Threshold behaviour of the maximum likelihood method in population decoding

We study the performance of the maximum likelihood (ML) method in population decoding as a function of the population size. Assuming uncorrelated noise in neural responses, the ML performance, quantified by the expected square difference between the estimated and the actual quantity, follows closely the optimal Cramer-Rao bound, provided that the population size is sufficiently large. However, when the population size decreases below a certain threshold, the performance of the ML method undergoes a rapid deterioration, experiencing a large deviation from the optimal bound. We explain the cause of such threshold behaviour, and present a phenomenological approach for estimating the threshold population size, which is found to be linearly proportional to the inverse of the square of the system's signal-to-noise ratio. If the ML method is used by neural systems, we expect the number of neurons involved in population coding to be above this threshold.