  The concept of feature selectivity in sensory signal processing can be
formalized as dimensionality reduction: in a stimulus space of very high
dimensions, neurons respond only to variations within some smaller, relevant
subspace. But if neural responses exhibit invariances, then the relevant
subspace typically cannot be reached by a Euclidean projection of the original
stimulus. We argue that, in several cases, we can make progress by appealing to
the simplest nonlinear construction, identifying the relevant variables as
quadratic forms, or "stimulus energies." Natural examples include
non-phase-locked cells in the auditory system, complex cells in visual cortex,
and motion-sensitive neurons in the visual system. Generalizing the idea of
maximally informative dimensions, we show that one can search for the kernels
of the relevant quadratic forms by maximizing the mutual information between
the stimulus energy and the arrival times of action potentials. Simple
implementations of this idea successfully recover the underlying properties of
model neurons even when the number of parameters in the kernel is comparable to
the number of action potentials and stimuli are completely natural. We explore
several generalizations that allow us to incorporate plausible structure into
the kernel and thereby restrict the number of parameters. We hope that this
approach will add significantly to the set of tools available for the analysis
of neural responses to complex, naturalistic stimuli.
