Dynamic state estimation based on Poisson spike trains—towards a theory of optimal encoding

Neurons in the nervous system convey information to higher brain regions by the generation of spike trains. An important question in the field of computational neuroscience is how these sensory neurons encode environmental information in a way which may be simply analyzed by subsequent systems. Many aspects of the form and function of the nervous system have been understood using the concepts of optimal population coding. Most studies, however, have neglected the aspect of temporal coding. Here we address this shortcoming through a filtering theory of inhomogeneous Poisson processes. We derive exact relations for the minimal mean squared error of the optimal Bayesian filter and, by optimizing the encoder, obtain optimal codes for populations of neurons. We also show that a class of non-Markovian, smooth stimuli are amenable to the same treatment, and provide results for the filtering and prediction error which hold for a general class of stochastic processes. This sets a sound mathematical framework for a population coding theory that takes temporal aspects into account. It also formalizes a number of studies which discussed temporal aspects of coding using time-window paradigms, by stating them in terms of correlation times and firing rates. We propose that this kind of analysis allows for a systematic study of temporal coding and will bring further insights into the nature of the neural code.