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Research Article  |   April 2002
Multiplied functions unify shapes of ganglion-cell receptive fields in retina of turtle
Author Affiliations
  • James R. Dearworth, Jr.
    Biological Sciences, University of Delaware, Newark, DE, USA
  • A. M. Granda
    Biological Sciences, University of Delaware, Newark, DE, USA
Journal of Vision April 2002, Vol.2, 1. doi:10.1167/2.3.1
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      James R. Dearworth, A. M. Granda; Multiplied functions unify shapes of ganglion-cell receptive fields in retina of turtle. Journal of Vision 2002;2(3):1. doi: 10.1167/2.3.1.

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Abstract

Retinal ganglion cells in the turtle were extracellularly recorded to define the shapes of their receptive fields by small moving light spots. To better define the geometries, spectral-light adaptations and vitreal injections of 2-amino-4-phosphonobutyric acid (APB) were used to disrupt balances in field organization along dimensions of wavelength, ON and OFF responses, and center/surround areas. Three-dimensional data plots were fit by Gaussian, Gabor, and cardioid functions to show that the shapes of receptive fields are predicted by combinations of these multiplied functions. Results indicate that Gaussian functions describe simple symmetrical receptive fields that are center-only; Gabor functions describe center/surround color-opponent receptive fields that have a ring of spike activity in the periphery; and directionally selective receptive fields, in contrast, which are asymmetrical, are described by cardioid functions adjoined to Gaussian or Gabor functions. The advantage of linking multiplied functions is that receptive fields are unified by a model that predicts progressively more complex field geometries derived from particular stimulating conditions.

Introduction
Spatial organization of center and surround in ganglion-cell receptive fields has been generalized as the most simple of visual receptive fields, one that can be modeled by difference of Gaussian (DOG) functions (Rodieck & Stone, 1965; Enroth-Cugell & Robson, 1966). 
In the retina of the turtle, Pseudemys scripta elegans, the shapes of ganglion-cell receptive fields can be complex, with the ability to process motion and color (Lipetz & Hill, 1970; Marchiafava, 1979; Bowling, 1980; Granda & Fulbrook, 1989; Ammermüller & Kolb, 1995; Ammermüller, Muller, & Kolb, 1995; Haverkamp, Eldred, Ottersen, Pow, & Ammermüller, 1997; Granda, Dearworth, & Subramaniam, 1999). Using adaptations to spectral light backgrounds, as well as vitreal injections of 2-amino-4-phosphonobutyric acid (APB) (Slaughter & Miller, 1981), we showed that field shapes are determined by balanced interactions along dimensions of wavelength, ON and OFF responses, and center/surround areas (Granda et al., 1999; Dearworth, 1999). The results support the idea of coextensive, center/surround receptive fields that interact with wavelength and ON/OFF sensitivities to organize the shapes of receptive fields. These dimensions are in equilibrium that can be influenced by stimulus-light sequence, and by application of APB. Weakening of one organizer of the field shape releases its counterpart to redraw the shape of the receptive field. 
Field shapes that are generated in turtle can be symmetrical, but when sensitivities also include preferences for particular directions of movement, asymmetries also result. Here we show that the array of shapes found in the turtle retina, which progresses from those that are symmetrical center-only to more complicated asymmetrical directional fields, is predictable by a serial linkage of mathematical functions. 
Methods
Details of extracellular recording, stimulation, and drug infusion were previously described in Granda et al. (1999); here they are briefly presented. 
Receptive fields of ganglion cells from intact turtles, Pseudemys scripta elegans, were defined by small light spots. Both light spots and background lights were projected onto a 1-m diameter hemispherical screen. Single-unit responses recorded in dark adaptation were compared to responses obtained after adaptations to spectral lights, and after infusions of APB. 
Stationary flashed light spots were used first to determine receptive field regions as ON, OFF, or ON/OFF. Cells were classified by their responses at receptive-field centers during dark adaptation. The shapes of the receptive fields were then defined by small light spots moving at 10 deg·s−1. The diameter of the light spot was 140 μm when projected onto the retinal surface. Wavelengths of light spots that were superimposed on dark and spectral backgrounds were 640 nm and 540 nm. The intensities of both light spots were 3.35 log photons·µm−2·s−1. Lighted backgrounds were either 650 nm at an intensity of 3.43 log photons·µm−2·s−1 or 550 nm at an intensity of 3.24 log photons·µm−2·s−1
Classification of Cells by Stationary, Flashed Light Spots
Light spots were flashed on for 4 s and off for 5 s at field centers to classify each cell as ON, OFF, or ON/OFF. The mean values of spikes, in response to 10 repeated presentations of flashed light spots, were plotted as time histograms and summed into bins of 25 ms. The standard error of the mean (SEM) was calculated for each bin. Cell types were quantitatively characterized by mean response modulation to compare responses of cells to spectral lights (cf., Granda et al., 1999). 
Three-Dimensional Data Plots Mapped by Moving Light Spots
Light spots were moved at a fixed speed of 10 deg·s−1 to map the shapes of receptive fields. A computer-controlled system correlated the spike’s field position with the location of the moving light spot. 
Receptive fields were sampled by raster scans, orthogonally (bi-directional horizontal and vertical scans) and angularly (bi-directional scans through receptive field centers). The latency of response occurring in each direction of scan was corrected, typically 50 ms. As a result of the correction to latency, responses in the forward direction of each scan were shifted 1 degree of visual angle to coincide with the peak responses in the reverse direction (100 ms is equal to 1 degree because the speed of the light spot was 10 deg·s−1). Overlap by scans at the center of angular raster scans (which generated over-sampling at the field center) was also considered and corrected. Mean values of spikes in response to raster scans were calculated for each bin of 1 degree, and assembled into three-dimensional bar graphs. 
For cells that were directionally selective, special care was taken to choose the axes of the orthogonal raster scans. Angular raster scans (bi-directional scans through receptive-field centers) were used to generate vector plots that identified each cell’s preferred angle of movement. The axes of orthogonal raster scans (bi-directionally horizontal and vertical scans) were aligned so that the scans were either normal or orthogonal to the cell’s preferred axis of movement. 
Fit of Functions
Mathematical functions were fitted to the data by the method-of-least-squares. Goodness-of-fit, the difference between observed and calculated responses, was by examination of the residuals (Fotheringham & Knudson, 1987). Goodness-of-fit was expressed as the profile-fitting error (PFE), the ratio between the sum of the squared deviations and the sum of the squared observed responses (Sun & Bonds, 1994). A perfect fit, PFE equal to 0%, occurred if predicted responses were equal to observed responses. 
Results
Shapes of ganglion-cell receptive fields (N = 74) were defined by light spots moving at 10 deg·s−1. Intervention by APB and spectral-light adaptations changed the shapes of fields when compared to the shapes determined under dark adaptation. Field shapes were described by combinations of mathematical functions: 26% of the cell population was described by Gaussians (19 cells, ON or OFF); 42% by Gabors (31 cells, ON/OFF); 9% by Gaussian-cardioids (7 cells, ON or OFF); and 23% by Gabor-cardioids (17 cells, ON/OFF). Table 1 shows examples with fitted parameters. 
Table 1
 
Parameters for combinations of Gaussian, Gabor, and cardioid functions fit to three-dimensional field plots
Table 1
 
Parameters for combinations of Gaussian, Gabor, and cardioid functions fit to three-dimensional field plots
Parameter values
Response typea A xc, yc σ F ϕ p k PFE
Functions (spikes) (°,°) (°) (cycles·deg−1) (radians) (°) (%)
+/green+ ON 4.8 0,34 1.4 - - - - 9
+/green+ ON 2.7 −35,−28 3.7 - - - - 18
red+/+ ON Gaussian 1.7 34,34 13.6 - - - - 17
red+/+ ON 2.9 −57,20 6.4 - - - - 13
red-/- OFF 2.7 −11,22 25.0 - - - - 32
red-/- ON/OFF 6.9 −1,−5 1.5 0.00 0 - - 22
red-/- ON/OFF 1.0 32,42 14.3 0.00 0 - - 18
red+/green- ON/OFF Gabor 2.0 −46,14 8.0 0.09 0 - - 34
red+/green- ON/OFF 4.3 −56,2 4.0 0.06 0 - - 37
red+/green- ON/OFF 1.0 −46,13 8.4 0.05 π - - 33
red-/- OFF Gaussian 0.7 −4,36 13.5 - - 0 0.5 34
cardioid
+/green+ ON/OFF 6.0 −30,33 5.0 0.00 0 226 0.5 20
red-/green+ ON/OFF Gabor 4.7 −29,19 4.4 0.00 0 72 0.4 25
red-/green+ ON/OFF cardioid 4.6 42,−15 5.3 0.00 0 240 0.6 20
red-/green+ ON/OFF 2.7 40,−16 9.0 0.12 0 240 0.7 36
 

A = maximum response in mean spike number; (xc, yc) = the field center on the hemispherical screen in degrees of visual angle; σ = radius of the receptive field in degrees of visual angle; f = spatial frequency; ϕ = phase constant; p = angle of preferred movement; k = gain constant; PFE = profile fitting error.

ON or OFF Cells: Gaussian Function
Spatial organizations of simple ON or OFF ganglion-cell receptive fields were modeled by Gaussian functions (Rodieck & Stone, 1965). These simple receptive fields in turtle retina were symmetrical, confined around a central point (Bowling, 1980; Granda & Fulbrook, 1989; Granda et al., 1999). Peak spectral sensitivities of these cells corresponded to the primary visual pigments found in this animal’s eye (Liebman & Granda, 1971). In dark adaptation, field types were center-only, with only ON or OFF responses in the field. 
Gaussian functions were fitted to five examples of center-only cells, for which parameters are listed in Table 1. An example of a positive Gaussian fitted to a receptive-field shape of a +/green + ON cell is shown in Figure 1A. It had the smallest field radius recorded (σ = 1.4°), with a PFE equal to 9%. A red-/-OFF cell that had the largest field radius recorded (σ = 25°) with a PFE equal to 32% was fitted by a negative Gaussian (Figure 1B). Small receptive fields were located close to the visual streak; large receptive fields were located more distally. 
Figure 1
 
Gaussian functions were fitted to two examples of receptive-field shapes measured during dark adaptation by light spots moving at 10 deg·s−1: (A) +/green+ ON cell defined by 640-nm light spots (red bar columns) and (B) red-/- OFF cell defined by 540-nm light spots (green bar columns). When moving light spots pass through the center of an OFF center receptive field, spikes do not occur until light spots leave the field center. Both plots show mean spike number as a function of field position in visual angle of degrees.
Figure 1
 
Gaussian functions were fitted to two examples of receptive-field shapes measured during dark adaptation by light spots moving at 10 deg·s−1: (A) +/green+ ON cell defined by 640-nm light spots (red bar columns) and (B) red-/- OFF cell defined by 540-nm light spots (green bar columns). When moving light spots pass through the center of an OFF center receptive field, spikes do not occur until light spots leave the field center. Both plots show mean spike number as a function of field position in visual angle of degrees.
ON/OFF Cells: Gabor Function
ON/OFF cells possessed center/surround, color-opponent interactions. When light spots moved across field centers, interactions between ON and OFF sensitivities were observed as a ring of spike activity in the periphery Figure 2). The position of the ring depended on the wavelength of stimulation, and on the adaptation state of the retina (see also Figure 9 in Granda et al., 1999). The differences in receptive-field shape caused by using different stimulation conditions are shown for a red+/green- ON/OFF in Figure 2. The left data plot in Figure 2 shows responses to a moving 640-nm light spot during dark adaptation. The shape of the field for the same cell is markedly different to a moving 540-nm light spot during adaptation to 550-nm light background (right data plot of Figure 2). Plus and minus symbols, arranged in concentric rings are superimposed on the data plots in Figure 2 to code regions of high (+) and low (−) spike activities. Gabor functions predicted these field shapes Figure 2 (bottom). Values of the parameters fitted to five examples are shown in Table 1
Figure 2
 
Receptive-field shapes of a single red+/green- ON/OFF cell. The cell’s response to 640-nm light spots during dark adaptation (red bar columns on black grid) is shown at figure left. Response to 540-nm light spots after adaptation to a 550-nm background light (green bar columns on green grid) is shown at figure right. Gabor functions were fitted to the receptive-field shapes
Figure 2
 
Receptive-field shapes of a single red+/green- ON/OFF cell. The cell’s response to 640-nm light spots during dark adaptation (red bar columns on black grid) is shown at figure left. Response to 540-nm light spots after adaptation to a 550-nm background light (green bar columns on green grid) is shown at figure right. Gabor functions were fitted to the receptive-field shapes
Directionally Selective Cells: Gaussian and Cardioid Functions
More than 40% of ganglion cells in turtle are reported to be sensitive to directional movement (Granda & Fulbrook, 1989). In this study, the number was close, 32%. 
A directionally selective cell is shown in Figure 3. This cell was the only directional cell found that was solely OFF responsive. The cell was classed as red-/- OFF most sensitive to a 540-nm light spot moving at an angle of 0°. The field shape was arranged around the field center, but not symmetrically so. Responses to flashed, stationary light spots are shown on either side of the field center in histograms. At the rear of the receptive field, a strong OFF response was clustered. 
Figure 3
 
Analysis of a red-/- OFF cell that was directionally selective. Vector plot (top center) shows that the cell was most sensitive to 540-nm light spots (green) moving from left to right at 0°. Dashed arrows point to field areas equidistant from the field center that show histograms of spike responses to stationary light spots flashed in these regions. Black vertical bars show the SEM (N = 10) for each 25-ms bin of spikes. OFF responses are clustered in the wake of the preferred direction (large black arrow) to define a receptive field with an asymmetric cardioid shape.
Figure 3
 
Analysis of a red-/- OFF cell that was directionally selective. Vector plot (top center) shows that the cell was most sensitive to 540-nm light spots (green) moving from left to right at 0°. Dashed arrows point to field areas equidistant from the field center that show histograms of spike responses to stationary light spots flashed in these regions. Black vertical bars show the SEM (N = 10) for each 25-ms bin of spikes. OFF responses are clustered in the wake of the preferred direction (large black arrow) to define a receptive field with an asymmetric cardioid shape.
To account for this asymmetry, a cardioid function was added to the Gaussian function (Figure 4) (cf., Oyster, 1990). The model is shown in Figure 4 with a minus symbol (−) to emphasize the location of the OFF responses clustered at the cleft of the cardioid. The preferred angle (p) was 0° (cf., Table 1). Remarkably, the fitted value for the function’s preferred angle predicted the one measured by the vector plot (cf., Figure 3). The preferred angle for the vector plots was calculated by vector addition of observed spike numbers in response to each angle of stimulation. 
Figure 4
 
A Gaussian function multiplied by a cardioid function fitted to the directionally selective OFF cell’s receptive-field shape shown in Figure 3. The minus sign (−) is positioned at the cleft of the cardioid to symbolize the peak location of OFF sensitivity. As the light spot moves into the OFF cleft, the cell is silent. After the trailing edge of the light spot leaves the OFF cleft, the cell fires.
Figure 4
 
A Gaussian function multiplied by a cardioid function fitted to the directionally selective OFF cell’s receptive-field shape shown in Figure 3. The minus sign (−) is positioned at the cleft of the cardioid to symbolize the peak location of OFF sensitivity. As the light spot moves into the OFF cleft, the cell is silent. After the trailing edge of the light spot leaves the OFF cleft, the cell fires.
Effects of APB on the organization of the cleft in the cardioid
To further investigate the idea that OFF response regions could be involved in organizing the cleft of the cardioid, we compared responses before and after application of APB in four directionally selective ON/OFF cells. Figure 5 shows a representative response from one cell. This +/green+ ON/OFF cell was directionally selective to movement at 96° as calculated by the vector sum of the vector plots. APB reduced the preferred vector’s magnitude to as little as 5% of the magnitude observed under dark adaptation, but did not change the vector’s angle of directional preference. Before APB, the field shape was asymmetric, rising steeply at the rear of the field and falling gradually toward the preferred direction of movement (Figure 5, top). After infusion, APB eliminated ON responses and increased the number of OFF responses, which in turn produced a field in the shape of an annulus, but asymmetrically positioned (Figure 5, bottom). The significance of the results using APB is that the hole of the annulus was asymmetrically located, at the back of receptive field, in the wake of preferred directional movement, suggesting that the asymmetry is defined by a strengthened OFF region, the cleft part of a cardioid. 
Figure 5
 
Vector plots (left) and three-dimensional receptive-field plots (right) for a directional-selective +/green+ ON/OFF cell. Before (top) and after (bottom) infusion of 1-mM APB during dark adaptation. Plots show responses to 640-nm moving light spots (red bar columns). White circles are superimposed on the plots to show that the hole of the annulus generated after application of APB is asymmetrically positioned at the rear of the field as defined before application of APB.
Figure 5
 
Vector plots (left) and three-dimensional receptive-field plots (right) for a directional-selective +/green+ ON/OFF cell. Before (top) and after (bottom) infusion of 1-mM APB during dark adaptation. Plots show responses to 640-nm moving light spots (red bar columns). White circles are superimposed on the plots to show that the hole of the annulus generated after application of APB is asymmetrically positioned at the rear of the field as defined before application of APB.
Directionally Selective Cells: Gabor and Cardioid Functions
Interactions among dimensions of spectral sensitivity (red/green) and of position (center/surround) could be observed in directionally selective cells. Backgrounds of spectral lights were used to selectively adapt regional sensitivities, which shifted the ring of spike activity produced by light spots passing through field centers (Figure 6). An example is shown for a directionally selective red-/green+ ON/OFF cell that preferred light spots moving through the field center at 240° (Figure 6, vector plot, center). 
Figure 6
 
Receptive-field plots for a directionally preferring red-/green+ ON/OFF cell during adaptation to different background lights. The vector plot (center) shows the responses of the cell under dark adaptation to 640-nm light spots moving at different angles through the receptive-field center. Top trio of three-dimensional bar graphs shows responses to 640-nm light spots (red bar columns); bottom trio of three-dimensional bar graphs shows responses to 540-nm light spots (green bar columns). Asymmetry for this cell’s receptive field was most apparent in response to 540-nm light spots under adaptation to 550-nm light background (right plot, bottom).
Figure 6
 
Receptive-field plots for a directionally preferring red-/green+ ON/OFF cell during adaptation to different background lights. The vector plot (center) shows the responses of the cell under dark adaptation to 640-nm light spots moving at different angles through the receptive-field center. Top trio of three-dimensional bar graphs shows responses to 640-nm light spots (red bar columns); bottom trio of three-dimensional bar graphs shows responses to 540-nm light spots (green bar columns). Asymmetry for this cell’s receptive field was most apparent in response to 540-nm light spots under adaptation to 550-nm light background (right plot, bottom).
A cardioid function joined to the Gabor function described directionally selective cells that possessed fields with asymmetrical rings of spike activity. Four examples with their parametric values are shown in Table 1. Functions were fitted to field plots measured under 550-nm background light for the red-/green+ ON/OFF cell shown in Figure 6 (top right and bottom right). These responses to 640-nm and 540-nm light spots are superimposed in Figure 7 (top) to emphasize the coextensiveness of the sensitivities in the field. Linked functions easily predicted changes in field shape that were derived under different stimulating conditions. 
Figure 7
 
Receptive-field plots for the directionally preferring red-/green+ ON/OFF cell measured under 550-nm backgrounds (top plot with green grid). Responses to 640-nm (red bar columns) and 540-nm (green bar columns) light spots are superimposed. Linked Gabor and cardioid functions were fitted to the field shapes. The white model (bottom left) is fitted to responses of 640 nm. The gray model (bottom right) is fitted to responses of 540 nm. Black arrows point in the direction of preferred movement.
Figure 7
 
Receptive-field plots for the directionally preferring red-/green+ ON/OFF cell measured under 550-nm backgrounds (top plot with green grid). Responses to 640-nm (red bar columns) and 540-nm (green bar columns) light spots are superimposed. Linked Gabor and cardioid functions were fitted to the field shapes. The white model (bottom left) is fitted to responses of 640 nm. The gray model (bottom right) is fitted to responses of 540 nm. Black arrows point in the direction of preferred movement.
Effects of spectral adaptations on the organization of the cleft in the cardioid
Similar to the effects of APB, adaptations to spectral backgrounds did not change the angle of preferred movement, but did change the spike activity distribution and its organization around the cardioid cleft (Figure 6). Under dark adaptation, there was a center/surround assembly to both 640-nm and 540-nm lights (Figure 6, center, top and bottom). Adaptation by 650-nm light weakened the center, thereby releasing the surround as shown by an increase in peripheral spike activity (Figure 6, left, top and bottom). Adaptation to background light of 550 nm weakened the surround, which released the receptive field center (Figure 6, right, top and bottom), making the asymmetry for this cell’s receptive field shape most apparent under this condition. The response to 640-nm light rose abruptly at the rear of the field, and then fell off gradually toward the front (Figure 6, top right). The collar of response to 540-nm light had a break that was located toward the rear of the field, on the trailing side of the preferred direction of movement (Figure 6, bottom right). The ring of spike activity in response to 540-nm light on 550-nm background is shaped similar to a horseshoe pointing nicely in the direction of preference at 240°, and matching the value of preferred direction that was calculated by vector addition from the vector plot (Figure 6, bottom right). 
Discussion
A serial multiplication of functions can predict ganglion-cell receptive fields in retina of turtle. The advantage of multiplied functions is that field formations are unified by a model that predicts field geometries derived from particular stimulating conditions. In support of this arrangement, under dark adaptation, some fields that appear to be center-only can be adapted by spectral lights to reveal particular surrounds (Granda et. al, 1999). Also, receptive fields that are non-directional can be transformed into fields possessing properties of directional selectivity (Pan & Slaughter, 1991). The linked model predicts a range of receptive-field shapes displaying center/surround interactions and directional selectivity, which in turn depend on particular stimulating conditions. 
Gabor Function
Gabor (1944) representation, a Fourier transform, is used to describe a signal in both frequency and spatial domains by mathematical expansion into its symmetrical (cosine) and antisymmetrical (sine) elements. A linkage of two functions by its own definition, the Gabor function is a Gaussian modulated by a complex sinusoid, possessing a real-valued part and an imaginary part. The real-valued part is the form often used in vision research (e.g., Daugman, 1985; cf., Klein & Beutter, 1992), and as part of the Gabor function used here. Because moving stimuli inherently possess temporal and spatial properties, it is appropriate to suggest that Gabor functions could predict receptive-field shapes defined by moving light spots. Gabor functions have been used to predict cortical responses to gratings as spatial frequency filters (for review see Shapley & Lennie, 1985; Jones & Palmer, 1987). And although receptive-field shapes of center-only cells under dark adaptation have been predicted by Gaussian functions (Rodieck, 1965), and center/surround fields by difference of Gaussians (DOGs) (Rodieck & Stone, 1965), here we show center/surround receptive fields displaying a spatial ring in response to moving light spots modeled by Gabor functions (Figure 2). 
Several models that include DOGs (e.g., Hawken & Parker, 1987), difference of offset Gaussians (DOOGs) (De Monasterio, 1978), and Gaussian derivatives (Young, 1987) are argued to predict the data of center/surround receptive fields best and potentially could be introduced into our scheme of unification, but these models have shortcomings of their own. For example, DOGs and DOOGs suffer from the ambiguity generated by combining separate parameters of center and surround into a single receptive field, and Gaussian derivatives can have only purely odd or even symmetry (Young, 1991). The Gabor function is able to describe multiple periodic responses of excitation and inhibition in a single parameter (e.g., Ikeda & Wright, 1972; Mullikin, Jones, & Palmer, 1984) and is more appropriate for describing the spatial ring of spikes that form in the peripheries of receptive field in response to moving stimuli. 
The rings of responses observed in center/surround fields occur in response to leading and trailing edges of light spots; these in turn derive from interactions between coextensive spectrally defined ON and OFF sensitivities located in the centers and surrounds of receptive fields (cf., Rodieck, 1965; Teeters, Jacobs, & Werblin, 1997; Granda et al., 1999). A ring of response is formed by the history of these interactions. As the light spot leaves the inhibitory surround (after first passing through the center), the cell is disinhibited and forms the spatial ring in the receptive-field periphery. 
In this study, we show data that were measured at constant speed (10 deg·s−1), a rate that the majority of cells in turtle are apparently most sensitive (Granda & Fulbrook, 1989). We did test the general effects of speed on the position of the ring for a few cells. At faster speeds (20, 30, and 50 deg·s−1), the spacing between the center response and ring was reduced but with increased number of spikes; at slower speeds (1 and 5 deg·s−1), the spacing was increased with a decrease in spike number (cf., Teeters et al., 1997). It appears that changes in position of the ring response are affected by speed and can be predicted by Gabor functions. 
Cardioid Functions
Cardioid functions are rectified cosine functions that have been used by others to predict directional responses of visual neurons (Oyster, 1968, 1990; Rosenberg & Ariel, 1991, 1998). We show that the spatial asymmetry observed in directionally selective ganglion-cell receptive fields in turtle is likewise predicted by a cardioid function when joined to Gaussian or Gabor functions. Modulation by the cardioid generates functions that become unimodal in the preferred direction of motion. By their fit to the asymmetrical shapes seen in the data, cardioids predict the cells’ preferred directions. 
Wyatt and Daw (1975) measured inhibition in their work on directionally selective cells of rabbit, using two light spots that were moved in opposite directions. In their work, the field shape of inhibition generated by a spot moving in the null direction that affected the response to a spot moving in the preferred direction was described as being like a cardioid, for which the cleft pointed in the null direction, a conclusion opposite to the findings here. In our work, excitatory spike responses were measured to single light spots, with field shapes fitted by cardioid functions having clefts pointing in the preferred direction of travel. In the wake of preferred direction, OFF response sensitivity defines the cleft of the cardioid. 
Cardioid functions, those whose clefts point in the preferred direction, have been used to describe vector-plot responses in directionally selective ganglion cells for rabbit (Oyster, 1968, 1990), and in basal optic nucleus (BON) cells for turtle (Rosenberg & Ariel, 1991, 1998). Here cardioid functions are used to predict the essential three-dimensional shape of ganglion-cell receptive fields (Figures 4 and 7). 
APB helps reveal the organization of the cardioid
Infusion of APB did not block directionality (cf., Kittila & Massey, 1995), but APB better defines the location of OFF responses in the shape of the cardioid. After administration of APB, ON responses were effectively eliminated, and OFF responses were enhanced (Slaughter & Miller, 1981; Arkin & Miller, 1987; Vitanova, Popova, Kupenova, Mitova, & Belcheva, 1993; Granda et al., 1999), the latter creating a hole of inhibition located at the cleft of the cardioid-shaped field (Figure 5, bottom). 
Effects of center/surround interactions on the cardioid
Backgrounds of spectral light shifted the distribution of spike activities observed in directionally selective cells. Shifts in response distribution rely on the relative strengths of overlapping spectral sensitivities (Granda et al., 1999). In the example shown in Figure 6, the center is most sensitive to 640-nm light. Adaptation by background light of 650 nm weakens that center and thereby releases surround responses, observed as increased spike activity in the periphery (Figure 6, left, top and bottom). Adaptation by background light of 550 nm weakens the surround thereby releasing the center of the field, observed as tight formations of spike responses (Figure 6, right, top and bottom). When the surround is inhibited, the center becomes disinhibited (or enhanced) causing the cardioid shape to be more apparent. 
Flexibility of the Model
The serial multiplication of three functions—Gaussian, Gabor, and cardioid—can predict the changes of field shapes occurring under particular states of light adaptation. An example of this flexibility is shown for the directionally selective ON/OFF cell in Figure 7. The Gabor function combined with the cardioid function predicts the directionally selective response to 540-nm light spots under adaptation to background light of 550 nm (Figure 7, bottom right). In the same cell, the Gabor function that predicts the response to 640-nm light spots (Figure 7, bottom, left) has a spatial frequency of 0.00 cycles·deg−1. This simplifies to a combination of a Gaussian with a cardioid function (cf., OFF cell in Figure 4). 
Possible Mechanism
In receptive fields, balances among spectral sensitivities, ON and OFF responses, and center/surround regions define interactions that operate in a push-pull format (cf., Sterling, 1983) to produce complex spike patterns responding to moving light spots in the forms of simple geometries (Granda et al., 1999; Dearworth, 1999). Similar patterns have been reported for ON/OFF ganglion cells in tiger salamander (Werblin, 1991; Teeters et al., 1997). Because moving stimuli combine spectral, temporal, and spatial properties together, receptive-field shapes result from mixtures of these interactions that are processed by the underlying retinal circuitry. 
Jacobs and Werblin (1998) have dealt directly with responses in space and time using flashed stimuli. Ganglion-cell responses, stimulated by light squares flashed at ∼2000 different regions, were replayed simultaneously to describe spatio-temporal patterns as “expanding ridges.” In tiger salamander, it is thought that feedback synapses from amacrine to bipolar cells at γ-aminobutyric acid-C (GABAC) receptors mediate the expanding ridges by modulation of the feed-forward pathway onto ganglion cells. 
Rodieck (1965) used the principle of superposition to interpret responses to moving light spots as equal to spatially summed responses to flashed light spots. This assumption can explain responses that are linear but not responses that are nonlinear (cf., Enroth, Cugell, & Robson, 1966). Here we focussed on field shapes defined by moving stimuli, and our purpose was not to delineate linear from nonlinear responses. Even so, the circuitry described by Jacobs and Werblin (1998) (cf., the synapse boxed by dotted lines in our |Figure 8), could be, in part, responsible for a delayed inhibition that generates a ring of spike activity observed in the responses to moving light spots. Lateral spread of the signal in turn could be carried by electrically coupled bipolar cells and modulated by lateral inhibitory units (Arkin & Miller, 1987; Vitanova et al., 1993; Granda et al., 1999). 
Figure 8
 
Hypothetical circuit for ganglion cells displaying progressively more complex receptive-field shapes in response to moving light spots. Bipolar cells are electrically coupled and are excitatory onto ganglion cells using N-methyl-D-aspartate (NMDA) glutamatergic receptors. An amacrine cell that feeds back onto the bipolar via GABAC receptors could produce the spatial ring of response that is observed in ganglion cells (stippled) and predicted by Gabor functions. The two-asymmetric pathways model is incorporated to explain directionally selective ganglion cells that display asymmetric spatial rings in the shape of a cardioid (Grzywacz, Merwine, & Amthor, 1998).
Figure 8
 
Hypothetical circuit for ganglion cells displaying progressively more complex receptive-field shapes in response to moving light spots. Bipolar cells are electrically coupled and are excitatory onto ganglion cells using N-methyl-D-aspartate (NMDA) glutamatergic receptors. An amacrine cell that feeds back onto the bipolar via GABAC receptors could produce the spatial ring of response that is observed in ganglion cells (stippled) and predicted by Gabor functions. The two-asymmetric pathways model is incorporated to explain directionally selective ganglion cells that display asymmetric spatial rings in the shape of a cardioid (Grzywacz, Merwine, & Amthor, 1998).
Directional-selective cells prefer movement in one direction, but are inhibited to movement in the opposite direction (Barlow & Levick, 1965). Our data show that directionally selective receptive fields are asymmetrically shaped and can be predicted by a cardioid function. The cleft of the cardioid is positioned at the rear of the field in the wake of preferred movement and becomes more apparent after application of APB (Figure 5, bottom). 
Whereas GABAC may mediate the symmetrically delayed inhibition observed as a ring of spike activity, another GABA receptor subtype may be involved in an asymmetric inhibition that produces directional responses. In tiger salamander, the favored subtype receptors are GABAB (Pan & Slaughter, 1991; Smith, Grzywacz, & Borg-Grahm, 1996). Baclofen, a GABAB receptor agonist, enhanced normal directional responses in some directionally selective cells in that animal and even induced 30% of those that were not normally directionally selective to become so (Pan & Slaughter, 1991). They showed that the effects of baclofen were especially strong with concomitant application of APB: interesting to note, because APB in this work enhanced a regional asymmetrical inhibition. 
The favored subtype receptor for directionality in turtle retina is GABAA, because picrotoxin and bicuculline block directionally to about the same degree (Ariel & Adolph, 1985; Smith et al. 1996); GABAB is insensitive to picrotoxin. GABAC receptors are not favored because they have been shown to be insensitive to bicuculline (Feigenspan, Wassle, & Bormann, 1993; Qian & Dowling, 1993; Lukasiewicz & Werblin, 1994). 
A circuit for directional units, but one with feedback, could explain units that possess asymmetrical rings of spike activity (Figure 6). A hypothetical circuit is shown in Figure 8, with a two-asymmetric-pathway model for directional selectivity developed by Grzywacz et al. (1998)
Conclusions
Gaussian functions describe center-only receptive fields, a spatial envelope that decays from center to periphery; Gabor functions describe center/surround mechanisms observed as a ring-of-spike activity, and cardioid functions define asymmetric fields that are directionally selective. These linked functions predict progressively more sophisticated, ganglion-cell receptive-field shapes in the retina of turtle congruent with its complex synaptic connectivity. The significance of this model is that it predicts a range of receptive-field shapes displaying center/surround interactions and directional selectivity, which in turn depend on particular stimulating conditions. A retina equipped with these ganglion-cell receptive fields provides a fairly simple but efficient mechanism for defining movements in space. 
Acknowledgments
The authors thank Paul D. R. Gamlin and Frankin R. Amthor for their comments after reading this manuscript. Commercial Relationships: None. 
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Figure 1
 
Gaussian functions were fitted to two examples of receptive-field shapes measured during dark adaptation by light spots moving at 10 deg·s−1: (A) +/green+ ON cell defined by 640-nm light spots (red bar columns) and (B) red-/- OFF cell defined by 540-nm light spots (green bar columns). When moving light spots pass through the center of an OFF center receptive field, spikes do not occur until light spots leave the field center. Both plots show mean spike number as a function of field position in visual angle of degrees.
Figure 1
 
Gaussian functions were fitted to two examples of receptive-field shapes measured during dark adaptation by light spots moving at 10 deg·s−1: (A) +/green+ ON cell defined by 640-nm light spots (red bar columns) and (B) red-/- OFF cell defined by 540-nm light spots (green bar columns). When moving light spots pass through the center of an OFF center receptive field, spikes do not occur until light spots leave the field center. Both plots show mean spike number as a function of field position in visual angle of degrees.
Figure 2
 
Receptive-field shapes of a single red+/green- ON/OFF cell. The cell’s response to 640-nm light spots during dark adaptation (red bar columns on black grid) is shown at figure left. Response to 540-nm light spots after adaptation to a 550-nm background light (green bar columns on green grid) is shown at figure right. Gabor functions were fitted to the receptive-field shapes
Figure 2
 
Receptive-field shapes of a single red+/green- ON/OFF cell. The cell’s response to 640-nm light spots during dark adaptation (red bar columns on black grid) is shown at figure left. Response to 540-nm light spots after adaptation to a 550-nm background light (green bar columns on green grid) is shown at figure right. Gabor functions were fitted to the receptive-field shapes
Figure 3
 
Analysis of a red-/- OFF cell that was directionally selective. Vector plot (top center) shows that the cell was most sensitive to 540-nm light spots (green) moving from left to right at 0°. Dashed arrows point to field areas equidistant from the field center that show histograms of spike responses to stationary light spots flashed in these regions. Black vertical bars show the SEM (N = 10) for each 25-ms bin of spikes. OFF responses are clustered in the wake of the preferred direction (large black arrow) to define a receptive field with an asymmetric cardioid shape.
Figure 3
 
Analysis of a red-/- OFF cell that was directionally selective. Vector plot (top center) shows that the cell was most sensitive to 540-nm light spots (green) moving from left to right at 0°. Dashed arrows point to field areas equidistant from the field center that show histograms of spike responses to stationary light spots flashed in these regions. Black vertical bars show the SEM (N = 10) for each 25-ms bin of spikes. OFF responses are clustered in the wake of the preferred direction (large black arrow) to define a receptive field with an asymmetric cardioid shape.
Figure 4
 
A Gaussian function multiplied by a cardioid function fitted to the directionally selective OFF cell’s receptive-field shape shown in Figure 3. The minus sign (−) is positioned at the cleft of the cardioid to symbolize the peak location of OFF sensitivity. As the light spot moves into the OFF cleft, the cell is silent. After the trailing edge of the light spot leaves the OFF cleft, the cell fires.
Figure 4
 
A Gaussian function multiplied by a cardioid function fitted to the directionally selective OFF cell’s receptive-field shape shown in Figure 3. The minus sign (−) is positioned at the cleft of the cardioid to symbolize the peak location of OFF sensitivity. As the light spot moves into the OFF cleft, the cell is silent. After the trailing edge of the light spot leaves the OFF cleft, the cell fires.
Figure 5
 
Vector plots (left) and three-dimensional receptive-field plots (right) for a directional-selective +/green+ ON/OFF cell. Before (top) and after (bottom) infusion of 1-mM APB during dark adaptation. Plots show responses to 640-nm moving light spots (red bar columns). White circles are superimposed on the plots to show that the hole of the annulus generated after application of APB is asymmetrically positioned at the rear of the field as defined before application of APB.
Figure 5
 
Vector plots (left) and three-dimensional receptive-field plots (right) for a directional-selective +/green+ ON/OFF cell. Before (top) and after (bottom) infusion of 1-mM APB during dark adaptation. Plots show responses to 640-nm moving light spots (red bar columns). White circles are superimposed on the plots to show that the hole of the annulus generated after application of APB is asymmetrically positioned at the rear of the field as defined before application of APB.
Figure 6
 
Receptive-field plots for a directionally preferring red-/green+ ON/OFF cell during adaptation to different background lights. The vector plot (center) shows the responses of the cell under dark adaptation to 640-nm light spots moving at different angles through the receptive-field center. Top trio of three-dimensional bar graphs shows responses to 640-nm light spots (red bar columns); bottom trio of three-dimensional bar graphs shows responses to 540-nm light spots (green bar columns). Asymmetry for this cell’s receptive field was most apparent in response to 540-nm light spots under adaptation to 550-nm light background (right plot, bottom).
Figure 6
 
Receptive-field plots for a directionally preferring red-/green+ ON/OFF cell during adaptation to different background lights. The vector plot (center) shows the responses of the cell under dark adaptation to 640-nm light spots moving at different angles through the receptive-field center. Top trio of three-dimensional bar graphs shows responses to 640-nm light spots (red bar columns); bottom trio of three-dimensional bar graphs shows responses to 540-nm light spots (green bar columns). Asymmetry for this cell’s receptive field was most apparent in response to 540-nm light spots under adaptation to 550-nm light background (right plot, bottom).
Figure 7
 
Receptive-field plots for the directionally preferring red-/green+ ON/OFF cell measured under 550-nm backgrounds (top plot with green grid). Responses to 640-nm (red bar columns) and 540-nm (green bar columns) light spots are superimposed. Linked Gabor and cardioid functions were fitted to the field shapes. The white model (bottom left) is fitted to responses of 640 nm. The gray model (bottom right) is fitted to responses of 540 nm. Black arrows point in the direction of preferred movement.
Figure 7
 
Receptive-field plots for the directionally preferring red-/green+ ON/OFF cell measured under 550-nm backgrounds (top plot with green grid). Responses to 640-nm (red bar columns) and 540-nm (green bar columns) light spots are superimposed. Linked Gabor and cardioid functions were fitted to the field shapes. The white model (bottom left) is fitted to responses of 640 nm. The gray model (bottom right) is fitted to responses of 540 nm. Black arrows point in the direction of preferred movement.
Figure 8
 
Hypothetical circuit for ganglion cells displaying progressively more complex receptive-field shapes in response to moving light spots. Bipolar cells are electrically coupled and are excitatory onto ganglion cells using N-methyl-D-aspartate (NMDA) glutamatergic receptors. An amacrine cell that feeds back onto the bipolar via GABAC receptors could produce the spatial ring of response that is observed in ganglion cells (stippled) and predicted by Gabor functions. The two-asymmetric pathways model is incorporated to explain directionally selective ganglion cells that display asymmetric spatial rings in the shape of a cardioid (Grzywacz, Merwine, & Amthor, 1998).
Figure 8
 
Hypothetical circuit for ganglion cells displaying progressively more complex receptive-field shapes in response to moving light spots. Bipolar cells are electrically coupled and are excitatory onto ganglion cells using N-methyl-D-aspartate (NMDA) glutamatergic receptors. An amacrine cell that feeds back onto the bipolar via GABAC receptors could produce the spatial ring of response that is observed in ganglion cells (stippled) and predicted by Gabor functions. The two-asymmetric pathways model is incorporated to explain directionally selective ganglion cells that display asymmetric spatial rings in the shape of a cardioid (Grzywacz, Merwine, & Amthor, 1998).
Table 1
 
Parameters for combinations of Gaussian, Gabor, and cardioid functions fit to three-dimensional field plots
Table 1
 
Parameters for combinations of Gaussian, Gabor, and cardioid functions fit to three-dimensional field plots
Parameter values
Response typea A xc, yc σ F ϕ p k PFE
Functions (spikes) (°,°) (°) (cycles·deg−1) (radians) (°) (%)
+/green+ ON 4.8 0,34 1.4 - - - - 9
+/green+ ON 2.7 −35,−28 3.7 - - - - 18
red+/+ ON Gaussian 1.7 34,34 13.6 - - - - 17
red+/+ ON 2.9 −57,20 6.4 - - - - 13
red-/- OFF 2.7 −11,22 25.0 - - - - 32
red-/- ON/OFF 6.9 −1,−5 1.5 0.00 0 - - 22
red-/- ON/OFF 1.0 32,42 14.3 0.00 0 - - 18
red+/green- ON/OFF Gabor 2.0 −46,14 8.0 0.09 0 - - 34
red+/green- ON/OFF 4.3 −56,2 4.0 0.06 0 - - 37
red+/green- ON/OFF 1.0 −46,13 8.4 0.05 π - - 33
red-/- OFF Gaussian 0.7 −4,36 13.5 - - 0 0.5 34
cardioid
+/green+ ON/OFF 6.0 −30,33 5.0 0.00 0 226 0.5 20
red-/green+ ON/OFF Gabor 4.7 −29,19 4.4 0.00 0 72 0.4 25
red-/green+ ON/OFF cardioid 4.6 42,−15 5.3 0.00 0 240 0.6 20
red-/green+ ON/OFF 2.7 40,−16 9.0 0.12 0 240 0.7 36
 

A = maximum response in mean spike number; (xc, yc) = the field center on the hemispherical screen in degrees of visual angle; σ = radius of the receptive field in degrees of visual angle; f = spatial frequency; ϕ = phase constant; p = angle of preferred movement; k = gain constant; PFE = profile fitting error.

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