  Neural decoding may be formulated as dynamic state estimation (filtering)
based on point process observations, a generally intractable problem. Numerical
sampling techniques are often practically useful for the decoding of real
neural data. However, they are less useful as theoretical tools for modeling
and understanding sensory neural systems, since they lead to limited conceptual
insight about optimal encoding and decoding strategies. We consider sensory
neural populations characterized by a distribution over neuron parameters. We
develop an analytically tractable Bayesian approximation to optimal filtering
based on the observation of spiking activity, that greatly facilitates the
analysis of optimal encoding in situations deviating from common assumptions of
uniform coding. Continuous distributions are used to approximate large
populations with few parameters, resulting in a filter whose complexity does
not grow with the population size, and allowing optimization of population
parameters rather than individual tuning functions. Numerical comparison with
particle filtering demonstrates the quality of the approximation. The analytic
framework leads to insights which are difficult to obtain from numerical
algorithms, and is consistent with biological observations about the
distribution of sensory cells' preferred stimuli.
