The interspike interval of a cable model neuron with white noise input

The firing time of a cable model neuron in response to white noise current injection is investigated with various methods. The Fourier decomposition of the depolarization leads to partial differential equations for the moments of the firing time. These are solved by perturbation and numerical methods, and the results obtained are in excellent agreement with those obtained by Monte Carlo simulation. The convergence of the random Fourier series is found to be very slow for small times so that when the firing time is small it is more efficient to simulate the solution of the stochastic cable equation directly using the two different representations of the Green's function, one which converges rapidly for small times and the other which converges rapidly for large times. The shape of the interspike interval density is found to depend strongly on input position. The various shapes obtained for different input positions resemble those for real neurons. The coefficient of variation of the interspike interval decreases monotonically as the distance between the input and trigger zone increases. A diffusion approximation for a nerve cell receiving Poisson input is considered and input/output frequency relations obtained for different input sites. The cases of multiple trigger zones and multiple input sites are briefly discussed.