Paper ID: 349

Testing efficient coding: projective (not receptive) fields are the
key theoretical prediction

Eizaburo Doi (edoi@cns.nyu.edu), Greg Field (gfield@salk.edu),
Jeffrey Gauthier (authier@salk.edu), Alexander Sher
(sasha@scipp.ucsc.edu), Martin Greschner (greschner@salk.edu),
Jonathon Shlens (shlens@salk.edu), Timothy Machado
(tam2138@columbia.edu), Liam Paninski (liam@stat.columbia.edu),
Debrah Gunning (dgunning@physics.gla.ac.uk), Keith Mathieson
(k.mathieson@physics.gla.ac.uk), Alan Litke
(alan.litke@cern.ch), E.J. Chichilnisky (ej@salk.edu), Eero
Simoncelli (eero.simoncelli@nyu.edu)

A fundamental principle for understanding the structure of early
sensory information processing is that of efficient coding:
information about the external world transmitted to the brain should
be maximized, subject to limitations on resources such as firing rate
and neural population size. Previous work has suggested that the
receptive field structure of retinal ganglion cells (RGCs), the
output neurons of the retina, is efficient in this sense [Atick &
Redlich, 1990; van Hateren, 1992]. Assuming a population of
identical, equally-spaced, circularly symmetric receptive fields,
efficient coding principles were shown to account qualitatively for
the spatial frequency responses of retinal ganglion cells at
different mean illumination levels. However, previous work did not
include known aspects of neural architecture, such as inhomogeneities
in the photoreceptor lattice and in the spacing and structure of
ganglion cell receptive fields, that can have a major impact on the
theoretical predictions.

Here we examine efficient coding in a framework with fewer
assumptions: inhomogeneous populations of linear input and output
neurons with different density and independent Gaussian noise [Campa
et al., 1994; Doi et al., Cosyne08]. We find that the efficient
coding principle does not uniquely specify the receptive fields of
output neurons, but instead places a strong constraint on the
"projective fields" of the input neurons, that is, the strength of
the connection between a given input and all the output cells.
Specifically, efficient coding uniquely predicts the inner products
of the projective fields of all pairs of input neurons. An
experimental test of efficient coding therefore requires measurement
of the pattern of connectivity between the full set of input neurons
and a complete collection of output neurons.

Recent advances in large-scale, high-resolution recording techniques
have made such measurements possible, and consequently, allow the
first quantitative test of the theory. We examine a data set in which
receptive fields of complete collections of RGCs covering a region of
retina were mapped at high resolution, revealing the strength of the
inputs of each cone photoreceptor to each RGC, and report three
principal findings. (1) The spatial pattern of the predicted inner
products is similar to the measured pattern. (2) The predicted inner
products explain only 16% of the variance of the measured values, in
particular the norms are highly variable in the data. Adjusting these
norms improves the prediction, allowing the theory to account for 51%
of the variance. (3) Information transmission of the measured
projective fields is approximately 90% of the theoretical limit.

We conclude that the linear-Gaussian form of the efficient coding
theory fails to match the data in detail, but predicts the efficiency
of the retinal network for encoding natural scenes fairly accurately.
A key feature suggested by this study is that the cone photoreceptors
are not utilized as uniformly as the theory predicts. Nonlinear
transformations and/or non-Gaussian forms of noise or image priors
may explain the discrepancies.