Estimating Neural Firing Rates: An Empirical Bayes Approach

Abstract

A lot of neurophysiological findings rely on accurate estimates of firing rates. In order to estimate an underlying rate function from sparse observations, i.e., spike trains, it is necessary to perform temporal smoothing over a short time window at each time point. In the empirical Bayes method, in which the assumption for the smoothness is incorporated in the Bayesian prior probability of underlying rate, the time scale of the temporal average, or the degree of smoothness, can be optimized by maximizing the marginal likelihood. Here, the marginal likelihood is obtained by marginalizing the complete-data likelihood over all possible latent rate processes. We carry out this marginalization using a path integral method. We show that there exists a lower bound of rate fluctuations below which the optimal smoothness parameter diverges. We also show that the optimal smoothness parameter obeys asymptotic scaling laws, the exponent of which depends on the smoothness of underlying rate processes.