Abstract

The difference-of-Gaussians (DOG) filter is a widely used model for the receptive field of neurons in the retina and lateral geniculate nucleus (LGN) and is a potential model in general for responses modulated by an excitatory center with an inhibitory surrounding region. A DOG filter is defined by three standard parameters: the center and surround sigmas (which define the variance of the radially symmetric Gaussians) and the balance (which defines the linear combination of the two Gaussians). These parameters are not directly observable and are typically determined by nonlinear parameter estimation methods applied to the frequency response function. DOG filters show both low-pass (optimal response at zero frequency) and bandpass (optimal response at a nonzero frequency) behavior. This paper reformulates the DOG filter in terms of a directly observable parameter, the zero-crossing radius, and two new (but not directly observable) parameters. In the two-dimensional parameter space, the exact region corresponding to bandpass behavior is determined. A detailed description of the frequency response characteristics of the DOG filter is obtained. It is also found that the directly observable optimal frequency and optimal gain (the ratio of the response at optimal frequency to the response at zero frequency) provide an alternate coordinate system for the bandpass region. Altogether, the DOG filter and its three standard implicit parameters can be determined by three directly observable values. The two-dimensional bandpass region is a potential tool for the analysis of populations of DOG filters (for example, populations of neurons in the retina or LGN), because the clustering of points in this parameter space may indicate an underlying organizational principle. This paper concentrates on circular Gaussians, but the results generalize to multidimensional radially symmetric Gaussians and are given as an appendix.

© 2013 Optical Society of America

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  1. R. W. Rodieck, “Quantitative analysis of cat retinal ganglion cell response to visual stimuli,” Vis. Res. 5, 583–601 (1965).
    [CrossRef]
  2. C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. 187, 517–552 (1966).
  3. E. Kaplan, S. Marcus, and Y. T. So, “Effects of dark adaptation on spatial and temporal properties of receptive fields in cat lateral geniculate nucleus,” J. Physiol. 294, 561–580 (1979).
  4. Y. T. So and R. Shapley, “Spatial tuning of cells in and around lateral geniculate nucleus of the cat: X and Y relay cells and perigeniculate interneurons,” J. Neurophysiol. 45, 107–120 (1981).
  5. A. M. Derrington and P. Lennie, “Spatial and temporal contrast sensitivities of neurons in lateral geniculate nucleus of macaque,” J. Physiol. 357, 219–240 (1984).
  6. G. E. Irvin, V. A. Casagrande, and T. T. Norton, “Center/surround relationships of magnocellular, parvocellular, and koniocellular relay cells in primate lateral geniculate nucleus,” Vis. Neurosci. 10, 363–373 (1993).
    [CrossRef]
  7. P. D. Spear, R. J. Moore, C. B. Y. Kim, J.-T. Xue, and N. Tumosa, “Effects of aging on the primate visual system: spatial and temporal processing by lateral geniculate neurons in young adult and old rhesus monkeys,” J. Neurophysiol. 72, 402–420 (1994).
  8. H. Cheng, Y. M. Chino, E. L. Smith, J. Hamamoto, and K. Yoshida, “Transfer characteristics of lateral geniculate nucleus X neurons in the cat: effects of spatial frequency and contrast,” J. Neurophysiol. 74, 2548–2557 (1995).
  9. L. P. O’Keefe, J. B. Levitt, D. C. Kiper, R. M. Shapley, and J. A. Movshon, “Functional organization of owl monkey lateral geniculate nucleus and visual cortex,” J. Neurophysiol. 80, 594–609 (1998).
  10. A. J. R. White, S. G. Solomon, and P. R. Martin, “Spatial properties of koniocellular cells in the lateral geniculate nucleus of the marmoset Callithrix jacchus,” J. Physiol. 533, 519–535 (2001).
    [CrossRef]
  11. M. S. Grubb and I. D. Thompson, “Quantitative characterization of visual response properties in the mouse dorsal lateral geniculate nucleus,” J. Neurophysiol. 90, 3594–3607 (2003).
    [CrossRef]
  12. V. Bonin, V. Mante, and M. Carandini, “The suppressive field of neurons in lateral geniculate nucleus,” J. Neurosci. 25, 10844–10856 (2005).
    [CrossRef]
  13. G. H. Jacobs and R. L. Yolton, “Center-surround balance in receptive fields of cells in the lateral geniculate nucleus,” Vis. Res. 10, 1127–1144 (1970).
    [CrossRef]
  14. T. P. Hicks, B. B. Lee, and T. R. Vidyasagar, “The responses of cells in macaque lateral geniculate nucleus to sinusoidal gradings,” J. Physiol. 337, 183–200 (1983).
  15. W. M. Usrey and R. C. Reid, “Visual physiology of the lateral geniculate nucleus in two species of new world monkey: Saimiri sciureus and Aotus trivirgatis,” J. Physiol. 523, 755–769 (2000).
    [CrossRef]
  16. S. D. Van Hooser, J. Alexander, F. Heimel, and S. B. Nelson, “Receptive field properties and laminar organization of lateral geniculate nucleus in the gray squirrel (Sciurus carolinesis),” J. Neurophysiol. 90, 3398–3418 (2003).
    [CrossRef]
  17. H. J. Alitto, B. D. Moore, D. L. Rathburn, and W. M. Ursey, “A comparison of visual responses in the lateral geniculate nucleus of alert and anaesthetized macaque monkeys,” J. Physiol. 589, 87–99 (2011).
    [CrossRef]
  18. J. Kremers and S. Weiss, “Receptive field dimensions of lateral geniculate cells in the common marmoset (Callithrix jacchus),” Vis. Res. 37, 2171–2181 (1997).
    [CrossRef]
  19. J. B. Levitt, R. A. Schumer, S. M. Sherman, P. D. Spear, and J. A. Movshon, “Visual response properties of neurons in the LGN of normally reared and visually deprived macaque monkeys,” J. Neurophysiol. 85, 2111–2129 (2001).
  20. X. Xu, A. B. Bonds, and V. A. Casagrande, “Modeling receptive-field structure of koniocellular, magnocellular, and parvocellular LGN cells in the owl monkey (Aotus trivigatus),” Vis. Neurosci. 19, 703–711 (2002).
  21. I. Ohzawa, G. C. DeAngelis, and R. D. Freeman, “Encoding of binocular disparity by simple cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).

2011 (1)

H. J. Alitto, B. D. Moore, D. L. Rathburn, and W. M. Ursey, “A comparison of visual responses in the lateral geniculate nucleus of alert and anaesthetized macaque monkeys,” J. Physiol. 589, 87–99 (2011).
[CrossRef]

2005 (1)

V. Bonin, V. Mante, and M. Carandini, “The suppressive field of neurons in lateral geniculate nucleus,” J. Neurosci. 25, 10844–10856 (2005).
[CrossRef]

2003 (2)

M. S. Grubb and I. D. Thompson, “Quantitative characterization of visual response properties in the mouse dorsal lateral geniculate nucleus,” J. Neurophysiol. 90, 3594–3607 (2003).
[CrossRef]

S. D. Van Hooser, J. Alexander, F. Heimel, and S. B. Nelson, “Receptive field properties and laminar organization of lateral geniculate nucleus in the gray squirrel (Sciurus carolinesis),” J. Neurophysiol. 90, 3398–3418 (2003).
[CrossRef]

2002 (1)

X. Xu, A. B. Bonds, and V. A. Casagrande, “Modeling receptive-field structure of koniocellular, magnocellular, and parvocellular LGN cells in the owl monkey (Aotus trivigatus),” Vis. Neurosci. 19, 703–711 (2002).

2001 (2)

J. B. Levitt, R. A. Schumer, S. M. Sherman, P. D. Spear, and J. A. Movshon, “Visual response properties of neurons in the LGN of normally reared and visually deprived macaque monkeys,” J. Neurophysiol. 85, 2111–2129 (2001).

A. J. R. White, S. G. Solomon, and P. R. Martin, “Spatial properties of koniocellular cells in the lateral geniculate nucleus of the marmoset Callithrix jacchus,” J. Physiol. 533, 519–535 (2001).
[CrossRef]

2000 (1)

W. M. Usrey and R. C. Reid, “Visual physiology of the lateral geniculate nucleus in two species of new world monkey: Saimiri sciureus and Aotus trivirgatis,” J. Physiol. 523, 755–769 (2000).
[CrossRef]

1998 (1)

L. P. O’Keefe, J. B. Levitt, D. C. Kiper, R. M. Shapley, and J. A. Movshon, “Functional organization of owl monkey lateral geniculate nucleus and visual cortex,” J. Neurophysiol. 80, 594–609 (1998).

1997 (1)

J. Kremers and S. Weiss, “Receptive field dimensions of lateral geniculate cells in the common marmoset (Callithrix jacchus),” Vis. Res. 37, 2171–2181 (1997).
[CrossRef]

1996 (1)

I. Ohzawa, G. C. DeAngelis, and R. D. Freeman, “Encoding of binocular disparity by simple cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).

1995 (1)

H. Cheng, Y. M. Chino, E. L. Smith, J. Hamamoto, and K. Yoshida, “Transfer characteristics of lateral geniculate nucleus X neurons in the cat: effects of spatial frequency and contrast,” J. Neurophysiol. 74, 2548–2557 (1995).

1994 (1)

P. D. Spear, R. J. Moore, C. B. Y. Kim, J.-T. Xue, and N. Tumosa, “Effects of aging on the primate visual system: spatial and temporal processing by lateral geniculate neurons in young adult and old rhesus monkeys,” J. Neurophysiol. 72, 402–420 (1994).

1993 (1)

G. E. Irvin, V. A. Casagrande, and T. T. Norton, “Center/surround relationships of magnocellular, parvocellular, and koniocellular relay cells in primate lateral geniculate nucleus,” Vis. Neurosci. 10, 363–373 (1993).
[CrossRef]

1984 (1)

A. M. Derrington and P. Lennie, “Spatial and temporal contrast sensitivities of neurons in lateral geniculate nucleus of macaque,” J. Physiol. 357, 219–240 (1984).

1983 (1)

T. P. Hicks, B. B. Lee, and T. R. Vidyasagar, “The responses of cells in macaque lateral geniculate nucleus to sinusoidal gradings,” J. Physiol. 337, 183–200 (1983).

1981 (1)

Y. T. So and R. Shapley, “Spatial tuning of cells in and around lateral geniculate nucleus of the cat: X and Y relay cells and perigeniculate interneurons,” J. Neurophysiol. 45, 107–120 (1981).

1979 (1)

E. Kaplan, S. Marcus, and Y. T. So, “Effects of dark adaptation on spatial and temporal properties of receptive fields in cat lateral geniculate nucleus,” J. Physiol. 294, 561–580 (1979).

1970 (1)

G. H. Jacobs and R. L. Yolton, “Center-surround balance in receptive fields of cells in the lateral geniculate nucleus,” Vis. Res. 10, 1127–1144 (1970).
[CrossRef]

1966 (1)

C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. 187, 517–552 (1966).

1965 (1)

R. W. Rodieck, “Quantitative analysis of cat retinal ganglion cell response to visual stimuli,” Vis. Res. 5, 583–601 (1965).
[CrossRef]

Alexander, J.

S. D. Van Hooser, J. Alexander, F. Heimel, and S. B. Nelson, “Receptive field properties and laminar organization of lateral geniculate nucleus in the gray squirrel (Sciurus carolinesis),” J. Neurophysiol. 90, 3398–3418 (2003).
[CrossRef]

Alitto, H. J.

H. J. Alitto, B. D. Moore, D. L. Rathburn, and W. M. Ursey, “A comparison of visual responses in the lateral geniculate nucleus of alert and anaesthetized macaque monkeys,” J. Physiol. 589, 87–99 (2011).
[CrossRef]

Bonds, A. B.

X. Xu, A. B. Bonds, and V. A. Casagrande, “Modeling receptive-field structure of koniocellular, magnocellular, and parvocellular LGN cells in the owl monkey (Aotus trivigatus),” Vis. Neurosci. 19, 703–711 (2002).

Bonin, V.

V. Bonin, V. Mante, and M. Carandini, “The suppressive field of neurons in lateral geniculate nucleus,” J. Neurosci. 25, 10844–10856 (2005).
[CrossRef]

Carandini, M.

V. Bonin, V. Mante, and M. Carandini, “The suppressive field of neurons in lateral geniculate nucleus,” J. Neurosci. 25, 10844–10856 (2005).
[CrossRef]

Casagrande, V. A.

X. Xu, A. B. Bonds, and V. A. Casagrande, “Modeling receptive-field structure of koniocellular, magnocellular, and parvocellular LGN cells in the owl monkey (Aotus trivigatus),” Vis. Neurosci. 19, 703–711 (2002).

G. E. Irvin, V. A. Casagrande, and T. T. Norton, “Center/surround relationships of magnocellular, parvocellular, and koniocellular relay cells in primate lateral geniculate nucleus,” Vis. Neurosci. 10, 363–373 (1993).
[CrossRef]

Cheng, H.

H. Cheng, Y. M. Chino, E. L. Smith, J. Hamamoto, and K. Yoshida, “Transfer characteristics of lateral geniculate nucleus X neurons in the cat: effects of spatial frequency and contrast,” J. Neurophysiol. 74, 2548–2557 (1995).

Chino, Y. M.

H. Cheng, Y. M. Chino, E. L. Smith, J. Hamamoto, and K. Yoshida, “Transfer characteristics of lateral geniculate nucleus X neurons in the cat: effects of spatial frequency and contrast,” J. Neurophysiol. 74, 2548–2557 (1995).

DeAngelis, G. C.

I. Ohzawa, G. C. DeAngelis, and R. D. Freeman, “Encoding of binocular disparity by simple cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).

Derrington, A. M.

A. M. Derrington and P. Lennie, “Spatial and temporal contrast sensitivities of neurons in lateral geniculate nucleus of macaque,” J. Physiol. 357, 219–240 (1984).

Enroth-Cugell, C.

C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. 187, 517–552 (1966).

Freeman, R. D.

I. Ohzawa, G. C. DeAngelis, and R. D. Freeman, “Encoding of binocular disparity by simple cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).

Grubb, M. S.

M. S. Grubb and I. D. Thompson, “Quantitative characterization of visual response properties in the mouse dorsal lateral geniculate nucleus,” J. Neurophysiol. 90, 3594–3607 (2003).
[CrossRef]

Hamamoto, J.

H. Cheng, Y. M. Chino, E. L. Smith, J. Hamamoto, and K. Yoshida, “Transfer characteristics of lateral geniculate nucleus X neurons in the cat: effects of spatial frequency and contrast,” J. Neurophysiol. 74, 2548–2557 (1995).

Heimel, F.

S. D. Van Hooser, J. Alexander, F. Heimel, and S. B. Nelson, “Receptive field properties and laminar organization of lateral geniculate nucleus in the gray squirrel (Sciurus carolinesis),” J. Neurophysiol. 90, 3398–3418 (2003).
[CrossRef]

Hicks, T. P.

T. P. Hicks, B. B. Lee, and T. R. Vidyasagar, “The responses of cells in macaque lateral geniculate nucleus to sinusoidal gradings,” J. Physiol. 337, 183–200 (1983).

Irvin, G. E.

G. E. Irvin, V. A. Casagrande, and T. T. Norton, “Center/surround relationships of magnocellular, parvocellular, and koniocellular relay cells in primate lateral geniculate nucleus,” Vis. Neurosci. 10, 363–373 (1993).
[CrossRef]

Jacobs, G. H.

G. H. Jacobs and R. L. Yolton, “Center-surround balance in receptive fields of cells in the lateral geniculate nucleus,” Vis. Res. 10, 1127–1144 (1970).
[CrossRef]

Kaplan, E.

E. Kaplan, S. Marcus, and Y. T. So, “Effects of dark adaptation on spatial and temporal properties of receptive fields in cat lateral geniculate nucleus,” J. Physiol. 294, 561–580 (1979).

Kim, C. B. Y.

P. D. Spear, R. J. Moore, C. B. Y. Kim, J.-T. Xue, and N. Tumosa, “Effects of aging on the primate visual system: spatial and temporal processing by lateral geniculate neurons in young adult and old rhesus monkeys,” J. Neurophysiol. 72, 402–420 (1994).

Kiper, D. C.

L. P. O’Keefe, J. B. Levitt, D. C. Kiper, R. M. Shapley, and J. A. Movshon, “Functional organization of owl monkey lateral geniculate nucleus and visual cortex,” J. Neurophysiol. 80, 594–609 (1998).

Kremers, J.

J. Kremers and S. Weiss, “Receptive field dimensions of lateral geniculate cells in the common marmoset (Callithrix jacchus),” Vis. Res. 37, 2171–2181 (1997).
[CrossRef]

Lee, B. B.

T. P. Hicks, B. B. Lee, and T. R. Vidyasagar, “The responses of cells in macaque lateral geniculate nucleus to sinusoidal gradings,” J. Physiol. 337, 183–200 (1983).

Lennie, P.

A. M. Derrington and P. Lennie, “Spatial and temporal contrast sensitivities of neurons in lateral geniculate nucleus of macaque,” J. Physiol. 357, 219–240 (1984).

Levitt, J. B.

J. B. Levitt, R. A. Schumer, S. M. Sherman, P. D. Spear, and J. A. Movshon, “Visual response properties of neurons in the LGN of normally reared and visually deprived macaque monkeys,” J. Neurophysiol. 85, 2111–2129 (2001).

L. P. O’Keefe, J. B. Levitt, D. C. Kiper, R. M. Shapley, and J. A. Movshon, “Functional organization of owl monkey lateral geniculate nucleus and visual cortex,” J. Neurophysiol. 80, 594–609 (1998).

Mante, V.

V. Bonin, V. Mante, and M. Carandini, “The suppressive field of neurons in lateral geniculate nucleus,” J. Neurosci. 25, 10844–10856 (2005).
[CrossRef]

Marcus, S.

E. Kaplan, S. Marcus, and Y. T. So, “Effects of dark adaptation on spatial and temporal properties of receptive fields in cat lateral geniculate nucleus,” J. Physiol. 294, 561–580 (1979).

Martin, P. R.

A. J. R. White, S. G. Solomon, and P. R. Martin, “Spatial properties of koniocellular cells in the lateral geniculate nucleus of the marmoset Callithrix jacchus,” J. Physiol. 533, 519–535 (2001).
[CrossRef]

Moore, B. D.

H. J. Alitto, B. D. Moore, D. L. Rathburn, and W. M. Ursey, “A comparison of visual responses in the lateral geniculate nucleus of alert and anaesthetized macaque monkeys,” J. Physiol. 589, 87–99 (2011).
[CrossRef]

Moore, R. J.

P. D. Spear, R. J. Moore, C. B. Y. Kim, J.-T. Xue, and N. Tumosa, “Effects of aging on the primate visual system: spatial and temporal processing by lateral geniculate neurons in young adult and old rhesus monkeys,” J. Neurophysiol. 72, 402–420 (1994).

Movshon, J. A.

J. B. Levitt, R. A. Schumer, S. M. Sherman, P. D. Spear, and J. A. Movshon, “Visual response properties of neurons in the LGN of normally reared and visually deprived macaque monkeys,” J. Neurophysiol. 85, 2111–2129 (2001).

L. P. O’Keefe, J. B. Levitt, D. C. Kiper, R. M. Shapley, and J. A. Movshon, “Functional organization of owl monkey lateral geniculate nucleus and visual cortex,” J. Neurophysiol. 80, 594–609 (1998).

Nelson, S. B.

S. D. Van Hooser, J. Alexander, F. Heimel, and S. B. Nelson, “Receptive field properties and laminar organization of lateral geniculate nucleus in the gray squirrel (Sciurus carolinesis),” J. Neurophysiol. 90, 3398–3418 (2003).
[CrossRef]

Norton, T. T.

G. E. Irvin, V. A. Casagrande, and T. T. Norton, “Center/surround relationships of magnocellular, parvocellular, and koniocellular relay cells in primate lateral geniculate nucleus,” Vis. Neurosci. 10, 363–373 (1993).
[CrossRef]

O’Keefe, L. P.

L. P. O’Keefe, J. B. Levitt, D. C. Kiper, R. M. Shapley, and J. A. Movshon, “Functional organization of owl monkey lateral geniculate nucleus and visual cortex,” J. Neurophysiol. 80, 594–609 (1998).

Ohzawa, I.

I. Ohzawa, G. C. DeAngelis, and R. D. Freeman, “Encoding of binocular disparity by simple cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).

Rathburn, D. L.

H. J. Alitto, B. D. Moore, D. L. Rathburn, and W. M. Ursey, “A comparison of visual responses in the lateral geniculate nucleus of alert and anaesthetized macaque monkeys,” J. Physiol. 589, 87–99 (2011).
[CrossRef]

Reid, R. C.

W. M. Usrey and R. C. Reid, “Visual physiology of the lateral geniculate nucleus in two species of new world monkey: Saimiri sciureus and Aotus trivirgatis,” J. Physiol. 523, 755–769 (2000).
[CrossRef]

Robson, J. G.

C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. 187, 517–552 (1966).

Rodieck, R. W.

R. W. Rodieck, “Quantitative analysis of cat retinal ganglion cell response to visual stimuli,” Vis. Res. 5, 583–601 (1965).
[CrossRef]

Schumer, R. A.

J. B. Levitt, R. A. Schumer, S. M. Sherman, P. D. Spear, and J. A. Movshon, “Visual response properties of neurons in the LGN of normally reared and visually deprived macaque monkeys,” J. Neurophysiol. 85, 2111–2129 (2001).

Shapley, R.

Y. T. So and R. Shapley, “Spatial tuning of cells in and around lateral geniculate nucleus of the cat: X and Y relay cells and perigeniculate interneurons,” J. Neurophysiol. 45, 107–120 (1981).

Shapley, R. M.

L. P. O’Keefe, J. B. Levitt, D. C. Kiper, R. M. Shapley, and J. A. Movshon, “Functional organization of owl monkey lateral geniculate nucleus and visual cortex,” J. Neurophysiol. 80, 594–609 (1998).

Sherman, S. M.

J. B. Levitt, R. A. Schumer, S. M. Sherman, P. D. Spear, and J. A. Movshon, “Visual response properties of neurons in the LGN of normally reared and visually deprived macaque monkeys,” J. Neurophysiol. 85, 2111–2129 (2001).

Smith, E. L.

H. Cheng, Y. M. Chino, E. L. Smith, J. Hamamoto, and K. Yoshida, “Transfer characteristics of lateral geniculate nucleus X neurons in the cat: effects of spatial frequency and contrast,” J. Neurophysiol. 74, 2548–2557 (1995).

So, Y. T.

Y. T. So and R. Shapley, “Spatial tuning of cells in and around lateral geniculate nucleus of the cat: X and Y relay cells and perigeniculate interneurons,” J. Neurophysiol. 45, 107–120 (1981).

E. Kaplan, S. Marcus, and Y. T. So, “Effects of dark adaptation on spatial and temporal properties of receptive fields in cat lateral geniculate nucleus,” J. Physiol. 294, 561–580 (1979).

Solomon, S. G.

A. J. R. White, S. G. Solomon, and P. R. Martin, “Spatial properties of koniocellular cells in the lateral geniculate nucleus of the marmoset Callithrix jacchus,” J. Physiol. 533, 519–535 (2001).
[CrossRef]

Spear, P. D.

J. B. Levitt, R. A. Schumer, S. M. Sherman, P. D. Spear, and J. A. Movshon, “Visual response properties of neurons in the LGN of normally reared and visually deprived macaque monkeys,” J. Neurophysiol. 85, 2111–2129 (2001).

P. D. Spear, R. J. Moore, C. B. Y. Kim, J.-T. Xue, and N. Tumosa, “Effects of aging on the primate visual system: spatial and temporal processing by lateral geniculate neurons in young adult and old rhesus monkeys,” J. Neurophysiol. 72, 402–420 (1994).

Thompson, I. D.

M. S. Grubb and I. D. Thompson, “Quantitative characterization of visual response properties in the mouse dorsal lateral geniculate nucleus,” J. Neurophysiol. 90, 3594–3607 (2003).
[CrossRef]

Tumosa, N.

P. D. Spear, R. J. Moore, C. B. Y. Kim, J.-T. Xue, and N. Tumosa, “Effects of aging on the primate visual system: spatial and temporal processing by lateral geniculate neurons in young adult and old rhesus monkeys,” J. Neurophysiol. 72, 402–420 (1994).

Ursey, W. M.

H. J. Alitto, B. D. Moore, D. L. Rathburn, and W. M. Ursey, “A comparison of visual responses in the lateral geniculate nucleus of alert and anaesthetized macaque monkeys,” J. Physiol. 589, 87–99 (2011).
[CrossRef]

Usrey, W. M.

W. M. Usrey and R. C. Reid, “Visual physiology of the lateral geniculate nucleus in two species of new world monkey: Saimiri sciureus and Aotus trivirgatis,” J. Physiol. 523, 755–769 (2000).
[CrossRef]

Van Hooser, S. D.

S. D. Van Hooser, J. Alexander, F. Heimel, and S. B. Nelson, “Receptive field properties and laminar organization of lateral geniculate nucleus in the gray squirrel (Sciurus carolinesis),” J. Neurophysiol. 90, 3398–3418 (2003).
[CrossRef]

Vidyasagar, T. R.

T. P. Hicks, B. B. Lee, and T. R. Vidyasagar, “The responses of cells in macaque lateral geniculate nucleus to sinusoidal gradings,” J. Physiol. 337, 183–200 (1983).

Weiss, S.

J. Kremers and S. Weiss, “Receptive field dimensions of lateral geniculate cells in the common marmoset (Callithrix jacchus),” Vis. Res. 37, 2171–2181 (1997).
[CrossRef]

White, A. J. R.

A. J. R. White, S. G. Solomon, and P. R. Martin, “Spatial properties of koniocellular cells in the lateral geniculate nucleus of the marmoset Callithrix jacchus,” J. Physiol. 533, 519–535 (2001).
[CrossRef]

Xu, X.

X. Xu, A. B. Bonds, and V. A. Casagrande, “Modeling receptive-field structure of koniocellular, magnocellular, and parvocellular LGN cells in the owl monkey (Aotus trivigatus),” Vis. Neurosci. 19, 703–711 (2002).

Xue, J.-T.

P. D. Spear, R. J. Moore, C. B. Y. Kim, J.-T. Xue, and N. Tumosa, “Effects of aging on the primate visual system: spatial and temporal processing by lateral geniculate neurons in young adult and old rhesus monkeys,” J. Neurophysiol. 72, 402–420 (1994).

Yolton, R. L.

G. H. Jacobs and R. L. Yolton, “Center-surround balance in receptive fields of cells in the lateral geniculate nucleus,” Vis. Res. 10, 1127–1144 (1970).
[CrossRef]

Yoshida, K.

H. Cheng, Y. M. Chino, E. L. Smith, J. Hamamoto, and K. Yoshida, “Transfer characteristics of lateral geniculate nucleus X neurons in the cat: effects of spatial frequency and contrast,” J. Neurophysiol. 74, 2548–2557 (1995).

J. Neurophysiol. (8)

Y. T. So and R. Shapley, “Spatial tuning of cells in and around lateral geniculate nucleus of the cat: X and Y relay cells and perigeniculate interneurons,” J. Neurophysiol. 45, 107–120 (1981).

P. D. Spear, R. J. Moore, C. B. Y. Kim, J.-T. Xue, and N. Tumosa, “Effects of aging on the primate visual system: spatial and temporal processing by lateral geniculate neurons in young adult and old rhesus monkeys,” J. Neurophysiol. 72, 402–420 (1994).

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Figures (9)

Fig. 1.
Fig. 1.

Diagram illustrating cross sections of three DOG field functions (1, 2, and 3) with different values of the balance (βCS) parameter. Curve (1): αC2=0.2459, αS2=0.9837 (μ=4.0, ν=1.1, βCS=0.871, strong bandpass). Curve (2): αC2=0.2164, αS2=0.8656 (μ=4.0, ν=1.25, βCS=0.707, moderate bandpass). Curve (3): αC2=0.1288, αS2=0.5152 (μ=4.0, ν=2.1, βCS=0.218, low pass).

Fig. 2.
Fig. 2.

Diagram illustrating the reformulation and graphical representation of the DOG filter typically parameterized using σC, σS, and βCS, reduced to a two-dimensional parameter space using αC2, αS2 coordinates. The lower and upper bounds of region AB are designated respectively by Curve A and Curve B (thick blue lines). Region AB is partitioned by the βCS-contours (thin blue lines) such that each point of the region lies on one and only one βCS contour for 0<βCS<1. The boundary curves A and B both correspond to βCS=1. The region and its boundary are defined by the totality of contours, 0<βCS1. There is also a one-to-one correspondence between the points of region AB and ordered pairs of auxiliary parameters (μ,ν), shown as dashed lines, where the boundary curves A and B correspond to the bounds μ=1 and ν=1, respectively. The gridlines for fixed μ are rays with slope μ through the origin since αS2=μαC2. On each gridline with fixed ν, βCS decreases monotonically from 1 to 0 as μ increases. Points labeled (1), (2), and (3) correspond to three DOG filters whose space- and frequency-domain representations are illustrated in Figs. 1 and 3, respectively.

Fig. 3.
Fig. 3.

Diagram illustrating the frequency response functions of the DOG filters (1), (2), and (3), whose space-domain cross sections appear in Fig. 1. Note the progression from strongly bandpass (1) to low-pass (3) frequency response as the value of the balance parameter (βCS) decreases. The maximum of each bandpass case defines the optimal linear frequency sLIN. Notice the optimal frequencies satisfy the bound πρ0sLIN1. Parameter values are given in Fig. 1.

Fig. 4.
Fig. 4.

Diagram illustrating normalized optimal-frequency contours (πρ0sLIN; thin pink lines) of DOG filters within region ABC. Curve C (thick pink line) indicates a boundary condition where the optimal frequency becomes zero, and the frequency response function becomes low-pass.

Fig. 5.
Fig. 5.

Diagram showing region ABC with sample gLIN contours (green lines). Notice that the gLIN contours partition region ABC. For a fixed value of μ, gLIN ranges from infinity on the boundary ν=1 (Curve B) and strictly decreases to unity at ν=2 (Curve C). For a fixed value of ν, gLIN strictly decreases from the value exp(ν2)/(ν1)>1 at μ=1 (Curve A) to one as μ increases.

Fig. 6.
Fig. 6.

Diagram illustrating the combination of the sLIN and gLIN contours from Figs. 4 and 5. The contours form a coordinate grid for region ABC; that is, there is a one-to-one correspondence between parameter pairs (αC2,αS2) and the pairs (sLIN,gLIN), which are directly observable.

Fig. 7.
Fig. 7.

Converse to Fig. 6 showing αC2, αS2 contours in coordinates log10(gLIN), (πρ0sLIN)2. The thick solid line is a boundary curve corresponding to Curve A in Fig. 6. The dashed lines are contours for αC2=0.15,0.20, 0.25, 0.30, and 0.35. For 0<αC20.25, the contours extend to the origin (log10(gLIN)=0), but for 0.25<αC2<0.5, the contours meet the boundary curve at nonzero values of log10(gLIN), behavior that is just visible on the plot. The solid lines are contours for αS2=0.45 (unmarked and just visible) and for αS2=0.5, 1.0, 2.0. For αS20.5, the contours extend to infinity (log10(gLIN)=), but for 0<αS2<0.5, the contours meet the boundary curve at finite values of log10(gLIN), behavior that is just visible in the plot.

Fig. 8.
Fig. 8.

Diagram illustrating Curve D together with βCS contours and πρ0sLIN contours. Note that the maximum optimal frequency on a given balance contour occurs at the point on the contour that is tangent to an optimal-frequency contour.

Fig. 9.
Fig. 9.

Optimal frequency and gain contours for dimension N=1. The plot shows region A1B1C1 as a subregion of region A1B1 with contours for πρ0sLIN=0.60, 0.65, 0.70, 0.725 and gLIN=1.25, 2.0, 4.0, 8.0, and a balance contour for βCS=0.85. The optimal frequency is bounded by 0<πρ0sLIN0.75. The dashed line is ν=3/2, where πρ0sLIN is a maximum for fixed μ and where the πρ0sLIN contour is tangent to the corresponding ray through the origin, as is evident in the plot.

Equations (43)

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f(r)12πσC2exp(r22σC2)βCS2πσS2exp(r22σS2).
F(s1,s2)R×Rf(x1,x2)exp(2πi(s1x1+s2x2))dx1dx2,
F(s)=exp(2π2σC2s2)βCSexp(2π2σS2s2).
0<σC<σSand0<βCS1.
αCσC/ρ0andαSσS/ρ0.
βCS=αS2αC2exp(12αS212αC2).
μ=αS2αC2,ν=αS2αC22αC2αS2ln(αS2/αC2)orαC2=μ12ln(μ)1μν,αS2=μ12ln(μ)1νwith1<μ<,1<ν<.
βCS=μ1ν.
CurveA:αS2=αC2forμ=1,
CurveB:αC2=μ12ln(μ)1μ,αS2=μαC2for1<μ<,ν=1.
F(s)=exp(2αC2π2ρ02s2)βCSexp(2αS2π2ρ02s2).
(πρ0sLIN)2=ln(αS2βCS/αC2)2(αS2αC2)=ν(2ν)μ(ln(μ)μ1)2.
CurveC:αC2=μ14ln(μ)1μ,αS2=μαC2for1<μ<,ν=2.
0πρ0sLIN1.
ν=1+(11μ(μ1ln(μ))2(πρ0sLIN)2)1/2(πρ0sLIN)2=μ0(ln(μ0)μ01)2,
gLINF(sLIN)/F(0).
gLIN=exp(ln(αS2/αC2)ln(βCS)αS2/αC21)1αC2/αS21βCS=exp((ν2)ln(μ)μ1)μ1μμ2ν.
gLIN=exp(ln(βCSμ)μ1)11/μ1βCS.
gLIN=1atμ=1/βCSandgLIN11βCSasμ.
(πρ0sLIN)2=((ln(μ))2(ln(βCS))2)μ(μ1)2.
πρ0sLIN=0atμ=1/βCSandπρ0sLIN0asμ.
(ln(μCS))2(ln(βCS))2=2μCS1μCS+1ln(μCS),
(πρ0sCS)2=2μCSμCS21ln(μCS).
f(r)1(2π)N/2σCNexp(r22σC2)βCS(2π)N/2σSNexp(r22σS2).
F(s1,,sN)RNf(x1,,xN)exp(2πi(s1x1++sNxN))dx1dxN,
F(s)=exp(2π2σC2s2)βCSexp(2π2σS2s2).
αCσC/ρ0andαSσS/ρ0.
βCS=αSNαCNexp(12αS212αC2).
μ:=αS2/αC2andναS2αC2NαC2αS2ln(αS2/αC2).
βCS=μ(1ν)N/2,
αC2=μ1Nln(μ)1μνandαS2=μ1Nln(μ)1ν.
αC2=μ1Nμ1ln(μ)2ln(βCS)/NandαS2=μ1N1ln(μ)2ln(βCS)/N.
(πρ0sLIN)2=(N2)2ν(2N+1ν)μ(ln(μ)μ1)2,ln(gLIN)=N2(ν12N)ln(μ)μ1+ln(11/μ1μ(1ν)N/2).
{(μ,ν):1<μ<,1<ν<1+2N}and{(μ,βCS):1<μ<,μ1<βCS<1}.
xln(μ),yln(1/βCS).
(πρ0sLIN)2=(N2x+y)(xy)ex(ex1)2,ln(gLIN)=yxex1+ln(1ex)ln(1ey).
dydx=xyexeyey1ex1ex.
1(πρ0sLIN)2ddx(πρ0sLIN)2=N+2dydxNx+2y+1dydxxyex+1ex1,
1(πρ0sLIN)2ddx(πρ0sLIN)2=2Nx+(2N)y(Nx+2y)(xy)ex+1ex1+(2N)x4y(Nx+2y)(xy)dydx,
ddx(πρ0sLIN)2=(πρ0sLIN)2ex+y/2E(Nx+2y)(exey)(ex1),
Eexy/2[(Nx+2y)(xy)(Nx+2y)N(xy)]ey/2[2(xy)2(Nx+2y)N(xy)]+ey/2[2(xy)2+(Nx+2y)+N(xy)]ex+y/2[(Nx+2y)(xy)+(Nx+2y)+N(xy)].
E=4(N+Na+22a)xsinh(x/2)((ax/2)cosh(ax/2)sinh(ax/2))+2(N+2)ax2(cosh(x/2)sinh(ax/2)asinh(x/2)cosh(ax/2)).
f(z)zcosh(z)sinh(z)>0forz>0g(a,z)cosh(z)sinh(az)asinh(z)cosh(az)>0forz>0,0<a<1.

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