  The sigmoidal tuning curve that maximizes the mutual information for a
Poisson neuron, or population of Poisson neurons, is obtained. The optimal
tuning curve is found to have a discrete structure that results in a
quantization of the input signal. The number of quantization levels undergoes a
hierarchy of phase transitions as the length of the coding window is varied. We
postulate, using the mammalian auditory system as an example, that the presence
of a subpopulation structure within a neural population is consistent with an
optimal neural code.
