Efficient coding is consistent with the irregular shapes of retinal ganglion cell receptive fields E Doi 1 , J Gauthier 3 , GD Field 3 , A Sher 4 , J Shlens 3 , M Greschner 3 , K Mathieson 5 , D Gunning 5 , AM Litke 4 , L Paninski 6 , EJ Chichilnisky 3 , EP Simoncelli 1,2 1 Center for Neural Science & 2 Howard Hughes Medical Institute, New York University, 3 The Salk Institute, 4 Santa Cruz Institute for Particle Physics, 5 University of Glasgow, 6 Center for Theoretical Neuroscience, Columbia University Physiology Theory Difference ON-Parasol OFF-Parasol ON-midget OFF-midget Simultaneous recording Error A single matrix #33 14.5% #46 7.4% #56 16.3% 7.1% #60 12.7% #27 12.5% #49 8.2% #50 12.7% #67 #10 27.2% #12 8.4% #13 8.1% #14 11.1% #5 #8 17.2% 25.4% #11 #12 16.0% 19.5% 0 + − Background Efficient coding is a fundamental principle underlying sensory coding. [Barlow 61] Earlier studies: [Linsker 89, Atick & Redlich 90, van Hateren 92, Ruderman 94, Doi & Lewicki 07] • Assume Gaussian and linear model (Fig. a) • Incorporate natural image statistics (b) • Seek the optimal receptive field (RF) shape (= a convolution kernel) • Results: - Concentric center-surround RF organization (c) - Transition from low-pass to band-pass filtering with the mean light level (d) Direct comparison with physiology has been hampered by simplifying assumptions. • Cone : retinal ganglion cell (RGC) ratio is not 1 : 1 (Fig. e) • Cone sampling lattice is not regular (f ) • Individual RF shapes are not identical nor circularly symmetric (g) • Roughly 20 RGC types with varying RF size x W u δ r � noisy signal signal linear transform optical blur representation noisy representation channel noise � sensory noise ν s H Amplitude Spatial frequency Data Average a b c d e a b c d e Model Comparison HΣ s H T = EΛE T W = PΩQ T We seek an M×N matrix W that maximizes mutual information between signal and representation: Subject to a constraint on the total power of neural responses: P tr[ W(HΣ s H T + σ 2 ν I N )W T ] ≤P 0 Using SVD, , and we proved that the solution is given by The optimal W eff does the following: : the eigenvectors (PCA basis) of the blurred signal covariance : the diagonal matrix whose entries can be found using the KKT condition : any orthogonal matrix is of blurred signal with different photoreceptor SNR with different neural SNR is unconstrained with different population size 20 dB 10 dB 0 dB −10 dB Modulation Eigenvector index 1 2 4 0 10 100 1 10 100 1 10 100 20 dB 10 dB 0 dB 10× 1× 0.1× Evaluate how much the measured RF population is consistent with the theory I (s; r )= 1 2 log |WHΣ s H T W T + σ 2 ν WW T + σ 2 δ I M | |σ 2 ν WW T + σ 2 δ I M | How to compare with physiology? • Select an orthogonal matrix P fit that minimizes the distance between W eff and W mea • Solution manifold W eff ( P ) Random manifold W rnd( P ) Measured RFs W mea E Pixelated image Cone mosaic RGCs N=69 N=5 N=6 N=26 N=32 Theory Human psychophysics Efficient coding is in a good agreement with the population consisting of ON/OFF midget/parasol cells. Appendix C Fitting errors for different subsets of a RGC population Appendix A Finding subject to where L is the number of RGCs in the region-of-interest. • Note that compensates for scale ambiguity between W eff and W mea. • If L = M, we can find the solution in closed form. • Otherwise, we can employ numerical optimization. Appendix B How much of the data variance explained by the model is due to efficient coding? This suggests that ~50% of the variance explained is due to the efficient coding solution, not the fitting with P . • 665 cones and 145 RGCs = 4.6 cones/RGC • 27 degrees of retinal eccentricity • Cone-weights W are visualized with the optical blur of the human eye • We show the contour of RFs at 0.3 × peak height • 62 natural images [Doi et al. 03] • 6.2 × 10 6 data points to calculate the data covariance • Human’s optical blur at 30 deg. of eccentricity [Navarro et al. 93] • 10 dB photoreceptor SNR • 10 dB neural SNR [Borst & Theunissen 99] Fovea 40 deg 0 1 2 RGC : Cone ratio We generalize the model to resolve these limitations and compare it with physiology. References K. M. Ahmad, K. Klug, S. Herr, P. Sterling, and S. Schein. Cell density ratios in a foveal patch in macaque retina. Visual Neuroscience, 20:189–209, 2003. J. J. Atick and A. N. Redlich. Towards a theory of early visual processing. Neural Computation, 2:308–320, 1990. J. J. Atick and A. N. Redlich. 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A theory of retinal population coding. In Advances in NIPS 19 (NIPS*2006), pp. 353–360. MIT Press, 2007. J. Gauthier, G. D. Field, A. Sher, M. Greschner, J. Shlens, A. M. Litke, & E. J. Chichilnisky. Fine-scale functional coordination enhances population encoding in large neural circuits. submitted. J. Gauthier, G. D. Field, A. Sher, M. Greschner, J. Shlens, D. Gunning, K. Mathieson, A. M. Litke, & E. J. Chichilnisky. Functional identification of individual cones in the receptive fields of primate retinal ganglion cells. SfN2008 Poster 365.5/HH20, 2008. J. H. van Hateren. A theory of maximizing sensory information. Biol. Cybern. 68:23–29, 1992. R. Linsker. An application of the principle of maximum information preservation to linear systems. In Advances in NIPS 1 (NIPS*1988), pp. 186–194. MIT Press, 1989. R. Navarro, P. Artal, and D. R. Williams. Modulation transfer of the human eye as a function of retinal eccentricity. J. Opt. Soc. Am. A, 10:201–212, 1993. A. Roorda and D. R. 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Network, 5:147–155, 1994. = 34.32% (13.44% in the blurred plot) Fig. (a) (b) (c) (d) (e) (f) (g) Future directions: • Filling-in missing RGCs • Trichromatic RFs and cone-type specificity • More RGC types Limiting factor of the current method: • Boundary of the cone mosaic • Noise model and measurement • Adaptation • Temporal RFs • RGC nonlinearity • Non-Gaussian image statistics • Rod system PP T = I M α • How much does the solution constrain the result? • Compare with a random , which defines , and find . • Error = 83.44 ± 0.11% with different (compare to 34.32% with ) ΩQ T Individual cell-type ON-parasol OFF-parasol parasol midget ON OFF ON-midget OFF-midget All parasol or midget All ON or OFF 44.37% 48.57% 42.56% 38.02% 52.44% 47.57% 50.04% 45.43% Efficient coding solution is closest to the RFs of a whole population. This suggest that the simplistic view of ON and OFF types as the mirror of each other should be too simplistic (see “All ON or OFF”), and that an alternative view in which those four types of RGCs jointly constitute a single information channel is a better description. Q eff = E Ω eff P Ω eff Ω eff Ω eff {α t , P fit } = arg min α,P �W mea − αI L×M P eff Q T eff � 2 F P fit Ω eff Q T eff P fit �W m a − W ff (P fit )� 2 F �W mea � 2 F Acknowledgements: This work is supported by NIH #EY018003 (EJC, LP, EPS) and HHMI (EPS) W eff = � representation with M -D orthogonal basis � ◦ � modulating x in the eigenspace � ◦ � representing x in the eigenspace � W (P fit ) All cell-types (total error 34.32%) ON-parasol OFF-parasol ON-midget OFF-midget 45.64% 28.78% 33.05% 35.73% [Atick & Redlich 90] [Atick & Redlich 92] [Curcio & Allen 90, Ahmad et al. 03] See also Gauthier et al. SfN08 [Curcio et al. 91] [Roorda & Williams 99] [Gauthier et al. submitted] W eff ( P fit) W rnd (P)= PΩ rnd Q T rnd Ω rnd Q T rnd Ω eff Q T eff Q eff P