Natural Ensembles and Sensory Signal Processing.

In this thesis we explore the idea that sensory systems in biology are well matched to the natural signals they encode. This would imply that the optimal design of a sensory system depends on the statistical structure of its stimuli. Further, the interpretation of sensory data is a statistically defined task: The best signal reconstruction algorithm relies on the statistics of the stimuli and of the noise. We discuss a few instances from the broad class of statistical problems in sensory signal processing which depend on the statistics of natural stimuli. In formalizing these problems, we find that the methods of statistical mechanics are ideally suited toward their solution. First, we demonstrate the importance of prior statistical knowledge in signal reconstruction from an array of noisy detectors. Reconstruction error due to "aliasing," in which two or more Fourier components become confounded, is reduced when knowledge of the ensemble statistics is applied. Next, we consider which design of the visual system encodes the most information about natural images. Since information is a statistical concept, the structure of natural scenes plays a central role in the optimal visual system's design. To lowest order in the signal-to-noise ratio, the only important statistic is the ensemble power spectrum of natural scenes. The optimal linear visual filter is found to solve a Schroedinger equation whose potential is the power spectrum. We find that many of the qualitative features found in mammalian visual systems fall out of a simple linear model: multi -scale processing, orientation selectivity, and the qualitative change in filter shape as a function of signal-to-noise ratio. Finally, we explore the statistics of natural scenes themselves. For an ensemble we gather images from the woods in springtime. We find that they possess a very salient form of scale-invariance: the power spectrum is a power-law, and histograms of local quantities of a given length scale retain their shape as the scale is changed. Further, the images are highly non-Gaussian. These images are thus analogous to a critical theory at a non-Gaussian fixed point. An attempt to "Gaussianize" the images produces a non-linear filtering scheme in which the images are split into Gaussian "variance-normalized" images and images of the local variance. The statistics of the local variance images are nearly identical to those of the original images. Further, this process can be iterated on the variance images to produce yet another set of Gaussians and variances with the same statistics. This remarkable fact implies that the structure of natural scenes is very hierarchical, with variances of variances having the same statistics as the variances themselves. As a final calculation, we use the power spectrum of natural scenes to estimate the information available in the encoding of images by the photoreceptors in the retina. We place an upper bound of a few bits per image per photoreceptor, even at signal-to-noise ratios as high as 10^3. This also places an upper limit to the demands on the optic nerve, which relays images to the brain.