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1Department of Physics and 2Department of Neurobiology, Pharmacology, and Physiology, University of Chicago, Chicago, Illinois
Submitted 18 October 2004; accepted in final form 5 March 2005
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ABSTRACT |
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INTRODUCTION |
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). Within each hypercolumn, information about the orientation, spatial frequency, and drift direction of image components is encoded by overlapping groups of neurons (Bonhoeffer and Grinvald 1991; Hubel and Wiesel 1977; Hubener et al. 1997; Issa et al. 2000; Shmuel and Grinvald 1996; Weliky et al. 1996). These hypercolumns tile the cortical surface, making interlacing maps of response properties.
However, other than direction, these parameters give no information about image motion. In a recent experiment, Basole et al. (2003) showed that changing the temporal properties of a moving image can change which orientation domains are activated by the image. These shifts in orientation responses cannot be predicted from the cortical organization of spatial parameters such as orientation and spatial frequency preference alone. This suggests that the classic description of cortical organization is not sufficient to describe the distributed responses of the cortex to moving images.
The response properties of primary visual cortex are important determinants of how moving images are perceived. For example, it is hypothesized that the tuning properties of V1 cells limit dynamic acuity (Levi 1996). Similarly, the perception of motion streaks that trail a quickly moving image seems to be encoded in the firing pattern of V1 neurons (Geisler 1999; Geisler et al. 2001). To understand how motion is encoded in the visual cortex, both spatial and temporal response properties must be characterized. The spatio-temporal energy model of human motion vision has been used as a framework for characterizing the response properties of individual cortical neurons (Adelson and Bergen 1985; DeAngelis et al. 1993b; De Valois et al. 1979; Gizzi et al. 1990; McLean and Palmer 1989; van Santen and Sperling 1985; Watson and Ahumada 1985). Two stages of processing constitute the spatio-temporal energy model. The first stage is a linear filtering of image properties including orientation, spatial frequency, and temporal frequency. In the second stage, the outputs of the first stage are combined in quadrature to remove phase-sensitivity from neuronal responses. Recently, the spatio-temporal energy model has been suggested as an appropriate framework for characterizing the organization of cortical maps (Basole et al. 2003; Mante and Carandini 2003).
In this paper we describe the functional organization of the primary visual cortex in terms of the linear filters of the spatio-temporal energy model and present it in the context of the hypercolumn model. We show that the distributed pattern of activity in the cortex generated in response to moving, complex images can be characterized by three maps of separable tuning properties: orientation, spatial frequency, and temporal frequency tuning. This model is sufficient to quantitatively predict the activity patterns previously measured in response to moving complex images (Basole et al. 2003). In addition, it predicts transitions in the responses of different spatial frequency domains that might underlie specific psychophysical phenomena.
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METHODS |
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We use a series of three linear filters to model the population responses of primary visual cortex, similar to the first stage of the spatio-temporal energy models proposed for human motion perception (Adelson and Berger 1985; van Santen and Sperling 1985; Watson and Ahumada 1985). These filters can be described by six separable tuning properties: orientation preference and bandwidth, spatial frequency preference and bandwidth, and temporal frequency preference and bandwidth. In this formulation, only a subset of image components is represented by neural activity. Image components whose spatial frequency and/or temporal frequency fall outside the band-pass of the spatio-temporal filters of primary visual cortex are not encoded by neural activity.
According to this model, the spatial pattern of cortical activity induced by a stimulus depends on how these six parameters are organized. We will present simulations for four different spatial arrangements; each layout is representative of a different region of V1, a different species, or a different developmental stage.
In the simplest arrangement, only orientation preference varies within a hypercolumn. Preliminary reports and recently published low-resolution maps suggest that this might be the case in ferret Area 17 (Basole et al. 2004; Schwartz 2003; Yu et al. 2002). On this small spatial scale, spatial frequency and temporal frequency tuning properties are assumed to be constant. A map of orientation preference is combined with a single spatial frequency tuning curve and a single temporal frequency tuning curve to simulate population responses. We compare the results of simulations of this cortical layout to data from measurements made over a small region of ferret Area 17 in response to drifting bar textures (Basole et al. 2003).
In the second arrangement, we assume a gradual change in the preferred spatial frequency over several millimeters of cortex, consistent with the variation in retinal sampling density with eccentricity (Basole et al. 2004). As in the first arrangement, we assume there is little to no change within a hypercolumn. Maps of orientation and spatial frequency preference and a single temporal frequency tuning curve are used to predict how orientation domains at different eccentricities respond to moving images.
The third arrangement simulates responses within cat Area 17. As in the previous case, maps of orientation and spatial frequency preference combined with a single temporal frequency tuning curve are used to make predictions about cortical response patterns. However, in this case, the spatial frequency map varies within a hypercolumn. There have been differing descriptions of the organization of spatial frequency preference in cat Area 17 (Everson et al. 1998; Hubener et al. 1997; Issa et al. 2000; Shoham et al. 1997; Tolhurst and Thompson 1982; Tootell et al. 1981), but all share the common finding that spatial frequency preference varies systematically within the space of a hypercolumn. We use one of the proposed layout of spatial frequency preference (Issa et al. 2000) as an example, but our simulations do not rely on a specific configuration—all that is required is that spatial frequency preference varies on a small spatial scale.
Finally we consider the layout in which temporal frequency preference varies in conjunction with spatial frequency preference. While temporal frequency preference does not seem to depend on spatial frequency preference in young cats, there is a clear dependence in Area 17 of adult cats (Baker 1990; DeAngelis et al. 1993a). Both extracellular recording and intrinsic signal imaging experiments suggest an inverse relationship between spatial frequency and temporal frequency preference (Baker 1990; DeAngelis et al. 1993a; Shoham et al. 1997).
Spatio-temporal filters
The three filters are described by standard orientation, spatial frequency, and temporal frequency tuning curves. The tuning curves are defined by Gaussians as
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 | (2) |
 | (3) |
the orientation coordinate, and 
the temporal frequency coordinate. Each tuning curve is characterized by a peak value and a width. In Eq. 1, Sp is the peak spatial frequency, and 
S is proportional to the spatial frequency bandwidth. In Eq. 2, Tp is the peak temporal frequency, and T is proportional to the temporal frequency bandwidth. The orientation tuning curve (Eq. 3) is defined as a wrapped Gaussian, with a peak orientation of 
and a characteristic width of 
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For ease of simulation, all the tuning curves and stimuli are calculated in spatial frequency space. However, this is not a global spatial frequency transform. Each functional domain is treated as though it has a spatio-temporal receptive field. The spatial frequency and orientation tuning curves define an effective receptive field size and orientation for each functional domain (reviewed in Dayan and Abbott 2001): the width of the effective receptive field is inversely proportional to the spatial frequency bandwidth (S), whereas the length of the receptive field is proportional to the ratio of the width and the orientation bandwidth (). The orientation of the receptive field is the same as the peak of the orientation tuning curve. Similarly, the temporal frequency tuning curve of a domain defines the temporal receptive field properties.
We simulate responses to rigid motion of an image (that is, all the components of the image move with the same velocity). The temporal frequency of each image component (i) can be determined from its spatial frequency and the drift velocity of the image along the orientation of the image component
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d is the drift angle of the image, p is the spatial frequency of the image component, and is the orientation of the image component defined as orientation in spatial frequency coordinates, so a vertical grating has an orientation of 0° (see Fig. 1). The absolute value is taken because we ignore direction selectivity. Substituting this expression for in Eq. 2 gives the projection of the temporal frequency tuning curve onto the spatial frequency plane.
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Population responses, R, within a functional domain (defined as small region of cortex that has a given set of orientation, spatial frequency and temporal frequency preferences) are modeled as the integral over all spatial frequencies and orientations of the stimulus, scaled by the spatial frequency, temporal frequency, and orientation tuning curves
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. By reducing the variables to spatial frequency and orientation, Eq. 5 allows one to visualize the effects of three separable filters in two dimensions.
In our analysis, we have limited stimuli to objects in an image moving at a single velocity. However, it can be generalized to account for several objects moving at different speeds and directions. Because it is a linear model, responses in a functional domain to a stimulus consisting of objects with different velocities would be determined by a summation of its responses to each object individually
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di).
An equivalent way of calculating these responses is to keep the temporal frequency coordinate in the model. The net response in a domain would then be given by
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() is the spatial Fourier transform of the spatial component of the receptive field, and the function T() is the temporal Fourier transform of the temporal component of the receptive field. Simulations
We simulated responses in functional domains to complex stimuli moving through the visual field. Stimuli are presented over 80 x 80° (full field), which is represented in a 512 x 512 pixel array (0.157°/pixel). To represent the stimuli in terms of their spatial frequency components, the two-dimensional spatial Fourier transform is taken (Fig. 2). Likewise, the tuning curves are represented in the two-dimensional spatial Fourier plane, as shown in Fig. 1. The integral shown in Eq. 5 is solved numerically by summing over a 3.2 cycles/deg range of spatial frequencies. In Figs. 3–6 and 9, 90° have been added to to give orientation domains as often defined by responses to oriented gratings (that is, horizontal grating have an orientation of 0° and vertical bars have an orientation of 90°).
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) and are similar to the values reported electrophysiologically (Movshon et al. 1978). We estimated the average temporal frequency preference (Tp), spatial frequency bandwidth (S), and temporal frequency bandwidth (T) for ferret Area 17 from fits to the data of Basole et al. (2003).
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RESULTS |
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Can linear spatio-temporal filtering account for the distributed pattern of cortical activity produced in response to complex, moving images? To address this question, we first compared the predictions of the spatio-temporal filtering model to measurements made in ferret primary visual cortex of activity driven by drifting texture stimuli (Basole et al. 2003). Assuming that only orientation preference varies across a hypercolumn, the simulations provide a quantitative description of the orientation domain activation patterns observed in the ferret cortex (Basole et al. 2003). We generalized this approach to other cortical areas by incorporating the organization of different parameters, such as spatial frequency and temporal frequency tuning. This model is used to make predictions about the qualitative differences in response patterns that cortical areas with different organizations should have.
Activity patterns in a hypercolumn in which only orientation preference varies
To test the spatio-temporal filtering model, we compared simulation results to measurements made in the ferret visual cortex by Basole et al. (2003). Because the measurements of Basole et al. (2003) were made over a small area of cortex, we restricted simulations to a region containing just one or a few hypercolumns. On this scale, orientation preference is arrayed in pinwheels (Chapman et al. 1996; Rao et al. 1997; Weliky et al. 1995), and we assumed that spatial frequency preference does not change substantially (schematized in Fig. 1, A–D).
The responses of an orientation domain to a moving image can be predicted from the domain’s spatial and temporal filtering properties. Individually, the spatio-temporal filters can be represented by the tuning curves for orientation, spatial frequency, and temporal frequency (Fig. 1, B–D). To visualize the predicted activity patterns, we will display the tuning curves in spatial Fourier coordinates (Fig. 1, E–G). Both the spatial frequency tuning curve and the orientation tuning curve have simple representations in the spatial Fourier plane: each depends on only one of the polar axes of the spatial Fourier plane, p (spatial frequency), or (orientation; see Fig. 1, E and F).
The representation of the temporal frequency filter in the spatial Fourier plane may be less intuitive. Because a rigid object moves through a visual scene at a single velocity, the temporal frequency of each component of the image is proportional to its spatial frequency: components with high spatial frequencies move with higher temporal frequencies than do components with low spatial frequencies (Eq. 4). As a result, the temporal frequency tuning curve of a domain limits the range of spatial frequencies that it can encode.
In a slowly moving image, a component can have any of a wide range of spatial frequencies and still have a temporal frequency within the temporal band-pass of the neuron. For example, the bright regions in Fig. 1H show the range of spatial frequencies that fall within the temporal frequency tuning curve for an image moving at a slow speed of 10°/s. In comparison, a quickly moving image has a much narrower range of spatial frequencies that a component can have and still fall within the temporal frequency band-pass; 70°/s drift speed (Fig. 1J). Because the temporal frequency tuning curve can be projected onto the spatial frequency plane, we can visualize how changing temporal properties of a stimulus can affect responses in different domains of primary visual cortex. In the following four sub-sections, we explore how specific changes in image drift velocity and the spatial statistics of an image affect the population responses in a small region of the ferret visual cortex.
DRIFT ANGLE. First we consider the effect of changing the stimulus drift angle on the population response. In this case, the spatial properties of the stimulus do not change. Figure 2A shows an example of the texture stimuli used in these simulations. Elongated bars that move with a fixed velocity are randomly distributed across the visual field. The spatial Fourier transform of the stimulus (Fig. 2B) shows that the components of the image cover a range of orientations and spatial frequencies. This distribution does not change with image motion. However, the temporal frequency of each component does depend on the drift angle, as given in Eq. 4. As a result, thetemporal frequency filter passes image components with different orientations depending on drift angle.
Figure 3 shows simulated responses in different orientation domains (orientation response curves) for four different drift angles of a texture stimulus. For motion perpendicular to the long axis of the bars, the texture activates the same orientation domains that would be activated by a grating of the same orientation (Fig. 3A, top row; Fig. 3B, solid line). For motion that is not perpendicular to the long axis of the bars, the orientation domain most strongly activated is not the same as the orientation of the bars.
Varying the drift angle changes which portion of the two-dimensional Fourier spectrum of the stimulus falls within the spatio-temporal tuning envelope of the neurons that make up the orientation domain. To show this, the activity in each of four orientation domains for four drift angles is shown in Fig. 3A (see also Supplementary Materials1 ). Each panel shows the product of all three tuning curves (as in Fig. 1, E–G) and the stimulus (Fig. 2B). The predicted response in an orientation domain is the integral of the filtered stimulus components over the spatial Fourier plane (Eq. 5). The brightness of the panel is proportional to the activity in the domain.
For horizontal drift of vertical bars, the 90° orientation domain passes more of the power in the stimulus than does any other domain (compare the top left panel to the other panels in the top row). For a drift angle of 
/2, the strongest response is in the 0° orientation domain. In the example shown in Fig. 3, a /4 (45°) clockwise shift in the axis of motion changed the distribution of population activity such that it peaked at 68°. In a symmetrical fashion, a /4 counterclockwise shift in texture motion evokes a response peaked at 112°.
As the drift angle changes, different image components are filtered out by the temporal frequency, spatial frequency, and orientation tuning curves. As Fig. 1, H–J and Eq. 4 show, drift angle is one of the main factors that determines which image components fall within the temporal frequency band-pass of cortical neurons. The representation of the temporal frequency tuning curve in two-dimensional spatial Fourier coordinates is rotated by a change in the drift angle of the stimulus. With the rotation, the stimulus components that fall within the temporal and spatial frequency tuning curves have orientations that differ from the orientation of the long axis of the bars.
DRIFT SPEED.
Changing the speed at which a stimulus drifts changes the peak orientation response (Basole et al. 2003). Figure 4A gives an impression of the effect of drift speed on responses in four orientation domains. At a slow drift speed (10 °/s), the peak response sits between 90 and 135°, as suggested by the nearly equal brightness of the two top left panels. It is more difficult to estimate by eye the peak orientation for high drift speeds (Fig. 4A, bottom row), but the numerical integral over each of the planes shows that the peak response is between 45 and 90°. The animation in Supplementary Materials 1 gives a better visual impression of how responses shift among orientation domains as a function of drift speed.
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, our simulations show that, for motion not orthogonal to the long edges of the bar, the peak response orientation changes smoothly with drift speed (Fig. 4). Again, the shift in orientation responses can be explained by changes in the projection of the temporal frequency tuning curve onto spatial frequency space (Fig. 1, H–J). As drift speed increases, the stimulus components that contribute to the response are limited to a narrower spatial frequency band along the direction of motion. Changing the drift speed alters which stimulus orientations fall within the temporal and spatial frequency tuning curves of the cortical area. This is in contrast to a full field grating stimulus whose image components are all of a single orientation; because no other orientations exist in the stimulus, there is no shift in the orientation domains that are activated as the drift velocity changes.
There is a unique drift speed (the crossing speed) at which the population response peaks at the dominant orientation of the texture (the long axis of the component bars). The value of the crossing speed depends on both the spatial and temporal frequency tuning curves (as well as the aspect ratio). The crossing speed measured in the ferret primary visual cortex is about 45°/s [for 4 x 1° vertical bars drifting at /4 radians, see Fig. 3 in Basole et al. (2003)]. Given an average spatial frequency preference of 0.25 cycles/° (Baker et al. 1998), the crossing speed is consistent with a peak temporal frequency preference of 6 to 7 cycles/s (Fig. 4C).
The shift in peak orientation response is not always a smooth one. When the bars are so short that they are essentially dots, the transition at the crossing speed is abrupt: the peak response shifts from one orientation domain to the orthogonal orientation domain without going through intermediate orientations. Figure 5 shows the results of two simulations in which random dot fields drift at a speed on either side of the crossing speed. Consistent with the response profiles in supplementary Fig. 4 of Basole et al. (2003), slowly drifting dot fields activate orientation domains determined by the drift angle (the 135° orientation domain in Fig. 5), but at high speeds, the dot field activates the orthogonal orientation (the 45° orientation domain in Fig. 5).
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Figure 6 shows the effect of increasing bar length on the predicted shift in peak orientation response. The shift is greatest for dots and progressively decreases with increasing aspect ratio (Fig. 6C). The reduction in shift occurs whether the drift speed is greater than (Fig. 6C, 
) or less than (Fig. 6C, 
) the crossing speed.
In the simulations shown in Fig. 6C, the peak orientations do not asymptote at 90° with increasing aspect ratio, as might be expected by the fact that full field vertical gratings best activate the 90° orientation domain. Instead, for a drift speed of 5°/s, the peak response approaches an asymptote of 110° as bar length increases, and the response peak approaches 83° for a drift speed of 100°/s. The offset of the asymptotes from 90° is due to the short edges of the bars (at the orthogonal orientation) that drift across the visual field even when the bar length is longer than the full visual field. The peak orientation response would be 90° if none of the short edges of the bars drift through the field of vision. In our simulation this occurs once the stimulus bars reached a length twice the height of the visual field (160°).
Using the estimated ferret parameters, the population response evoked by a stimulus of vertical bars of length 2, 4, or 10° (and 1° wide) are peaked at 124, 117, and 109°, respectively, when the drift speed is 5°/s. Although these shifts are qualitatively similar to those seen in Fig. 2A of Basole et al. (2003), with the estimated ferret parameters, the model underestimates the shift with the shortest bars by about 10° and overestimates the shift by 5° for the longest bar [equivalent values in Fig. 2A of Basole et al. (2003) are 135, 119, and 104°].
SHIFTS WITH ORTHOGONAL MOTION.
The smooth transitions among different orientation domains have all been observed for diagonal motion (motion not orthogonal to the long axis of the bars; Basole et al. 2003). However, the model also predicts transitions between orientation domains for orthogonal motion. For example, vertical bars drifting horizontally slowly should activate vertical orientation domains, but the same bars drifting quickly should activate horizontal orientation domains.
The speed at which this transition should occur depends both on the tuning properties of the cortex and on the characteristics of the stimulus. The transition speed increases as the aspect ratio of the bars increases (Fig. 7). For ferret Area 17 parameters, the transition for random dots is at 33°/s, but for bars with an aspect ratio of 4 the transition is around 70°/s. No such transition is predicted for gratings moving at various speeds.
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While we assume spatial frequency preference is relatively constant on the scale of an orientation pinwheel in ferret V1, spatial frequency preference does seem to change over larger spatial scales (Basole et al. 2004; Schwartz 2003). In central visual fields, spatial frequency preference is higher than in peripheral visual fields (Basole et al. 2004; Schwartz 2003). This cortical arrangement is schematized in Fig. 8, in which orientation preference repeats many times over several millimeters of cortex, but spatial frequency preference increases gradually and monotonically from the peripheral to the central visual field. The spatio-temporal filtering model predicts how transitions described in Figs. 3, 4, and 6 occur at different eccentricities. Simulation results are shown in Fig. 9.
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Changing aspect ratios also differentially affects activity patterns in the central and peripheral regions of V1. At low drift speeds (below the crossing speed) these effects are subtle (Fig. 9C, filled symbols), but are accentuated at higher drift speeds (Fig. 9C, open symbols). As the aspect ratio of the bars in a texture increases, power is lost at all orientations other than the dominant one (the orientation of the long axis of the bars). Interestingly, the areas of low spatial frequency preference are more sensitive to this shift than the areas of high spatial frequency preference for high drift speeds. This can be attributed to the effect of the temporal frequency tuning curve which at high speeds filters out all high spatial frequency components except for a narrow band oriented orthogonal to the drift angle (see Supplementary Materials 3 and 41).
The larger the difference in spatial frequency preference between cortical zones, the greater the effects outlined in Fig. 9. Because spatial frequency preference only varies by a factor of 2 over the dorsal part of ferret V1 (Basole et al. 2004), different locations should show only a small difference in the peak orientation of responses. A wider spread of spatial frequency preferences will produce more substantial differences in response patterns across spatial frequency domains.
Activity patterns within a hypercolumn of cat Area 17
In cat Area 17, spatial frequency preference varies systematically within the space of a hypercolumn (Everson et al. 1998; Hubener et al. 1997; Issa et al. 2000; Shoham et al. 1997; Tolhurst and Thompson 1982; Tootell et al. 1981). As a result, identifiable regions within a hypercolumn encode different combinations of stimulus orientation and spatial frequency. The layout shown in Fig. 10 summarizes the organization of spatial frequency preference as reported by Issa et al. (2000), in which the spatial frequency map is coupled to the orientation map.
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As an example, we compared the predicted responses to drifting sine wave and square wave gratings in different spatial frequency domains. A square wave grating is made up of a series sinusoidal components covering a range of spatial frequencies all at the same orientation. Each component of the drifting square wave grating has a different temporal frequency, and as the drift speed increases, the temporal frequencies increase. As Fig. 11, C and D shows, the high spatial frequency components of a slowly drifting square wave grating should effectively drive the visual cortex. With increasing speeds, however, the high spatial frequency components fall outside of the temporal frequency tuning curve. At high drift speeds, therefore, square wave gratings should not be distinguishable from sine wave gratings of the same frequency (Fig. 11, A–D).
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In the previous section, we assumed that temporal frequency preference is constant across spatial frequency domains. In young cats, this seems to be the case; De Angelis et al. (1993a) found that temporal frequency preference seems to be independent of spatial frequency tuning in cats younger than 8 wk old. In adult cats, however, temporal frequency preference is inversely proportional to spatial frequency preference (Baker 1990; DeAngelis et al. 1993a). In this section, we studied how the relationship between temporal frequency and spatial frequency preferences, schematized in Fig. 13, should affect population responses.
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Orientation responses should also be affected by the relationship between temporal and spatial frequency preferences. For example, the transitions between orientation domains shown in Fig. 12 depend on temporal frequency tuning. Because low spatial frequency domains in the adult cat have a high temporal frequency cut-off, they should be more resistant to the transition than low spatial frequency domains in young cats (cf. 
and 
in Fig. 12). Conversely, because high spatial frequency domains in the adult have a low temporal frequency cut-off, they are expected to be more sensitive to drift speed (cf. and in Fig. 12).
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DISCUSSION |
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showed that these two spatial variables are not enough to describe distributed cortical responses to moving images. We showed that the combined organization of spatial and temporal tuning properties in ferret can account for distributed population responses to both grating stimuli and more complex stimuli such as drifting bar textures. This result is consistent with the preliminary report of Mante and Carandini (2003, 2004). We also used the spatio-temporal filtering model to make predictions about activity patterns in cat Area 17, in which spatial and temporal tuning properties are distributed differently than in ferret V1. Implications for cortical mapping
These simulations reconcile the traditional idea that stimulus parameters are represented more or less independently across the cortical surface with the findings of Basole et al. (2003) that suggest that there is a single map of spatio-temporal energy that determines cortical responses to moving images. The spatio-temporal map of the cortex can be characterized by combining the traditionally defined maps of orientation and spatial frequency preference with a map of temporal frequency preference.
Each point on the cortical surface encodes a small range of spatial frequencies and orientations in a stimulus—if this combination of spatial frequency and orientation does not exist in the stimulus, that part of cortex will not be activated, regardless of the speed or angle at which an image moves. However, even if a specific component is present in the stimulus, it may not be represented if it is moving too quickly or too slowly to fall within the temporal frequency band-pass of the visual cortex.
It is useful to consider cortical maps as maps of receptive field parameters rather than stimulus features (Issa and Stryker 2004). The average receptive field properties of a cortical domain determine the spatio-temporal tuning curves measured in response to drifting gratings. Activity in a given domain signals the presence of a specific combination of features in the stimulus that are consistent with the spatio-temporal receptive field properties of the domain.
How complex images are represented in different primary visual areas depends on the structure of the cortical maps in the different species. One surprising result from measurements in ferret primary visual cortex was that population responses to drifting textures looked like responses generated by grating stimuli (Basole et al. 2003). The model presented here suggests that this should only be the case in species in which spatial frequency preference does not vary across the cortical surface. A simple explanation for the ferret results of Basole et al. (2003) is that spatial frequency preference does not vary significantly across the small area of cortical surface sampled. If spatial frequency preference does not vary significantly on the scale of a hypercolumn in ferret Area 17, only stimulus orientations would be encoded in spatially segregated domains.
However, in a primary visual area in which spatial frequency preference varies systematically across the cortical surface, as is the case in cat Area 17, complex stimuli should recruit different domains than do sine wave gratings. For example, square wave gratings should stimulate a wider range of spatial frequency domains than do sine wave gratings. The tangential organization of spatial frequency preference provides a second variable with which stimulus information can be encoded by position on the cortical surface.
Describing the organization of cortex in terms of maps of separable tuning properties simplifies the job of predicting responses to complex stimuli. If, for example, the tuning curves are not separable one would have to measure responses to every combination of spatial frequency, orientation, and temporal frequency to characterize the spatio-temporal responses of cortex. If we need n measurements to densely sample one dimension (for example, 8 orientations to sample all orientations), we would need n3 measurements to specify the spatio-temporal response properties of any point. If, however, you can assume separability, only 3n measurements are needed: n orientations for the orientation tuning curve, n for spatial frequency, and n for temporal frequency. Because a model of separable tuning curves predicts the responses presented in Basole et al. (2003), mapping the cortical organization of tuning properties is a simple way to summarize the cortical responses to complex images.
Model assumptions and limitations
This linear model ignores several aspects of cortical organization that may contribute to the distributed responses to moving images. For example, maps of direction selectivity have been found in ferret Area 17 (Weliky et al. 1996) and cat Area 18 (Shmuel and Grinvald 1996) but are not incorporated in our model. Our simulations show that direction selectivity is not necessary to characterize the cortical responses to texture stimuli measured in ferret Area 17 (Basole et al. 2003). Ocular dominance columns are also ignored, because the structure of both the orientation and spatial frequency maps are the same whether constructed using monocular or binocular stimuli (Crair et al. 1997; Hubener et al. 1997; Issa et al. 2000).
The effects of several well-described nonlinearities in cortical responses (reviewed in Carandini et al. 1999; Dayan and Abbott 2001; De Valois and De Valois 1990) are not considered in this analysis. Some of the nonlinearities, for example, the compressive nonlinearity that describes response saturation with increasing contrast, would change response amplitudes but not the pattern of activity across functional domains. Other nonlinearities, however, will alter the pattern of activity. Specifically, suppression of cortical responses has been characterized for stimuli containing multiple orientations or multiple spatial frequencies (Bonds 1989; Ringach et al. 2002). In ongoing experiments, we are studying the effects of these nonlinearities on the distributed pattern of cortical responses.
The quantitative match between our simulations and the measurements of Basole et al. (2003) rely on the estimated spatio-temporal parameters for the ferret. One recent study that measured temporal frequency tuning in 32 neurons in ferret Area 17 (Alitto and Usrey 2004) found several neurons with a peak temporal frequency near our estimate; however, the majority of units had a lower peak temporal frequency than we estimated. Different temporal frequency tuning properties would affect the magnitude of the shifts. For example, a lower average temporal frequency preference would produce a lower crossing speed than shown in Fig. 4. However, the qualitative trends presented in Figs. 3–7 would be preserved.
Implications for psychophysics
The spatio-temporal energy model has been used to predict a variety of psychophysical phenomena. Combining the model with a description of cortical organization allows us to predict the distributed responses that correspond to such phenomena.
Psychophysical experiments show that visual acuity changes with image speed (dynamic acuity, Brown 1972; Chung and Bedell 2003; Levi 1996; Westheimer and McKee 1975). As the speed of image motion increases, less detail is perceived. The predictions shown in Fig. 11 suggest that increasing drift speed should shift cortical responses to lower spatial frequency domains. If these predictions are born out experimentally, they will confirm the hypothesis that the tuning properties of primary visual cortex limit dynamic acuity.
Motion streaks are another perceptual phenomenon associated with quickly moving images—the streaks are perceived as lines trailing a moving object. Motion streaks can be used by observers to determine motion direction (Geisler 1999), and individual neurons in V1 can respond to a quickly moving dot as though presented with a bar oriented along the axis of motion (Geisler et al. 2001). The 90° flip in peak orientation response observed by Basole et al. (2003) and simulated in Figs. 5, 7, and 12 is consistent with the perception of motion streaks. Above a certain speed, the dominant orientation encoded in the pattern of cortical activity is determined by the axis of motion, not the spatial orientation of image components.
The simulations also suggest that the speed at which these transitions occur depends on the spatial frequency and temporal frequency tuning properties of the cortical domain (as in Figs. 9 and 12). High spatial frequency channels are more susceptible to the transition than are low spatial frequency channels, even when their temporal tuning properties are identical. This suggests that motion streaks should be perceived first in high spatial frequency domains as image velocity is increased.
The effects of temporal frequency tuning on orientation domain activity should act through a similar mechanism as the spatial frequency tuning properties. That is, temporal frequency tuning restricts which spatial frequencies can contribute to cortical responses. When temporal frequency tuning is constant across spatial frequency domains, it produces a loss of high spatial frequency information at high drift speeds. With the inverse relationship between temporal frequency and spatial frequency preferences (Baker 1990; DeAngelis et al. 1993a), the loss of high spatial frequency information is accentuated. One side effect of this is that peak orientation responses in high spatial frequency domains shift at very low speeds. The inverse relationship between spatial and temporal frequency preference has the effect of further decoupling orientation responses in different spatial frequency channels—the greater the difference in temporal frequency tuning among spatial frequency domains, the less likely it is that the same orientation domains are activated in the different spatial frequency channels.
In conclusion, a model of spatio-temporal filtering predicts that the pattern of cortical responses driven by moving images should depend on how filter parameters are mapped across the cortical surface. Six separable parameters, consisting of spatial and temporal preferences and bandwidth, specify the spatio-temporal filters. Variations in these parameters define the standard orientation and spatial frequency preference maps, along with a map of temporal frequency preference. For the simple organization of filtering parameters in which only orientation preference varies, the spatio-temporal filtering model can quantitatively account for the surprising shifts observed in orientation domains activated in response to drifting textures (Basole et al. 2003). In addition, the model predicts qualitatively different activation patterns for visual cortical areas that differ in how spatial frequency and temporal frequency tuning properties are mapped.
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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1 The Supplementary Material for this article (four animations) is available online at http://jn.physiology.org/cgi/content/full/01093.2004/DC1. 
Address for reprint requests and other correspondence: Naoum P. Issa, 947 E. 58th St., MC0926, Dept. of Neurobiology, Pharmacology, and Physiology, Univ. of Chicago, Chicago, IL 60637 (E-mail: nissa{at}drugs.bsd.uchicago.edu)
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REFERENCES |
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