Spatial receptive field structure of double-opponent cells in macaque V1

Abstract

The spatial processing of color is important for visual perception. Double-opponent (DO) cells likely contribute to this processing by virtue of their spatially opponent and cone-opponent receptive fields (RFs). However, the representation of visual features by DO cells in the primary visual cortex of primates is unclear because the spatial structure of their RFs has not been fully characterized. To fill this gap, we mapped the RFs of DO cells in awake macaques with colorful, dynamic white noise patterns. The spatial RF of each neuron was fitted with a Gabor function and three versions of the difference of Gaussians (DoG) function. The Gabor function provided the more accurate description for most DO cells, a result that is incompatible with a center-surround RF organization. A nonconcentric version of the DoG function, in which the RFs have a circular center and a crescent-shaped surround, performed nearly as well as the Gabor model thus reconciling results from previous reports. For comparison, we also measured the RFs of simple cells. We found that the superiority of the Gabor fits over DoG fits was slightly more decisive for simple cells than for DO cells. The implications of these results on biological image processing and visual perception are discussed.

NEW & NOTEWORTHY Double-opponent cells in macaque area V1 respond to spatial chromatic contrast in visual scenes. What information they carry is debated because their receptive field organization has not been characterized thoroughly. Using white noise analysis and statistical model comparisons, De and Horwitz show that many double-opponent receptive fields can be captured by either a Gabor model or a center-with-an-asymmetric-surround model but not by a difference of Gaussians model.

INTRODUCTION

The spatial layout of chromatic signals influences color perception (1–5). A fundamental goal of visual neuroscience is to understand how the spatial processing of chromaticity by neurons mediate these effects. Double-opponent (DO) cells in primate primary visual cortex (V1) serve as a prime substrate for such an investigation because they respond to spatial chromatic contrast (6–9). The information that DO cells carry can be gleaned from analysis of their spatial receptive fields (RFs), but the spatial RF structure of DO cells is controversial in part due to disagreements regarding the definition of double opponency.

Neurons were originally defined to be DO if they were cone opponent (that is, if they receive antagonistic input from at least two types of cone photoreceptor in individual regions of their RFs) and if they had opposite spectral sensitivity in different parts of their RFs (7, 9–15). Early studies of macaque V1 neurons with these characteristics assumed that their RFs had a circularly symmetric organization (7, 9, 12–14). Later investigations adopting the same definition reported the surround to be asymmetric with respect to the center (6, 16).

Other studies classified neurons as DO if they had bandpass spatial frequency tuning to drifting equiluminant colored gratings (8, 17–19). These studies reported DO neurons to be orientation tuned, inconsistent with a center-surround RF organization. However, this definition of double opponency admits complex cells that respond at equiluminance as well as classically defined DO cells (16, 20). Complex cells respond to contrasts of both polarity within individual parts of the RF and therefore do not have opposite spectral sensitivity in different parts of the RF, in violation of the classical definition of double opponency (7).

Classically defined DO cells can be distinguished from complex cells on the basis of the response phase to drifting, cone-isolating gratings. Using this approach, Johnson et al. (17) showed that many classically defined DO cells are orientation tuned. This result rules out a symmetric, center-surround RF organization but does not reveal the RF organization because many two-dimensional (2-D) RFs could underlie any given one-dimensional (1-D) orientation tuning curve.

Mapping the complete 2-D spatial structure of DO cell RFs is important for three reasons. First, it is necessary for the construction of image-computable models, which can generalize to any image as input (21). Second, it facilitates comparison with normative theories of image encoding (22–25). Third, it elucidates the link between neurophysiology and perception. For example, center-surround DO RFs could contribute to differences in form perception defined by chromatic or luminance contrast (2, 26–29). Gabor-like RFs could contribute, additionally, to shape-from-shading—the ability to estimate the three-dimensional (3-D) shapes of objects from shading cues (30, 31).

In this study, we stimulated V1 neurons with white noise that varied independently in two spatial dimensions, as well as in color and time. We analyzed the data by spike-triggered averaging to identify DO cells and to measure their spatial RFs. We fit each spatial RF with four models: a Gabor function (8, 32, 33), a difference of Gaussians (DoG) function (34–38), an elliptical DoG model that breaks the constraint of circular symmetry, and a model that allows a crescent-shaped surround (6, 16). We compared model fits between DO cells and simple cells—a benchmark cell type whose RF is well known to be better fit by a Gabor than by a DoG function (7, 39–42).

The Gabor model outperformed the DoG model (including the elliptical DoG) for most DO cells we studied. The goodness of fit of the Gabor model was similar for simple and DO cells. A variant of the DoG model that allows for crescent-shaped surrounds performed nearly as well as the Gabor model for DO cells, but poorly for simple cells. Together, these results show that simple and DO RFs are both well described by the Gabor model, they are poorly described by the DoG model, and a center-crescent surround model is a reasonable description for many DO cells.

METHODS

General

All protocols conformed to the guidelines provided by the US National Institutes of Health and were approved by the University of Washington Animal Care and Use Committee. Data were collected from two male and two female rhesus macaques (Macaca mulatta) weighing 7–13 kg. Each monkey was surgically implanted with a titanium headpost and a recording chamber (Crist Instruments) over area V1. Eye position was continuously monitored using either an implanted monocular scleral search coil or an optical eye-tracking system (SMI iView X Hi-Speed Primate, SensoMotoric Instruments).

Monitor Calibration

Stimuli were presented on a cathode-ray tube (CRT) monitor (Dell Trinitron Ultrascan P991) with a refresh rate of 75 Hz against a uniform gray background (x = 0.3, y = 0.3, Y = 43–83 cd/m²). Monitor calibration routines were adapted from those included in the MATLAB Psychophysics Toolbox (43). The emission spectrum and voltage-intensity relationship of each monitor phosphor were characterized using a spectroradiometer (PR650, PhotoResearch, Inc.). Stimuli were gamma-corrected to compensate for the nonlinear voltage-intensity relationship. The color resolution of each channel was increased from 8 to 14 bits using a Bits++ video signal processor (Cambridge Research Systems, Ltd.) at the expense of spatial resolution; each pixel was twice as wide as it was tall.

Task

Monkeys sat in a primate chair 0.7–1.0 m from a CRT monitor in a dark room during the experiments. The monkeys were trained to fixate a centrally located dot measuring 0.2° × 0.2° and to maintain their gaze within a square 1.0°–2.0° fixation window. Successful fixation was rewarded, and fixation breaks aborted trials.

Electrophysiological Recordings

We recorded spike waveforms from well-isolated V1 neurons using extracellular tungsten microelectrodes (Frederick Haer, Inc.) that were lowered through dura mater by hydraulic microdrive (Narishige, Inc. or Stoelting Co.). Electrical signals were amplified and digitized at 40 kHz (Plexon, Inc.) and stored in a PC.

Visual Stimuli and Experimental Protocol

Each neuron was stimulated binocularly with white noise chromatic checkerboards (44, 45). Each stimulus frame was a grid of 10 × 10 stimulus elements (stixels), and each stixel subtended 0.2° × 0.2°. The stimulus changed on every screen refresh. The intensity of each phosphor was modulated independently according to a Gaussian distribution with a standard deviation of 5%–15% of the physically achievable range. The space-time averaged intensity of each phosphor was equal to its contribution to the background. This stimulus produces correlated Gaussian contrasts in each of the three cone types (Supplemental Fig. S1; all Supplemental material available at https://doi.org/10.6084/m9.figshare.13519907).

Neuronal responses to the white noise stimuli were analyzed using spike-triggered averaging (Fig. 1A). Neurons that did not have clear spike-triggered averages (STAs) were passed over for data collection.

Figure 1.

Derivation of cone weights and spatial weighting function. A: calculation of the weighted spike-triggered average (STA) (the weighted sum of the peak STA frame and two flanking frames) (right) from spike-triggered white noise stimuli (left). B: singular value decomposition (SVD) of the weighted STA reveals cone weights and spatial weighting function. C: reconstructing a low-rank approximation of the weighted STA by multiplying cone weights and spatial weighting function. Subtracting the weighted STA from the low-rank approximation yields the residual, which has little structure. D: percent explained variance plotted against the three sets of singular vectors for the example cell and the population (means ± SD). Percent explained variance was derived from the singular values using SVD over the entire 10 stixels× 10 stixels of spatial weighting function (black circles) or omitting stixels outside of the RF (black squares). RF, receptive field.

Cone Weights and Spatial RF

For each cell, we identified the frame from the STA that differed most from the background, based on the sum of squared red, green, and blue stixel intensities (negative intensities were defined as those below the contribution to the background). We then took the weighted average of the peak and the two flanking frames to create a 10 stixels × 10 stixels × 3 color channels tensor. The weight of each frame was proportional to the square root of sum of squared red, green, and blue stixel intensities. We reshaped the tensor into a 100 × 3 matrix and used a singular value decomposition to separate this weighted STA into a color weighting function and a spatial weighting function, defined as the first row and column singular vectors, respectively (Fig. 1B) (46). The color weighting function and the spatial weighting function captured most of the variance in the weighted STAs (Fig. 1, C–D).

The color weighting function, which quantifies neuronal sensitivity to modulations of the red, green, and blue phosphors, was converted to cone weights that are assumed to act on cone contrast signals (47). Cone weights were normalized such that the sum of their absolute values was 1 (16, 17, 46, 48). We analyzed only cells that were spatially opponent (see Cell Screening). As a result, each cell had cone weights with different signs in different RF subregions.

Cell Screening

We recorded from 393 V1 neurons and omitted 188 from the analyses on the basis of four criteria. Every neuron was required to have an STA with 1) high signal-to-noise ratio (SNR), 2) interpretable structure, 3) spatial opponency, and 4) cone weights that were either clearly opponent or clearly nonopponent. Below, we explain the rationale for each criterion and how it was implemented.

We excluded cells with low SNR because noisy STAs could lead to inaccurate estimates of color and spatial weighting functions. SNR was computed by comparing the peak STA frame to first STA frame and was defined as:

$$\mathrm{S}\mathrm{N}\mathrm{R}=\frac{1}{N}\stackrel{N}{\sum _{i=1}}{\left(\frac{I_i}{\mathrm{\sigma }}\right)}^2,$$ (1)

where N is the total number of elements within a frame: 10 stixels × 10 stixels × 3 color channels = 300 elements, I is the intensity of each element in the peak STA frame relative to the background, and σ is the standard deviation of the 300 elements that compose the first STA frame. The intensity of each element was divided by this standard deviation so that each element had (approximately) a standard normal distribution under the null hypothesis of no signal. We squared and summed these normalized intensity values and omitted from analysis the 60 cells for which this sum failed to reach a statistical threshold (P < 0.0001, χ² test, df = 300).

We excluded cells that combine cone inputs nonlinearly because their STAs do not reflect their stimulus tuning accurately (44). We identified nonlinear neurons using a nonlinearity index (NLI) (45). The NLI uses the STA and spike-triggered covariance to find the maximally informative stimulus dimension under a multivariate Gaussian assumption (49). For each cell, we projected the stimuli shown in the experiment onto the maximally informative dimension and binned the projections, excluding the upper and lower 5% to avoid the influence of outliers. We calculated the average firing rate across the stimuli within each bin. The relationship between firing rate and stimulus projection was fit with the following three regression equations:

$$y_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}=b_0+b_{1}x,$$ (2)

$$y_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}}=b_0+b_1{x}^2,$$ (3)

$$y_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}=b_0+b_{1}x+b_2{x}^2.$$ (4)

The goodness of fit of each regression was quantified with an R² statistic. The NLI is defined as follows:

$$\mathrm{N}\mathrm{L}\mathrm{I}=\frac{R_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}}^2-R_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}^2}{R_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}^2}.$$ (5)

The NLI attains its theoretical maximal value of 1 when the inclusion of a linear term does not improve the regression fit. This would be the case, for example, for a V1 complex cell whose response is invariant to contrast polarity. NLI attains its theoretical minimum value of −1 when the inclusion of a quadratic term does not improve the regression fit as would be the case for a purely linear cell. We used the NLI to identify neurons that were excited by light increments and decrements in the same part of the RF. Output nonlinearities of such neurons can be modeled as quadratic but not as odd-symmetric functions, which motivates the squaring in the definition of NLI. Twenty-six cells were excluded on the basis that their NLI was > 0.

We excluded cells that were spatially nonopponent because these cells can be neither DO nor simple. We identified spatially nonopponent cells by analyzing the power spectrum of their spatial weighting functions. Spatially nonopponent cells, by definition, had maximal power in the lowest spatial frequency bin, which included power from 0 to ∼0.7 cycles/°. This criterion excluded 53 cells. Other, stricter criteria excluded more cells but did not affect the main results.

We segregated simple cells from DO cells on the basis of cone weights, and we excluded neurons outside of these categories. Cells were classified as simple if their L- and M-cone weights had the same sign, accounted for 80% of the total cone weight, and individually accounted for at least 10%. Cells were classified as DO if they had large magnitude cone weights of opposite sign. DO_LM-opponent cells were defined as those that had L- and M-cone weights of opposite sign that together accounted for 80% and individually accounted for at least 20% of the total cone weight. DO_{S-cone sensitive} cells were cone opponent and had an S-cone weight that accounted for at least 20% of the total. Forty-nine cells that were not categorized as simple, DO_LM-opponent, or DO_{S-cone sensitive} were omitted from the analyses. These criteria were implemented to exclude neurons that were neither clearly cone-opponent nor cone nonopponent. However, other criteria produced results similar to those reported in the results (Supplemental Figs. S5–S8).

A total of 205 neurons contributed to the final pool (monkey 1: 43 simple, 57 DO_LM-opponent, 37 DO_{S-cone sensitive}; monkey 2: 11 simple, 11 DO_LM-opponent, 1 DO_{S-cone sensitive}; monkey 3: 18 simple, 14 DO_LM-opponent, 6 DO_{S-cone sensitive}; monkey 4: 1 simple, 4 DO_LM-opponent, 2 DO_{S-cone sensitive}).

Model Fitting of the Spatial Weighting Function

We fit the spatial weighting function of each neuron with four models. Fitting was performed using the inbuilt MATLAB fmincon function to minimize the sum of squared errors between the spatial weighting function and the model fit. We describe each of the models below.

Gabor model.

The Gabor model was defined as follows:

$$f\left(x\prime ,y\prime \right)=A{e}^{-\frac{\left(x^{\prime 2}+{\gamma }^2{y}^{\prime 2}\right)}{2{\sigma }^2}}cos\left(\left(\frac{2\pi y^{\prime }}{\lambda }\right)-\mathrm{\varphi }\right),$$ (6)

where (x′, y′) is obtained by translating the original coordinate frame to the RF center, (x_c, y_c), and rotating it by an angle θ:

$$x^{\prime }=\left(x-x_c\right)cos\left(\mathrm{\theta }\right)+\left(y-y_c\right)sin\left(\mathrm{\theta }\right)$$ (7)

$$y^{\prime }=-\left(x-x_c\right)sin\left(\mathrm{\theta }\right)+\left(y-y_c\right)cos\left(\mathrm{\theta }\right).$$ (8)

λ is the spatial period of the cosine component in °/cycle, and ϕ is the spatial phase. A spatial phase of ϕ = 0° produces an even-symmetric RF, whereas spatial phase of ϕ = 90° produces an odd-symmetric RF. The two axes of the Gaussian envelope align with the x′ and the y′ axes. The parameter A is the amplitude, γ is the aspect ratio, and σ is the standard deviation of the Gaussian envelope along the x′ axis.

DoG model.

The difference of Gaussians (DoG) model can be written as follows:

$$f\left(x,y\right)=\frac{A_c}{\sqrt{2\mathrm{\pi }{\mathrm{\sigma }}_{\mathrm{c}}^2}}exp\left(\frac{-\left(\right(x-x_c{)}^2+(y-y_c{)}^2)}{{\mathrm{\sigma }}_c^2}\right)-\frac{A_s}{\sqrt{2\mathrm{\pi }{\mathrm{\sigma }}_s^2}}exp\left(\frac{-\left(\right(x-x_c{)}^2+(y-y_c{)}^2)}{{\mathrm{\sigma }}_s^2}\right),$$ (9)

where A_c and A_s are the amplitudes of the center and surround. σ_c and σ_s are the standard deviations of the center and surround.

Elliptical DoG model.

The elliptical DoG model can be written as follows:

$$f\left(x,y\right)=\frac{A_c}{\sqrt{2\mathrm{\pi }{\mathrm{\sigma }}_c^2}}exp\left(\frac{-\left(x^{\prime 2}+\mathrm{\tau }{y}^{\prime 2}\right)}{{\mathrm{\sigma }}_c^2}\right)-\frac{A_s}{\sqrt{2\mathrm{\pi }{\mathrm{\sigma }}_s^2}}exp\left(\frac{-\left(x^{\prime 2}+\mathrm{\tau }{y}^{\prime 2}\right)}{{\mathrm{\sigma }}_s^2}\right),$$ (10)

$$x^{\prime }=\left(x-x_c\right)cos\left(\mathrm{\theta }\right)+\left(y-y_c\right)sin\left(\mathrm{\theta }\right),$$ (11)

$$y^{\prime }=-\left(x-x_c\right)sin\left(\mathrm{\theta }\right)+\left(y-y_c\right)cos\left(\mathrm{\theta }\right).$$ (12)

The elliptical DoG model is similar to the DoG model but has two additional parameters that allow for elongation (τ) and rotation (θ).

Nonconcentric DoG model.

The nonconcentric DoG model is identical to the DoG model but has two additional parameters (x_s, y_s) that allow the circularly-symmetric surround to be offset from the circularly-symmetric center (50):

$$f\left(x,y\right)=\frac{A_c}{\sqrt{2\mathrm{\pi }{\mathrm{\sigma }}_c^2}}exp\left(\frac{-\left(\right(x-x_c{)}^2+(y-y_c{)}^2)}{{\sigma }_c^2}\right)-\frac{A_s}{\sqrt{2\mathrm{\pi }{\mathrm{\sigma }}_s^2}}exp\left(\frac{-\left(\right(x-x_s{)}^2+(y-y_s{)}^2)}{{\mathrm{\sigma }}_s^2}\right).$$ (13)

Note that the non-concentric DoG model differs from the elliptical DoG model in two ways. First, the fitted RFs are constrained to be circularly symmetric in the nonconcentric DoG model and not the elliptical DoG model. Second, the center and surround are constrained to be concentric in the elliptical DoG and not the nonconcentric DoG model.

Evaluating Goodness of Model Fit: R

We evaluated the quality of model fits by calculating Pearson’s correlation coefficient (R) between the data and the model predictions. To foster a fair comparison between model fits, we used fivefold cross-validation, fitting the model with 80% of the data, and testing it on the remaining 20%. This approach tests how well a model is able to predict the portion of the data that was not used for fitting the model. This procedure avoids overfitting, but it augments the natural bias of R toward zero because the training set is smaller than the actual data set. We report the averaged R across the five folds.

Evaluating Goodness of Model Fit: Fraction of Variance Unexplained

We evaluated the fraction of the variance unexplained by model fits. The fraction of unexplained variance was defined as the ratio of the residual sum of squared errors and the total sum of errors. For this calculation, we fit the models to the entire data set for each neuron.

Evaluating Goodness of Model Fit: Bayesian Information Criterion

Model fits were further quantified using the Bayesian information criterion (BIC). Assuming that the model errors are independent and identically distributed according to a normal distribution, the BIC can be written as follows:

$$\mathrm{B}\mathrm{I}\mathrm{C}=n\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{R}\mathrm{S}\mathrm{S}}{n}+k\mathrm{l}\mathrm{o}\mathrm{g}\left(n\right),$$ (14)

where n is the number of data points (n = 100), RSS is the residual sum of squared errors, and k is the number of model parameters.

Evaluating Goodness of Model Fit: Sum of Squared Errors

Model fits were further compared by calculating the sum of squared errors between the data and the fitted model. We used fivefold cross-validation and reported the averaged sum of squared errors across the five folds.

Evaluating Goodness of Model Fit: Prediction of Spike-Triggering Stimuli

Model fits were further compared by calculating the ability of the models to predict spiking responses to white noise stimuli. Using fivefold cross-validation, the model was fit to 80% of the data and tested on the remaining 20%. We assessed the ability of the fitted model to classify movie segments that evoked a spike versus those that did not by projecting stimulus frames onto the 10 × 10 stixels (space) × 15 frames (time) spatial-temporal RF. The spatial-temporal RF was derived by combining the fitted spatial RF with the empirical time course of the STA. To compute classification performance, we constructed a receiver operating characteristic (ROC) from the spike and nonspike distributions of projections (51). We report the averaged area under the ROC across the five folds.

Spatial Opponency Index

We defined a spatial opponency index (SOI) that quantifies the degree of antagonism across the RF as follows:

$$\mathrm{S}\mathrm{O}\mathrm{I}=1-|\left|\frac{P-N}{P+N}|\right|,$$ (15)

P was defined as the sum of positive values in the spatial weighting function. N was defined as the sum of negative values. If the sum of positive and negative values were matched, then P and N would be equal, and SOI would be 1. On the contrary, if the RF consisted of a single subregion, then either P or N would equal 0, and so would SOI.

RESULTS

We analyzed the responses of 205 V1 neurons from four macaque monkeys that met our inclusion criteria (see methods). RFs of neurons ranged in eccentricity from 1.7° to 8.4° (median = 4.7°).

Cone Weights

We classified neurons that met our inclusion criteria as simple cells or DO cells on the basis of spatial opponency and cone weights (Fig. 2). Simple cells had large magnitude, L- and M-cone weights of the same sign that, together, accounted for 80% of the total cone weight (n = 73). Neurons that were cone opponent and spatially opponent were classified as DO cells. DO cells were further classified as LM-opponent (n = 86) or S-cone sensitive (n = 46) based on cone weight magnitudes and signs. Of the 46 DO_{S-cone sensitive} neurons recorded, 16 were S-(L + M), 26 were (S + M)-L, and 4 were (S + L)-M. The overrepresentation of (S + M)-L neurons relative to (S + L)-M neurons in V1 has been noted previously (6, 45, 52).

Figure 2.

Normalized cone weights of simple (black), DO_LM-opponent (red), DO_{S-cone sensitive} (blue), and unclassified (gray) cells. The 49 “unclassified” cells survived all of the inclusion criteria except for the requirement of clear cone opponency or nonopponency. M-cone weights were constrained to be positive. Points closer to the origin have larger S-cone weights than those far from the origin. DO, double opponent; DO_LM-opponent, cells with large, opponent L- and M-cone weights; DO_{S-cone sensitive}, cells that were cone opponent and had an S-cone weight that accounted for at least 20% of the total.

Model Comparison: Gabor versus DoG

STAs of six example neurons illustrate patterns that we observed in the data (Fig. 3, first row). Statistical tests performed on individual phosphor channels, which are independent (Fig. 3, second to fourth rows), reveal the color- and spatial-opponency of the four DO cells (Fig. 3, C–F). Sensitivity to the three phosphors was converted to cone weights (Fig. 3, fifth row). Simple cell RFs consisted of adjacent ON and OFF regions (Fig. 3, A and B). Most simple cell RFs were elongated and clearly oriented (Fig. 3A), but others were less so (Fig. 3B). RFs of DO cells displayed similar features: some were clearly oriented (Fig. 3, C and E), whereas others had nearly circular RF centers and diffuse surrounds (Fig. 3, D and F).

Figure 3.

Gabor and difference of Gaussians (DoG) model fits to spatial weighting functions of six example cells. Each spike-triggered average (top row) has been decomposed into red, green, and blue channel components. Significant stixels (Z test, P < 0.05) have been colored on the basis of their sign (red = positive, blue = negative). The quality of each model fit was quantified using cross-validated R. A: a simple cell with R_Gabor = 0.77 and R_DoG = 0.45. B: a simple cell with R_Gabor = 0.67 and R_DoG = 0.59. C: a DO_LM-opponent with R_Gabor = 0.45 and R_DoG = 0.30. D: a DO_LM-opponent cell with R_Gabor = 0.90 and R_DoG = 0.91. E: a DO_{S-cone sensitive} cell with R_Gabor = 0.68 and R_DoG = 0.44. F: a DO_{S-cone sensitive} cell with R_Gabor = 0.78 and R_DoG = 0.75 ; DO_LM-opponent, cells with large, opponent L- and M-cone weights; DO_{S-cone sensitive}, cells that were cone opponent and had an S-cone weight that accounted for at least 20% of the total; R_Gabor and R_DoG, cross-validated Pearson’s correlation coefficient, R, between the data and the prediction from the Gabor model and the difference of Gaussians model, respectively; STA, spike-triggered average.

To compare the spatial RF organization of simple and DO cells quantitatively, we converted the STAs to grayscale spatial weighting functions (see methods; Fig. 3, sixth row) and fit them with a Gabor model (Fig. 3, seventh row) and a DoG model (Fig. 3, eighth row). Goodness of fit was quantified with cross-validated R between the data and the model predictions, a measure that allows fair comparison between models with different numbers of parameters (the Gabor model has 8 parameters; the DoG model has 6 parameters).

The Gabor model outperformed the DoG model for most of the cells tested (139/204, R_Gabor > R_DoG). The superiority of the Gabor model was evident for both simple cells (P < 0.001; Wilcoxon signed-rank test; Fig. 4A) and DO cells (P = 0.001; Wilcoxon signed-rank test). This trend was maintained within DO subtypes: DO_LM-opponent (P = 0.07; Fig. 4B) and DO_{S-cone sensitive} (P = 0.006; Fig. 4C). These results show that DO cells, like simple cells, have RFs that are more accurately described by Gabor functions than DoG functions. However, the spatial RFs of simple and DO cells were not identical. The difference between R_Gabor and R_DoG was larger for simple cells than for DO_LM-opponent or DO_{S-cone sensitive} cells (P < 0.001 for each comparison; Mann–Whitney U tests). The difference between R_Gabor and R_DoG was similar for DO_LM-opponent and DO_{S-cone sensitive} cells (P = 0.25; Mann–Whitney U test). Analysis of fraction of variance unexplained by the model fits produced a similar result (see Evaluating Goodness of Model Fit: Fraction of Variance Unexplained, Gabor model = 10.25% ± 1.24%, DoG model = 15.63% ± 0.97%).

Figure 4.

Comparison of Gabor and DoG model fits. Cross-validated R is plotted from Gabor fits and from DoG fits for simple (A), DO_LM-opponent (B), and DO_{S-cone sensitive} cells (C). Five example STAs are shown in each panel to illustrate the diversity of RF structures observed and their relationship to R. DO, double opponent; DoG, difference of Gaussians; DO_LM-opponent, cells with large, opponent L- and M-cone weights; DO_{S-cone sensitive}, cells that were cone opponent and had an S-cone weight that accounted for at least 20% of the total; RF, receptive field; STA, spike-triggered average

We considered the possibility that systematic differences in SNR between DO cell STAs and simple cell STAs affected the model fits. For example, a neuron that was driven strongly by the stimulus might have a clearer STA than one that was driven weakly, which could lead to one model fitting better than the other. We, therefore, investigated the relationship between SNR of the peak STA frame and R_Gabor, the goodness of fit of the Gabor model (see methods for the definition of SNR). As SNR increased, so did R_Gabor, which was similar across the three cell types (P = 0.14, Kruskal–Wallis test; Fig. 5A). This result shows that much of the error in the model fits is due to noise in the STAs, not to systematic errors in the Gabor model fits.

Figure 5.

Analyses of Gabor and DoG model fits. A: scatterplot of cross-validated R of Gabor fits vs. signal-to-noise ratio (SNR) of peak STA frames for simple cells (black), DO_LM-opponent cells (red), and DO_{S-cone sensitive} cells (blue). B: identical to A but plotted for DoG fits. DO, double opponent; DoG, difference of Gaussians; DO_LM-opponent, cells with large, opponent L- and M-cone weights; DO_{S-cone sensitive}, cells that were cone opponent and had an S-cone weight that accounted for at least 20% of the total; STA, spike-triggered average.

A different result was obtained when SNR was compared to R_DoG, the goodness of fit of the DoG model. R_DoG was lower for simple cells than for DO cells (Fig. 5B, median for simple cells, 0.38 vs. median for DO_LM-opponent, 0.44 vs. median for DO_{S-cone sensitive} cells, 0.44, P = 0.07; Kruskal–Wallis test). This difference was clearest for cells with high SNR (P < 0.0001, Kruskal–Wallis test on R_DoG values for cells with SNRs above the median). A linear regression also confirmed that the relationship between (Fisher’s Z-transformed) R_DoG and log₁₀(SNR) differed across cell types (F test, P < 0.0001).

To dissect the differences between simple cell and DO cell RFs more finely, we asked whether simple cell RFs are more frequently odd symmetric or more elongated than those of DO cells. Either of these properties could degrade the quality of the DoG model fits relative to Gabor fits because DoG fits are constrained to be radially symmetric. First, we analyzed the spatial phase of the best-fitting Gabor function, which makes the RF odd symmetric, even symmetric, or intermediate (ϕ, see methods). Most simple cell RFs had ϕ > 45°, indicating a bias for odd symmetry (Fig. 6A; mean = 57.2°) as did DO_LM-opponent (Fig. 6B; mean = 52.0°) and DO_{S-cone sensitive} cells (Fig. 6C; mean = 52.0°). Preferred spatial phases did not differ significantly across the three cell types (P = 0.53, Kruskal–Wallis test). The results did not change when we restricted our analysis to cells that were better fit by the Gabor model than the DoG model (P = 0.80, Kruskal–Wallis test).

Figure 6.

Analyses of Gabor model parameters for all cells (white), cells that are better fit by the Gabor model than the DoG model (black), and cells that are better fit by the DoG model than the Gabor model (green). A: best fitting phase (ϕ) of Gabor fits to simple cell spatial weighting functions. B and C: identical to A but for DO_LM-opponent cells and DO_{S-cone sensitive} cells, respectively. D: best fitting aspect ratio (γ) of Gabor fits to simple cell spatial weighting functions. The median is plotted for all simple cell RFs (open triangle), for those that were better fit by the Gabor model (closed black triangle), and also for those that were better fit by the DoG model (green triangle). E and F: identical to D but for DO_LM-opponent cells and DO_{S-cone sensitive} cells, respectively. DO, double opponent; DoG, difference of Gaussians; DO_LM-opponent, cells with large, opponent L- and M-cone weights; DO_{S-cone sensitive}, cells that were cone opponent and had an S-cone weight that accounted for at least 20% of the total; RF, receptive field.

Second, we analyzed the aspect ratio, which determines how elongated an RF is (γ, see methods). Aspect ratios were broadly distributed and differed across cell types (P = 0.05, Kruskal–Wallis test). Simple cells (Fig. 6D; median = 1.33) had aspect ratios that were similar to those of DO_LM-opponent cells (Fig. 6E; median = 1.09; P = 0.15; Mann–Whitney U test) but larger than those of DO_{S-cone sensitive} cells (Fig. 6F; median = 0.93; P = 0.01; Mann–Whitney U test). This difference in aspect ratios reflects a tendency of simple cells to have a slightly more elongated RF than DO cells. Restricting analysis to cells that were better fit by the Gabor model (R_Gabor > R_DoG) agreed qualitatively with the above results (Fig. 6, A–F, black histograms). These results indicate that subtle differences exist between simple and DO RFs.

Many RFs in our data set were not circularly symmetric, which contributed to the superiority of the Gabor fits over the DoG fits. However, the DoG model can be extended to fit elongated RFs. We relaxed the constraint of circular symmetry by adding two parameters to the DoG model (see Eqs. 10–12). This elliptical DoG model captured the elongated RFs of simple and DO cells better than the circularly symmetric DoG model did (R_{EllipticalDoG} > R_DoG, P < 0.001; Wilcoxon signed-rank test; Supplemental Fig. S2). However, the improvement of fits afforded by the elliptical DoG model still fell short of the Gabor fits (R_Gabor > R_{EllipticalDoG}, P < 0.001 for simple cells, P = 0.05 for DO cells; Wilcoxon signed-rank tests). We conclude that even this generalized, elliptical DoG model fails to describe simple and DO RFs as well as the Gabor model does.

Nonconcentric DoG Model

Gabor and DoG models are classic descriptions of DO RFs, but a crescent-shaped surround has also been proposed (6, 16). Some DO RFs in our data set appeared to have this geometry by eye. To analyze DO RFs more incisively, we formalized the center crescent-shaped surround description by modifying the DoG model to allow the center and surround Gaussians to be nonconcentric (50). This model captured many of the diverse RF structures in our data set (Fig. 7A).

Figure 7.

Comparison of nonconcentric DoG and Gabor model fits. A: nonconcentric DoG fits to data from the six example cells in Fig. 3. R_{nonconcentricDoG} = 0.54, 0.66, 0.41, 0.90, 0.60, and 0.78 from left to right. R from Gabor fits are plotted against R from nonconcentric DoG fits for simple (B), DO_LM-opponent cells (C), and DO_{S-cone sensitive} cells (D). DO, double opponent; DoG, difference of Gaussians; DO_LM-opponent, cells with large, opponent L- and M-cone weights; DO_{S-cone sensitive}, cells that were cone opponent and had an S-cone weight that accounted for at least 20% of the total; RF, receptive field; R_{nonconcentricDoG}, cross-validated Pearson’s correlation coefficient, R, between the data and the prediction from the nonconcentric difference of Gaussians model.

The nonconcentric DoG model fit simple and DO RFs better than the DoG model (P < 0.05; Wilcoxon signed-rank tests; Supplemental Fig. S3. However, it did not fit simple cell RFs as well as the Gabor model (P < 0.001; Wilcoxon signed-rank test; Fig. 7B) and fit DO_LM-opponent and DO_{S-cone sensitive} RFs similarly to the Gabor model (P > 0.5; Wilcoxon signed-rank tests; Fig. 7, C and D). The nonconcentric DoG model is thus a reasonable description of DO cell RFs, but a Gabor model is superior for simple cell RFs. These quantitive results support previous qualitative descriptions of DO RF structure (6, 16).

Analysis of Spatial Opponency

The antagonistic subfields of simple cell RFs in our data set were more nearly balanced than those of the DO cells. Spatial opponency indices (SOIs) were greater for simple cells (Fig. 8A; median = 0.91) than for DO_LM-opponent cells (Fig. 8B; median = 0.72) or DO_{S-cone sensitive} cells (Fig. 8C; median = 0.78) (P < 0.001, Kruskal–Wallis test). This result shows that simple cells have greater spatial antagonism than DO cells.

Figure 8.

Analysis of spatial opponency. A: histogram of spatial opponency indices (SOIs) for simple cells. The median SOI is plotted for all simple cell RFs (open triangle) and also of those that were better fit by the Gabor model (filled triangle). B and C: identical to A but for DO_LM-opponent cells and DO_{S-cone sensitive} cells, respectively. D: difference in R between Gabor and DoG fits is plotted against the SOI for simple (black), DO_LM-opponent (red) and DO_{S-cone sensitive} (blue) cells. DO, double opponent; DoG, difference of Gaussians DO_LM-opponent, cells with large, opponent L- and M-cone weights; DO_{S-cone sensitive}, cells that were cone opponent and had an S-cone weight that accounted for at least 20% of the total; RF, receptive field.

As the SOI increased, so did the difference in goodness of fit of the Gabor model and the DoG model (r = 0.39, P < 0.001, Spearman’s correlation between R_Gabor-R_DoG and SOI; Fig. 8D). This trend was not due to an increase in SNR (r = −0.08, P = 0.25, Spearman’s correlation between SNR and SOI). The difference between R_Gabor and R_DoG was larger for simple cells than for DO cells even when analysis was restricted to the subset of cells with strong spatial opponency (P < 0.001, Kruskal–Wallis test on R_Gabor-R_DoG values for cells with SOIs above the median). These results suggest that the superiority of the Gabor fits to simple cell RFs is not simply a consequence of their greater spatial opponency relative to DO cells.

Effect of Eye Movements

Eye movements cannot depend on which type of cell is recorded, but they could potentially favor one model over the other. To investigate whether this was the case in our data, we computed the median eye displacement from the average fixating eye position for each neuronal recording and checked whether eye movements biased the model comparison results. The difference between R_Gabor and R_DoG was not significantly correlated with the magnitude of the eye displacement for any of the cell types (r = −0.08, P = 0.49, simple cells; r = −0.03, P = 0.76, DO_LM-opponent cells; r = 0.12, P = 0.41, DO_{S-cone sensitive} cells; Spearman’s correlation between R_Gabor-R_DoG and median eye displacement; Fig. 9A). Similarly, the difference between R_Gabor and R_{nonconcentricDoG} was not significantly correlated with the magnitude of the eye displacement for any of the cell types (P > 0.45, Spearman’s correlation between R_Gabor-R_{nonconcentricDoG} and median eye displacement).

Figure 9.

Effect of eye movements. A: difference in R between Gabor and DoG fits is plotted against the median eye displacement for simple (black), DO_LM-opponent (red), and DO_{S-cone sensitive} (blue) cells. B: spatial opponency index is plotted against the median eye displacement for simple (black), DO_LM-opponent (red), and DO_{S-cone sensitive} (blue) cells. DO, double opponent; DoG, difference of Gaussians; DO_LM-opponent, cells with large, opponent L- and M-cone weights; DO_{S-cone sensitive}, cells that were cone opponent and had an S-cone weight that accounted for at least 20% of the total.

We also asked whether eye movements affected the spatial opponency we measured. We did not find any significant relationship between the SOI and the magnitude of eye movement (r = −0.06, P = 0.59, simple cells; r = −0.02, P = 0.80, DO_LM-opponent cells; r = −0.12, P = 0.43, DO_{S-cone sensitive} cells; Spearman’s correlation between SOI and median eye displacement; Fig. 9B). We conclude that our results on the spatial RFs of DO and simple cells are robust to eye movements.

DISCUSSION

We measured the spatial RFs of macaque V1 DO and simple cells under identical conditions and compared them with rigorous statistical techniques. We report three new results. First, DO RFs, like simple cell RFs, were more accurately described by a Gabor function than a DoG function. Second, DO cells tend to have odd-symmetric RFs, similarly to simple cells. Third, DO RFs are more weakly spatially opponent than simple cell RFs. In summary, our results show that most DO cells lack a center-surround RF organization, the spatial RFs of DO and simple cells are broadly similar, and a center-crescent surround spatial structure describes DO cell RFs nearly as accurately as a Gabor function.

Below, we compare our results to those of previous studies. We speculate on the neural wiring underlying simple and DO cells and the relationship between the neurons we studied and functional organization for color in V1. We then discuss the robustness of our results to the statistics used to compare model fits. Finally, we discuss the potential roles of DO cells in image processing and how our findings have constrained these roles.

Comparison with Previous Studies

Different studies have reached different conclusions about the spatial RF structure of DO cells in monkey V1 (6–9, 12, 14, 16–18). Early investigations, mostly using circular spots of light, assumed DO cells to have a concentric center-surround RF organization (7, 9, 12, 14). In some of these experiments, extremely large (>10° diameter) stimuli were tested and these produced no response from DO cells. Such stimuli presumably recruit suppressive mechanisms from beyond the classical RF, which reconciles the lack of response to large, uniform stimuli with our observation that many DO RFs had imbalanced subfields. Later investigations using sparse noise stimuli measured 2-D RF structure and found that DO cell RFs have circular centers and crescent-shaped surrounds (6, 16).

Sparse noise stimuli have the advantage of stimulating different parts of the RF independently and thus make no assumptions about the spatial structure of the RF (6, 16). However, if every frame in a sparse noise stimulus consists of a pair of spots that are equal and opposite in contrast, then STAs, which are sums of these frames, will necessarily consist of equal parts contrast-increment and contrast-decrement. If care is not taken to relate the responses to an accurate baseline (16), this may introduce the appearance of spatial opponency where none exists (54). The fact that our stimulus was not constrained to have a mean of zero across space likely provided a clearer picture of DO RF organization than has previously been available.

Some other studies used drifting and rapidly flashed sinusoidal gratings to characterize the spatial RFs of DO cells (8, 17, 18). DO cells were identified based on their bandpass spatial frequency tuning to equiluminant color gratings and found to be orientation tuned. Orientation tuning is compatible with a Gabor-like RF but does not make a strong case for it as revealed by direct 2-D RF mapping of DO cells (6).

These studies also included complex cells in the population of DO cells (8, 17, 18). Complex cells do not abide by the classical definition of double-opponency because they do not have opposite color preferences in different parts of their RFs (16, 20). Some V1 neurons encode spatial phase and others do not. This distinction has proven to be useful in the achromatic domain (e.g., for understanding the signal transformation between simple and complex cells) (56, 57). Such a distinction is probably useful in the chromatic domain as well.

The cone weights we measured from DO V1 neurons, agree qualitatively with those measured by Conway and Livingstone (16) and Johnson et al. (17) [compare Fig. 3 to Fig. 7B of Conway and Livingstone (16) and Fig. 4 of Johnson et al. (17)]. S-cone weights measured by Johnson et al. (17) were generally smaller than those measured by Conway and Livingstone (16) or by us, presumably due to the relatively low S-cone contrast Johnson et al. (17) used. The consistency of cone weights measured by us and by Conway and Livingstone (16) is striking given substantial differences in the stimuli and data analytical techniques. A feature that distinguishes our study from that of Conway and Livingstone (16) is the inclusion of cone-opponent and cone-nonopponent neurons, which facilitated a fair comparison between DO and simple cell RFs.

We found that simple cell RFs are usually odd-symmetric, consistent with previous studies (41, 42). A novel contribution of the current study is the extension of this result to DO cells. The existence of odd-symmetric, chromatic edge detectors in the primate visual system was predicted on the basis of psychophysical experiments (58, 59).

Link between DO Cells and Functional Organization in V1

The recording techniques we used do not allow us to determine locations of DO cells in V1 relative to each other or to the cytochrome oxidase (CO) blobs. Our results should therefore be interpreted as describing the properties of the average, randomly sampled V1 cell, without regard to anatomical subcompartment. These subcompartments are important; classic electrophysiological and recent optical imaging results show unequivocally that the spectral and spatial sensitivity of V1 neurons is tightly linked to their location relative to these compartments, and they provide some guidance regarding the locations of DO cells, at least in the superficial layers of V1. Neurons in the CO blobs tend to be color opponent and poorly tuned for orientation, whereas many neurons adjacent to CO blobs are color opponent and orientation tuned (60). One possibility is that DO cells are preferentially enriched near, but not at, the centers of CO blobs. Single-opponent neurons nearer to blob centers may represent a distinct color pathway that has not yet been characterized with the rigor applied to DO cells.

Are DO Cells Cone-Opponent Simple Cells?

DO and simple cell RFs differ in detail but are similar in many ways. This similarity motivates the hypothesis that the primary difference between them is the sign of input they receive (indirectly) from the three cone photoreceptor classes. Indeed, the models proposed to underlie simple cell RFs can also be applied to some DO cells with only a minor change in the wiring (Fig. 10).

Figure 10.

Schematic diagram of the circuitry proposed to underlie simple cell and DO cell RFs. A: a simple cell RF constructed from parvocellular lateral geniculate nucleus (LGN) afferents. The ON subregion (L + M) is excited by L-ON and M-ON LGN cells and is inhibited by L-OFF and M-OFF LGN cells. Similarly, the OFF subregion (-L-M) is excited by L-OFF and M-OFF LGN cells and is inhibited by L-ON and M-ON LGN cells. B: construction of a DO cell RF using the same set of parvocellular LGN afferents that provide input to a simple cell. The L-M subregion is excited by L-ON and M-OFF and is inhibited by L-OFF and M-ON, whereas the M-L subregion is excited by L-OFF and M-ON and is inhibited by L-ON and M-OFF. DO, double opponent; DoG, difference of Gaussians; RF, receptive field.

A hallmark of simple cells is spatial linearity, a property mediated in part by push-pull excitation and inhibition (61–64). Some DO cells exhibit push-pull responses, consistent with the proposed similarly between them and simple cells (16). However, whether the departures from linearity observed in DO cells exceeds expectations provided by the benchmark of simple cells is unclear. To answer this question, a useful next step is to compare quantitatively the degree of spatial linearity between DO and simple cells.

Accuracy of RF Structure and Size

We mapped V1 RFs using a reverse correlation technique in awake macaque monkeys, similar to numerous previous studies (6, 16, 44–46). Eye movements made by well-trained monkeys during fixation blur measured RF maps, but this blurring would be expected to reduce or eliminate spatial structure, not to create it where is does not exist. All of the neurons we studied had spatially structured STAs. We analyzed the effects of eye movement on model fits and found no evidence that eye movements favored one model over another (Fig. 9). The distribution of eye positions was weakly anisotropic, showing more variance along the vertical than the horizontal axis, but the distribution of STA orientations did not exhibit a similar anisotropy. In addition, these STA orientation preferences matched closely those measured directly with drifting sinusoidal gratings (data not shown), consistent with a previous study from our group (45) and inconsistent with the idea that eye movements produced artifactual, oriented STAs.

The RFs we measured were larger on average than those reported in anesthetized macaques at matched eccentricities (Supplemental Fig. S4) (65, 66). Eye movements surely contribute to this discrepancy as does the low contrast of the white noise stimulus and the large stixels in the white noise stimulus (67, 68). Neurons with very small, spatially opponent RFs would be unlikely to respond to the stimulus and therefore would not have been studied.

Effects of Cell Categorization Criteria

We distinguished simple cells from DO cells on the basis of cone weights. We applied a more lenient criterion to L- and M-cone weights to categorize a cell as simple than as DO_LM-opponent—a fact that is visible from the greater spread of L- and M-cone weights for simple cells than DO_LM-opponent cells (Fig. 2). The rationale for this decision is the greater variability in estimated cone weights for nonopponent cells (45). Nevertheless, our results are robust to this decision as other cone weight thresholds did not change the central conclusions (Supplemental Figs. S5–S8).

Nonlinear neurons were excluded from the pool of double-opponent cells because analysis was limited to STAs, and STAs alone do not describe the stimulus tuning of nonlinear neurons. In principle, a nonlinear neuron that fails to conform to the classical definition of double opponency (opposite responses for opponent colors in different parts of the RF) could have an STA that appears DO. Restricting our analysis to linear neurons allowed us to focus on neurons that conformed to the classical definition of double opponency and facilitated a meaningful comparison with simple cells, which are the most linear neurons in V1 by definition (69).

Alternative Metrics for Model Comparison

We compared models using cross-validated correlation between held-out data and model fits, but our results are robust to this choice. We repeated the model comparisons using the Bayesian information criterion, cross-validated sum of squared errors and cross-validated prediction of spike-triggering stimuli. The results from all of these analyses agreed; RFs of DO and simple cells were more accurately described by a Gabor model than a DoG model (Supplemental Fig. S9), and the nonconcentric DoG model provided a reasonable description of DO cell RFs but not simple cell RFs (Supplemental Fig. S10). We conclude that our conclusions are robust to the metric used to compare model fits.

Role of DO Cells in Image Processing

Our results show that some DO cells carry information about the phase and orientation of local chromatic variations. This information is useful for at least two visual computations. The first is shape-from-shading. Extraction of chromatic orientation flows in 2-D images is critical for accurate perception of 3-D shapes (30, 31, 70). In some displays, alignment of chromatic and luminance edges suppresses the percept of 3-D form whereas misalignment enhances the 3-D percept (30). We speculate that signals from DO cells are integrated with those from simple cells to infer 3-D structure from 2-D retinal images. The similarity of RF structure between simple cells and DO cells may facilitate downstream integration of their responses. Second, DO cells might aid in inferring whether an edge in a visual scene is caused by the same material under different lighting conditions or by two different materials under the same lighting condition. An edge produced by a shadow falling across one-half of a uniform material is a nearly pure intensity difference. On the contrary, an edge between two different materials under the same illumination produces spatially coincident intensity and spectral differences. A comparison of simple cell and DO cell responses could help to disambiguate material edges from illumination edges (71–74).

GRANTS

A. De was supported by a training grant from the National Institutes of Health (R90 DA033461). Additional support for the research came from National Institutes of Health/National Eye Institutes (R01 EY018849) and Office of Research Infrastructure Programs (P51 OD010425).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

A.D. and G.D.H. conceived and designed research; A.D. and G.D.H. performed experiments; A.D. analyzed data; A.D. and G.D.H. interpreted results of experiments; A.D. prepared figures; A.D. drafted manuscript; A.D. and G.D.H. edited and revised manuscript; A.D. and G.D.H. approved final version of manuscript.