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 the visual cortex, and motion-sensitive neurons in the visual system. Generalizing the idea of maximally informative dimensions, we show that one can search for 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.
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