Multinomial pulse-number distributions for neural spikes in primary auditory fibers: theory

We previously reported experimental short- and long-counting-time pulse-number distributions (PND's) for the neural spike train in cat primary auditory nerve fibers. Data were obtained for spontaneous activity, pure-tone stimuli with a wide range of frequencies and intensity levels, and Gaussian noise. The irregular shapes of the PND's are an indication of the presence of spike clusters of various sizes in the neural impulse train. We develop a family of theoretical cluster counting distributions and examine their suitability for describing the experimental PND's. The reduced-quintinomial distribution provides theoretical results that describe the characteristics of the PND's quite well, accounting for the smooth or scalloped behavior of short-counting-time data, the jagged nature of long-counting-time data, and the Poisson-like character of very-short-counting-time data. This family of distributions admits values for the spike-number mean-to-variance ratio that are independent of stimulus level, in agreement with experimental observation. A number of procedures for fitting the theoretical distributions to the experimental PND's are studied. These include the use of a minimum mean-square error criterion, the factorial moments of the data, and the discrete Fourier transform of the PND. The first of these techniques appears to be the most useful.