  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.
