Abstract

A major open problem in systems neuroscience is to understand the relationship between behavior and the detailed spiking properties of neural populations. We assess how faithfully velocity information can be decoded from a population of spiking model retinal neurons whose spatiotemporal receptive fields and ensemble spike train dynamics are closely matched to real data. We describe how to compute the optimal Bayesian estimate of image velocity given the population spike train response and show that, in the case of global translation of an image with known intensity profile, on average the spike train ensemble signals speed with a fractional standard deviation of about 2% across a specific set of stimulus conditions. We further show how to compute the Bayesian velocity estimate in the case where we only have some a priori information about the (naturalistic) spatial correlation structure of the image but do not know the image explicitly. As expected, the performance of the Bayesian decoder is shown to be less accurate with decreasing prior image information. There turns out to be a close mathematical connection between a biologically plausible “motion energy” method for decoding the velocity and the Bayesian decoder in the case that the image is not known. Simulations using the motion energy method and the Bayesian decoder with unknown image reveal that they result in fractional standard deviations of 10% and 6%, respectively, across the same set of stimulus conditions. Estimation performance is rather insensitive to the details of the precise receptive field location, correlated activity between cells, and spike timing.

© 2009 Optical Society of America

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  1. M. Meister, L. Lagnado, and D. Baylor, “Concerted signaling by retinal ganglion cells,” Science 270, 1207-1210 (1995).
    [CrossRef] [PubMed]
  2. S. Nirenberg, S. Carcieri, A. Jacobs, and P. Latham, “Retinal ganglion cells act largely as independent encoders,” Nature 411, 698-701 (2002).
    [CrossRef]
  3. E. Chichilnisky and R. Kalmar, “Functional asymmetries in ON and OFF ganglion cells of primate retina,” J. Neurosci. 22, 2737-2747 (2002).
    [PubMed]
  4. E. Frechette, A. Sher, M. Grivich, D. Petrusca, A. Litke, and E. Chichilnisky, “Fidelity of the ensemble code for visual motion in the primate retina,” J. Neurophysiol. 94, 119-135 (2005).
    [CrossRef]
  5. E. Schneidman, M. Berry, R. Segev, and W. Bialek, “Weak pairwise correlations imply strongly correlated network states in a neural population,” Nature 440, 1007-1012 (2006).
    [CrossRef] [PubMed]
  6. J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
    [CrossRef] [PubMed]
  7. J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
    [CrossRef] [PubMed]
  8. E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
    [CrossRef]
  9. A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
    [CrossRef]
  10. R. Segev, J. Goodhouse, J. Puchalla, and M. Berry, “Recording spikes from a large fraction of the ganglion cells in a retinal patch,” Nat. Neurosci. 7, 1154-1161 (2004).
    [CrossRef] [PubMed]
  11. D.Knill and W.Richards, eds., Perception as Bayesian Inference (Cambridge Univ. Press, 1996).
  12. E. P. Simoncelli, “Distributed analysis and representation of visual motion,” Ph.D. thesis (Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1993). Also available as MIT Media Laboratory Vision and Modeling Technical Report #209.
  13. D. Ascher and N. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427-3434 (2000).
    [CrossRef] [PubMed]
  14. Y. Weiss, E. Simoncelli, and E. Adelson, “Motion illusions as optimal percepts,” Nat. Neurosci. 5, 598-604 (2002).
    [CrossRef] [PubMed]
  15. E. P. Simoncelli, “Local analysis of visual motion,” in The Visual Neurosciences, L.M.Chalupa and J.S.Werner, eds. (MIT Press, 2003), Chap. 109, pp. 1616-1623.
  16. A. Stocker and E. Simoncelli, “Noise characteristics and prior expectations in human visual speed perception,” Nat. Neurosci. 9, 578-585 (2006).
    [CrossRef] [PubMed]
  17. A. E. Welchman, J. M. Lam, and H. H. Bulthoff, “Bayesian motion estimation accounts for a surprising bias in 3D vision,” Proc. Natl. Acad. Sci. U.S.A. 105, 12087-12092 (2008).
    [CrossRef] [PubMed]
  18. F. Hurlimann, D. Kiper, and M. Carandini, “Testing the Bayesian model of perceived speed,” Vision Res. 42, 2253-2257 (2002).
    [CrossRef] [PubMed]
  19. P. Thompson, K. Brooks, and S. Hammett, “Speed can go up as well as down at low contrast: Implications for models of motion perception,” Vision Res. 46, 782-786 (2005).
    [CrossRef] [PubMed]
  20. A. Thiel, M. Greschner, C. Eurich, J. Ammermüller, and J. Kretzberg, “Contribution of individual retinal ganglion cell responses to velocity and acceleration encoding,” J. Neurophysiol. 98, 2285-2296 (2007).
    [CrossRef] [PubMed]
  21. J. Kretzberg, I. Winzenborg, and A. Thiel, “Bayesian analysis of the encoding of constant and changing stimulus velocities by retinal ganglion cells,” presented at Frontiers in Neuroinformatics 2008, Stockholm, September 7-9, 2008.
  22. D. Brillinger, “Maximum likelihood analysis of spike trains of interacting nerve cells,” Biol. Cybern. 59, 189-200 (1988).
    [CrossRef] [PubMed]
  23. P. McCullagh and J. Nelder, Generalized Linear Models (Chapman & Hall, 1989).
  24. L. Paninski, “Maximum likelihood estimation of cascade point-process neural encoding models,” Network Comput. Neural Syst. 15, 243-262 (2004).
    [CrossRef]
  25. W. Truccolo, U. Eden, M. Fellows, J. Donoghue, and E. Brown, “A point process frame-work for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects,” J. Neurophysiol. 93, 1074-1089 (2005).
    [CrossRef]
  26. L. Paninski, J. Pillow, and J. Lewi, “Statistical models for neural encoding, decoding, and optimal stimulus design,” in Computational Neuroscience: Progress in Brain Research, P.Cisek, T.Drew, and J.Kalaska, eds. (Elsevier, 2007).
  27. D. Snyder and M. Miller, Random Point Processes in Time and Space (Springer-Verlag, 1991).
    [CrossRef]
  28. D. Field, “Relations between the statistics of natural images and the response profiles of cortical cells,” J. Opt. Soc. Am. A 4, 2379-2394 (1987).
    [CrossRef] [PubMed]
  29. D. H. Brainard, D. R. Williams, and H. Hofer, “Trichromatic reconstruction from the interleaved cone mosaic: Bayesian model and the color appearance of small spots,” J. Vision 8, 1-23 (2008).
    [CrossRef]
  30. R. Kass and A. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90, 773-795 (1995).
    [CrossRef]
  31. E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson, “A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells,” J. Neurosci. 18, 7411-7425 (1998).
    [PubMed]
  32. W. Bialek and A. Zee, “Coding and computation with neural spike trains,” J. Stat. Phys. 59, 103-115 (1990).
    [CrossRef]
  33. S. Koyama and S. Shinomoto, “Empirical Bayes interpretations of random point events,” J. Phys. A 38, 531-537 (2005).
    [CrossRef]
  34. J. Pillow, Y. Ahmadian, and L. Paninski, “Model-based decoding, information estimation, and change-point detection in multi-neuron spike trains,” submitted to Neural Comput.
    [PubMed]
  35. Y. Ahmadian, J. Pillow, and L. Paninski, “Efficient Markov chain Monte Carlo methods for decoding neural spike trains,” submitted to Neural Comput.
    [PubMed]
  36. E. Adelson and J. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284-99 (1985).
    [CrossRef] [PubMed]
  37. E. Chichilnisky and R. Kalmar, “Temporal resolution of ensemble visual motion signals in primate retina,” J. Neurosci. 23, 6681-6689 (2003).
    [PubMed]
  38. W. Bialek (Princeton University, bbrinker@princeton.edu) and R. de Ruyter van Steveninck (Indiana University, deruyter@indiana.edu) (personal communication, 2003).
  39. V. Perry and A. Cowey, “The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors,” Vision Res. 25, 1795-1810 (1985).
    [CrossRef] [PubMed]
  40. S. Ullman, The Interpretation of Visual Motion (MIT Press, 1979).
  41. P. Thompson, “Perceived rate of movement depends on contrast,” Vision Res. 22, 377-380 (1982).
    [CrossRef] [PubMed]
  42. L. Stone and P. Thompson, “Human speed perception is contrast dependent,” Vision Res. 32, 1535-1549 (1992).
    [CrossRef] [PubMed]
  43. D. C. Bradley and M. S. Goyal, “Velocity computation in the primate visual system,” Nat. Rev. Neurosci. 9, 686-695 (2008).
    [CrossRef]
  44. M. Potters and W. Bialek, “Statistical mechanics and visual signal processing,” J. Phys. I France 4, 1755-1775 (1994).
    [CrossRef]
  45. S. McKee, G. Silvermann, and K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609-619 (1986).
    [CrossRef] [PubMed]
  46. M. Blakemore and R. Snowden, “The effect of contrast upon perceived speed: a general phenomenon?” Perception 28, 33-48 (1999).
    [CrossRef]
  47. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge Univ. Press, 1992).
  48. N. Shephard and M. Pitt, “Likelihood analysis of non-Gaussian measurement time series,” Biometrika 84, 653-667 (1997).
    [CrossRef]
  49. R. Davis and G. Rodriguez-Yam, “Estimation for state-space models: an approximate likelihood approach,” Stat. Sin. 15, 381-406 (2005).
  50. L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

2008 (4)

J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
[CrossRef] [PubMed]

A. E. Welchman, J. M. Lam, and H. H. Bulthoff, “Bayesian motion estimation accounts for a surprising bias in 3D vision,” Proc. Natl. Acad. Sci. U.S.A. 105, 12087-12092 (2008).
[CrossRef] [PubMed]

D. H. Brainard, D. R. Williams, and H. Hofer, “Trichromatic reconstruction from the interleaved cone mosaic: Bayesian model and the color appearance of small spots,” J. Vision 8, 1-23 (2008).
[CrossRef]

D. C. Bradley and M. S. Goyal, “Velocity computation in the primate visual system,” Nat. Rev. Neurosci. 9, 686-695 (2008).
[CrossRef]

2007 (1)

A. Thiel, M. Greschner, C. Eurich, J. Ammermüller, and J. Kretzberg, “Contribution of individual retinal ganglion cell responses to velocity and acceleration encoding,” J. Neurophysiol. 98, 2285-2296 (2007).
[CrossRef] [PubMed]

2006 (3)

E. Schneidman, M. Berry, R. Segev, and W. Bialek, “Weak pairwise correlations imply strongly correlated network states in a neural population,” Nature 440, 1007-1012 (2006).
[CrossRef] [PubMed]

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

A. Stocker and E. Simoncelli, “Noise characteristics and prior expectations in human visual speed perception,” Nat. Neurosci. 9, 578-585 (2006).
[CrossRef] [PubMed]

2005 (5)

E. Frechette, A. Sher, M. Grivich, D. Petrusca, A. Litke, and E. Chichilnisky, “Fidelity of the ensemble code for visual motion in the primate retina,” J. Neurophysiol. 94, 119-135 (2005).
[CrossRef]

S. Koyama and S. Shinomoto, “Empirical Bayes interpretations of random point events,” J. Phys. A 38, 531-537 (2005).
[CrossRef]

P. Thompson, K. Brooks, and S. Hammett, “Speed can go up as well as down at low contrast: Implications for models of motion perception,” Vision Res. 46, 782-786 (2005).
[CrossRef] [PubMed]

W. Truccolo, U. Eden, M. Fellows, J. Donoghue, and E. Brown, “A point process frame-work for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects,” J. Neurophysiol. 93, 1074-1089 (2005).
[CrossRef]

R. Davis and G. Rodriguez-Yam, “Estimation for state-space models: an approximate likelihood approach,” Stat. Sin. 15, 381-406 (2005).

2004 (4)

L. Paninski, “Maximum likelihood estimation of cascade point-process neural encoding models,” Network Comput. Neural Syst. 15, 243-262 (2004).
[CrossRef]

E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
[CrossRef]

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

R. Segev, J. Goodhouse, J. Puchalla, and M. Berry, “Recording spikes from a large fraction of the ganglion cells in a retinal patch,” Nat. Neurosci. 7, 1154-1161 (2004).
[CrossRef] [PubMed]

2003 (1)

E. Chichilnisky and R. Kalmar, “Temporal resolution of ensemble visual motion signals in primate retina,” J. Neurosci. 23, 6681-6689 (2003).
[PubMed]

2002 (4)

S. Nirenberg, S. Carcieri, A. Jacobs, and P. Latham, “Retinal ganglion cells act largely as independent encoders,” Nature 411, 698-701 (2002).
[CrossRef]

E. Chichilnisky and R. Kalmar, “Functional asymmetries in ON and OFF ganglion cells of primate retina,” J. Neurosci. 22, 2737-2747 (2002).
[PubMed]

Y. Weiss, E. Simoncelli, and E. Adelson, “Motion illusions as optimal percepts,” Nat. Neurosci. 5, 598-604 (2002).
[CrossRef] [PubMed]

F. Hurlimann, D. Kiper, and M. Carandini, “Testing the Bayesian model of perceived speed,” Vision Res. 42, 2253-2257 (2002).
[CrossRef] [PubMed]

2000 (1)

D. Ascher and N. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427-3434 (2000).
[CrossRef] [PubMed]

1999 (1)

M. Blakemore and R. Snowden, “The effect of contrast upon perceived speed: a general phenomenon?” Perception 28, 33-48 (1999).
[CrossRef]

1998 (1)

E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson, “A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells,” J. Neurosci. 18, 7411-7425 (1998).
[PubMed]

1997 (1)

N. Shephard and M. Pitt, “Likelihood analysis of non-Gaussian measurement time series,” Biometrika 84, 653-667 (1997).
[CrossRef]

1995 (2)

R. Kass and A. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90, 773-795 (1995).
[CrossRef]

M. Meister, L. Lagnado, and D. Baylor, “Concerted signaling by retinal ganglion cells,” Science 270, 1207-1210 (1995).
[CrossRef] [PubMed]

1994 (1)

M. Potters and W. Bialek, “Statistical mechanics and visual signal processing,” J. Phys. I France 4, 1755-1775 (1994).
[CrossRef]

1992 (1)

L. Stone and P. Thompson, “Human speed perception is contrast dependent,” Vision Res. 32, 1535-1549 (1992).
[CrossRef] [PubMed]

1990 (1)

W. Bialek and A. Zee, “Coding and computation with neural spike trains,” J. Stat. Phys. 59, 103-115 (1990).
[CrossRef]

1988 (1)

D. Brillinger, “Maximum likelihood analysis of spike trains of interacting nerve cells,” Biol. Cybern. 59, 189-200 (1988).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

S. McKee, G. Silvermann, and K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609-619 (1986).
[CrossRef] [PubMed]

1985 (2)

V. Perry and A. Cowey, “The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors,” Vision Res. 25, 1795-1810 (1985).
[CrossRef] [PubMed]

E. Adelson and J. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284-99 (1985).
[CrossRef] [PubMed]

1982 (1)

P. Thompson, “Perceived rate of movement depends on contrast,” Vision Res. 22, 377-380 (1982).
[CrossRef] [PubMed]

Adelson, E.

Y. Weiss, E. Simoncelli, and E. Adelson, “Motion illusions as optimal percepts,” Nat. Neurosci. 5, 598-604 (2002).
[CrossRef] [PubMed]

E. Adelson and J. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284-99 (1985).
[CrossRef] [PubMed]

Ahmadian, Y.

L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

Y. Ahmadian, J. Pillow, and L. Paninski, “Efficient Markov chain Monte Carlo methods for decoding neural spike trains,” submitted to Neural Comput.
[PubMed]

J. Pillow, Y. Ahmadian, and L. Paninski, “Model-based decoding, information estimation, and change-point detection in multi-neuron spike trains,” submitted to Neural Comput.
[PubMed]

Ammermüller, J.

A. Thiel, M. Greschner, C. Eurich, J. Ammermüller, and J. Kretzberg, “Contribution of individual retinal ganglion cell responses to velocity and acceleration encoding,” J. Neurophysiol. 98, 2285-2296 (2007).
[CrossRef] [PubMed]

Ascher, D.

D. Ascher and N. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427-3434 (2000).
[CrossRef] [PubMed]

Baylor, D.

M. Meister, L. Lagnado, and D. Baylor, “Concerted signaling by retinal ganglion cells,” Science 270, 1207-1210 (1995).
[CrossRef] [PubMed]

Bergen, J.

Berry, M.

E. Schneidman, M. Berry, R. Segev, and W. Bialek, “Weak pairwise correlations imply strongly correlated network states in a neural population,” Nature 440, 1007-1012 (2006).
[CrossRef] [PubMed]

R. Segev, J. Goodhouse, J. Puchalla, and M. Berry, “Recording spikes from a large fraction of the ganglion cells in a retinal patch,” Nat. Neurosci. 7, 1154-1161 (2004).
[CrossRef] [PubMed]

Bezayiff, N.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Bialek, W.

E. Schneidman, M. Berry, R. Segev, and W. Bialek, “Weak pairwise correlations imply strongly correlated network states in a neural population,” Nature 440, 1007-1012 (2006).
[CrossRef] [PubMed]

M. Potters and W. Bialek, “Statistical mechanics and visual signal processing,” J. Phys. I France 4, 1755-1775 (1994).
[CrossRef]

W. Bialek and A. Zee, “Coding and computation with neural spike trains,” J. Stat. Phys. 59, 103-115 (1990).
[CrossRef]

W. Bialek (Princeton University, bbrinker@princeton.edu) and R. de Ruyter van Steveninck (Indiana University, deruyter@indiana.edu) (personal communication, 2003).

Blakemore, M.

M. Blakemore and R. Snowden, “The effect of contrast upon perceived speed: a general phenomenon?” Perception 28, 33-48 (1999).
[CrossRef]

Bradley, D. C.

D. C. Bradley and M. S. Goyal, “Velocity computation in the primate visual system,” Nat. Rev. Neurosci. 9, 686-695 (2008).
[CrossRef]

Brainard, D. H.

D. H. Brainard, D. R. Williams, and H. Hofer, “Trichromatic reconstruction from the interleaved cone mosaic: Bayesian model and the color appearance of small spots,” J. Vision 8, 1-23 (2008).
[CrossRef]

Brillinger, D.

D. Brillinger, “Maximum likelihood analysis of spike trains of interacting nerve cells,” Biol. Cybern. 59, 189-200 (1988).
[CrossRef] [PubMed]

Brooks, K.

P. Thompson, K. Brooks, and S. Hammett, “Speed can go up as well as down at low contrast: Implications for models of motion perception,” Vision Res. 46, 782-786 (2005).
[CrossRef] [PubMed]

Brown, E.

W. Truccolo, U. Eden, M. Fellows, J. Donoghue, and E. Brown, “A point process frame-work for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects,” J. Neurophysiol. 93, 1074-1089 (2005).
[CrossRef]

E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson, “A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells,” J. Neurosci. 18, 7411-7425 (1998).
[PubMed]

Bulthoff, H. H.

A. E. Welchman, J. M. Lam, and H. H. Bulthoff, “Bayesian motion estimation accounts for a surprising bias in 3D vision,” Proc. Natl. Acad. Sci. U.S.A. 105, 12087-12092 (2008).
[CrossRef] [PubMed]

Carandini, M.

F. Hurlimann, D. Kiper, and M. Carandini, “Testing the Bayesian model of perceived speed,” Vision Res. 42, 2253-2257 (2002).
[CrossRef] [PubMed]

Carcieri, S.

S. Nirenberg, S. Carcieri, A. Jacobs, and P. Latham, “Retinal ganglion cells act largely as independent encoders,” Nature 411, 698-701 (2002).
[CrossRef]

Chichilnisky, E.

J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
[CrossRef] [PubMed]

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

E. Frechette, A. Sher, M. Grivich, D. Petrusca, A. Litke, and E. Chichilnisky, “Fidelity of the ensemble code for visual motion in the primate retina,” J. Neurophysiol. 94, 119-135 (2005).
[CrossRef]

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

E. Chichilnisky and R. Kalmar, “Temporal resolution of ensemble visual motion signals in primate retina,” J. Neurosci. 23, 6681-6689 (2003).
[PubMed]

E. Chichilnisky and R. Kalmar, “Functional asymmetries in ON and OFF ganglion cells of primate retina,” J. Neurosci. 22, 2737-2747 (2002).
[PubMed]

Chichilnisky, E. J.

E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
[CrossRef]

Cowey, A.

V. Perry and A. Cowey, “The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors,” Vision Res. 25, 1795-1810 (1985).
[CrossRef] [PubMed]

Cunningham, W.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Dabrowski, W.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Davis, R.

R. Davis and G. Rodriguez-Yam, “Estimation for state-space models: an approximate likelihood approach,” Stat. Sin. 15, 381-406 (2005).

de Ruyter van Steveninck, R.

W. Bialek (Princeton University, bbrinker@princeton.edu) and R. de Ruyter van Steveninck (Indiana University, deruyter@indiana.edu) (personal communication, 2003).

Donoghue, J.

W. Truccolo, U. Eden, M. Fellows, J. Donoghue, and E. Brown, “A point process frame-work for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects,” J. Neurophysiol. 93, 1074-1089 (2005).
[CrossRef]

Eden, U.

W. Truccolo, U. Eden, M. Fellows, J. Donoghue, and E. Brown, “A point process frame-work for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects,” J. Neurophysiol. 93, 1074-1089 (2005).
[CrossRef]

Eurich, C.

A. Thiel, M. Greschner, C. Eurich, J. Ammermüller, and J. Kretzberg, “Contribution of individual retinal ganglion cell responses to velocity and acceleration encoding,” J. Neurophysiol. 98, 2285-2296 (2007).
[CrossRef] [PubMed]

Fellows, M.

W. Truccolo, U. Eden, M. Fellows, J. Donoghue, and E. Brown, “A point process frame-work for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects,” J. Neurophysiol. 93, 1074-1089 (2005).
[CrossRef]

Ferreira, D.

L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

Field, D.

Field, G.

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge Univ. Press, 1992).

Frank, L.

E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson, “A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells,” J. Neurosci. 18, 7411-7425 (1998).
[PubMed]

Frechette, E.

E. Frechette, A. Sher, M. Grivich, D. Petrusca, A. Litke, and E. Chichilnisky, “Fidelity of the ensemble code for visual motion in the primate retina,” J. Neurophysiol. 94, 119-135 (2005).
[CrossRef]

Frechette, E. S.

E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
[CrossRef]

Gauthier, J.

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

Goodhouse, J.

R. Segev, J. Goodhouse, J. Puchalla, and M. Berry, “Recording spikes from a large fraction of the ganglion cells in a retinal patch,” Nat. Neurosci. 7, 1154-1161 (2004).
[CrossRef] [PubMed]

Goyal, M. S.

D. C. Bradley and M. S. Goyal, “Velocity computation in the primate visual system,” Nat. Rev. Neurosci. 9, 686-695 (2008).
[CrossRef]

Greschner, M.

A. Thiel, M. Greschner, C. Eurich, J. Ammermüller, and J. Kretzberg, “Contribution of individual retinal ganglion cell responses to velocity and acceleration encoding,” J. Neurophysiol. 98, 2285-2296 (2007).
[CrossRef] [PubMed]

Grillo, A.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Grivich, M.

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

E. Frechette, A. Sher, M. Grivich, D. Petrusca, A. Litke, and E. Chichilnisky, “Fidelity of the ensemble code for visual motion in the primate retina,” J. Neurophysiol. 94, 119-135 (2005).
[CrossRef]

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Grivich, M. I.

E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
[CrossRef]

Grybos, P.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Grzywacz, N.

D. Ascher and N. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427-3434 (2000).
[CrossRef] [PubMed]

Hammett, S.

P. Thompson, K. Brooks, and S. Hammett, “Speed can go up as well as down at low contrast: Implications for models of motion perception,” Vision Res. 46, 782-786 (2005).
[CrossRef] [PubMed]

Hofer, H.

D. H. Brainard, D. R. Williams, and H. Hofer, “Trichromatic reconstruction from the interleaved cone mosaic: Bayesian model and the color appearance of small spots,” J. Vision 8, 1-23 (2008).
[CrossRef]

Hottowy, P.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Hurlimann, F.

F. Hurlimann, D. Kiper, and M. Carandini, “Testing the Bayesian model of perceived speed,” Vision Res. 42, 2253-2257 (2002).
[CrossRef] [PubMed]

Jacobs, A.

S. Nirenberg, S. Carcieri, A. Jacobs, and P. Latham, “Retinal ganglion cells act largely as independent encoders,” Nature 411, 698-701 (2002).
[CrossRef]

Kachiguine, S.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Kalmar, R.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

E. Chichilnisky and R. Kalmar, “Temporal resolution of ensemble visual motion signals in primate retina,” J. Neurosci. 23, 6681-6689 (2003).
[PubMed]

E. Chichilnisky and R. Kalmar, “Functional asymmetries in ON and OFF ganglion cells of primate retina,” J. Neurosci. 22, 2737-2747 (2002).
[PubMed]

Kalmar, R. S.

E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
[CrossRef]

Kass, R.

R. Kass and A. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90, 773-795 (1995).
[CrossRef]

Kiper, D.

F. Hurlimann, D. Kiper, and M. Carandini, “Testing the Bayesian model of perceived speed,” Vision Res. 42, 2253-2257 (2002).
[CrossRef] [PubMed]

Koyama, S.

S. Koyama and S. Shinomoto, “Empirical Bayes interpretations of random point events,” J. Phys. A 38, 531-537 (2005).
[CrossRef]

L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

Kretzberg, J.

A. Thiel, M. Greschner, C. Eurich, J. Ammermüller, and J. Kretzberg, “Contribution of individual retinal ganglion cell responses to velocity and acceleration encoding,” J. Neurophysiol. 98, 2285-2296 (2007).
[CrossRef] [PubMed]

J. Kretzberg, I. Winzenborg, and A. Thiel, “Bayesian analysis of the encoding of constant and changing stimulus velocities by retinal ganglion cells,” presented at Frontiers in Neuroinformatics 2008, Stockholm, September 7-9, 2008.

Lagnado, L.

M. Meister, L. Lagnado, and D. Baylor, “Concerted signaling by retinal ganglion cells,” Science 270, 1207-1210 (1995).
[CrossRef] [PubMed]

Lam, J. M.

A. E. Welchman, J. M. Lam, and H. H. Bulthoff, “Bayesian motion estimation accounts for a surprising bias in 3D vision,” Proc. Natl. Acad. Sci. U.S.A. 105, 12087-12092 (2008).
[CrossRef] [PubMed]

Latham, P.

S. Nirenberg, S. Carcieri, A. Jacobs, and P. Latham, “Retinal ganglion cells act largely as independent encoders,” Nature 411, 698-701 (2002).
[CrossRef]

Lewi, J.

L. Paninski, J. Pillow, and J. Lewi, “Statistical models for neural encoding, decoding, and optimal stimulus design,” in Computational Neuroscience: Progress in Brain Research, P.Cisek, T.Drew, and J.Kalaska, eds. (Elsevier, 2007).

Litke, A.

J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
[CrossRef] [PubMed]

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

E. Frechette, A. Sher, M. Grivich, D. Petrusca, A. Litke, and E. Chichilnisky, “Fidelity of the ensemble code for visual motion in the primate retina,” J. Neurophysiol. 94, 119-135 (2005).
[CrossRef]

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Litke, A. M.

E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
[CrossRef]

Mathieson, K.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

McCullagh, P.

P. McCullagh and J. Nelder, Generalized Linear Models (Chapman & Hall, 1989).

McKee, S.

S. McKee, G. Silvermann, and K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609-619 (1986).
[CrossRef] [PubMed]

Meister, M.

M. Meister, L. Lagnado, and D. Baylor, “Concerted signaling by retinal ganglion cells,” Science 270, 1207-1210 (1995).
[CrossRef] [PubMed]

Miller, M.

D. Snyder and M. Miller, Random Point Processes in Time and Space (Springer-Verlag, 1991).
[CrossRef]

Nakayama, K.

S. McKee, G. Silvermann, and K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609-619 (1986).
[CrossRef] [PubMed]

Nelder, J.

P. McCullagh and J. Nelder, Generalized Linear Models (Chapman & Hall, 1989).

Nirenberg, S.

S. Nirenberg, S. Carcieri, A. Jacobs, and P. Latham, “Retinal ganglion cells act largely as independent encoders,” Nature 411, 698-701 (2002).
[CrossRef]

Paninski, L.

J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
[CrossRef] [PubMed]

L. Paninski, “Maximum likelihood estimation of cascade point-process neural encoding models,” Network Comput. Neural Syst. 15, 243-262 (2004).
[CrossRef]

J. Pillow, Y. Ahmadian, and L. Paninski, “Model-based decoding, information estimation, and change-point detection in multi-neuron spike trains,” submitted to Neural Comput.
[PubMed]

Y. Ahmadian, J. Pillow, and L. Paninski, “Efficient Markov chain Monte Carlo methods for decoding neural spike trains,” submitted to Neural Comput.
[PubMed]

L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

L. Paninski, J. Pillow, and J. Lewi, “Statistical models for neural encoding, decoding, and optimal stimulus design,” in Computational Neuroscience: Progress in Brain Research, P.Cisek, T.Drew, and J.Kalaska, eds. (Elsevier, 2007).

Perry, V.

V. Perry and A. Cowey, “The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors,” Vision Res. 25, 1795-1810 (1985).
[CrossRef] [PubMed]

Petrusca, D.

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

E. Frechette, A. Sher, M. Grivich, D. Petrusca, A. Litke, and E. Chichilnisky, “Fidelity of the ensemble code for visual motion in the primate retina,” J. Neurophysiol. 94, 119-135 (2005).
[CrossRef]

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
[CrossRef]

Pillow, J.

J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
[CrossRef] [PubMed]

L. Paninski, J. Pillow, and J. Lewi, “Statistical models for neural encoding, decoding, and optimal stimulus design,” in Computational Neuroscience: Progress in Brain Research, P.Cisek, T.Drew, and J.Kalaska, eds. (Elsevier, 2007).

Y. Ahmadian, J. Pillow, and L. Paninski, “Efficient Markov chain Monte Carlo methods for decoding neural spike trains,” submitted to Neural Comput.
[PubMed]

J. Pillow, Y. Ahmadian, and L. Paninski, “Model-based decoding, information estimation, and change-point detection in multi-neuron spike trains,” submitted to Neural Comput.
[PubMed]

Pitt, M.

N. Shephard and M. Pitt, “Likelihood analysis of non-Gaussian measurement time series,” Biometrika 84, 653-667 (1997).
[CrossRef]

Potters, M.

M. Potters and W. Bialek, “Statistical mechanics and visual signal processing,” J. Phys. I France 4, 1755-1775 (1994).
[CrossRef]

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge Univ. Press, 1992).

Puchalla, J.

R. Segev, J. Goodhouse, J. Puchalla, and M. Berry, “Recording spikes from a large fraction of the ganglion cells in a retinal patch,” Nat. Neurosci. 7, 1154-1161 (2004).
[CrossRef] [PubMed]

Quirk, M.

E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson, “A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells,” J. Neurosci. 18, 7411-7425 (1998).
[PubMed]

Raftery, A.

R. Kass and A. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90, 773-795 (1995).
[CrossRef]

Rahman, M.

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

Rahnama, K.

L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

Rodriguez-Yam, G.

R. Davis and G. Rodriguez-Yam, “Estimation for state-space models: an approximate likelihood approach,” Stat. Sin. 15, 381-406 (2005).

Schneidman, E.

E. Schneidman, M. Berry, R. Segev, and W. Bialek, “Weak pairwise correlations imply strongly correlated network states in a neural population,” Nature 440, 1007-1012 (2006).
[CrossRef] [PubMed]

Segev, R.

E. Schneidman, M. Berry, R. Segev, and W. Bialek, “Weak pairwise correlations imply strongly correlated network states in a neural population,” Nature 440, 1007-1012 (2006).
[CrossRef] [PubMed]

R. Segev, J. Goodhouse, J. Puchalla, and M. Berry, “Recording spikes from a large fraction of the ganglion cells in a retinal patch,” Nat. Neurosci. 7, 1154-1161 (2004).
[CrossRef] [PubMed]

Shephard, N.

N. Shephard and M. Pitt, “Likelihood analysis of non-Gaussian measurement time series,” Biometrika 84, 653-667 (1997).
[CrossRef]

Sher, A.

J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
[CrossRef] [PubMed]

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

E. Frechette, A. Sher, M. Grivich, D. Petrusca, A. Litke, and E. Chichilnisky, “Fidelity of the ensemble code for visual motion in the primate retina,” J. Neurophysiol. 94, 119-135 (2005).
[CrossRef]

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
[CrossRef]

Shinomoto, S.

S. Koyama and S. Shinomoto, “Empirical Bayes interpretations of random point events,” J. Phys. A 38, 531-537 (2005).
[CrossRef]

Shlens, J.

J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
[CrossRef] [PubMed]

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

Silvermann, G.

S. McKee, G. Silvermann, and K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609-619 (1986).
[CrossRef] [PubMed]

Simoncelli, E.

J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
[CrossRef] [PubMed]

A. Stocker and E. Simoncelli, “Noise characteristics and prior expectations in human visual speed perception,” Nat. Neurosci. 9, 578-585 (2006).
[CrossRef] [PubMed]

Y. Weiss, E. Simoncelli, and E. Adelson, “Motion illusions as optimal percepts,” Nat. Neurosci. 5, 598-604 (2002).
[CrossRef] [PubMed]

Simoncelli, E. P.

E. P. Simoncelli, “Distributed analysis and representation of visual motion,” Ph.D. thesis (Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1993). Also available as MIT Media Laboratory Vision and Modeling Technical Report #209.

E. P. Simoncelli, “Local analysis of visual motion,” in The Visual Neurosciences, L.M.Chalupa and J.S.Werner, eds. (MIT Press, 2003), Chap. 109, pp. 1616-1623.

Snowden, R.

M. Blakemore and R. Snowden, “The effect of contrast upon perceived speed: a general phenomenon?” Perception 28, 33-48 (1999).
[CrossRef]

Snyder, D.

D. Snyder and M. Miller, Random Point Processes in Time and Space (Springer-Verlag, 1991).
[CrossRef]

Stocker, A.

A. Stocker and E. Simoncelli, “Noise characteristics and prior expectations in human visual speed perception,” Nat. Neurosci. 9, 578-585 (2006).
[CrossRef] [PubMed]

Stone, L.

L. Stone and P. Thompson, “Human speed perception is contrast dependent,” Vision Res. 32, 1535-1549 (1992).
[CrossRef] [PubMed]

Tang, D.

E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson, “A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells,” J. Neurosci. 18, 7411-7425 (1998).
[PubMed]

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge Univ. Press, 1992).

Thiel, A.

A. Thiel, M. Greschner, C. Eurich, J. Ammermüller, and J. Kretzberg, “Contribution of individual retinal ganglion cell responses to velocity and acceleration encoding,” J. Neurophysiol. 98, 2285-2296 (2007).
[CrossRef] [PubMed]

J. Kretzberg, I. Winzenborg, and A. Thiel, “Bayesian analysis of the encoding of constant and changing stimulus velocities by retinal ganglion cells,” presented at Frontiers in Neuroinformatics 2008, Stockholm, September 7-9, 2008.

Thompson, P.

P. Thompson, K. Brooks, and S. Hammett, “Speed can go up as well as down at low contrast: Implications for models of motion perception,” Vision Res. 46, 782-786 (2005).
[CrossRef] [PubMed]

L. Stone and P. Thompson, “Human speed perception is contrast dependent,” Vision Res. 32, 1535-1549 (1992).
[CrossRef] [PubMed]

P. Thompson, “Perceived rate of movement depends on contrast,” Vision Res. 22, 377-380 (1982).
[CrossRef] [PubMed]

Truccolo, W.

W. Truccolo, U. Eden, M. Fellows, J. Donoghue, and E. Brown, “A point process frame-work for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects,” J. Neurophysiol. 93, 1074-1089 (2005).
[CrossRef]

Ullman, S.

S. Ullman, The Interpretation of Visual Motion (MIT Press, 1979).

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge Univ. Press, 1992).

Vidne, M.

L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

Vogelstein, J.

L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

Weiss, Y.

Y. Weiss, E. Simoncelli, and E. Adelson, “Motion illusions as optimal percepts,” Nat. Neurosci. 5, 598-604 (2002).
[CrossRef] [PubMed]

Welchman, A. E.

A. E. Welchman, J. M. Lam, and H. H. Bulthoff, “Bayesian motion estimation accounts for a surprising bias in 3D vision,” Proc. Natl. Acad. Sci. U.S.A. 105, 12087-12092 (2008).
[CrossRef] [PubMed]

Williams, D. R.

D. H. Brainard, D. R. Williams, and H. Hofer, “Trichromatic reconstruction from the interleaved cone mosaic: Bayesian model and the color appearance of small spots,” J. Vision 8, 1-23 (2008).
[CrossRef]

Wilson, M.

E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson, “A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells,” J. Neurosci. 18, 7411-7425 (1998).
[PubMed]

Winzenborg, I.

J. Kretzberg, I. Winzenborg, and A. Thiel, “Bayesian analysis of the encoding of constant and changing stimulus velocities by retinal ganglion cells,” presented at Frontiers in Neuroinformatics 2008, Stockholm, September 7-9, 2008.

Wu, W.

L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

Zee, A.

W. Bialek and A. Zee, “Coding and computation with neural spike trains,” J. Stat. Phys. 59, 103-115 (1990).
[CrossRef]

Biol. Cybern. (1)

D. Brillinger, “Maximum likelihood analysis of spike trains of interacting nerve cells,” Biol. Cybern. 59, 189-200 (1988).
[CrossRef] [PubMed]

Biometrika (1)

N. Shephard and M. Pitt, “Likelihood analysis of non-Gaussian measurement time series,” Biometrika 84, 653-667 (1997).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

A. Litke, N. Bezayiff, E. Chichilnisky, W. Cunningham, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher, “What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity,” IEEE Trans. Nucl. Sci. 51, 1434-1440 (2004).
[CrossRef]

J. Am. Stat. Assoc. (1)

R. Kass and A. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90, 773-795 (1995).
[CrossRef]

J. Neurophysiol. (3)

W. Truccolo, U. Eden, M. Fellows, J. Donoghue, and E. Brown, “A point process frame-work for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects,” J. Neurophysiol. 93, 1074-1089 (2005).
[CrossRef]

A. Thiel, M. Greschner, C. Eurich, J. Ammermüller, and J. Kretzberg, “Contribution of individual retinal ganglion cell responses to velocity and acceleration encoding,” J. Neurophysiol. 98, 2285-2296 (2007).
[CrossRef] [PubMed]

E. Frechette, A. Sher, M. Grivich, D. Petrusca, A. Litke, and E. Chichilnisky, “Fidelity of the ensemble code for visual motion in the primate retina,” J. Neurophysiol. 94, 119-135 (2005).
[CrossRef]

J. Neurosci. (4)

J. Shlens, G. Field, J. Gauthier, M. Grivich, D. Petrusca, A. Sher, A. Litke, and E. Chichilnisky, “The structure of multi-neuron firing patterns in primate retina,” J. Neurosci. 26, 8254-8266 (2006).
[CrossRef] [PubMed]

E. Chichilnisky and R. Kalmar, “Functional asymmetries in ON and OFF ganglion cells of primate retina,” J. Neurosci. 22, 2737-2747 (2002).
[PubMed]

E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson, “A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells,” J. Neurosci. 18, 7411-7425 (1998).
[PubMed]

E. Chichilnisky and R. Kalmar, “Temporal resolution of ensemble visual motion signals in primate retina,” J. Neurosci. 23, 6681-6689 (2003).
[PubMed]

J. Opt. Soc. Am. A (2)

J. Phys. A (1)

S. Koyama and S. Shinomoto, “Empirical Bayes interpretations of random point events,” J. Phys. A 38, 531-537 (2005).
[CrossRef]

J. Phys. I France (1)

M. Potters and W. Bialek, “Statistical mechanics and visual signal processing,” J. Phys. I France 4, 1755-1775 (1994).
[CrossRef]

J. Stat. Phys. (1)

W. Bialek and A. Zee, “Coding and computation with neural spike trains,” J. Stat. Phys. 59, 103-115 (1990).
[CrossRef]

J. Vision (2)

E. S. Frechette, M. I. Grivich, R. S. Kalmar, A. M. Litke, D. Petrusca, A. Sher, and E. J. Chichilnisky, “Retinal motion signals and limits on speed discrimination,” J. Vision 4, 570 (2004).
[CrossRef]

D. H. Brainard, D. R. Williams, and H. Hofer, “Trichromatic reconstruction from the interleaved cone mosaic: Bayesian model and the color appearance of small spots,” J. Vision 8, 1-23 (2008).
[CrossRef]

Nat. Neurosci. (3)

R. Segev, J. Goodhouse, J. Puchalla, and M. Berry, “Recording spikes from a large fraction of the ganglion cells in a retinal patch,” Nat. Neurosci. 7, 1154-1161 (2004).
[CrossRef] [PubMed]

A. Stocker and E. Simoncelli, “Noise characteristics and prior expectations in human visual speed perception,” Nat. Neurosci. 9, 578-585 (2006).
[CrossRef] [PubMed]

Y. Weiss, E. Simoncelli, and E. Adelson, “Motion illusions as optimal percepts,” Nat. Neurosci. 5, 598-604 (2002).
[CrossRef] [PubMed]

Nat. Rev. Neurosci. (1)

D. C. Bradley and M. S. Goyal, “Velocity computation in the primate visual system,” Nat. Rev. Neurosci. 9, 686-695 (2008).
[CrossRef]

Nature (3)

J. Pillow, J. Shlens, L. Paninski, A. Sher, A. Litke, E. Chichilnisky, and E. Simoncelli, “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature 454, 995-999 (2008).
[CrossRef] [PubMed]

E. Schneidman, M. Berry, R. Segev, and W. Bialek, “Weak pairwise correlations imply strongly correlated network states in a neural population,” Nature 440, 1007-1012 (2006).
[CrossRef] [PubMed]

S. Nirenberg, S. Carcieri, A. Jacobs, and P. Latham, “Retinal ganglion cells act largely as independent encoders,” Nature 411, 698-701 (2002).
[CrossRef]

Network Comput. Neural Syst. (1)

L. Paninski, “Maximum likelihood estimation of cascade point-process neural encoding models,” Network Comput. Neural Syst. 15, 243-262 (2004).
[CrossRef]

Neural Comput. (2)

J. Pillow, Y. Ahmadian, and L. Paninski, “Model-based decoding, information estimation, and change-point detection in multi-neuron spike trains,” submitted to Neural Comput.
[PubMed]

Y. Ahmadian, J. Pillow, and L. Paninski, “Efficient Markov chain Monte Carlo methods for decoding neural spike trains,” submitted to Neural Comput.
[PubMed]

Perception (1)

M. Blakemore and R. Snowden, “The effect of contrast upon perceived speed: a general phenomenon?” Perception 28, 33-48 (1999).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

A. E. Welchman, J. M. Lam, and H. H. Bulthoff, “Bayesian motion estimation accounts for a surprising bias in 3D vision,” Proc. Natl. Acad. Sci. U.S.A. 105, 12087-12092 (2008).
[CrossRef] [PubMed]

Science (1)

M. Meister, L. Lagnado, and D. Baylor, “Concerted signaling by retinal ganglion cells,” Science 270, 1207-1210 (1995).
[CrossRef] [PubMed]

Stat. Sin. (1)

R. Davis and G. Rodriguez-Yam, “Estimation for state-space models: an approximate likelihood approach,” Stat. Sin. 15, 381-406 (2005).

Vision Res. (7)

S. McKee, G. Silvermann, and K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609-619 (1986).
[CrossRef] [PubMed]

V. Perry and A. Cowey, “The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors,” Vision Res. 25, 1795-1810 (1985).
[CrossRef] [PubMed]

P. Thompson, “Perceived rate of movement depends on contrast,” Vision Res. 22, 377-380 (1982).
[CrossRef] [PubMed]

L. Stone and P. Thompson, “Human speed perception is contrast dependent,” Vision Res. 32, 1535-1549 (1992).
[CrossRef] [PubMed]

F. Hurlimann, D. Kiper, and M. Carandini, “Testing the Bayesian model of perceived speed,” Vision Res. 42, 2253-2257 (2002).
[CrossRef] [PubMed]

P. Thompson, K. Brooks, and S. Hammett, “Speed can go up as well as down at low contrast: Implications for models of motion perception,” Vision Res. 46, 782-786 (2005).
[CrossRef] [PubMed]

D. Ascher and N. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427-3434 (2000).
[CrossRef] [PubMed]

Other (11)

P. McCullagh and J. Nelder, Generalized Linear Models (Chapman & Hall, 1989).

E. P. Simoncelli, “Local analysis of visual motion,” in The Visual Neurosciences, L.M.Chalupa and J.S.Werner, eds. (MIT Press, 2003), Chap. 109, pp. 1616-1623.

J. Kretzberg, I. Winzenborg, and A. Thiel, “Bayesian analysis of the encoding of constant and changing stimulus velocities by retinal ganglion cells,” presented at Frontiers in Neuroinformatics 2008, Stockholm, September 7-9, 2008.

W. Bialek (Princeton University, bbrinker@princeton.edu) and R. de Ruyter van Steveninck (Indiana University, deruyter@indiana.edu) (personal communication, 2003).

L. Paninski, J. Pillow, and J. Lewi, “Statistical models for neural encoding, decoding, and optimal stimulus design,” in Computational Neuroscience: Progress in Brain Research, P.Cisek, T.Drew, and J.Kalaska, eds. (Elsevier, 2007).

D. Snyder and M. Miller, Random Point Processes in Time and Space (Springer-Verlag, 1991).
[CrossRef]

D.Knill and W.Richards, eds., Perception as Bayesian Inference (Cambridge Univ. Press, 1996).

E. P. Simoncelli, “Distributed analysis and representation of visual motion,” Ph.D. thesis (Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1993). Also available as MIT Media Laboratory Vision and Modeling Technical Report #209.

S. Ullman, The Interpretation of Visual Motion (MIT Press, 1979).

L. Paninski, Y. Ahmadian, D. Ferreira, S. Koyama, K. Rahnama, M. Vidne, J. Vogelstein, and W. Wu, “A new look at state-space models for neural data,” J. Comput. Neurosci. (to be published). Epub ahead of print, doi 10.1007/s10827-009-0179-x.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge Univ. Press, 1992).

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Figures (9)

Fig. 1
Fig. 1

Ensemble motion signals. (A) Moving bar stimulus and cell layout. Cells in the left most column are numbered 1–10 from top to bottom, cells in the second column are numbered 11–20 from top to bottom, etc. (B) Raw responses from the ON cells for a moving bar with speed 14.4 ° s . Each tick represents one spike and each row represents the response of a different cell. (C)–(E) Same spike trains circularly shifted by an amount equal to the time required for a stimulus with the indicated putative speed to move from an arbitrary reference location to the receptive field center. Responses from OFF cells were also included in this procedure.

Fig. 2
Fig. 2

Optimal linear spike train filter w L G for a range of velocities from 0.2 ° s to 28.8 ° s (bottom) in exponential steps. The y axes are scaled in dimensionless units for clarity here. As discussed in Subsection 2B3, there are three time scales that determine the time scale of our filter w L G . At low velocities, shown in the upper panels, the width of w ( t ) is determined by the two scales x k v and x corr v and is thus quite large (since the denominator v is small). At the higher velocities shown in the lower panels, the optimal filter width is dominated by the time scale of the receptive field τ k , and is of the order of τ k , which is 10 20 ms . For even higher velocities the shape of this filter remains essentially the same as in the bottom panel.

Fig. 3
Fig. 3

Effect of filter width τ w on the standard deviation of velocity estimates [obtained using the signal defined in Eq. (22)] across 100 presentations of a black bar moving at a speed of 28.8 ° s across a gray background. Note that a filter width of τ w 10 ms is optimal, in agreement with the findings of [4] and with the width of the optimal filter shown in Fig. 2.

Fig. 4
Fig. 4

Optimal decoder leads to the most precise velocity estimates, while the marginal decoder outperforms the energy-based “net motion signal” method, in terms of precision. (A) Posterior for the optimal decoder. (B) Log marginal likelihood for the marginal decoder. (C) Net motion signal N as a function of putative stimulus speed v for spike trains generated using a stimulus with speed 36.0 ° s and contrast 0.5 for a trial where all methods successfully estimate the stimulus speed. It can be seen that the nonmarginal likelihood (A) is more sharply peaked around the stimulus speed than the marginal likelihood (B) and the net motion signal (C). Distribution of speed estimates across 100 presentations of a bar moving at a speed of 36.0 ° s using (D) the optimal posterior probability, (E) the marginal likelihood, and (F) the net motion signal. Also plotted are Gaussian fits to the distributions with mean ± SD of 35.76 ± 0.81 for the optimal decoder, 37.1 ± 3.05 for the marginal decoder, and 36 ± 6.09 for the net motion signal.

Fig. 5
Fig. 5

Fractional standard deviation of speed estimates versus (A) stimulus speed and (B) stimulus contrast for the Bayesian decoder with full image information (optimal decoder), the Bayesian decoder with incomplete image information (marginal decoder), and the energy method. (C), (D) plot the difference between the mean estimated speed v and the true stimulus speed v normalized by v against the true stimulus speed and stimulus contrast, respectively. Note that the Bayesian decoder provides more precise estimates than the energy method at all levels, with performance improving with prior image information. Furthermore, it should be noted that for contrasts lower than ± 0.5 , particularly at high speeds, the dearth of information in the spike train ensemble resulted in the estimate from an inordinate number of trials being either the highest or lowest input putative speed. Accordingly, performance for s in this range were not calculated and not plotted.

Fig. 6
Fig. 6

Effect of decreasing image uncertainty on accuracy of Bayesian velocity estimation. See Subsection 2C for a detailed description of this simulation. (A) The solid curve with error bars shows the drop in the fractional rms error of the velocity estimate for an a priori unknown image as the number of preview flashes increases. The dashed curve is the fractional error for the case of an a priori known image. The true velocity was 28.8 ° sec and the bar contrast 0.6. (B) The plots show the maximum a posteriori estimate of the image luminance profile (solid curve) in four trials with different numbers of preview flashes (indicated below each plot). The gray areas indicate the marginal uncertainty of the estimated luminance, and the dashed curve shows the actual image profile.

Fig. 7
Fig. 7

Effect of correlated activity and spike timing structure on speed estimates. Fractional SD of speed estimates using shuffled responses plotted as a function of that obtained using regular simulated data for (A) the Bayesian decoder with full image information, (B) the Bayesian decoder with incomplete image information, and (C) the energy method. Fractional SD of speed estimate using resampled spike trains plotted as a function of that obtained using regular simulated data for (D) the Bayesian decoder with full image information, (E) the Bayesian decoder with incomplete image information, and (F) the energy method. Diagonal lines indicate equality. Each circle represents a different one of the 48 speed-by-contrast stimulus conditions. Note that the performance of the decoders is relatively unaffected by these rather drastic manipulations of spike timing.

Fig. 8
Fig. 8

(A) Simple rectangular grid cell arrangement. (B) Jittered cell arrangement.

Fig. 9
Fig. 9

Histograms illustrating the velocity estimates over 100 trials for a stimulus with velocity 28.8 ° s and contrast of 1 using the regular cell arrangement and uniform baseline log firing rates (left column) and the jittered cell arrangement and random baseline log firing rates (middle column). The top row represents the performance of the optimal Bayesian decoder, the middle row represents that of the marginal decoder, and the bottom row presents that of the energy method. Similar performance was obtained with both the rectangular-grid and randomized spatial layouts for all three methods. The right column illustrates the improved estimation performance obtained for all three methods by doubling the baseline log firing rates from 2 and 3   spikes s to 4 and 6   spikes s for the ON and OFF cells respectively.

Equations (50)

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r i ( t ) = α δ ( t t i , α ) ,
λ i ( t ) f ( b i + J i ( t ) + j , β h i j ( t t j , β ) ) ,
J i ( t ) = k i ( t τ , n ) I ( τ , n ) d 2 n d τ ,
log p ( r | I ) i , α log λ ( t i , α ) i 0 T λ i ( t ) d t .
I ( t , n ) = x ( n v t ) ,
J i ( t ) = K i , v ( t ; n ) x ( n ) d 2 n ,
K i , v ( t ; n ) k i ( t τ , n + v τ ) d τ .
p ( v | r ) = p ( r | v ) p v ( v ) v p ( r | v ) p v ( v ) .
p ( v | r , x ) = p ( r | x , v ) p v ( v ) v p ( r | x , v ) p v ( v ) .
p ( r | v ) = p ( r , x | v ) d x = p ( r | x , v ) p x ( x ) d x .
L ( x , r , v ) log p x ( x ) + log p ( r | x , v ) + 1 2 log ( 2 π ) d | C x | ,
p ( r | v ) = 1 ( 2 π ) d | C x | e L ( x , r , v ) d x .
L ( x , r , v ) = 1 2 x T C x 1 x + i [ α log λ i ( t i , α ; x , r ) λ i ( t ; x , r ) d t ] ,
L ( x , r ) L ( x MAP , r , v ) 1 2 ( x x MAP ) T H ( r , v ) ( x x MAP ) ,
H ( r , v ) | x x L ( x , r , v ) | x = x MAP ,
e L ( x , r , v ) p ( x | r , v ) N ( x MAP ( r , v ) , C x ( r , v ) ) ,
p ( r | v ) e L ( x MAP ( r , v ) , r , v ) | C x H ( r , v ) |
log p ( r | v ) L ( x MAP ( r , v ) , r , v ) 1 2 log | C x H ( r , v ) | .
x MAP ( n ; r , v ) = d 2 n C x ( n , n ) i K i , v ( t ; n ) [ r i ( t ) λ i ( t ; x MAP , r ) ] d t .
r ̃ i ( t ) = w r i = α e ( t t i , α ) 2 2 τ 2 .
E ( v , r ) = i , j r ̃ i ( t + n i v ) r ̃ j ( t + n j v ) d t = [ i r ̃ i ( t + n i v ) ] 2 d t .
N ( v , r ) E ( v , r ) E ( v , r ) .
r i = b i + K i , v x + ϵ i ,
log p LG ( r | v ) = 1 2 i , j R i ( t + n i v ) R j ( t + n j v ) d t + A ( v ) ,
R i = w LG ( r i b i ) ,
p LG ( r | x , v ) = i N ( b i + K i , v x , Σ ) .
L LG ( x , r , v ) log [ p x ( x ) ] + log [ p LG ( r | x , v ) ] = 1 2 x T C x 1 x 1 2 i ( r i b i K i , v x ) T Σ 1 ( r i b i K i , v x ) + const.
p LG ( r | v ) = e L LG ( x , r , v ) d x ,
x MAP ( r , v ) = H ( v ) 1 i K i , v T Σ 1 ( r i b i ) .
H ( v ) = x x L LG = C x 1 + i K i , v T Σ 1 K i , v ,
L LG ( x , r , v ) = 1 2 ( x x MAP ) T H ( v ) ( x x MAP ) 1 2 i δ r i T Σ 1 δ r i + 1 2 i j X i T C x ( v ) X j + const. ,
X i K i , v T Σ 1 δ r i .
log p LG ( r | v ) = 1 2 i j X i T C x ( v ) X j 1 2 log | C x H ( v ) | + const.
E LG ( v , r ) 1 2 i j X i T C x ( v ) X j = 1 2 i j X i ( n 1 ) C x ( n 1 , n 2 ; v ) X j ( n 2 ) d 2 n 1 d 2 n 2 .
X i ( n ) = 1 σ 2 d t d τ k i ( t τ , τ v + n ) δ r i ( t ) .
E LG ( v , r ) = 1 2 i j X ̃ i ( n 1 ) C x ( n 1 , n 2 ) X ̃ j ( n 2 ) d n 1 d n 2 ,
X ̃ i ( n ) X i ( n ) d n = 1 σ 2 d t d τ k ̃ i ( t τ , τ v + n ) δ r i ( t ) ,
q i ( t , n ) k ̃ i ( t , n + n i ) = k i ( t , n + n i ) d n ,
E LG ( v , r ) = 1 2 v 2 i j R ̃ i ( t 1 + n i v ) C x ( v t 1 , v t 2 ) R ̃ j ( t 2 + n j v ) d t 1 d t 2 .
R ̃ i ( t 1 ) X ̃ i ( n i v t 1 ) = 1 σ 2 d t d τ k ̃ i ( t τ , v τ v t 1 + n i ) δ r i ( t ) = 1 σ 2 d t d τ q i ( t τ , v ( τ t 1 ) ) δ r i ( t ) .
C x ( n 1 n 2 ) = B x ( n 1 n ) B x ( n 2 n ) d n .
E LG ( v , r ) = 1 2 v i j R ̃ i ( t 1 + n i v ) B x ( v ( t 1 t ) ) B x ( v ( t 2 t ) ) R ̃ j ( t 2 + n j v ) d t 1 d t 2 d t = 1 2 v i j R ̃ i ( t 1 ) B x ( v ( t + n i v t 1 ) ) B x ( v ( t + n j v t 2 ) ) R ̃ j ( t 2 ) d t 1 d t 2 d t .
R i ( t ) 1 v B x ( v ( t t 1 ) ) R ̃ i ( t 1 ) d t 1 ,
E LG ( v , r ) = 1 2 i j R i ( t + n i v ) R j ( t + n j v ) d t .
R i ( t ) = w LG ( t t ) δ r i ( t ) d t ,
R i ( t ) = 1 σ 2 v d t 1 d t d τ B x ( v ( t t 1 ) ) q i ( t τ , v ( τ t 1 ) ) δ r i ( t ) ,
= 1 σ 2 v d t 1 d t d τ B x ( v ( t t t 1 ) ) q i ( τ , v ( τ t 1 ) ) δ r i ( t ) ,
w LG ( t ) = 1 σ 2 v d t 1 d τ B x ( v ( t t 1 ) ) q i ( τ , v ( τ t 1 ) ) .
H ( x ) = C x 1 + i , t J i , t ( x ) ,
J i , t ( n 1 , n 2 ; x ) = K i , v ( t ; n 1 ) K i , v ( t ; n 2 ) λ i ( t ; x ) d t ,

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