useful simulation methods for the solution of the extended spatial model using properties of stochastic integrals involving eigenfunctions to obtain one-dimensional representations which are easily implemented.