Abstract

When an oriented bar or grating is drifted across the receptive field of a cortical neuron at various orientations, the tuning function reflects both, and thus confounds the orientation (ORI) and the direction-of-motion (DIR) selectivity of the cell. Since ORI (or DIR), by definition, has a period of 180(or360)degcycle, a popular method for separating these two components, due to Wörgötter and Eysel [Biol. Cybern. 57, 349 (1987)] , is to Fourier decompose the neuron’s response along the angular direction and then identify the first and the second harmonic with DIR and ORI, respectively (the SDO method). Zhang [Biol. Cybern. 63, 135 (1990)] pointed out that this interpretation is misconceived—all odd harmonics (not just the first harmonic) reflect the DIR component, whereas all even harmonics (including the second harmonic) contain contributions from both DIR and ORI. Here, a simplified procedure is proposed to accomplish the goal of unconfounding ORI and DIR. We first construct the sum of all odd harmonics of the overall tuning curve, denoted ODDSUM, by calculating the difference in the neuronal response to opposite drifting directions. Then we construct ODDSUM+ODDSUM and identify it with DIR (here denotes the absolute value). Subtracting DIR, that is ODDSUM+ODDSUM, from the overall tuning curve gives ORI. Our method ensures that (i) the reconstructed DIR contains only one, positive peak at the preferred direction and can have power in all harmonics, and (ii) the reconstructed ORI has two peaks separated by 180° and has zero power for all odd harmonics. Using this procedure, we have unconfounded orientation and direction components for a considerable sample of macaque striate cortical cells, and compared the results with those obtained using Wörgötter and Eysel’s SDO method. We found that whereas the estimate of the peak angle of ORI remains largely unaffected, Wörgötter and Eysel’s method considerably overestimated the relative strength of ORI. To conclude, a simple method is provided for appropriately separating the orientation and directional tuning in a neuron’s response that is confounded as a result of the use of drifting oriented stimuli.

© 2005 Optical Society of America

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  1. F. Wörgötter, U. T. Eysel, “Quantitative determination of orientational and directional components in the response of visual cortical cells to moving stimuli,” Biol. Cybern. 57, 349–355 (1987).
    [CrossRef] [PubMed]
  2. F. Wörgötter, T. Muche, E. T. Eysel, “Correlations between directional and orientational tuning of cells in cat striate cortex,” Exp. Brain Res. 83, 665–669 (1991).
    [CrossRef] [PubMed]
  3. F. Wörgötter, U. T. Eysel, “Topographical aspects of intracortical excitation and inhibition contributing to orientation specificity in area 17 of the cat visual cortex,” Eur. J. Neurosci. 3, 1232–1244 (1991).
    [CrossRef] [PubMed]
  4. F. Wörgötter, O. Gründel, U. T. Eysel, “Quantification and comparison of cell properties in cat’s striate cortex determined by different types of stimuli,” Eur. J. Neurosci. 2, 928–941 (1990).
    [CrossRef]
  5. D. O’Carroll, “Feature-detecting neurons in dragonflies,” Nature (London) 362, 541–543 (1993).
    [CrossRef]
  6. B. Chapman, M. P. Stryker, “Development of orientation selectivity in ferret visual-cortex and effects of deprivation,” J. Neurosci. 13, 5251–5262 (1993).
    [PubMed]
  7. D. Malonek, R. B.H. Tootell, A. Grinvald, “Optical imaging reveals the functional architecture of neurons processing shape and motion in owl monkey area MT,” Proc. R. Soc. London, Ser. B 258, 109–119 (1994).
    [CrossRef]
  8. A. Shmuel, A. Grinvald, “Functional organization for direction of motion and its relationship to orientation maps in cat area 18,” J. Neurosci. 16, 6945–6964 (1996).
    [PubMed]
  9. M. Weliky, L. C. Katz, “Disruption of orientation tuning in visual cortex by artificially correlated neuronal activity,” Nature (London) 386, 680–685 (1997).
    [CrossRef]
  10. A. Schmidt, J. Engelage, H. J. Bischof, “Single cell responses from the optic tectum of the zebra finch (Taeniopygia guttata castanotis Gould),” J. Comp. Physiol. A 185, 69–79 (1999).
    [CrossRef]
  11. F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nat. Neurosci. 2, 727–732 (1999).
    [CrossRef] [PubMed]
  12. T. Yousef, T. Bonhoeffer, D.-S. Kim, U. T. Eysel, É. Tóth, Z. F. Kisvárday, “Orientation topography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area 18),” Eur. J. Neurosci. 11, 4291–4308 (1999).
    [CrossRef] [PubMed]
  13. J. Sharma, A. Angelucci, M. Sur, “Induction of visual orientation modules in auditory cortex,” Nature (London) 404, 841–847 (2000).
    [CrossRef]
  14. K. Krug, C. J. Akerman, I. D. Thompson, “Responses of neurons in neonatal cortex and thalamus to patterned visual stimulation through the naturally closed lids,” J. Neurophysiol. 85, 1436–1443 (2001).
    [PubMed]
  15. A. Schmidt, H. J. Bischof, “Neurons with complex receptive fields in the stratum griseum centrale of the zebra finch (Taeniopygia guttata castanotis Gould) optic tectum” J. Comp. Physiol., A 187, 913–924 (2001).
    [CrossRef]
  16. C. J. Akerman, D. Smyth, I. D. Thompson, “Visual experience before eye-opening and the development of the retinogeniculate pathway,” Neuron 36, 869–879 (2002).
    [CrossRef] [PubMed]
  17. F. Sengpiel, F. T. Bonhoeffer, “Orientation specificity of contrast adaptation in visual cortical pinwheel centres and iso-orientation domains,” Eur. J. Neurosci. 15, 876–886 (2002).
    [CrossRef] [PubMed]
  18. C. Monier, F. Chavane, P. Baudot, L. J. Graham, Y. Frégnac, “Orientation and direction selectivity of synaptic inputs in visual cortical neurons: a diversity of combinations produces spike tuning,” Neuron 37, 663–680 (2003).
    [CrossRef] [PubMed]
  19. T. H. Schwartz, “Optical imaging of epileptiform events in visual cortex in response to patterned photic stimulation,” Cereb. Cortex 13, 1287–1298 (2003).
    [CrossRef] [PubMed]
  20. A. G. Leventhal, K. G. Thompson, D. Liu, Y. Zhou, S. J. Ault, “Concomitant sensitivity to orientation, direction, and color of cells in layers 2, 3, and 4 of monkey striate cortex,” J. Neurosci. 15, 1808–1818 (1995).
    [PubMed]
  21. T. D. Shou, A. G. Leventhal, K. G. Thompson, Y. F. Zhou, “Direction biases of X-type and Y-type retinal ganglion-cells in the cat,” J. Neurophysiol. 73, 1414–1421 (1995).
    [PubMed]
  22. Y. F. Zhou, A. G. Leventhal, K. G. Thompson, “Visual deprivation does not affect the orientation and direction sensitivity of relay cells in the lateral geniculate nucleus of the cat,” J. Neurosci. 15, 689–698 (1995).
    [PubMed]
  23. A. G. Leventhal, Y. C. Wang, M. T. Schmolesky, Y. F. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
    [CrossRef]
  24. A. Shmuel, A. Grinvald, “Coexistence of linear zones and pinwheels within orientation maps in cat visual cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 5568–5573 (2000).
    [CrossRef] [PubMed]
  25. E. Batschelet, Circular Statistics in Biology (Academic, 1981).
  26. J. Zhang, “How to unconfound the directional and orientational information in visual neuron’s response,” Biol. Cybern. 63, 135–142 (1990).
    [CrossRef]
  27. Note that here and below the designation of “first” and “second” harmonics refers to using a fundamental period of 360 deg∕cycle in Fourier analysis. If one were to Fourier decompose, say ORI, using a fundamental period of 180 deg∕cycle, then the second harmonic mentioned above should be renamed the first harmonic (in describing a periodic function with a period of 180 deg∕cycle).
  28. In fact, higher-order harmonics were consistently present even in these authors’ own data; see Ref. [2]. However they were dismissed based on the percentage of total power they contributed. This was misguided, because any single-peaked DIR will have decreasing power in its higher harmonics. In fact, Zhang[26] showed that under quite mild restriction, the strength of the kth harmonic of DIR is proportional to sin(kα)∕k, with α characterizing the bandwidth. So the systematic presence of powers with decreasing strength in higher harmonics cannot be explained away simply as noise; rather it reflects narrow tuning curves of DIR.
  29. Sinusoidal tuning may describe directional selectivity in ganglion cells; the reader is directed to S. G. He et al. , “Distinguishing direction selectivity from orientation selectivity in the rabbit retina,” Visual Neurosci. 15, 439–447 (1998).This is where the vector sum method (and also the SDO analysis) would work well. In estimating the orientation (and by analogy, direction) tuning, the vector sum or SDO method is equivalent to calculating the least-squares fit to a sinusoidal curve, as pointed out by N. V. Swindale, “Orientation tuning curves: empirical description and estimation of parameters,” Biol. Cybern. 78, 45–56 (1998).
    [CrossRef] [PubMed]
  30. R. L. De Valois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
    [CrossRef] [PubMed]
  31. B. Li, Y. Wang, Y. C. Diao, “Quantification of directional and orientational selectivities of visual neurons to moving stimuli,” Biol. Cybern. 70, 282–290 (1994).
    [CrossRef]
  32. The reason we do not identify DIR(θ) as [ODDSUM(θ)+∣ODDSUM(θ)∣]∕2 is because all odd harmonics of DIR(θ) are required to be exactly the same as all odd harmonics of R(θ). The reason we do not write DIR(θ) as ODDSUM(θ)+λ∣ODDSUM(θ)∣ for some 0<λ<1 is because we want to completely cancel its negative lobe/peak.
  33. J. Zhang, R. L. De Valois (1998), “Unconfounding orientation and direction tuning in cortical neuron’s response,” Invest. Ophthalmol. Visual Sci. 39, s324 (1998).
  34. R. L. De Valois, N. P. Cottaris, “Inputs to directionally selective simple cells in macaque striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 95, 14488–14493 (1998).
    [CrossRef] [PubMed]
  35. Note that our analysis applies only to the use of drifting oriented stimuli. When, for instance, cells are driven by flashing oriented stimuli, the use of the circular variance measure in analogy to Eq. (2) to characterize orientation tuning would not be subject to our criticism described here; the reader is directed to Ringach et al. , “Dynamics of orientation tuning in macaque primary visual cortex,” Nature 387, 281–284 (1997) Ringach et al. , “Orientation selectivity in macaque V1: diversity and laminar dependence,” J. Neurol. Sci. 22, 5639–5651 (2002).Also, our analysis will not apply if the stimulus is a drifting random dot pattern because, under certain conditions, the cell’s directional tuning would exhibit a bifurcation of peaks, as shown by Skottun et al. , “On the direction selectivity of cortical neurons to drifting dot patterns,” Visual Neurosci. 11, 885–897 (1994).
    [CrossRef] [PubMed]
  36. For an approach using a particular parametric form for orientation and direction tuning curves, the reader is directed to Swindale et al. , “The spatial pattern of response magnitude and selectivity for orientation and direction in cat visual cortex,” Cereb. Cortex 13, 225–238 (2003), and the reference by Swindale cited in Ref. [29].
    [CrossRef] [PubMed]
  37. F. Wörgötter, U. T. Eysel, “Axial responses in visual cortical cells: Spatial-temporal mechanisms quantified by Fourier components of cortical tuning curves,” Exp. Brain Res. 83, 656–664 (1991).
    [CrossRef]
  38. If indeed such higher-order tuning mechanism exists, then the correct way of testing it is to assume the quadruple-lobed mechanism Q(θ) to have a period of 90° and hence Fourier decomposable as Q(θ)=q0+q1cos[4(θ−θq)]+q2cos[8(θ−θq)]+⋯. One can then proceed as before and relate (η4,ζ4) to d4, o2, and q1.
  39. This includes, for example, Ref. [9], which reported the weakening of orientation selectivity in the primary visual cortex (V1) of ferrets after chronic electric stimulation of the optic nerve during their early, postnatal visual development.

2003

C. Monier, F. Chavane, P. Baudot, L. J. Graham, Y. Frégnac, “Orientation and direction selectivity of synaptic inputs in visual cortical neurons: a diversity of combinations produces spike tuning,” Neuron 37, 663–680 (2003).
[CrossRef] [PubMed]

T. H. Schwartz, “Optical imaging of epileptiform events in visual cortex in response to patterned photic stimulation,” Cereb. Cortex 13, 1287–1298 (2003).
[CrossRef] [PubMed]

For an approach using a particular parametric form for orientation and direction tuning curves, the reader is directed to Swindale et al. , “The spatial pattern of response magnitude and selectivity for orientation and direction in cat visual cortex,” Cereb. Cortex 13, 225–238 (2003), and the reference by Swindale cited in Ref. [29].
[CrossRef] [PubMed]

2002

C. J. Akerman, D. Smyth, I. D. Thompson, “Visual experience before eye-opening and the development of the retinogeniculate pathway,” Neuron 36, 869–879 (2002).
[CrossRef] [PubMed]

F. Sengpiel, F. T. Bonhoeffer, “Orientation specificity of contrast adaptation in visual cortical pinwheel centres and iso-orientation domains,” Eur. J. Neurosci. 15, 876–886 (2002).
[CrossRef] [PubMed]

2001

K. Krug, C. J. Akerman, I. D. Thompson, “Responses of neurons in neonatal cortex and thalamus to patterned visual stimulation through the naturally closed lids,” J. Neurophysiol. 85, 1436–1443 (2001).
[PubMed]

A. Schmidt, H. J. Bischof, “Neurons with complex receptive fields in the stratum griseum centrale of the zebra finch (Taeniopygia guttata castanotis Gould) optic tectum” J. Comp. Physiol., A 187, 913–924 (2001).
[CrossRef]

2000

J. Sharma, A. Angelucci, M. Sur, “Induction of visual orientation modules in auditory cortex,” Nature (London) 404, 841–847 (2000).
[CrossRef]

A. Shmuel, A. Grinvald, “Coexistence of linear zones and pinwheels within orientation maps in cat visual cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 5568–5573 (2000).
[CrossRef] [PubMed]

1999

A. Schmidt, J. Engelage, H. J. Bischof, “Single cell responses from the optic tectum of the zebra finch (Taeniopygia guttata castanotis Gould),” J. Comp. Physiol. A 185, 69–79 (1999).
[CrossRef]

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nat. Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

T. Yousef, T. Bonhoeffer, D.-S. Kim, U. T. Eysel, É. Tóth, Z. F. Kisvárday, “Orientation topography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area 18),” Eur. J. Neurosci. 11, 4291–4308 (1999).
[CrossRef] [PubMed]

1998

Sinusoidal tuning may describe directional selectivity in ganglion cells; the reader is directed to S. G. He et al. , “Distinguishing direction selectivity from orientation selectivity in the rabbit retina,” Visual Neurosci. 15, 439–447 (1998).This is where the vector sum method (and also the SDO analysis) would work well. In estimating the orientation (and by analogy, direction) tuning, the vector sum or SDO method is equivalent to calculating the least-squares fit to a sinusoidal curve, as pointed out by N. V. Swindale, “Orientation tuning curves: empirical description and estimation of parameters,” Biol. Cybern. 78, 45–56 (1998).
[CrossRef] [PubMed]

A. G. Leventhal, Y. C. Wang, M. T. Schmolesky, Y. F. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

J. Zhang, R. L. De Valois (1998), “Unconfounding orientation and direction tuning in cortical neuron’s response,” Invest. Ophthalmol. Visual Sci. 39, s324 (1998).

R. L. De Valois, N. P. Cottaris, “Inputs to directionally selective simple cells in macaque striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 95, 14488–14493 (1998).
[CrossRef] [PubMed]

1997

Note that our analysis applies only to the use of drifting oriented stimuli. When, for instance, cells are driven by flashing oriented stimuli, the use of the circular variance measure in analogy to Eq. (2) to characterize orientation tuning would not be subject to our criticism described here; the reader is directed to Ringach et al. , “Dynamics of orientation tuning in macaque primary visual cortex,” Nature 387, 281–284 (1997) Ringach et al. , “Orientation selectivity in macaque V1: diversity and laminar dependence,” J. Neurol. Sci. 22, 5639–5651 (2002).Also, our analysis will not apply if the stimulus is a drifting random dot pattern because, under certain conditions, the cell’s directional tuning would exhibit a bifurcation of peaks, as shown by Skottun et al. , “On the direction selectivity of cortical neurons to drifting dot patterns,” Visual Neurosci. 11, 885–897 (1994).
[CrossRef] [PubMed]

M. Weliky, L. C. Katz, “Disruption of orientation tuning in visual cortex by artificially correlated neuronal activity,” Nature (London) 386, 680–685 (1997).
[CrossRef]

1996

A. Shmuel, A. Grinvald, “Functional organization for direction of motion and its relationship to orientation maps in cat area 18,” J. Neurosci. 16, 6945–6964 (1996).
[PubMed]

1995

A. G. Leventhal, K. G. Thompson, D. Liu, Y. Zhou, S. J. Ault, “Concomitant sensitivity to orientation, direction, and color of cells in layers 2, 3, and 4 of monkey striate cortex,” J. Neurosci. 15, 1808–1818 (1995).
[PubMed]

T. D. Shou, A. G. Leventhal, K. G. Thompson, Y. F. Zhou, “Direction biases of X-type and Y-type retinal ganglion-cells in the cat,” J. Neurophysiol. 73, 1414–1421 (1995).
[PubMed]

Y. F. Zhou, A. G. Leventhal, K. G. Thompson, “Visual deprivation does not affect the orientation and direction sensitivity of relay cells in the lateral geniculate nucleus of the cat,” J. Neurosci. 15, 689–698 (1995).
[PubMed]

1994

D. Malonek, R. B.H. Tootell, A. Grinvald, “Optical imaging reveals the functional architecture of neurons processing shape and motion in owl monkey area MT,” Proc. R. Soc. London, Ser. B 258, 109–119 (1994).
[CrossRef]

B. Li, Y. Wang, Y. C. Diao, “Quantification of directional and orientational selectivities of visual neurons to moving stimuli,” Biol. Cybern. 70, 282–290 (1994).
[CrossRef]

1993

D. O’Carroll, “Feature-detecting neurons in dragonflies,” Nature (London) 362, 541–543 (1993).
[CrossRef]

B. Chapman, M. P. Stryker, “Development of orientation selectivity in ferret visual-cortex and effects of deprivation,” J. Neurosci. 13, 5251–5262 (1993).
[PubMed]

1991

F. Wörgötter, T. Muche, E. T. Eysel, “Correlations between directional and orientational tuning of cells in cat striate cortex,” Exp. Brain Res. 83, 665–669 (1991).
[CrossRef] [PubMed]

F. Wörgötter, U. T. Eysel, “Topographical aspects of intracortical excitation and inhibition contributing to orientation specificity in area 17 of the cat visual cortex,” Eur. J. Neurosci. 3, 1232–1244 (1991).
[CrossRef] [PubMed]

F. Wörgötter, U. T. Eysel, “Axial responses in visual cortical cells: Spatial-temporal mechanisms quantified by Fourier components of cortical tuning curves,” Exp. Brain Res. 83, 656–664 (1991).
[CrossRef]

1990

J. Zhang, “How to unconfound the directional and orientational information in visual neuron’s response,” Biol. Cybern. 63, 135–142 (1990).
[CrossRef]

F. Wörgötter, O. Gründel, U. T. Eysel, “Quantification and comparison of cell properties in cat’s striate cortex determined by different types of stimuli,” Eur. J. Neurosci. 2, 928–941 (1990).
[CrossRef]

1987

F. Wörgötter, U. T. Eysel, “Quantitative determination of orientational and directional components in the response of visual cortical cells to moving stimuli,” Biol. Cybern. 57, 349–355 (1987).
[CrossRef] [PubMed]

1982

R. L. De Valois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef] [PubMed]

Akerman, C. J.

C. J. Akerman, D. Smyth, I. D. Thompson, “Visual experience before eye-opening and the development of the retinogeniculate pathway,” Neuron 36, 869–879 (2002).
[CrossRef] [PubMed]

K. Krug, C. J. Akerman, I. D. Thompson, “Responses of neurons in neonatal cortex and thalamus to patterned visual stimulation through the naturally closed lids,” J. Neurophysiol. 85, 1436–1443 (2001).
[PubMed]

Angelucci, A.

J. Sharma, A. Angelucci, M. Sur, “Induction of visual orientation modules in auditory cortex,” Nature (London) 404, 841–847 (2000).
[CrossRef]

Ault, S. J.

A. G. Leventhal, K. G. Thompson, D. Liu, Y. Zhou, S. J. Ault, “Concomitant sensitivity to orientation, direction, and color of cells in layers 2, 3, and 4 of monkey striate cortex,” J. Neurosci. 15, 1808–1818 (1995).
[PubMed]

Batschelet, E.

E. Batschelet, Circular Statistics in Biology (Academic, 1981).

Baudot, P.

C. Monier, F. Chavane, P. Baudot, L. J. Graham, Y. Frégnac, “Orientation and direction selectivity of synaptic inputs in visual cortical neurons: a diversity of combinations produces spike tuning,” Neuron 37, 663–680 (2003).
[CrossRef] [PubMed]

Bischof, H. J.

A. Schmidt, H. J. Bischof, “Neurons with complex receptive fields in the stratum griseum centrale of the zebra finch (Taeniopygia guttata castanotis Gould) optic tectum” J. Comp. Physiol., A 187, 913–924 (2001).
[CrossRef]

A. Schmidt, J. Engelage, H. J. Bischof, “Single cell responses from the optic tectum of the zebra finch (Taeniopygia guttata castanotis Gould),” J. Comp. Physiol. A 185, 69–79 (1999).
[CrossRef]

Bonhoeffer, F. T.

F. Sengpiel, F. T. Bonhoeffer, “Orientation specificity of contrast adaptation in visual cortical pinwheel centres and iso-orientation domains,” Eur. J. Neurosci. 15, 876–886 (2002).
[CrossRef] [PubMed]

Bonhoeffer, T.

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nat. Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

T. Yousef, T. Bonhoeffer, D.-S. Kim, U. T. Eysel, É. Tóth, Z. F. Kisvárday, “Orientation topography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area 18),” Eur. J. Neurosci. 11, 4291–4308 (1999).
[CrossRef] [PubMed]

Chapman, B.

B. Chapman, M. P. Stryker, “Development of orientation selectivity in ferret visual-cortex and effects of deprivation,” J. Neurosci. 13, 5251–5262 (1993).
[PubMed]

Chavane, F.

C. Monier, F. Chavane, P. Baudot, L. J. Graham, Y. Frégnac, “Orientation and direction selectivity of synaptic inputs in visual cortical neurons: a diversity of combinations produces spike tuning,” Neuron 37, 663–680 (2003).
[CrossRef] [PubMed]

Cottaris, N. P.

R. L. De Valois, N. P. Cottaris, “Inputs to directionally selective simple cells in macaque striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 95, 14488–14493 (1998).
[CrossRef] [PubMed]

De Valois, R. L.

J. Zhang, R. L. De Valois (1998), “Unconfounding orientation and direction tuning in cortical neuron’s response,” Invest. Ophthalmol. Visual Sci. 39, s324 (1998).

R. L. De Valois, N. P. Cottaris, “Inputs to directionally selective simple cells in macaque striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 95, 14488–14493 (1998).
[CrossRef] [PubMed]

R. L. De Valois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef] [PubMed]

Diao, Y. C.

B. Li, Y. Wang, Y. C. Diao, “Quantification of directional and orientational selectivities of visual neurons to moving stimuli,” Biol. Cybern. 70, 282–290 (1994).
[CrossRef]

Engelage, J.

A. Schmidt, J. Engelage, H. J. Bischof, “Single cell responses from the optic tectum of the zebra finch (Taeniopygia guttata castanotis Gould),” J. Comp. Physiol. A 185, 69–79 (1999).
[CrossRef]

Eysel, E. T.

F. Wörgötter, T. Muche, E. T. Eysel, “Correlations between directional and orientational tuning of cells in cat striate cortex,” Exp. Brain Res. 83, 665–669 (1991).
[CrossRef] [PubMed]

Eysel, U. T.

T. Yousef, T. Bonhoeffer, D.-S. Kim, U. T. Eysel, É. Tóth, Z. F. Kisvárday, “Orientation topography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area 18),” Eur. J. Neurosci. 11, 4291–4308 (1999).
[CrossRef] [PubMed]

F. Wörgötter, U. T. Eysel, “Topographical aspects of intracortical excitation and inhibition contributing to orientation specificity in area 17 of the cat visual cortex,” Eur. J. Neurosci. 3, 1232–1244 (1991).
[CrossRef] [PubMed]

F. Wörgötter, U. T. Eysel, “Axial responses in visual cortical cells: Spatial-temporal mechanisms quantified by Fourier components of cortical tuning curves,” Exp. Brain Res. 83, 656–664 (1991).
[CrossRef]

F. Wörgötter, O. Gründel, U. T. Eysel, “Quantification and comparison of cell properties in cat’s striate cortex determined by different types of stimuli,” Eur. J. Neurosci. 2, 928–941 (1990).
[CrossRef]

F. Wörgötter, U. T. Eysel, “Quantitative determination of orientational and directional components in the response of visual cortical cells to moving stimuli,” Biol. Cybern. 57, 349–355 (1987).
[CrossRef] [PubMed]

Frégnac, Y.

C. Monier, F. Chavane, P. Baudot, L. J. Graham, Y. Frégnac, “Orientation and direction selectivity of synaptic inputs in visual cortical neurons: a diversity of combinations produces spike tuning,” Neuron 37, 663–680 (2003).
[CrossRef] [PubMed]

Graham, L. J.

C. Monier, F. Chavane, P. Baudot, L. J. Graham, Y. Frégnac, “Orientation and direction selectivity of synaptic inputs in visual cortical neurons: a diversity of combinations produces spike tuning,” Neuron 37, 663–680 (2003).
[CrossRef] [PubMed]

Grinvald, A.

A. Shmuel, A. Grinvald, “Coexistence of linear zones and pinwheels within orientation maps in cat visual cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 5568–5573 (2000).
[CrossRef] [PubMed]

A. Shmuel, A. Grinvald, “Functional organization for direction of motion and its relationship to orientation maps in cat area 18,” J. Neurosci. 16, 6945–6964 (1996).
[PubMed]

D. Malonek, R. B.H. Tootell, A. Grinvald, “Optical imaging reveals the functional architecture of neurons processing shape and motion in owl monkey area MT,” Proc. R. Soc. London, Ser. B 258, 109–119 (1994).
[CrossRef]

Gründel, O.

F. Wörgötter, O. Gründel, U. T. Eysel, “Quantification and comparison of cell properties in cat’s striate cortex determined by different types of stimuli,” Eur. J. Neurosci. 2, 928–941 (1990).
[CrossRef]

He, S. G.

Sinusoidal tuning may describe directional selectivity in ganglion cells; the reader is directed to S. G. He et al. , “Distinguishing direction selectivity from orientation selectivity in the rabbit retina,” Visual Neurosci. 15, 439–447 (1998).This is where the vector sum method (and also the SDO analysis) would work well. In estimating the orientation (and by analogy, direction) tuning, the vector sum or SDO method is equivalent to calculating the least-squares fit to a sinusoidal curve, as pointed out by N. V. Swindale, “Orientation tuning curves: empirical description and estimation of parameters,” Biol. Cybern. 78, 45–56 (1998).
[CrossRef] [PubMed]

Hepler, N.

R. L. De Valois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef] [PubMed]

Katz, L. C.

M. Weliky, L. C. Katz, “Disruption of orientation tuning in visual cortex by artificially correlated neuronal activity,” Nature (London) 386, 680–685 (1997).
[CrossRef]

Kim, D.-S.

T. Yousef, T. Bonhoeffer, D.-S. Kim, U. T. Eysel, É. Tóth, Z. F. Kisvárday, “Orientation topography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area 18),” Eur. J. Neurosci. 11, 4291–4308 (1999).
[CrossRef] [PubMed]

Kisvárday, Z. F.

T. Yousef, T. Bonhoeffer, D.-S. Kim, U. T. Eysel, É. Tóth, Z. F. Kisvárday, “Orientation topography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area 18),” Eur. J. Neurosci. 11, 4291–4308 (1999).
[CrossRef] [PubMed]

Krug, K.

K. Krug, C. J. Akerman, I. D. Thompson, “Responses of neurons in neonatal cortex and thalamus to patterned visual stimulation through the naturally closed lids,” J. Neurophysiol. 85, 1436–1443 (2001).
[PubMed]

Leventhal, A. G.

A. G. Leventhal, Y. C. Wang, M. T. Schmolesky, Y. F. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

Y. F. Zhou, A. G. Leventhal, K. G. Thompson, “Visual deprivation does not affect the orientation and direction sensitivity of relay cells in the lateral geniculate nucleus of the cat,” J. Neurosci. 15, 689–698 (1995).
[PubMed]

A. G. Leventhal, K. G. Thompson, D. Liu, Y. Zhou, S. J. Ault, “Concomitant sensitivity to orientation, direction, and color of cells in layers 2, 3, and 4 of monkey striate cortex,” J. Neurosci. 15, 1808–1818 (1995).
[PubMed]

T. D. Shou, A. G. Leventhal, K. G. Thompson, Y. F. Zhou, “Direction biases of X-type and Y-type retinal ganglion-cells in the cat,” J. Neurophysiol. 73, 1414–1421 (1995).
[PubMed]

Li, B.

B. Li, Y. Wang, Y. C. Diao, “Quantification of directional and orientational selectivities of visual neurons to moving stimuli,” Biol. Cybern. 70, 282–290 (1994).
[CrossRef]

Liu, D.

A. G. Leventhal, K. G. Thompson, D. Liu, Y. Zhou, S. J. Ault, “Concomitant sensitivity to orientation, direction, and color of cells in layers 2, 3, and 4 of monkey striate cortex,” J. Neurosci. 15, 1808–1818 (1995).
[PubMed]

Malonek, D.

D. Malonek, R. B.H. Tootell, A. Grinvald, “Optical imaging reveals the functional architecture of neurons processing shape and motion in owl monkey area MT,” Proc. R. Soc. London, Ser. B 258, 109–119 (1994).
[CrossRef]

Monier, C.

C. Monier, F. Chavane, P. Baudot, L. J. Graham, Y. Frégnac, “Orientation and direction selectivity of synaptic inputs in visual cortical neurons: a diversity of combinations produces spike tuning,” Neuron 37, 663–680 (2003).
[CrossRef] [PubMed]

Muche, T.

F. Wörgötter, T. Muche, E. T. Eysel, “Correlations between directional and orientational tuning of cells in cat striate cortex,” Exp. Brain Res. 83, 665–669 (1991).
[CrossRef] [PubMed]

O’Carroll, D.

D. O’Carroll, “Feature-detecting neurons in dragonflies,” Nature (London) 362, 541–543 (1993).
[CrossRef]

Ringach,

Note that our analysis applies only to the use of drifting oriented stimuli. When, for instance, cells are driven by flashing oriented stimuli, the use of the circular variance measure in analogy to Eq. (2) to characterize orientation tuning would not be subject to our criticism described here; the reader is directed to Ringach et al. , “Dynamics of orientation tuning in macaque primary visual cortex,” Nature 387, 281–284 (1997) Ringach et al. , “Orientation selectivity in macaque V1: diversity and laminar dependence,” J. Neurol. Sci. 22, 5639–5651 (2002).Also, our analysis will not apply if the stimulus is a drifting random dot pattern because, under certain conditions, the cell’s directional tuning would exhibit a bifurcation of peaks, as shown by Skottun et al. , “On the direction selectivity of cortical neurons to drifting dot patterns,” Visual Neurosci. 11, 885–897 (1994).
[CrossRef] [PubMed]

Schmidt, A.

A. Schmidt, H. J. Bischof, “Neurons with complex receptive fields in the stratum griseum centrale of the zebra finch (Taeniopygia guttata castanotis Gould) optic tectum” J. Comp. Physiol., A 187, 913–924 (2001).
[CrossRef]

A. Schmidt, J. Engelage, H. J. Bischof, “Single cell responses from the optic tectum of the zebra finch (Taeniopygia guttata castanotis Gould),” J. Comp. Physiol. A 185, 69–79 (1999).
[CrossRef]

Schmolesky, M. T.

A. G. Leventhal, Y. C. Wang, M. T. Schmolesky, Y. F. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

Schwartz, T. H.

T. H. Schwartz, “Optical imaging of epileptiform events in visual cortex in response to patterned photic stimulation,” Cereb. Cortex 13, 1287–1298 (2003).
[CrossRef] [PubMed]

Sengpiel, F.

F. Sengpiel, F. T. Bonhoeffer, “Orientation specificity of contrast adaptation in visual cortical pinwheel centres and iso-orientation domains,” Eur. J. Neurosci. 15, 876–886 (2002).
[CrossRef] [PubMed]

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nat. Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

Sharma, J.

J. Sharma, A. Angelucci, M. Sur, “Induction of visual orientation modules in auditory cortex,” Nature (London) 404, 841–847 (2000).
[CrossRef]

Shmuel, A.

A. Shmuel, A. Grinvald, “Coexistence of linear zones and pinwheels within orientation maps in cat visual cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 5568–5573 (2000).
[CrossRef] [PubMed]

A. Shmuel, A. Grinvald, “Functional organization for direction of motion and its relationship to orientation maps in cat area 18,” J. Neurosci. 16, 6945–6964 (1996).
[PubMed]

Shou, T. D.

T. D. Shou, A. G. Leventhal, K. G. Thompson, Y. F. Zhou, “Direction biases of X-type and Y-type retinal ganglion-cells in the cat,” J. Neurophysiol. 73, 1414–1421 (1995).
[PubMed]

Smyth, D.

C. J. Akerman, D. Smyth, I. D. Thompson, “Visual experience before eye-opening and the development of the retinogeniculate pathway,” Neuron 36, 869–879 (2002).
[CrossRef] [PubMed]

Stawinski, P.

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nat. Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

Stryker, M. P.

B. Chapman, M. P. Stryker, “Development of orientation selectivity in ferret visual-cortex and effects of deprivation,” J. Neurosci. 13, 5251–5262 (1993).
[PubMed]

Sur, M.

J. Sharma, A. Angelucci, M. Sur, “Induction of visual orientation modules in auditory cortex,” Nature (London) 404, 841–847 (2000).
[CrossRef]

Swindale,

For an approach using a particular parametric form for orientation and direction tuning curves, the reader is directed to Swindale et al. , “The spatial pattern of response magnitude and selectivity for orientation and direction in cat visual cortex,” Cereb. Cortex 13, 225–238 (2003), and the reference by Swindale cited in Ref. [29].
[CrossRef] [PubMed]

Thompson, I. D.

C. J. Akerman, D. Smyth, I. D. Thompson, “Visual experience before eye-opening and the development of the retinogeniculate pathway,” Neuron 36, 869–879 (2002).
[CrossRef] [PubMed]

K. Krug, C. J. Akerman, I. D. Thompson, “Responses of neurons in neonatal cortex and thalamus to patterned visual stimulation through the naturally closed lids,” J. Neurophysiol. 85, 1436–1443 (2001).
[PubMed]

Thompson, K. G.

Y. F. Zhou, A. G. Leventhal, K. G. Thompson, “Visual deprivation does not affect the orientation and direction sensitivity of relay cells in the lateral geniculate nucleus of the cat,” J. Neurosci. 15, 689–698 (1995).
[PubMed]

A. G. Leventhal, K. G. Thompson, D. Liu, Y. Zhou, S. J. Ault, “Concomitant sensitivity to orientation, direction, and color of cells in layers 2, 3, and 4 of monkey striate cortex,” J. Neurosci. 15, 1808–1818 (1995).
[PubMed]

T. D. Shou, A. G. Leventhal, K. G. Thompson, Y. F. Zhou, “Direction biases of X-type and Y-type retinal ganglion-cells in the cat,” J. Neurophysiol. 73, 1414–1421 (1995).
[PubMed]

Tootell, R. B.H.

D. Malonek, R. B.H. Tootell, A. Grinvald, “Optical imaging reveals the functional architecture of neurons processing shape and motion in owl monkey area MT,” Proc. R. Soc. London, Ser. B 258, 109–119 (1994).
[CrossRef]

Tóth, É.

T. Yousef, T. Bonhoeffer, D.-S. Kim, U. T. Eysel, É. Tóth, Z. F. Kisvárday, “Orientation topography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area 18),” Eur. J. Neurosci. 11, 4291–4308 (1999).
[CrossRef] [PubMed]

Wang, Y.

B. Li, Y. Wang, Y. C. Diao, “Quantification of directional and orientational selectivities of visual neurons to moving stimuli,” Biol. Cybern. 70, 282–290 (1994).
[CrossRef]

Wang, Y. C.

A. G. Leventhal, Y. C. Wang, M. T. Schmolesky, Y. F. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

Weliky, M.

M. Weliky, L. C. Katz, “Disruption of orientation tuning in visual cortex by artificially correlated neuronal activity,” Nature (London) 386, 680–685 (1997).
[CrossRef]

Wörgötter, F.

F. Wörgötter, T. Muche, E. T. Eysel, “Correlations between directional and orientational tuning of cells in cat striate cortex,” Exp. Brain Res. 83, 665–669 (1991).
[CrossRef] [PubMed]

F. Wörgötter, U. T. Eysel, “Topographical aspects of intracortical excitation and inhibition contributing to orientation specificity in area 17 of the cat visual cortex,” Eur. J. Neurosci. 3, 1232–1244 (1991).
[CrossRef] [PubMed]

F. Wörgötter, U. T. Eysel, “Axial responses in visual cortical cells: Spatial-temporal mechanisms quantified by Fourier components of cortical tuning curves,” Exp. Brain Res. 83, 656–664 (1991).
[CrossRef]

F. Wörgötter, O. Gründel, U. T. Eysel, “Quantification and comparison of cell properties in cat’s striate cortex determined by different types of stimuli,” Eur. J. Neurosci. 2, 928–941 (1990).
[CrossRef]

F. Wörgötter, U. T. Eysel, “Quantitative determination of orientational and directional components in the response of visual cortical cells to moving stimuli,” Biol. Cybern. 57, 349–355 (1987).
[CrossRef] [PubMed]

Yousef, T.

T. Yousef, T. Bonhoeffer, D.-S. Kim, U. T. Eysel, É. Tóth, Z. F. Kisvárday, “Orientation topography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area 18),” Eur. J. Neurosci. 11, 4291–4308 (1999).
[CrossRef] [PubMed]

Yund, E. W.

R. L. De Valois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef] [PubMed]

Zhang, J.

J. Zhang, R. L. De Valois (1998), “Unconfounding orientation and direction tuning in cortical neuron’s response,” Invest. Ophthalmol. Visual Sci. 39, s324 (1998).

J. Zhang, “How to unconfound the directional and orientational information in visual neuron’s response,” Biol. Cybern. 63, 135–142 (1990).
[CrossRef]

Zhou, Y.

A. G. Leventhal, K. G. Thompson, D. Liu, Y. Zhou, S. J. Ault, “Concomitant sensitivity to orientation, direction, and color of cells in layers 2, 3, and 4 of monkey striate cortex,” J. Neurosci. 15, 1808–1818 (1995).
[PubMed]

Zhou, Y. F.

A. G. Leventhal, Y. C. Wang, M. T. Schmolesky, Y. F. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

T. D. Shou, A. G. Leventhal, K. G. Thompson, Y. F. Zhou, “Direction biases of X-type and Y-type retinal ganglion-cells in the cat,” J. Neurophysiol. 73, 1414–1421 (1995).
[PubMed]

Y. F. Zhou, A. G. Leventhal, K. G. Thompson, “Visual deprivation does not affect the orientation and direction sensitivity of relay cells in the lateral geniculate nucleus of the cat,” J. Neurosci. 15, 689–698 (1995).
[PubMed]

Biol. Cybern.

F. Wörgötter, U. T. Eysel, “Quantitative determination of orientational and directional components in the response of visual cortical cells to moving stimuli,” Biol. Cybern. 57, 349–355 (1987).
[CrossRef] [PubMed]

J. Zhang, “How to unconfound the directional and orientational information in visual neuron’s response,” Biol. Cybern. 63, 135–142 (1990).
[CrossRef]

B. Li, Y. Wang, Y. C. Diao, “Quantification of directional and orientational selectivities of visual neurons to moving stimuli,” Biol. Cybern. 70, 282–290 (1994).
[CrossRef]

Cereb. Cortex

T. H. Schwartz, “Optical imaging of epileptiform events in visual cortex in response to patterned photic stimulation,” Cereb. Cortex 13, 1287–1298 (2003).
[CrossRef] [PubMed]

For an approach using a particular parametric form for orientation and direction tuning curves, the reader is directed to Swindale et al. , “The spatial pattern of response magnitude and selectivity for orientation and direction in cat visual cortex,” Cereb. Cortex 13, 225–238 (2003), and the reference by Swindale cited in Ref. [29].
[CrossRef] [PubMed]

Eur. J. Neurosci.

F. Wörgötter, U. T. Eysel, “Topographical aspects of intracortical excitation and inhibition contributing to orientation specificity in area 17 of the cat visual cortex,” Eur. J. Neurosci. 3, 1232–1244 (1991).
[CrossRef] [PubMed]

F. Wörgötter, O. Gründel, U. T. Eysel, “Quantification and comparison of cell properties in cat’s striate cortex determined by different types of stimuli,” Eur. J. Neurosci. 2, 928–941 (1990).
[CrossRef]

T. Yousef, T. Bonhoeffer, D.-S. Kim, U. T. Eysel, É. Tóth, Z. F. Kisvárday, “Orientation topography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area 18),” Eur. J. Neurosci. 11, 4291–4308 (1999).
[CrossRef] [PubMed]

F. Sengpiel, F. T. Bonhoeffer, “Orientation specificity of contrast adaptation in visual cortical pinwheel centres and iso-orientation domains,” Eur. J. Neurosci. 15, 876–886 (2002).
[CrossRef] [PubMed]

Exp. Brain Res.

F. Wörgötter, T. Muche, E. T. Eysel, “Correlations between directional and orientational tuning of cells in cat striate cortex,” Exp. Brain Res. 83, 665–669 (1991).
[CrossRef] [PubMed]

F. Wörgötter, U. T. Eysel, “Axial responses in visual cortical cells: Spatial-temporal mechanisms quantified by Fourier components of cortical tuning curves,” Exp. Brain Res. 83, 656–664 (1991).
[CrossRef]

Invest. Ophthalmol. Visual Sci.

J. Zhang, R. L. De Valois (1998), “Unconfounding orientation and direction tuning in cortical neuron’s response,” Invest. Ophthalmol. Visual Sci. 39, s324 (1998).

J. Comp. Physiol. A

A. Schmidt, J. Engelage, H. J. Bischof, “Single cell responses from the optic tectum of the zebra finch (Taeniopygia guttata castanotis Gould),” J. Comp. Physiol. A 185, 69–79 (1999).
[CrossRef]

J. Comp. Physiol., A

A. Schmidt, H. J. Bischof, “Neurons with complex receptive fields in the stratum griseum centrale of the zebra finch (Taeniopygia guttata castanotis Gould) optic tectum” J. Comp. Physiol., A 187, 913–924 (2001).
[CrossRef]

J. Neurophysiol.

K. Krug, C. J. Akerman, I. D. Thompson, “Responses of neurons in neonatal cortex and thalamus to patterned visual stimulation through the naturally closed lids,” J. Neurophysiol. 85, 1436–1443 (2001).
[PubMed]

T. D. Shou, A. G. Leventhal, K. G. Thompson, Y. F. Zhou, “Direction biases of X-type and Y-type retinal ganglion-cells in the cat,” J. Neurophysiol. 73, 1414–1421 (1995).
[PubMed]

J. Neurosci.

Y. F. Zhou, A. G. Leventhal, K. G. Thompson, “Visual deprivation does not affect the orientation and direction sensitivity of relay cells in the lateral geniculate nucleus of the cat,” J. Neurosci. 15, 689–698 (1995).
[PubMed]

A. G. Leventhal, K. G. Thompson, D. Liu, Y. Zhou, S. J. Ault, “Concomitant sensitivity to orientation, direction, and color of cells in layers 2, 3, and 4 of monkey striate cortex,” J. Neurosci. 15, 1808–1818 (1995).
[PubMed]

B. Chapman, M. P. Stryker, “Development of orientation selectivity in ferret visual-cortex and effects of deprivation,” J. Neurosci. 13, 5251–5262 (1993).
[PubMed]

A. Shmuel, A. Grinvald, “Functional organization for direction of motion and its relationship to orientation maps in cat area 18,” J. Neurosci. 16, 6945–6964 (1996).
[PubMed]

Nat. Neurosci.

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nat. Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

Nature

Note that our analysis applies only to the use of drifting oriented stimuli. When, for instance, cells are driven by flashing oriented stimuli, the use of the circular variance measure in analogy to Eq. (2) to characterize orientation tuning would not be subject to our criticism described here; the reader is directed to Ringach et al. , “Dynamics of orientation tuning in macaque primary visual cortex,” Nature 387, 281–284 (1997) Ringach et al. , “Orientation selectivity in macaque V1: diversity and laminar dependence,” J. Neurol. Sci. 22, 5639–5651 (2002).Also, our analysis will not apply if the stimulus is a drifting random dot pattern because, under certain conditions, the cell’s directional tuning would exhibit a bifurcation of peaks, as shown by Skottun et al. , “On the direction selectivity of cortical neurons to drifting dot patterns,” Visual Neurosci. 11, 885–897 (1994).
[CrossRef] [PubMed]

Nature (London)

J. Sharma, A. Angelucci, M. Sur, “Induction of visual orientation modules in auditory cortex,” Nature (London) 404, 841–847 (2000).
[CrossRef]

M. Weliky, L. C. Katz, “Disruption of orientation tuning in visual cortex by artificially correlated neuronal activity,” Nature (London) 386, 680–685 (1997).
[CrossRef]

D. O’Carroll, “Feature-detecting neurons in dragonflies,” Nature (London) 362, 541–543 (1993).
[CrossRef]

Neuron

C. J. Akerman, D. Smyth, I. D. Thompson, “Visual experience before eye-opening and the development of the retinogeniculate pathway,” Neuron 36, 869–879 (2002).
[CrossRef] [PubMed]

C. Monier, F. Chavane, P. Baudot, L. J. Graham, Y. Frégnac, “Orientation and direction selectivity of synaptic inputs in visual cortical neurons: a diversity of combinations produces spike tuning,” Neuron 37, 663–680 (2003).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A.

R. L. De Valois, N. P. Cottaris, “Inputs to directionally selective simple cells in macaque striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 95, 14488–14493 (1998).
[CrossRef] [PubMed]

A. Shmuel, A. Grinvald, “Coexistence of linear zones and pinwheels within orientation maps in cat visual cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 5568–5573 (2000).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. B

D. Malonek, R. B.H. Tootell, A. Grinvald, “Optical imaging reveals the functional architecture of neurons processing shape and motion in owl monkey area MT,” Proc. R. Soc. London, Ser. B 258, 109–119 (1994).
[CrossRef]

Vision Res.

R. L. De Valois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef] [PubMed]

Visual Neurosci.

Sinusoidal tuning may describe directional selectivity in ganglion cells; the reader is directed to S. G. He et al. , “Distinguishing direction selectivity from orientation selectivity in the rabbit retina,” Visual Neurosci. 15, 439–447 (1998).This is where the vector sum method (and also the SDO analysis) would work well. In estimating the orientation (and by analogy, direction) tuning, the vector sum or SDO method is equivalent to calculating the least-squares fit to a sinusoidal curve, as pointed out by N. V. Swindale, “Orientation tuning curves: empirical description and estimation of parameters,” Biol. Cybern. 78, 45–56 (1998).
[CrossRef] [PubMed]

A. G. Leventhal, Y. C. Wang, M. T. Schmolesky, Y. F. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

Other

E. Batschelet, Circular Statistics in Biology (Academic, 1981).

Note that here and below the designation of “first” and “second” harmonics refers to using a fundamental period of 360 deg∕cycle in Fourier analysis. If one were to Fourier decompose, say ORI, using a fundamental period of 180 deg∕cycle, then the second harmonic mentioned above should be renamed the first harmonic (in describing a periodic function with a period of 180 deg∕cycle).

In fact, higher-order harmonics were consistently present even in these authors’ own data; see Ref. [2]. However they were dismissed based on the percentage of total power they contributed. This was misguided, because any single-peaked DIR will have decreasing power in its higher harmonics. In fact, Zhang[26] showed that under quite mild restriction, the strength of the kth harmonic of DIR is proportional to sin(kα)∕k, with α characterizing the bandwidth. So the systematic presence of powers with decreasing strength in higher harmonics cannot be explained away simply as noise; rather it reflects narrow tuning curves of DIR.

The reason we do not identify DIR(θ) as [ODDSUM(θ)+∣ODDSUM(θ)∣]∕2 is because all odd harmonics of DIR(θ) are required to be exactly the same as all odd harmonics of R(θ). The reason we do not write DIR(θ) as ODDSUM(θ)+λ∣ODDSUM(θ)∣ for some 0<λ<1 is because we want to completely cancel its negative lobe/peak.

If indeed such higher-order tuning mechanism exists, then the correct way of testing it is to assume the quadruple-lobed mechanism Q(θ) to have a period of 90° and hence Fourier decomposable as Q(θ)=q0+q1cos[4(θ−θq)]+q2cos[8(θ−θq)]+⋯. One can then proceed as before and relate (η4,ζ4) to d4, o2, and q1.

This includes, for example, Ref. [9], which reported the weakening of orientation selectivity in the primary visual cortex (V1) of ferrets after chronic electric stimulation of the optic nerve during their early, postnatal visual development.

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Figures (6)

Fig. 1
Fig. 1

Illustration of the method of separating the DIR and ORI components, using a typical cell. (a) Overall tuning curve R ( θ ) of the cell, plotting its spike rate response (in ordinate) as a function of drifting angles θ (in abscissa) of a sinusoidal grating stimulus. Drifting angles are in 15° increments. (b) ODDSUM, denoted G ( θ ) in the text, is the sum of all odd Fourier harmonics of R ( θ ) . It can be easily calculated by shifting R ( θ ) by 180°, then subtracting it from the unshifted R ( θ ) , and dividing by 2. Note the antisymmetric shape of ODDSUM, and the presence of both the positive and the negative peak in it. (c) ∣ODDSUM∣, denoted G ( θ ) in the text, is the point-by-point absolute value of ODDSUM, obtained by flipping all negative values of the curve in (b). (d) The sum of all even Fourier harmonics of R ( θ ) plus the zeroth harmonic (the DC term). (e) Recovered DIR component, by adding the graph in (b) to the graph in (c). (f) Recovered ORI component, by subtracting DIR [graph in (e)] from R ( θ ) [graph in (a)]. Note that the graphs in (b) and (d) sum to the graph in (a); this is the Fourier decomposition of the cell’s overall tuning curve. Note also that the graphs in (e) and (f) sum to the graph in (a); this is the decomposition of the cell’s tuning curve into DIR and ORI. These graphs clearly show that ODDSUM DIR , EVENSUM ORI , contrary to the claim of Li et al.[31]

Fig. 2
Fig. 2

Twelve sample cells, with decomposed DIR and ORI components. The ordinate represents a cell’s response, the abscissa the drifting angles. The overall tuning curves are plotted as solid lines with solid square symbols representing the data points. The DIR (or ORI) component is plotted as dotted (or dashed) lines with diamond (or circle) symbols. Note that the DIR and ORI curves sum to the overall tuning curve for each cell. The alphanumeric string starting with “m” is a cell’s identification number.

Fig. 3
Fig. 3

Relative strength of ORI/DIR, when directional contamination of even harmonics has been corrected (the current method) or uncorrected (the SDO method). (a) The scatter plot of relative strengths of the population of 149 neurons, before correction (as abscissa) and after correction (as ordinate) by our method. Each square represents a cell. Note the overestimation of the orientation strength by SDO analysis. (b) The histogram of the distribution of the relative strengths in this population. Note the leftward shift of the distribution when corrections have been performed, again demonstrating the overestimation of the strength of the orientation component by SDO analysis.

Fig. 4
Fig. 4

Overestimation of ORI strength as a result of DIR contamination of the second harmonic. (a) Scatter plot of the amplitude of DIR’s second harmonic (ordinate, in logarithmic scale) versus the amplitude of its first harmonic (abscissa, in logarithmic scale). Note the positive correlation between the two amplitudes. (b) Scatter plot of the amplitude of the second harmonic of the overall tuning curve (before correction, abscissa) versus the amplitude of the second harmonic of the ORI component (after correction, ordinate), both in logarithmic scale. Note most data points fall below the positive diagonal. In both (a) and (b), each square represents a cell, and the amplitudes are all normalized by the DC component.

Fig. 5
Fig. 5

Comparison of the distributions of peak angles before (as abscissa) and after (as ordinate) correction for DIR contamination of the second harmonic. Each square represents a cell. Generally speaking, there does not appear to be a systematic difference between the peak angles estimated by the SDO method (before correction) and by our method (after correction).

Fig. 6
Fig. 6

Distribution of the difference between the peak angles of DIR and ORI components. The absolute value of difference in the two peak angles, Δ, for each cell, is calculated and segregated into six categories, ranging from almost perpendicular ( Δ = 0 ) to almost colinear ( Δ = 90 ° ) , with a precision of ± 15 ° on both sides.

Equations (40)

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R ( θ ) = S + D ( θ ) + O ( θ ) .
η l = 1 π k R ( θ k ) cos ( l θ k ) , ζ l = 1 π k R ( θ k ) sin ( l θ k ) .
θ d = arctan ( ζ 1 η 1 ) , r d = r 1 = ( ζ 1 ) 2 + ( η 1 ) 2 ,
θ o = arctan ( ζ 2 η 2 ) , r o = r 2 = ( ζ 2 ) 2 + ( η 2 ) 2 .
exp ( i θ d ) k R ( θ k ) exp ( i θ k ) = k [ R ( θ k ) cos ( θ k ) + i R ( θ k ) sin ( θ k ) ] η 1 + i ζ 1 ,
exp ( i 2 θ o ) k R ( θ k ) exp ( i 2 θ k ) = k [ R ( θ k ) cos ( 2 θ k ) + i R ( θ k ) sin ( 2 θ k ) ] η 2 + i ζ 2 .
R ( θ ) = r 0 + l = 1 2 [ η l cos ( l θ ) + ζ l sin ( l θ ) ] .
R ( θ ) = r 0 + r d cos ( θ θ d ) + r o cos [ 2 ( θ θ o ) ] ,
DIR ( θ ) = G ( θ ) + G ( θ ) ,
DIR ( θ ) = { 2 G ( θ ) if G ( θ ) 0 0 if G ( θ ) < 0 } .
G ( θ ) = 1 2 [ R ( θ ) R ( θ + π ) ] .
ORI ( θ ) = R ( θ ) DIR ( θ ) = R ( θ ) G ( θ ) G ( θ ) .
DIR ( θ ) = 1 2 [ R ( θ ) R ( θ + π ) + R ( θ ) R ( θ + π ) ] ,
ORI ( θ ) = 1 2 [ R ( θ ) + R ( θ + π ) R ( θ ) R ( θ + π ) ] .
R ( θ ) = η 0 + l = 1 N 2 1 [ η l cos ( l θ ) + ζ l sin ( l θ ) ] + η N 2 ,
η l = 1 π k = 1 N R ( θ k ) cos ( l θ k ) , ζ l = 1 π k = 1 N R ( θ k ) sin ( l θ k ) ,
η 0 = 1 π k = 1 N R ( θ k ) , η N 2 = 1 π k = 1 N ( 1 ) k R ( θ k ) .
G ( θ ) = l = 1 , 3 , [ η l cos ( l θ ) + ζ l sin ( l θ ) ] .
G ( θ ) = l = 2 , 4 , [ α l cos ( l θ ) + β l sin ( l θ ) ] ,
α l = 1 π k = 1 N G ( θ k ) cos ( l θ k ) , β l = 1 π k = 1 N G ( θ k ) sin ( l θ k ) .
DIR ( θ ) = l = 1 , 3 , [ η l cos ( l θ ) + ζ l sin ( l θ ) ] + l = 2 , 4 , [ α l cos ( l θ ) + β l sin ( l θ ) ] .
ORI ( θ ) = η 0 + l = 2 , 4 , [ η l cos ( l θ ) + ζ l sin ( l θ ) ] ,
η l = η l α l , ζ l = ζ l β l , for l = 2 , 4 , .
η 1 = 1 π k = 1 N R ( θ k ) cos ( θ k ) , ζ 1 = 1 π k = 1 N R ( θ k ) sin ( θ k ) ;
η 2 = 1 π k = 1 N R ( θ k ) cos ( 2 θ k ) , ζ 2 = 1 π k = 1 N R ( θ k ) sin ( 2 θ k ) .
α 2 = 1 2 π k = 1 N R ( θ k ) R ( θ k + π ) cos ( 2 θ k ) ,
β 2 = 1 2 π k = 1 N R ( θ k ) R ( θ k + π ) sin ( 2 θ k ) .
η 2 = η 2 α 2 , ζ 2 = ζ 2 β 2 .
θ d = arctan ( ζ 1 η 1 ) , r d = ( ζ 1 ) 2 + ( η 1 ) 2 ,
θ o = arctan ( ζ 2 η 2 ) , r o = ( ζ 2 ) 2 + ( η 2 ) 2 .
γ = r o r d .
R ( θ ) = F ( ORI ( θ ) , DIR ( θ ) ) .
R ( θ ) = ORI ( θ ) + DIR ( θ ) .
DIR ( θ ) = d 0 + d 1 cos ( θ θ d ) + d 2 cos [ 2 ( θ θ d ) ] +
ORI ( θ ) = o 0 + o 1 cos [ 2 ( θ θ o ) ] + o 2 cos [ 4 ( θ θ o ) ] + .
η k = d k cos ( k θ d ) , ζ k = d k sin ( k θ d ) , k = 1 , 3 , 5 , ,
η k = d k cos ( k θ d ) + o k 2 cos ( k θ o ) ,
ζ k = d k sin ( k θ d ) + o k 2 sin ( k θ o ) , k = 2 , 4 , 6 , ,
r 0 = d 0 + o 0 .
( r 2 ) 2 = ( o 1 ) 2 + ( d 2 ) 2 + 2 o 1 d 2 cos [ 2 ( θ d θ o ) ] .

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