Efficient coding of spatial information in the primate retina

Abstract

Sensory neurons have been hypothesized to efficiently encode signals from the natural environment subject to resource constraints. The predictions of this efficient coding hypothesis regarding the spatial filtering properties of the visual system have been found consistent with human perception, but they have not been compared directly with neural responses. Here, we analyze the information that retinal ganglion cells transmit to the brain about the spatial information in natural images subject to three resource constraints: the number of retinal ganglion cells, their total response variances, and their total synaptic strengths. We derive a model that optimizes the transmitted information and compare it directly with measurements of complete functional connectivity between cone photoreceptors and the four major types of ganglion cells in the primate retina, obtained at single-cell resolution. We find that the ganglion cell population exhibited 80% efficiency in transmitting spatial information relative to the model. Both the retina and the model exhibited high redundancy (~30%) among ganglion cells of the same cell type. A novel and unique prediction of efficient coding, the relationships between projection patterns of individual cones to all ganglion cells, was consistent with the observed projection patterns in the retina. These results indicate a high level of efficiency with near-optimal redundancy in visual signaling by the retina.

Figure 1.

Functional model of RGC responses, used for assessment of efficient coding theory. The model consists of retinal images of natural scenes, represented in the cone photoreceptor mosaic obtained from the data. A linear combination of these cone signals, specified by a connectivity matrix, W, governs model RGC responses. White Gaussian noise is added before and after the linear combination, with amplitude set in accordance with previous studies (Atick and Redlich, 1990; van Hateren, 1993). The set of connectivity weights arising from a single cone (red) constitute the projective field (PF) of that cone.

Figure 2.

Comparison of RGC receptive field spatial structure to theoretical predictions. a, A schematic of connectivity matrices. Each point in space corresponds to one connectivity matrix, and three such matrices are indicated. Note that PW_opt is the optimal connectivity with any choice of orthogonal matrix P, constituting a manifold of the optimal solution as illustrated by the ellipsoid. b, Measured RGC receptive fields, i.e., (the rows of) W_ret. c, The optimal connectivity closest to the retinal data, W_opt-fit. d, Arbitrarily chosen optimal connectivity, W_opt. Each panel in b–d shows contours of receptive fields of the four major RGC types at 30% of maximum, superimposed on the cone lattice. A contour from one cell per each panel is highlighted in orange for clarity.

Figure 3.

Comparison of cone PF spatial structure to theoretical predictions. a, Each panel shows the inner product of the PF of a single cone (yellow) with the PFs of all other cones. The diameter of each circle indicates the magnitude of the inner product; the color indicates the sign (black, negative; white, positive). Top panels show inner products for three cones from the retina; bottom panels show predictions of efficient coding theory for those cones. b, The PF inner product as a function of distance between cones. Solid lines indicate average values; shaded regions indicate the 5th to 95th percentile range. Values at zero separation indicate squared norms of individual PFs. The gap at small separations reflects the minimum separation between cones in the lattice.

Figure 4.

Comparison of RGC spatial redundancy to theoretical predictions. Each panel shows the spatial redundancy between pairs of RGCs as a function of the distance between them. Left panels shows results obtained from the measured retinal connectivity, W_ret. Right panel shows results obtained from the optimal connectivity closest to the data, W_opt-fit. Vertical axis shows the fraction of stimulus information conveyed by one cell that is captured by the other; a completely redundant cell pair would exhibit redundancy of 1. Each symbol corresponds to a pair of ON (magenta) or OFF (cyan), Parasol (square) or Midget (dot) RGCs.

Figure 5.

Receptive fields obtained from simulations of the developmental model. Panels show 30% maximum contours of receptive fields of each cell type (compare to Figure 2b).