  We examine how the prior probability distribution of a sensory variable in
the environment influences the optimal allocation of neurons and spikes in a
population that represents that variable. We start with a conventional response
model, in which the spikes of each neuron are drawn from a Poisson distribution
with a mean rate governed by an associated tuning curve. For this model, we
approximate the Fisher information in terms of the density and amplitude of the
tuning curves, under the assumption that tuning width varies inversely with
cell density. We consider a family of objective functions based on the expected
value, over the sensory prior, of a functional of the Fisher information. This
family includes lower bounds on mutual information and perceptual
discriminability as special cases. For all cases, we obtain a closed form
expression for the optimum, in which the density and gain of the cells in the
population are power law functions of the stimulus prior. Thus, the allocation
of these resources is uniquely specified by the prior. Since perceptual
discriminability may be expressed directly in terms of the Fisher information,
it too will be a power law function of the prior. We show that these results
hold for tuning curves of arbitrary shape and correlated neuronal variability.
This framework thus provides direct and experimentally testable predictions
regarding the relationship between sensory priors, tuning properties of neural
representations, and perceptual discriminability.
