Decoding Poisson spike trains by Gaussian filtering

The temporal waveform of neural activity is commonly estimated by low-pass filtering spike train data through convolution with a gaussian kernel. However, the criteria for selecting the gaussian width sigma are not well understood. Given an ensemble of Poisson spike trains generated by an instantaneous firing rate function lambda(t), the problem was to recover an optimal estimate of lambda(t) by gaussian filtering. We provide equations describing the optimal value of sigma using an error minimization criterion and examine how the optimal sigma varies within a parameter space defining the statistics of inhomogeneous Poisson spike trains. The process was studied both analytically and through simulations. The rate functions lambda(t) were randomly generated, with the three parameters defining spike statistics being the mean of lambda(t), the variance of lambda(t), and the exponent alpha of the Fourier amplitude spectrum 1/f(alpha) of lambda(t). The value of sigma(opt) followed a power law as a function of the pooled mean interspike interval I, sigma(opt) = aI(b), where a was inversely related to the coefficient of variation C(V) of lambda(t), and b was inversely related to the Fourier spectrum exponent alpha. Besides applications for data analysis, optimal recovery of an analog signal waveform lambda(t) from spike trains may also be useful in understanding neural signal processing in vivo.