  We propose a method that allows for a rigorous statistical analysis of neural
responses to natural stimuli which are non-Gaussian and exhibit strong
correlations. We have in mind a model in which neurons are selective for a
small number of stimulus dimensions out of a high dimensional stimulus space,
but within this subspace the responses can be arbitrarily nonlinear. Existing
analysis methods are based on correlation functions between stimuli and
responses, but these methods are guaranteed to work only in the case of
Gaussian stimulus ensembles. As an alternative to correlation functions, we
maximize the mutual information between the neural responses and projections of
the stimulus onto low dimensional subspaces. The procedure can be done
iteratively by increasing the dimensionality of this subspace. Those dimensions
that allow the recovery of all of the information between spikes and the full
unprojected stimuli describe the relevant subspace. If the dimensionality of
the relevant subspace indeed is small, it becomes feasible to map the neuron's
input-output function even under fully natural stimulus conditions. These ideas
are illustrated in simulations on model visual and auditory neurons responding
to natural scenes and sounds, respectively.
