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1Program in Neuroscience, University of California, San Francisco, California 94143–0444; 2Department of Psychology, University of California, Berkeley, California 94720–1650; and 3Department of Physiology, University of Cambridge, Cambridge CB2 3EG, United Kingdom
Submitted 8 August 2003; accepted in final form 21 January 2004
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ABSTRACT |
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INTRODUCTION |
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; Barlow 1959). Natural visual input contains features such as edges and homogenous color patches, which make the patterns of light falling on the retina highly redundant and which give rise to statistical dependencies between neighboring regions of the visual image (Field 1987). To reduce redundancy, the visual system might use these features as a basis for representing visual input (Barlow 1989).
Independent component analysis (ICA) (Comon 1994) is a widely used method for finding a redundancy-reducing encoding of data (such as natural visual scenes). Although the resulting independent components (ICs) are often only approximately independent, when ICA is applied to achromatic natural images, it produces ICs that are strikingly similar to the achromatic spatial receptive fields (RFs) of simple cells in primary visual cortex (V1) (Bell and Sejnowski 1997). Sparseness maximization (Olshausen and Field 1996) is conceptually similar to ICA and produces similar results; Ringach (2002) has argued that it better models the diversity of spatial RFs in V1. However, ICA has been validated by many rigorous comparisons with V1: van Hateren and van der Schaaf (1998) quantified the comparison between achromatic ICs and macaque V1 simple cells and found that the distributions of all RF measurements except optimal spatial frequency (SF) were well matched. When a temporal dimension is added and simple cells are compared with the IC filters of achromatic natural movies, the SF similarity improves (van Hateren and Ruderman 1998).
Because the ICA model successfully explains the spatial tuning of V1 simple cells to achromatic stimuli, the model may also explain other response properties such as color tuning. To test this hypothesis, ICA has been applied to colored natural scenes to produce spatiochromatic ICs (Doi et al. 2003; Hoyer and Hyvärinen 2000; Tailor et al. 2000; Wachtler et al. 2001). However, these studies used varying methodologies and found conflicting sets of ICs. Additionally, none of the studies has quantitatively compared ICs to a standard set of physiological measurements of V1. Finally, a richer understanding of V1 color coding has recently been developing: the view that color sensitivity is infrequent in V1 and restricted to weakly orientation-tuned neurons in cytochrome oxidase blobs (Livingstone and Hubel 1984; Lennie et al. 1990) has given way to a more diverse picture, in which oriented cells also have a rich variety of color sensitivities (Conway 2001; Johnson et al. 2001).
In this study, our goal was to rigorously test the hypothesis that ICA can account for the chromatic tuning of neurons in V1. We have controlled for the methodological variations between previous color ICA studies and have chosen the most biologically realistic set of ICs for analysis. Treating the ICs as though they were real neuronal RFs, we use the cone-opponent grating stimuli of Derrington et al. (1984) to determine their color tuning. We then directly, quantitatively compare the color tuning with the Lennie et al. (1990) classic V1 data. To further test the hypothesis, we compare the structure of ICs' model cone inputs with recent V1 data from Conway (2001) and Johnson et al. (2001). At first sight, the color tuning of ICs of natural scenes does not look very similar to V1 neurons, but we will show that, in fact, there is considerable similarity, once appropriate ICA methodology is employed, and the limitations of real experimental protocols are considered.
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METHODS |
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We analyzed a "hyperspectral" set of 25 distinct colored natural scenes (Párraga et al. 1998). Each 256 x 256-pixel scene was photographed through 31 filters covering the human visible spectrum (centered at wavelengths 400, 410... 700 nm, bandwidth: 
10 nm); thus it was not limited by the spectral sensitivity of a standard RGB (Red-Green-Blue) camera. Each plane was digitized to 12 bits of intensity. We aligned the color planes of each scene with subpixel accuracy by maximizing cross-correlation of adjacent planes; then, to ensure proper alignment, we averaged pixels together in 2 x 2 blocks.
We encoded the scenes in three ways that encompass previous work. Wachtler et al. (2001) transformed the Párraga et al. (1998) hyperspectral images into the human visual colorspace defined by the absorption spectra of L, M, and S cones. Hoyer and Hyvärinen (2000) used uncompressed images encoded in a digital camera's RGB colorspace, while Tailor et al. (2000) used JPEG-compressed RGB images taken from the Internet.
In the LMS condition, we used the Smith-Pokorny cone-sensitivity curves (Fig. 1E) (Smith and Pokorny 1975) to transform the 31-plane hyperspectral scenes into 3-plane LMS space. These human psycho-physical curves are consistent with, but more precise than, the physiological absorption spectra of macaque photoreceptors (Baylor et al. 1987). In the RGB condition, we converted the 31-plane hyperspectral scenes into 3-plane RGB images, using the sensitivity curves (Fig. 1F) of the red, green, and blue detectors of a typical digital camera (Nikon Coolpix 950; C.A. Párraga, unpublished observations). In the JPEG condition, we applied JPEG compression to these RGB-encoded images to mimic the Tailor et al. (2000) dataset. JPEG images were encoded at quality setting 90 [using Matlab's (the MathWorks) IM-WRITE function]. This is conservative compared with the quality setting of between 50 and 75, which we estimate (from the encoded file sizes) was used in the Tailor et al. (2000) image set. After performing ICA, we linearly transformed the resulting RGB ICs into the LMS colorspace for comparison with neurophysiological data.
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In the LMS and RGB conditions, each image fragment was log-transformed. The log transform is commonly applied as ICA preprocessing (e.g., van Hateren and van der Schaaf 1998) because it improves the convergence of ICA learning algorithms (see Willmore et al. 2000 for analysis of its effects). The raw luminance variation within our set of natural scenes, which covers three orders of magnitude, badly skews the data distribution and breaches the linear superposition model assumed by ICA (see next section). There are other remedies for this problem: while raw LMS images yield poorly converged ICs, standardizing the mean and variance of those images produces ICs similar to those of log-transformed images (data not shown). We chose to use the log transform, however, because it makes our results comparable with other studies.
From a functional point of view, the log transform has a clear biological correlate in retinal luminance adaptation (Field 1994; van Hateren and van der Schaaf 1998), a process that also compensates for the luminance changes (of many orders of magnitude) occurring in the natural environment. Our investigations of color tuning (in the following text) are conducted in the equivalent of laboratory conditions and always modulate cone contrast (at fixed mean luminance) rather than raw luminance. Therefore we did not apply log transformation to our "grating stimuli."
In the JPEG condition, we did not apply the log transform, because our aim was to replicate the results of Tailor et al. (2000), where the transform was not used.
ICA: algorithm and preprocessing
Each image fragment rj was converted into a one-dimensional vector, and these comprised the columns of the 432 x 86,400 data matrix R. The mean of each vector was set to zero. As a computational convenience for the ICA step, we whitened the data vectors; this transformation was inverted before the ICs were analyzed, so it has no effect on the final results. Our whitening method was based on principal component analysis, and produced a whitened data matrix
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The assumption behind the ICA algorithm is that each whitened image fragment xj is composed of a weighted sum of a fixed set of underlying source vectors ai, the activities of which are given by scalars si
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This can be rewritten as a matrix equation, x = As, where A is the matrix of sources and s is the vector of activities.
These sources, also called basis functions, are identified as independent sources of variance of the image data, which produce image fragments by mixing together linearly. Along with the basis functions, one can also derive a set of IC filters that reverse the process: they "unmix" the image fragment to yield activity values si, which correspond to individual sources and which should also be maximally independent. The set of filters produced by ICA is directly analogous to a population of V1 neurons, each analyzing part of the incoming visual stimulus.
There are a number of ICA algorithms, which make slightly different assumptions about the distribution of the underlying sources, but for ICA of colored natural scenes most algorithms yield similar results (Wachtler et al. 2001). We used an implementation of the information-maximization ICA algorithm with natural gradient feature (Bell and Sejnowski 1997; Delorme and Makeig 2003). This algorithm uses gradient ascent to find vectors in the distribution that give us the IC filter matrix F, which is the inverse of the sources: F = A-1.
IC filters were similar over multiple runs of the algorithm with different random seeds and different sets of image fragments. As a control for artifacts due to misalignment of the spectral planes, we performed extra analyses where we subsampled the LMS images in blocks of 2 x 2, 3 x 3, and 4 x 4 before extracting fragments; at each block size, the character of filters was preserved, which shows that the color planes were sufficiently aligned to prevent artifacts.
Spatial tuning
To find the best spatial stimulus for each IC, we exploited the equivalence between the Fourier transform of an RF and its responses to grating stimuli. We took the two-dimensional discrete Fourier transform of each IC, identified the highest amplitude Fourier component, and computed its orientation and spatial frequency.
These measurements were limited in precision by the 12 x 12 patch size because only discrete values corresponding to integer harmonics are possible. This patch size enabled us to measure orientation with precision of 10°, and to measure SF with precision of 1 octave at low SFs (near 0.1 cycles/pixel) and <0.3 octaves at high SFs (near 0.5 cycles/pixel).
We also assessed other spatial properties of each IC. Using the best achromatic stimulus, we computed orientation bandwidth, SF bandwidth, length, and width by measuring full width at half-maximum (FWHM) for band-pass functions or twice the half-width at half-maximum for low- and high-pass functions. SF bandwidth was computed as the log of the SF ratio between the half-heights in the amplitude spectrum along the radius corresponding to peak orientation following van Hateren and van der Schaaf (1998). Orientation tuning bandwidth was measured as the difference between half-heights along the circle corresponding to peak SF. For a small number of very high-SF oblique filters, some measurements fell outside the boundary of the Fourier transformed patch; in these cases, we took the measurement at the boundary, which caused a slight narrowing of their bandwidth. Aspect ratio was computed as the ratio of the length and width, where the length is the RF envelope FWHM in the low-pass direction and width is the envelope FWHM in the band-pass direction (Field and Tolhurst 1986).
Color tuning
Color tuning was quantified using the Lennie et al. (1990) cone-opponent stimuli in the "DKL" colorspace developed by Derrington et al. (1984) from the work of Macleod and Boynton (1979). In this three-dimensional colorspace, the xy plane is equiluminant, similar to the CIE colorspace, and so any changes along the x and y axes alter stimulus color without affecting luminance. Increasing x increases L-cone input while proportionally decreasing M-cone input (i.e., a stimulus becomes more "red" and less "green"), leaving S cones unaffected. Increasing y increases both L- and M-cone inputs while reducing S-cone input (resulting in more "yellow" and less "blue"). Increasing z increases input to all three cones, i.e., increases overall luminance. Lennie et al.'s white point was set to cone input values (0.311, 0.336, 0.353). Modulation from this point along the x axis to (1, 0, 0) corresponds to LMS cone input changes of (0.074, -0.14, 0), and modulation along the y axis to (0, 1, 0) corresponds to changes of (0, 0, -0.84). Modulation along the z axis causes a change in luminance without a change in color so that modulation to (0, 0, 1) doubles all three cones' inputs. Because of the overlap of L and M cone spectra (Fig. 1E), the effect of X axis modulation on L and M cones is small (compared with the effect of y axis modulation on S cones) so we followed Lennie et al. in scaling the x coordinate by 3.125.
In the DKL colorspace, as defined in Lennie et al. (1990), a linearly summating neuron's color tuning can be described using only two parameters. To do this, we treat each set of coordinates (x, y, z) as a vector and describe it in spherical coordinates (ø, 
, 
). The azimuth ø is the preferred color within the equiluminant xy plane: A neuron which is purely sensitive to R-G variation lies along the X axis and has an azimuth of 0°, whereas a neuron purely sensitive to B-Y variation lies along the y axis and has an azimuth of 90°. The elevation is the angle the vector makes with the xy plane, and it describes the neuron's response to luminance change relative to chromatic change. A neuron with elevation 0° responds best to chromatic stimuli, whereas a neuron with elevation 90° responds best to luminance stimuli. The stimuli are cone-opponent drifting gratings that are spatially modulated in color and luminance along the line defined by (ø, ) and crossing the sphere defined by fixed . Thus a cell's color tuning can be fully described using only azimuth and elevations in the upper hemisphere ( > 0°).
We computed the color tuning of each IC by finding the elevation and azimuth values that produce the maximal response, at a resolution of 1°. We converted each (ø, ) pair into (x, y, z) coordinates and then converted these coordinates into cone-specific luminance values (L, M, S). To measure an IC's response to a color grating, we multiplied each cone luminance by the IC's Fourier component for that cone, summed the components over cones, and then took the amplitude.
Complementary to this cone-opponency analysis, we measured the cone-isolating grating responses of single cone inputs. In each color plane, we found the highest amplitude Fourier component and measured its phase, orientation, and SF.
Comparing color tuning distributions
The color tuning of each IC was thus represented by a single point in the DKL colorspace, q(ø,). The distribution of all 432 ICs, Q(ø,), was then compared with the distribution P(ø,) of the 96 neurons described by Lennie et al. (1990). To compare two DKL color tuning distributions P(ø,) and Q(ø,), we needed a metric valid for arbitrary two-dimensional distributions. We used the Kullback-Leibler (K-L) distance from P to Q, which is also known as the relative entropy of P with respect to Q
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K-L distance is not symmetric, that is, in general D(P||Q)
D(Q||P). However, when comparing physiological distributions P(ø,) to IC distributions Q(ø,), we avoid this issue by consistently using it in one direction: D(P || Q).
We binned the color tuning distribution into 30 x 30° bins, centered at azimuths -30, 0,.. .150° and elevations 15, 45, and 75°. Some of the distributions had a small number n0 of zero bins (oriented cells had 7, IC filters had between 1 and 4 depending on condition); to avoid singularity, these bins must be assigned a count. For each distribution, we wanted to calculate a zero-bin count, which was as conservative (high) as possible. Therefore we computed the maximum probability of data in each zero bin, p0, such that the probability of no data falling in all n0 observed zero bins was <0.5 (because, in our single trial, we observed n0 0 bins). The relative order of K-L distances was robust, and our results did not change substantially for other bin sizes, center locations, and zero bin counts between 0.01 and 1.
Measurements of K-L distance, like other information-theoretic measurements, cannot be assigned statistical confidence without extensive prior information. Therefore these distances must be referenced to another distance value. We chose our reference to be the distance between Lennie et al.'s nonoriented and oriented cell distributions (shown in Fig. 2, C and D), which represents a substantial change between distributions, and is generally believed to represent a difference between cell classes. Because of the asymmetry of K-L distance, D(oriented||nonoriented) = 0.97 while D(nonoriented||oriented) = 1.69, so we took as our reference the average, 1.33. However, using either individual distance value does not affect our results.
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In our spatial and color tuning analyses, we considered the influence of artifactual ICs on the distribution of color tuning. Because ICA is a noisy optimization process with a finite data set, it will inevitably produce some artifactual ICs. However, using low variance as an indicator of artifactuality (as is commonly done) would introduce a bias against color because color is a smaller source of variance in natural images than luminance (Ruderman et al. 1998). Therefore we used the van Hateren and van der Schaaf (1998) criteria: artifactual ICs extend over only a few pixels and have nearly equal power in all four corners of the power spectrum. These criteria are entirely spatial and do not, a priori, introduce a color bias.
We also considered whether the limited size of image fragments could cause artifacts in the color-tuning data. If spatial or spatial frequency edge effects were a problem, we would expect to see more artifactual bias at the smaller size. Notably, we obtained identical results using two patch sizes: 8 x 8 and 12 x 12 (data not shown). This suggests that edge effects do not significantly alter color tuning.
Spatial-chromatic separability assumption
We replicated the Lennie et al. (1990) study of V1 neurons in two ways: under ideal conditions and under conditions that imitate their exact methods, including experimental limitations. In the ideal case, we simultaneously measured spatial tuning (optimal orientation and SF) and color tuning (azimuth and elevation). In the imitation case, we followed the Lennie et al. assumption that color and spatial tuning are separable (i.e., regardless of stimulus color, a cell will have the same optimal grating stimulus). Therefore in this case we first measured spatial tuning using achromatic gratings and then used the orientation and SF of the best achromatic grating to make chromatic gratings for color tuning measurement. Lennie et al. sometimes re-optimized spatial tuning with a colored grating and used colored gratings on occasions when achromatic ones did not work well. However, we excluded these heuristics, because they were not formalizable.
Effect of noise
A major difference between much modeling and real experiments is the presence of noise in the experiments. Therefore when imitating the Lennie et al. (1990) experiments, we tested the effect of noise on color tuning. We assumed that each IC's maximum amplitude of response modulation equaled 75 spikes/s, a value typical of Lennie et al.'s neuronal responses. For response modulations of that size, the amplitude SD is 5.25 spikes/s (Levine 1995). We simulated 25 different experiments in which each grating stimulus was presented for 20 trials. To each trial's "grating response" (Fourier amplitude), we added amplitude noise with a SD of 5.25 and a mean of zero. In each experiment, we averaged responses over all 20 trials and took the best stimulus as the color tuning. Then to assess whether noise caused an overall bias, we computed the spherical mean of these color tuning measurements across 25 experiments (Fisher 1987).
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RESULTS |
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), we constructed three different image sets with three color planes each (LMS-encoded, RGB-encoded, and JPEG-compressed RGB-encoded). We took each data set, performed ICA, and obtained 432 IC basis functions and IC filters. Figure 1, A–C, shows equivalent samples of ICA filters for the RGB, JPEG, and LMS image sets, whereas Fig. 1D shows a corresponding sample of LMS basis functions (see figure legend). Many of the filters look similar to the elongated Gabor function RFs of V1 simple cells (e.g., Ringach 2002), and many have apparent red-green (L-M) or blue-yellow [S-(L+M)] opponency. However, there are significant differences between the three sets of filters in spatial structure and color tuning. JPEG encoding creates artifacts in ICs
Both Tailor et al. (2000) and Hoyer and Hyvärinen (2000) have published ICA results from RGB images. Because the Tailor et al. images were encoded using JPEG compression, which is lossy (i.e., does not preserve all information in the image), we investigated whether this had affected the ICs by directly comparing the effect of the different encodings on IC filters derived from our data set.
In the RGB condition, our IC filters (Fig. 1A) corresponded fairly well to the basis functions of Hoyer and Hyvärinen, whereas in the JPEG condition, our filters (Fig. 1B) were similar to those found by Tailor et al. We found that the spatial structure of the ICs found by Tailor et al. was indeed affected by JPEG compression of the source images: JPEG filters contained a large number of blue-yellow and red-green checkerboard patterns (e.g., rows 3–4, and 9–10 in Fig. 1B) that do not correspond to any RGB filters. To ensure that these artifacts were the results of JPEG compression, and not the absence of the log transform, we also ran ICA on non-log-transformed RGB images. The results (data not shown) are spatially similar to the log-transformed RGB ICs, not the JPEG ICs.
These artifacts result from JPEG's compression algorithm, which divides images into 8 x 8 blocks and then discards information within blocks. Because the image fragments used for ICA are not aligned with block boundaries, the boundaries become artifacts. JPEG encoding also produces chromatic artifacts because it imitates an idealized human visual system by encoding images into luminance, red-green and blue-yellow color planes. Thus ICA of JPEG images reveals mechanisms of the JPEG compression algorithm rather than features of the natural visual world.
RGB encoding also biases ICs
We next investigated the effect of using uncompressed RGB-encoded images, compared with the more biologically realistic LMS encoding. Hoyer and Hyvärinen (2000) justify their use of RGB images from a digital camera by arguing that the colorspace choice will make no difference. Because RGB and LMS colorspaces are related by a linear transform, they argue, the correlations in the data remain unchanged, thus ICA should produce the same ICs for both image sets. However, there are several nonlinearities in the digital camera's imaging process, such as color balancing and gamma correction, and preprocessing also includes another nonlinear step, the log transform. Because all of these nonlinearities may cause discrepancies between the RGB and LMS conditions, we decided to measure their influence on spatial and chromatic tuning.
The spatial structures of IC filters in the RGB condition (Fig. 1A) and the LMS condition (Fig. 1C) were very similar, insofar as both sets contain mostly elongated Gabor-like features. There was noise in the LMS filters that was not present in the RGB filters, which was also true of the basis functions (data not shown). This may have resulted from the substantially greater overlap of L and M cone spectra (Fig. 1E) compared with R and G detectors (Fig. 1F): a given color image will have very similar L and M activations, and thus image noise within the nonoverlapping spectral regions will have an exaggerated effect on ICs in which L and M are opponent.
To compare color tuning in the RGB condition with the LMS condition, we linearly transformed RGB filters into the LMS colorspace, using the matrix of dot products between the RGB and LMS spectra in Fig. 1, F and E. Color tuning was summarized as the elevation (luminance sensitivity) and azimuth (chromatic preference) of the drifting sinusoidal grating that evoked the greatest response (see METHODS). The color tuning of RGB filters (Fig. 2A) differed substantially from LMS filters' color tuning (Fig. 2B): RGB filters fell into a small number of line-like, continuous clusters corresponding to yellow (30° azimuth), magenta (135°), and blue (90°). LMS filters clustered at red (0°) and blue-magenta (120°) and were much more diffusely clustered. The shift in the centers of azimuth clusters suggests that the difference between RGB and LMS is not merely due to noise and that, contrary to the claim of Hoyer and Hyvärinen (2000), RGB-encoded ICs are not appropriate after all for comparison with the visual system. Therefore an ICA model of the visual cortex should use the visual system's LMS encoding of the chromatic information in natural scenes.
Insofar as RGB filters are comparable to the visual cortex, RGB encoding has one provisional advantage over LMS encoding: it reduces noise in the ICs, as explained above. Consequently, although RGB encoding produces biased IC filters, any similarity to V1 cells bolsters our claim (in the following text) that IC filters resemble V1 cells.
LMS IC filters have similar spatial tuning to simple cells and achromatic ICs
To examine whether color information affects the spatial structure of ICs, we compared the spatial properties of LMS IC filters with the spatial properties of achromatic IC filters (van Hateren and van der Schaaf 1998) and macaque simple cells (De Valois et al. 1982; Parker and Hawken 1988). We obtained the distributions of best orientation, orientation bandwidth, spatial frequency, spatial frequency bandwidth, and aspect ratio (envelope length divided by envelope width), using the best achromatic stimulus for each IC.
We identified artifactual ICs following the van Hateren and van der Schaaf (1998) criteria; they constituted 29% (124/432) of our ICs. However, some of these artifactual ICs had high variances (i.e., they were robust sources for the image data), so we also show, but do not discuss, spatial analyses for the full set of ICs.
The orientation distribution of nonartifactual filters (Fig. 3A, 
) was similar to the distribution of achromatic filters, with peaks at 0° (vertical), 45°, 90° (horizontal), and 135°. The orientation bandwidth peak (Fig. 3B, ) was near 60° in our colored ICs, and the entire distribution seemed to be shifted toward higher values compared with achromatic ICs and older studies of macaque V1 (De Valois et al. 1982; Parker and Hawken 1988), Interestingly, more recent physiological studies have reported an orientation bandwidth distribution much like ours (Ringach et al. 2002); in fact, Ringach (2002) comments on the discrepancy between their data and achromatic ICA data, which color ICA apparently resolves. The distribution of color filter spatial frequencies (Fig. 3C) was qualitatively similar to achromatic filters in that it increased exponentially with SF and peaked at the Nyquist frequencies for both horizontal/vertical and diagonal filters. However, color filters were more diverse in preferred SF than achromatic, and better resemble the physiological data. The spatial frequency bandwidth distribution of color filters (Fig. 3D) was similar to both achromatic filters (van Hateren and van der Schaaf 1998) and simple cells (De Valois et al. 1982). The aspect ratio distribution of color filters (Fig. 3E) was centered at 1 (envelope length equal to envelope width) as with achromatic filters and simple cells (van Hateren and van der Schaaf 1998). Overall, with artifactual ICs removed, the distributions of the spatial parameters of colored IC filters were similar to the distributions reported for achromatic ICs and fit the simple cell distributions about as well. The fit between the spatial properties of colored IC filters and simple cells was somewhat better for SF, marginally worse for aspect ratio; and while colored IC filters fit the most recent orientation tuning bandwidth data, achromatic IC filters fit the older data.
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Because LMS IC filters are spatially similar to simple cells, we compared the filters' color tuning with V1. We used the data of Lennie et al. (1990), who divided V1 cells into three classes: oriented, nonoriented and complex. We first considered oriented cells, which roughly correspond to simple cells, but because many ICs are chromaticity sensitive, we also considered nonoriented cells, which Lennie et al. found to be most color sensitive.
Comparing the color tuning of LMS filters (Fig. 2B) to oriented V1 cells (Fig. 2C), we found similar classes of color tuning in both. The azimuth distributions were bimodal, with a red-green (L-M) opponent cluster near 0° azimuth and a slightly larger blue-yellow [S-(L+M)] opponent cluster near 90°. However, the azimuth distribution of chromaticity-tuned, S-cone-sensitive filters was actually centered near 120°; thus filters are closer to (S+L)-M cone opponency (135°). In both distributions, we found chromaticity-tuned and luminance-tuned RFs. Chromaticity-tuned RFs (at lower elevations) were bimodal with very few intermediates. However, filters seemed substantially more chromaticity tuned than oriented V1 cells; a large number of L-M opponent filters with azimuths near 0° were purely chromatic, with elevations near 0°.
According to the Lennie et al. (1990) strict criteria, which require a nonoriented cell to respond better to full-field modulation than to any grating, only one of our IC filters (a full-field noise patch) was nonoriented. Nevertheless, a moderately large population of filters had a center-surround-like organization reminiscent of blob cells (Livingstone and Hubel 1984), so we also compared the filters to Lennie et al.'s nonoriented (presumed blob) cells (Fig. 2D). Like filters, nonoriented cells were basically bimodal in azimuth with a red-green (L-M) cluster near 0° and a blue-yellow [S-(L+M)] cluster near 90°; however, the nonoriented cells' L-M mode was dominant, and azimuths between 30 and 60° were more common. The elevation distributions of nonoriented cells and filters were similar: both were bimodal, although nonoriented cells were less sharply so.
Previously, LMS IC basis functions (rather than LMS filters) have been compared with V1 (Hoyer and Hyvärinen 2000; Lee et al. 2002). Although there are a priori reasons why basis functions should not be analyzed this way (see DISCUSSION), we have also observed that the color tuning of basis functions (Fig. 2E) was markedly different from both V1 cells and filters: the distribution contained only a single cluster near azimuth 75° and elevation 75°. This is somewhat surprising because basis functions are apparently diverse in color; however, their cone inputs were typically poorly balanced. Full-field modulation, which is a poor stimulus for V1 cells, was the best stimulus for 73% of basis functions and <1% of filters; this empirically confirms that IC filters are the correct comparison for V1 cells.
Quantitative comparison between IC filters and V1 shows significant similarity once experimental biases are removed
The color tuning of the LMS filters showed some qualitative similarities with the Lennie et al. (1990) V1 cells; still, the raw filters contained two notably chromatic groups, near 0 and 120° azimuth, which were not strongly evident in the raw V1 data. However, comparing raw color tuning is misleading because it fails to take account of three discrepancies between the V1 data and the filters. First, the raw ICA data included a subpopulation of artifactual low-variance ICs. Second, the V1 data assumed that spatial tuning can be determined achromatically before determining color tuning; that is, it assumes the best chromatic stimulus is spatially identical to the best luminance stimulus. This spatial-chromatic separability assumption might have biased measurements of color tuning. Third, the firing rates of V1 cells are subject to noise; this could also have biased estimates of cells' color tuning. Because we could not test the effects of noise and separability on actual neurons, we measured their effects on the color tuning of the model IC filters.
Using the van Hateren and van der Schaaf (1998) criteria, we found that 29% of color ICs are artifactual compared to the 25% of achromatic ICs they report. In terms of color tuning, these artifactual ICs (Fig. 4A) tended to have very low elevations. Eliminating them made the color tuning of the remaining filters more similar to V1 in addition to improving spatial tuning.
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Noise in neuronal responses is another possible source of bias. In Lennie et al.'s spherical coordinate system, there is a nonlinear relationship between elevation and stimulus contrast. In essence, because z-axis modulation strength dominates the x and y axes, cone contrast changes little with changes at low elevations and much more with changes at medium elevations. For most cells, contrast determines the firing rate (and therefore noise level). Because noise increases with elevation, stimuli with higher elevations are more likely to be experimentally misidentified as the optimal stimulus, which could cause a systematic upward elevation bias.
We measured the distance between the color tuning of noiseless IC filters and their mean color tuning with noise added and found that noise biased the color tuning toward higher elevations (Fig. 4C), causing a mean elevation overestimate of 7.6° and a number of severe overestimates. Thus some of the higher elevations seen in the physiological data (Fig. 2, C and D) compared with the noiseless LMS IC filters (Fig. 2B) may have arisen from noise in the physiological responses.
Thus the assumption of spatial-chromatic separability, and the presence of noise in real V1 neuronal responses, both result in experimental biases that make cells appear dissimilar to IC filters. Furthermore, the inclusion of low-variance artifactual ICs in the comparison with real neurons also makes the IC filters look dissimilar to the real cells. Figure 4D shows how the color tuning of our IC filters (from Fig. 2B) changes once we account for these three biases; it shows more similarity to the color tuning distributions of real V1 neurons (Fig. 2, C and D).
To make the comparison between IC filters and V1 cells more rigorous, we measured the Kullback-Leibler (K-L) distance between the color tuning distribution of filters and the distributions of oriented cells (Fig. 2C), nonoriented cells (Fig. 2D), and both cell types pooled (i.e., all noncomplex V1 cells). K-L distances were normalized relative to a reference distance, which we took to be the distance between the distributions of oriented and nonoriented cells (see METHODS).
We measured these normalized K-L distances for raw IC filters, filters corrected for each individual bias, filters corrected for noise and separability, and filters corrected for all three biases. Although raw IC filters best resembled the pooled V1 cells, they were not especially similar to this group or to the oriented or nonoriented groups since all K-L distances were near 1 (Table 1, column 1). However, every single bias correction improved the fit between filters and all types of cells (Table 1, columns 2–4); correcting the noise bias had the largest effect. Noise and separability corrections appeared to be most important; subsequent to these, the effect of removing artifactual IC filters was only moderate (Table 1, column 5). When all three biases were taken into account (Table 1, column 6), the filters revealed a strong similarity to the pooled cells and substantial similarity to oriented and nonoriented cell groups.
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Ultimately, in all analyses of our filters, we found that the closest match (smallest K-L distance) was with the pooled V1 cells (Table 1) rather than with either subpopulation, such as the more chromaticity-preferring nonoriented cells or the more Gabor-like oriented cells. It seems that IC filters are a heterogeneous group most similar in color tuning to the pool of oriented and nonoriented V1 cells.
Dependence of chromaticity on orientation selectivity is consistent with experimental observations
Because it has been widely believed that responsiveness to purely chromatic stimuli is concentrated among nonoriented cells (Livingstone and Hubel 1984; Lennie et al. 1990), we examined the relation between orientation tuning bandwidth and chromaticity sensitivity (elevation in DKL space). Figure 5 shows that, indeed, filters with broad orientation tuning tend to be chromaticity sensitive and have low elevations. Filters with narrow orientation tuning are more diverse but have a tendency toward higher elevations. Nonartifactual filters are primarily responsible for this observation since almost all artifactual filters have elevations near zero. Because most filters have relatively sharp orientation tuning, we can thus account for the physiological observation that nonoriented cells tend to be more chromatic without inferring that oriented cells are necessarily achromatic. In fact, our results are also consistent with the Johnson et al. (2001) recent report that 79% of color-responsive neurons, stimulated with cone-isolating gratings (see following text), had oriented RFs. Because most filters are well tuned for orientation, it emerges that most chromaticity-preferring filters do have fairly narrow orientation tuning.
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To compare ICs to more recent physiological data (Conway 2001; Johnson et al. 2001), we simulated the LMS IC filters' responses to cone-isolating stimuli. These stimuli avoid some of the confounds of cone-opponent stimuli by modulating only one cone type at a time. For instance, in an L-cone isolating grating, the light stripe increases L-cone excitation relative to background, whereas the dark stripe decreases L-cone excitation. Although physiological cone-isolating measurements are often noisy and difficult to calibrate because of the overlap between cone absorption spectra (Conway 2001), in our model, this measurement is trivial: we simply look at the filter's three color planes, which represent the individual LMS inputs.
For each IC filter, we measured its response amplitude to the optimal L-, M-, and S-isolating gratings, which indicates the relative strength of different cone inputs. We also measured its response phase to each grating because this determines the spatial relationship between the cone inputs. For example, a cell could have L and M inputs with identical best orientation and SF, and if those inputs are in phase with each other (0° apart) the cell will be (L+M) or yellow sensitive. However, if they are 180° apart in phase, it will be L-M or red-green double-opponent. Figure 7A shows a schematic RF containing a double-opponent component (Livingstone and Hubel 1984; Michael 1978), which responds fairly specifically to a border between two colors (yellow and blue) because its inputs from the corresponding cones are opposite in sign.
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). Filters fell into three clusters: one elongated cluster with high S response, some M response, and little L response; another elongated cluster with L and M response but little S response; and a diffuse cluster near zero. These roughly correspond to the modes of the cone-opponency data in Fig. 2B.
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) physiological SF measurements.
Conway (2001) reported that, in a population of weakly oriented L-M cells studied with cone-isolating pixel stimuli, most S responses were in phase with M responses. These cells could therefore be called red-cyan [i.e., L-(M+S)]. Among our L-M responsive filters, the amplitude of S response indeed correlates with M response (lower cluster in Fig. 6A), but the M and S inputs were phase opponent (Fig. 6C) when measured at the M-isolating grating's best orientation and SF. Thus our filters tend to show magenta-green [(L+S)-M] opponency, which was atypical but occasionally seen in Conway's data. This tendency of our filters may arise from the spectral overlap of M and S cones, which ICA will attempt to decorrelate even within L-M opponent IC filters.
Although there is little physiological data on S-cone double-opponency in V1, we measured its prevalence in our filters. Among S-responsive filters (S amplitude >20 units), we found double-opponency between M and S when measured at the S-isolating grating's best orientation and SF (Fig. 6D). We might have expected that these S-cone driven filters would be opponent to yellow (L+M). However, in the majority of filters, L and S actually have similar phase (Fig. 6E), whereas the remainder are phase opponent. This is probably also due to the spectral overlap described in the preceding paragraph. Thus our strongly S-responsive filters are also mostly magenta-green opponent, although a few are red-cyan. This prediction of our model has not yet been explicitly tested, although it does appear consistent with Lennie et al.'s data.
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DISCUSSION |
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; Lennie et al. 1990; Livingstone and Hubel 1984), whereas ICs have been reported to be unbiologically chromatic and double-opponent (Tailor et al. 2000). However, it emerges naturally from a biologically realistic set of LMS IC filters when care is taken with the ICA methodology to make proper comparisons between V1 cells and ICA output and when the reality of physiological experiments (such as response noise) is taken into account.
It is methodologically essential to use LMS encoding of natural scenes (as was also done by Doi et al. 2003; Lee et al. 2002; Wachtler et al. 2001); this accurately represents the chromatic information available to the cortical, experience-dependent stages of the visual system. Colored images encoded in the usual ways for transmission over, say, the internet (RGB or even JPEG encoding) produce filters that are not appropriate for comparison; moreover, they are unlike real cells. Additionally, it is important to compare IC filters, not basis functions, with the physiological data. Although some previous studies (Hoyer and Hyvärinen 2000; Lee et al. 2002) have focused on basis functions, there is no theoretical justification for this (see following text); additionally, we find that basis functions are dominated by properties (such as their strong response to full-field illumination) that have no counterpart in visual neurons.
The other major factor in this comparison is the correction of biases both in the IC analysis and the physiological data. A simplistic, uncorrected comparison between the IC filters and V1 cells (Lennie et al. 1990) suggests only a weak similarity (Fig. 2, B and C). The ICs appear to be more chromatically tuned than the real neurons. However, we find that this comparison is skewed by several biases. Some of the IC filters are highly chromatic noisy structures; when these are eliminated, the remaining filters better resemble simple cells, spatially and chromatically. Similarly, we estimate that the Lennie et al. V1 data are biased by their assumption of spatial-chromatic separability and by the effect of cortical noise.
Cortical noise is a particularly important source of bias in the Lennie et al. physiological data. We find that, in a simulated neurophysiological experiment, luminance (high-elevation) stimuli, which modulate all cones in concert, tend to produce noisier responses than chromatic (low-elevation) stimuli, which modulate contrast between two cone types. This skews measurements of color tuning, so that even cells' averaged responses make them appear much less chromaticity sensitive than they actually are. Any method of finding the best color direction will be swayed by this noise, especially if it interpolates between stimuli (as do many experiments) or picks the best stimulus (as does our simulation). Johnson et al. (2001) speculate that one reason for the lack of chromaticity-tuned cells in Lennie et al.'s study is that their stimuli only weakly modulate chromatic neurons. For modulation strength to directly influence tuning in the DKL colorspace, however, a nonlinearity is necessary; we suggest that noise is a very significant source of nonlinearity.
Analysis of these biases suggests that simple cells, which are generally not considered the substrate for color coding, are in fact more chromatically sensitive than has often been depicted. This result is independent of the correctness of our model and aids in reconciling the Lennie et al. study with a growing body of recent neurophysiology (Conway 2001; Johnson et al. 2001).
The similarity between IC filters and V1 cells suggests that chromatic and spatial information are distributed across cells in V1 in a way that is compatible with theoretical principles of independence and sparse representation. In other words, the spatial and chromatic response properties of neurons in V1 can be accounted for by the redundancy-reduction hypothesis (Attneave 1954; Barlow 1959).
Some systematic differences between IC filters and V1 cells remain. Even the corrected filters shown in Fig. 4D are somewhat more chromaticity sensitive than the pool of oriented and non-oriented cells in V1. Filters are also probably more double-opponent than simple cells. Some limitations of our study, which may account for these differences, are discussed in the following text.
Limitations of the IC analysis
ICA is not a guaranteed method for finding an independent code for a data set. It is designed to discover independent sources of variability (basis functions), assuming these are linearly superposed to form the observed data. Natural images break this assumption, because they are not formed by a linear superposition of sources (occlusion would be a better model than superposition). Under these circumstances, ICA can only discover sources that appear (by their non-Gaussianity) to be independent; these may be unrelated to the underlying structure of the data set.
Another limitation of our study is that the number of IC filters is capped by the dimensionality of the input data, which can artificially limit our ability to capture the structure of the input distribution. This limitation may become more critical as dimensionality increases as when going from achromatic to chromatic images. Generating an overcomplete basis set (e.g., Lewicki and Sejnowski 2000; Olshausen and Field 1996) from colored natural scenes would likely change the distribution of spatial tuning and may also change the distribution of color tuning.
Although we consider the effects of noise on neurophysiological experiments, we do not consider its effects on the redundancy-reduction hypothesis. Atick and Redlich (1990) have shown that, under noisy conditions, the RFs need to be more correlated (less independent) to preserve information about the image. We do not yet have an ICA model that allows us to take this into account.
Finally, ICA produces a strictly linear encoding; real neurons exhibit output nonlinearities and contrast normalization (e.g., Heeger 1992a,b), which are not part of our ICA model. There is evidence that nonlinearities play a role in V1 color processing (De Valois et al. 2000; Hanazawa et al. 2000). However, these nonlinearities primarily seem to sharpen a cell's tuning across color stimuli (De Valois et al. 2000). We surmise that this kind of sharpening nonlinearity might enhance the effect of noise and separability bias in studies of V1.
Filters versus basis functions
To compare the ICs of natural visual input with the RFs of real neurons, it is necessary to decide whether the IC filters or the IC basis functions are comparable to real RFs. For achromatic images, the basis functions and filters are very similar to one another, and this decision is not very significant (van Hateren and van der Schaaf 1998). For color data, however, there are substantial differences between the spatial and chromatic structure of the basis functions and filters (compare the color tuning of filters in Fig. 2B to the equivalent basis functions in 2E), which makes the choice critical. There has been some disagreement about this question in the color ICA literature: Hoyer and Hyvärinen (2000) and Lee et al. (2002) focus on basis functions, whereas Tailor et al. (2000) and Doi et al. (2003) focus on filters.
The mathematical formulation of ICA is unambiguous that IC filters are the correct choice for comparison with neuronal RFs. Basis functions represent sources of variability in the data set; they are notional image patches that can be superposed to generate the data set. IC filters, on the other hand, are simply linear filters designed to extract information from the data set; in this way, they are completely compatible with the popular linear-filtering model of simple cells (e.g., Field 1987).
Retinal cone mosaic
There is a discrepancy between our results and those of Doi et al. (2003), who also analyzed LMS IC filters but used a model featuring a fixed preprocessing stage mimicking the distribution of cones in the retinal mosaic. Their results, however, compare rather poorly to their choice of V1 data (Hanazawa et al. 2000), in both the prevalence of color selectivity (54% in V1 vs. 5% in their filters) and color tuning distribution (broad in V1 vs. extremely peaked in their filters). In addition, their model's retinogeniculate stage developed many fewer S-preferring cells (3%) than koniocellular/parvocellular LGN (12%). The likely explanation for these differences is that Doi et al. only modeled the fovea (actually foveola), so that just 3% of the cone inputs to their model cortex were S-type. Most primate studies of color, including the data they use for comparison, have been done in the perifovea, where the proportion of S cones is much higher: Lennie et al. (1990) studied cells within the central 3° of the visual field, and in fact Hanazawa et al.'s cells were between 0.8 and 2.4°. Within the central 2° (excluding the foveola), S cones represent 
10% of the population (de Monasterio et al. 1985). Therefore the Doi et al. results may be most valid for minute foveolar RFs that are acutely sensitive to the structure of the cone mosaic.
Our present model, which assumes that S-cone density does not constrain RFs' spatial structure, compares better with existing physiological data. However, if the model were to incorporate a retinal mosaic stage with a normal perifoveal concentration of S cones, we would expect an increase in the proportion of luminance-preferring filters (given the Doi et al. results), and possibly improvement in the fit between the model and V1.
Double-opponency and separability
An interesting feature of our IC filters is that the vast majority is double-opponent. This double-opponency has two components: first, each cone type's excitatory and inhibitory inputs balance across the RF; second, the inputs from opponent cones are exactly opposite in phase. The balance of excitatory and inhibitory inputs is a natural outcome for ICA primarily because unbalanced filters would be correlated. The phase opponency, however, arises from a fixed property of the early visual system: the overlap in the absorption spectra of L and M cones, as well as S and M cones, forces the filters to decorrelate those input pairs as much as possible. The color statistics of natural scenes may also contribute significantly to opponency (Lee et al. 2002).
Double-opponency seemed rare in Lennie et al. (1990)'s study. This is likely to be an underestimate because they used achromatic gratings to measure the optimal SF and orientation of the cells in their sample. This reflects an assumption of spatial-chromatic separability and generates especially misleading results for double-opponent cells. Figure 7A shows a schematic double-opponent cell the preferred stimulus of which contains a yellow-blue edge. When an achromatic grating (of optimal orientation and SF) is presented at any position, its light stripe will drive the yellow (R+G) and blue (B) inputs in one subfield equally, causing them to cancel out. Its dark stripe will do the same, and thus the response will not modulate. As a result, the estimates of preferred SF and orientation will be inaccurate; achromatic gratings cannot be used to estimate the tuning of double-opponent cells.
This effect can be demonstrated in our ICA data. An IC filter that responds poorly to achromatic gratings, much like the schematic cell in Fig. 7A, is shown in Fig. 7B. Its true best color grating stimulus, which has the filter's dominant orientation and shows the influence of its L and M inputs, is shown in Fig. 7C. However, if one assumes separability and stimulates this filter with an achromatic grating, the orientation and SF measurement is confounded by the double-opponency, and the resulting "best" grating (Fig. 7D) bears no relation to the RF; it seems primarily influenced by spatial noise, and gives rise to arbitrary color tuning (with error shown by the open circle in Fig. 4B).
More recent studies, using reverse correlation (Conway 2001) and cone-isolating stimuli (Johnson et al. 2001), have found a population of double-opponent cells in V1. However, it is likely that the nearly obligatory double-opponency of our filters exceeds that of cortical cells. One possible reason is that the cortex, being subject to noise, does not decorrelate its RFs to the same degree as our noise-free IC filters. Another possible reason is that our image preprocessing excludes some known properties of the precortical visual system, such as chromatic aberration, the lower spatial resolution of the S-cone system, and the scattered color tuning of LGN cells. It may be that such considerations also help explain why our filters show magenta-green opponency rather than the red-cyan reported by Conway (2001).
Nonoriented cells
It has often been supposed that nonoriented cells in V1 are the substrate for color tuning rather than oriented simple and complex cells (Livingstone and Hubel 1984). However, ICA generates few nonoriented, center-surround filters because oriented features are more significant sources of variance in natural scenes than center-surround features. This would seem to make ICA a poor model for the center-surround color-selective cells in cytochrome oxidase blobs. However, we did find many small, chromatically double-opponent filters, especially among the lower-variance ICs. These resemble the side-by-side double-opponent subunits found by Johnson et al. (2001) and the wide-band orientation tuned cells of Conway (2001), both of which probably occur in blobs. Also, consistent with the traditional understanding of blobs, we found that preference for chromaticity correlates with weakness of orientation tuning. ICA might therefore be able to partially model color selectivity within blobs, however a complete model for V1 will require additional organizing principles such as topography (Barrow et al. 1996; Hyvärinen et al. 2001).
Conclusions
The similarity between the color-independent components of natural scenes and RFs in V1 suggests that redundancy reduction (by ICA in particular) provides a plausible account of spatiochromatic RF structure in V1. This similarity is not apparent from inspection largely because some filters that appear colored are often very sensitive to luminance variations. It is likely that a similar effect is present in V1, where simple cells are conventionally thought to be luminance-dominated, but we find that they are likely to have substantial color tuning that would be evident if their inputs were visualized.
We predict that, as our understanding of V1 color coding improves, the color sensitivity of oriented cells will assume a more important role. In place of a clear division between luminance and color coding, our results suggest that simple cells may multiplex spatial and color information. Similarly, our results highlight the importance of double-opponency, which is advantageous for coding of colored borders and, as we have shown, for redundancy reduction. Finally, the success of ICA in accounting for spatiochromatic RF structure in V1 suggests that redundancy reduction will also prove to be a fruitful hypothesis in other sensory systems.
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ACKNOWLEDGMENTS |
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GRANTS
This research was begun while M. S. Caywood was supported by a Frank Knox Fellowship to the University of Cambridge and was continued with support from an National Defense Science and Engineering Graduate Fellowship. B. Willmore was supported by a Biotechnology and Biological Sciences Research Council studentship.
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FOOTNOTES |
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Address for reprint requests and other correspondence: M. Caywood, Department of Physiology, 513 Parnassus Ave., Room S-762, University of California, San Francisco, CA 94143–0444 (E-mail: caywood{at}phy.ucsf.edu).
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ABSTRACT
INTRODUCTION
, distributions for all ICs; 
and —, the reported V1 distributions (De Valois et al. 1982
