Department of Ophthalmology and Visual Sciences, University of British Columbia, Vancouver, Canada V5Z 3N9, , 1 Department of Neurobiology, The Weizmann Institute of Science, Rehovot, 76100, Israel and , 2 Center for Magnetic Resonance Research, University of Minnesota, 2021 6th St. SE, Minneapolis, MN 55455, USA
Address correspondence to Dr Nicholas V. Swindale, Department of Ophthalmology, University of British Columbia, 2550 Willow St., Vancouver, BC, Canada, V5Z 3N9. Email: swindale{at}interchange.ubc.ca.
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Abstract |
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Introduction |
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While vector averaging is a simple and fast way of computing maps of orientation and direction angle from single-condition images, it has drawbacks. Direction tuning curves obtained by optical imaging are, as in single-unit data, bimodal, with two peaks with variable relative heights, positions and widths. If the two peaks are of nearly equal height, or not 180° apart, the vector resultant angle may bear little relation to the position of the higher peak, and will not accurately reflect the direction of motion to which that region of cortex was most responsive (Kim et al., 1999;Kisvardáy et al., 2001
). In addition, no information about the relative positions and heights of the two peaks in the direction tuning curve can be obtained, although these parameters are of interest. Vector averaging is probably an accurate way of estimating preferred orientation when applied to orientation tuning data (Swindale, 1998
). However the interpretation of vector magnitude is problematic, since a low magnitude may be caused by either a low overall responsiveness of the point in question, or broad orientation tuning width. Although it has been suggested that variations in vector magnitude following division by the cocktail blank should reflect only variations in tuning width (Bonhoeffer et al., 1995
), in practice low vector magnitude has generally been regarded as indicative of either low responsiveness or broad tuning, or a combination of both [e.g. Bonhoeffer and Grinvald (Bonhoeffer and Grinvald, 1991
)].
In this paper we have studied orientation and direction angle, tuning width and tuning height maps with an analysis method that avoids the limitations of vector averaging. It works by finding, at each location in the image, best-fitting model functions of the orientation or direction tuning responses. For orientation, the model is a circular normal (von Mises) function with parameters describing the position, height and width of the peak. For direction, the model is the sum of two such functions, which are unconstrained in their relative heights, widths and positions. We apply the method to optical data from area 18 of the cat and show that it reveals features not evident with vector averaging. Orientation and direction preference angles are always closely orthogonal and reversals in direction preference form lines that originate precisely in orientation singularities. Orientation tuning height is low in blob-shaped regions surrounding singularities. Bands of broad and narrow orientation tuning width, and of low and high direction selectivity, alternate in a spoke-like fashion around each singularity.
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Materials and Methods |
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Single-condition images were obtained from area 18 of four anesthetized, paralyzed cats, using standard methods in experiments reported elsewhere (Shmuel and Grinvald, 1996, 2000
). The stimuli were moving high-contrast square-wave gratings with a spatial frequency of 0.150.18 c/deg, and a temporal frequency of 2.56.0 Hz, presented binocularly. Sixteen directions of motion were used, spanning a 360° range at intervals of 22.5°. For three of the datasets (referred to as datasets 13), a limited amount of post-processing of the images was performed. Specifically, the images obtained during stimulation periods were divided by the images obtained during control periods in which an equi-luminant gray screen was presented. To remove slow noise of biological origin, a first frame analysis (Bonhoeffer and Grinvald, 1996
) was performed: the first frame of the resulting division (corresponding to data obtained during the 0.5 s period prior to the stimulus and prior to the acquisition of the control) was subtracted from all subsequent frames. Division by the cocktail blank was avoided, as this would normalize the tuning curves and remove information about tuning height. It was noted that the mean pixel value of each of the 16 single-condition images was different, and these differences were not systematically related to stimulus orientation or direction. This apparently random variability in image means could be caused by slow spatial and temporal variations of blood volume during the period of data acquisition. To remove this source of variability, image means were equalized by subtraction. Note that high-pass filtering would have done this in any case, since this removes the DC component of each image, in addition to other low frequencies. An additional justification for the procedure is that the overall goodness-of-fit values were substantially improved by it. The single-condition images were then smoothed by convolution with a Gaussian kernel with width = 50 µm. No other type of filtering was done. A fourth dataset (referred to as dataset 4) consisted of single-condition images pre-processed as described in Shmuel and Grinvald (Shmuel and Grinvald, 1996
). This included division by the cocktail blank followed by high-pass and low-pass filtering.
Data Analysis
For each pixel in the image, the data to be analyzed consist of an optical response, measured at position (i, j) to one of N directions of motion, n = 1 ... N. For the present data there were 16 such values for each pixel. For each of these tuning curves we defined a baseline value, bi, j, as the minimum of the response values for the pixel in question. We show below that this baseline, or stimulus non-specific response level, varies across the map, and has a structure that includes a spatially smooth, slowly varying component, as well as blood vessel artifacts, which are thereby removed from the analysis. Orientation and direction tuning parameters are thus defined in terms of the response evoked above this non-stimulus-specific background. Models in which the baseline was an independently variable parameter, fit to the tuning curves [as in Sharon and Grinvald (Sharon and Grinvald, 2002)] did not work well when tuning curves were broad (as was often the case with the present data) because tuning height and baseline values were insufficiently constrained by the data values and could be very variable.
To standardize the goodness-of-fit values, tuning curves were rescaled as follows: taking H(i, j) as the difference between the maximum and minimum values of the responses at position (i, j) in the map, we define Hmax as the largest value of H(i, j) taken across the whole map. Tuning curves were rescaled by setting
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Following this, every tuning curve in the map has a minimum value of zero, their shapes relative to each other remain unchanged, and all the data points comprising the tuning curves in the map lie between 0 and 100. Goodness-of-fit values can be interpreted as distances on this scale and compared between maps.
Analysis of Direction Tuning
Figure 1 (right panel) shows examples of optical direction tuning curves scaled as described in the preceding section. They are bimodal with peaks of varying relative height and width. Since orientation tuning curves are well described by a single circular normal (von Mises) function (Swindale, 1998
) we decided to model these bimodal tuning curves as the sum of two circular normal functions, whose positions, heights and widths are unconstrained. The model function was
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This has a value of zero if the two peaks are equal in height, and has a maximum value of 1 if only one peak is present in the response.
Analysis of Orientation Tuning
Stimulus orientation was defined as the angle , at right angles to the direction of motion, mod(180), i.e.
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Pairs of responses obtained to stimulus conditions 180° apart were averaged to obtain a single response at each of eight different stimulus orientations. The following model function was then fit to the data points:
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Least-squares Fitting
The LevenburgMarquardt method (Press et al., 1994) was used to fit the model functions to the data. Initial estimates of parameters were made as follows: for direction tuning data, A1 = the maximum value of the data, A2 = 0.8A1, k1 = k2 = 2.0 (
0.5 =50°),
1 was set equal to the angle for which response was maximum and
2 =
1 + 180°; for orientation data the estimates were made similarly, with k = 0.59 (
0.5 =50°). During the fits, the value(s) of k were constrained to be >0.1. This corresponds to a tuning curve that is almost flat. A fit was considered to have converged if the change in the goodness of fit,
f<10-5 for 10 successive iterations of the fitting algorithm. After the fits, checks were made to ensure that parameter values lay within acceptable bounds. The fit was rejected if any of the following applied: (i) convergence had not occurred after 500 iterations; (ii) if the goodness of fit, f > 10.0; (iii) if B (or A1 or A2) < 0; (iv) if
0.5 < 15° or
0.5 < 15°. When parameters fell outside these bounds, it was clear from inspection that the tuning curve data were either noisy or flat with ill-defined peaks. These points tended to occur near the edges of the maps. They were excluded from subsequent data analyses by compiling separate masking images for the orientation and direction tuning analysis data.
Analysis programs were written in Compaq Visual FortranTM and run under Windows 2000 on a 600 MHz Athlon PC. On average, it took 1.2 ms to fit a directional tuning function to a set of 16 data points, and
2030 s to do a complete set of fits for an individual map.
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Results |
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Figure 1 shows examples of fits to orientation and direction data taken from randomly selected sites in one map. Other examples can be found in Figure 9
. Table 1
gives averaged values of the goodness-of-fit measure, f, for individual maps. The means of these values for the four datasets studied were f = 2.9 for orientation and f = 3.5 for direction. Since pixel values were scaled to lie between 0 and 100, these values can be interpreted as distances on the same scale.
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Comparison with Estimates Obtained by Vector Averaging
The estimates of orientation and direction obtained with the model fits were compared with those obtained by vector averaging (Fig. 2a,b). There is good agreement between the estimates for orientation, but not for direction. Table 2
shows, for each of the four datasets analyzed, the means and standard deviations of the difference between the model and vector estimates of orientation and direction angle. In all cases the mean difference is close to zero, which is not surprising given the lack of any expected systematic difference between the two types of estimate (e.g. a clockwise orientation difference). The standard deviations of the differences show by how much the estimates tend to differ. For orientation, the differences are small, with estimates generally being within about ±6° of each other. For direction, the differences average about ±51°, indicating substantial disagreement between the methods. One reason for considering the model-fitting estimate to be the better of the two is that it corresponds more accurately to the angle of stimulus motion that gives the largest cortical response. Reasons for these large differences are considered further in the Discussion.
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As in previous studies (Malonek et al., 1994;Shmuel and Grinvald, 1996;
Weliky et al., 1996
), we found that the vector estimates of orientation and direction were often non-orthogonal (Fig. 2c
). The model estimates, however, were much closer to orthogonal (Fig. 2d
). Table 3
shows the means and standard deviations of the differences between direction and orientation angles for the vector and model estimates. The mean differences are close to 90° for both methods, which is expected given the lack of any systematic clockwise or anticlockwise bias, but the standard deviations for the model fits (
±9°) are substantially less than those obtained with vector averaging (
±41°).
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Figures 4 and 5 show maps obtained from model fits of orientation angle (Figs 4a and 5a
), orientation tuning width (Figs 4c and 5c
) and orientation tuning height (Figs 4e and 5e
). Positive and negative singularities in the orientation map are shown in Figures 4 and 5
as green (= positive) or red (= negative) asterisks or circles. Positive singularities are defined as those for which orientation values rotate in a clockwise direction as a clockwise circuit is made around the singularity; negative singularities are those where orientations rotate in an anticlockwise direction for a clockwise circuit. The tuning height maps (Figs 4e and 5e
) resembled maps of vector magnitude, and contained circular or elongated regions of low tuning height
200400 µm in diameter. An orientation singularity was present in many, although not all, of such regions. Singularities were never found outside regions of low tuning height.
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In theory, if orientation tuning curves of different widths and heights are normalized (by dividing by the sum of the responses, or the cocktail blank) the vector magnitude will depend only on the tuning width (Wörgötter and Eysel, 1987;Swindale, 1998
). A previous analysis that looked at variation in orientation tuning curve width using this method (Bonhoeffer et al., 1995
) showed only a local isotropic broadening of selectivity close to orientation singularities, unlike the changes observed in the present study. We confirmed that this difference is due to our subtraction of baseline activity from the single-condition maps before analysis. If we did not do this subtraction, images of the normalized vector magnitude showed a spatial pattern of variation similar to that observed by Bonhoeffer et al. (Bonhoeffer et al., 1995
). If the baseline was subtracted before normalization, the images of vector magnitude looked similar to those seen in the tuning width maps obtained with the model fitting method. We confirmed by calculations with model tuning curves that adding a constant baseline response to orientation tuning curves of different heights and widths re-introduces a relationship between tuning height and vector magnitude following normalization.
Spatial Layout of Direction Tuning Parameters
Angle maps of direction preference (Figs 4b and 5b
) show clearly defined iso-direction domains bounded in part by linear fractures across which direction preference changes by 180°. Figure 9b
shows that these lines are not discontinuities but represent points of equality in the heights of the two peaks in the direction tuning curve. As reported previously in cat areas 17 (Kim et al., 1999
) and 18 (Swindale et al., 1987;
Shmuel and Grinvald, 1996
), owl monkey MT (Malonek et al., 1994
) and ferret area 17 (Weliky et al., 1996
), these reversal lines appear to begin and end close to singularities. However here the relationship is even more clear, with odd numbers of fracture lines terminating quite precisely at singularity locations.
The map of direction selectivity (Figs 4d and 5d) shows linear regions of low selectivity (dark regions in the figure) that coincide with the fractures, and abut singularities. Regions of high selectivity lie in between the fracture lines, and, surprisingly, can often be found close to singularities, opposite a region of low selectivity. An increase in direction selectivity with decreasing distance from a singularity implies that the height of the smaller of the two peaks decreases at a faster rate than the other peak as the singularity is approached. Figure 9c
shows examples of direction tuning curves obtained close to a singularity.
Effects of Smoothing on Spatial Patterns of Tuning and Responsiveness
While the complex patterns of tuning and responsiveness within the pinwheel regions are of interest, they do not necessarily reflect the properties of individual neurons in the same regions or even the mean activities of neurons in the imaged location. Light scattering, blurring caused by signals from out-of-focus layers, the spatially smooth (or blurred) nature of the hemodynamic signal, and smoothing done after the images have been acquired might affect the signals from regions where properties are changing relatively rapidly. We investigated this possibility in two ways. First, we examined data where no post-acquisition smoothing of any kind had been done. As expected, the angle, height and width maps were noisier, but the pattern of orientation tuning height and width, and of direction selectivity, around the singularities and elsewhere was similar to that observed with the smoothed data. As a second test we took the angle maps of orientation and direction preference resulting from the model fits, and then generated simulated single-condition images assuming a fixed tuning height, a fixed tuning half-width at half-height of 30° and a fixed direction selectivity, D = 0.20. We then smoothed the simulated single-condition maps with a Gaussian kernel with widths, , varying from 20 to 300 µm. New tuning width and selectivity maps were then calculated. For intermediate amounts of smoothing (
150 µm) the control maps resembled the real ones in their general appearance (Fig. 10ad
). The similarity was further evaluated by calculating, for each pair of control and real maps, and a particular smoothing width: (i) the correlation between the orientation tuning width values; (ii) the slope of the correlation, and (iii) the root mean square (r.m.s.) of the difference in the values. A similar analysis was done for direction selectivity. Table 5
shows the results for tuning width for dataset 1 for a variety of smoothing widths. (Values for datasets 2 and 3 show smaller correlations, smaller slopes and larger r.m.s. difference values.) While strong positive correlations between the measured and control tuning width values were found, the slopes of the correlations were substantially less than 1 across a range of smoothing width values (Fig. 10e,f
). This was true for both orientation tuning width and direction selectivity values (Table 6
), in all three of the datasets that were tested. Additional tests with a range of initial selectivity values and smoothing widths were done (data not shown), and these gave no indication that other combinations of parameters would give closer similarity between the real and control maps. These results suggest that the observed variations in orientation tuning and direction selectivity can be only partly explained (or modeled) as the result of isotropic Gaussian blurring of signals. The remaining variations may be real, perhaps caused by network interactions that are not locally isotropic or are non-Gaussian in character.
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A priori it seems likely that the baseline signal, b (i.e. the minimum response value at each pixel; see Materials and Methods) might be a mixture of blood vessel artifacts common to all images, spatial fluctuations in the hemodynamic signal unrelated to the stimuli, and real variations in stimulusnon-specific responses. These possibilities are consistent with the appearance of the baseline images (Figs 6 and 11) inasmuch as they contain low-spatial frequency variations, and bright streaks that coincide with the blood vessels observed in the mapped regions (images not shown). Further examination of the images suggested a weak tendency for some singularities to be located in regions where there was a small local elevation in the baseline. Overall correlations (i.e. taken over an entire map) between orientation tuning height and baseline (Table 7
) were positive in two cases (datasets 1 and 3) and close to zero in the other (dataset 2). However these correlations represent trends averaged over the entire map, and an opposite trend might be present locally, in regions of low tuning height close to singularities. To test for that, we calculated the average of the baseline signal as a function of radial distance from a singularity, in the same way that we calculated the variation in tuning height and width. The resulting functions showed an increase in baseline with decreasing distance from singularities for all three non-normalized datasets (Fig. 7ac
). We observed that the baseline increase was often not clearly evident at individual singularities (e.g. Fig. 11
) and that there were many regions (e.g. the right-hand side of Fig. 6
) where pronounced height and width variations occurred that were not systematically related to the variation in baseline magnitude.
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Discussion |
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Vector averaging treats optical responses as vectors whose length is proportional to the response and whose angle relative to the origin is given by either twice the stimulus orientation, or by the direction of stimulus motion. In the latter case, the resultant angle gives the estimated direction preference, while for orientation the angle of the resultant is halved to find the estimated best orientation. Mathematically, these procedures are identical to performing a least-squares fit of a cosine function with variable phase and amplitude (Swindale, 1998). They are also identical to finding the phase angle of the first and second harmonic Fourier components of the data (Wörgötter and Eysel, 1987
). It has been shown that a cosine provides a much less good description of single-unit orientation tuning data than does the circular normal function (Swindale, 1998
). In the present study, the mean goodness of fit obtained with an angle-doubled cosine was only slightly worse than the value obtained by fitting the model function (Table 1
). Accordingly, estimates of preferred orientation did not differ substantially between the two methods, although it is reasonable to expect that the model estimate should be the better of the two. In contrast, a cosine is a priori a bad description of directional tuning data, since a function with one peak is being fit to data in which two peaks are generally present. When a cosine was fit to the direction tuning data, the mean goodness of fit was substantially worse than the value obtained by fitting the model function and was only slightly better than the value obtained by fitting a horizontal line (the r.m.s. value of the points in each tuning curve about their mean). Vector averaging gives reasonably accurate estimates of the position of the higher of the two peaks when one is substantially higher than the other, however in regions of low direction selectivity the angles returned may bear little relation to the position of either peak [Figs 1, 3a, 9b
, middle graph; see also Fig.10a
in Shmuel and Grinvald (Shmuel and Grinvald, 1996
), or Fig.2a
in Weliky et al. (Weliky et al., 1996
)]. Errors can occur even when the two peaks are dissimilar in height (e.g. Figure 1
, second graph from top in right-hand column). With the model fitting method the direction preference angle is almost always closely orthogonal to the angle of orientation preference. As a result, iso-direction domains have sharply defined boundaries that intersect the direction fracture lines at well defined angles (Figs 4b and 5b
).
Comparison of the Model Method with Previous Approaches
Other authors have pointed out the weakness of the vector method when applied to direction tuning data. Alternatives proposed involve interpolating between data points to estimate the maximum (Kim et al., 1999) or simply taking the direction for which response is maximum (Kisvardáy et al., 2001
). We believe that our solution to the problem may have advantages over these methods. First, neither method yields parameter values that correspond to aspects of tuning curve shape other than the position of the peak. Secondly, simply taking the position of the maximum response (Kisvardáy et al., 2001
) ignores information in the rest of the tuning curve data that could be used to establish the position of the peak more precisely. For this reason, the method is likely to perform poorly when tested with noisy data. The interpolation method (Kim et al., 1999
) does not have this problem. It has the advantage over our method of not requiring iterative least-squares fitting, and is similar inasmuch as it involves an implied fit to the tuning curve data of a model function that is the sum of a series of sine and cosine terms. A direct comparison of this method and ours would require comparing the goodness of fit of each model function to the data, the assumption being that the model giving the better fit gives the better estimation of direction preference. Neither of the previous methods addresses, as does ours, the issue of disambiguating variations in tuning height and tuning width. Our method has the further advantage of offering criteria for accepting or rejecting data from different regions of a mapped area. A suitable threshold can be chosen and, if the goodness of fit exceeds it, the data point in question can be excluded from subsequent analysis. If the fitting process does not converge, or returns parameters that are outside expected ranges it is likely that the data being analyzed are noisy or uninformative and should be excluded from subsequent analyses.
The Relationship Between Orientation and Direction Preference
Our method does not rigidly constrain orientation and direction preferences to be orthogonal but it may be biased to do so because we have defined, as is conventional, the direction of motion of the stimulus as orthogonal to its orientation. Whether or not orientation and direction preferences are orthogonal is arguably an empirical question that can only be answered by using stationary oriented stimuli to measure orientation preference, and moving dots to measure direction preference. Such data as exist suggest that these types of stimuli yield orientation and direction maps similar to those obtained with moving gratings. For example, orientation maps obtained with stationary, phase-reversing gratings are similar to those obtained with moving gratings (A. Shmuel and A. Grinvald, unpublished results). Likewise, the spatial pattern of regions responding differentially to directions of motion 180º apart is similar for moving dots, moving bars or moving grating stimuli [Fig. 5 in (Shmuel and Grinvald, 1996
)].
Directional tuning curves obtained in single cell recordings in which noise patterns are used as stimuli are often bimodal with peaks symmetrically placed on either side of the direction preference for gratings (Hammond and Reck, 1980). The spacing between these peaks increases with the velocity of the stimulus. This suggests that direction and orientation preference can vary independently, are often non-orthogonal and even, for direction, non-unique. However such bimodal responses can probably be explained by spatio-temporal filtering of random noise by an oriented receptive field in which temporally specific interactions are exerted symmetrically along the cells orientation axis (Skottun et al., 1994
). This symmetry would in fact guarantee orthogonal orientation and direction preferences for stationary versus moving gratings.
The Baseline Response
Measurement of orientation tuning width requires the establishment of a baseline in order that a width at half-height can be calculated. We defined the baseline as the part of the optical response that is common to all the stimuli, i.e. the minimum response taken across all the stimulus conditions for each pixel in the image. One reason for defining tuning parameters in this way is that a portion of the optical response to a stimulus is always non-specific and independent of the particular stimulus, i.e. the orientation or direction of motion (Bonhoeffer and Grinvald, 1996). This component of the optical signal, termed the non-mapping signal might be caused by synaptic activity (Logothetis et al., 2001
), or in part by neuronal spiking activity. It may also reflect spatially non-specific hemodynamic responses (Malonek and Grinvald, 1996
). However it is difficult if not impossible to separate these putative components. We suggest that the best way of dealing with this problem is to define the mapping signal as we have done here, in terms of the height above a common background.
Our analysis suggests that, on average, the baseline signal is larger near singularity locations than elsewhere (Figs 7 and 11). Possible reasons for this include: (i) hemodynamic and/or optical blurring, which would be expected to elevate the baseline in regions of high orientation gradient; (ii) increased broadening of tuning curve width near singularities, which may result in an artifactual elevation in the measured baseline, since if there is a response component at the orthogonal orientation it will be subtracted out by our method; (iii) sub-threshold neural responses which may be larger near singularities (Schummers et al., 2001
); and finally, (iv) an increase in the stimulus non-specific physiological spiking response at or near singularities. Regarding the latter, single-unit recording studies have demonstrated small responses to the non-preferred orientation in some neurons in cats (Azouz et al., 1997
) and monkeys (Ringach et al., 2002
); however, whether these responses increase near singularities is not yet known. The possible consequences for our results, if explanations (i) and (ii) are correct, are that, near singularities, tuning widths may be broader, and response heights larger, than our analysis suggests. It is important to note, however, that there is not an obvious, systematic relation between baseline height and tuning curve height and width that is clearly evident at all singularities. Thus it seems unlikely that the height and width variations that we see are distorted significantly by the subtraction of the baseline as we have defined it.
The Spatial Pattern of Orientation Tuning Width and Height
Our results show, we believe for the first time, that orientation tuning width and height vary independently across the surface of the cortex (Fig. 8). Tuning height correlates well with vector magnitude, and varies in the way previously described for it (Blasdel and Salama, 1986;
Bonhoeffer and Grinvald, 1991
, 1993;
Blasdel, 1992
). It is low in blob-shaped regions
500 µm in diameter (Figs 46
), nearly all of which contain an orientation singularity. Tuning width varies in a pattern that has not been reported before. It is broadest in narrow punctate or elongated regions that coincide exactly with the singularities, around which regions of narrow and wide tuning width often alternate in a spoke-like fashion (Figs 4c, 5c, 8 and 9a
).
The finding of regions of high orientation selectivity close to singularities (i.e. within 200 µm), albeit with low response magnitudes, is consistent with single-unit studies that have reported that the selectivities of neurons close to singularities are as high as they are elsewhere (Maldonado et al., 1997). Systematic variations in tuning width also occur in regions distant from the singularities. The strong correlation between width and the orientation gradient (Fig. 3b
) suggests that blurring might be a cause of width variations, since it would be expected to broaden tuning curves most in regions where orientation gradient was largest. Control calculations confirmed that isotropic Gaussian blurring can reproduce some, although not all, of the observed variation in tuning width (Fig. 10
). Blur could be caused by signals from out of focus layers and by the local averaging of neural activity that determines the hemodynamic response that ultimately gives rise to the optical signal. Local physiological interactions may also mimic blur. These interactions could be mediated by the dense network of local (<400 µm) intracortical connections which has been shown to be unselective for orientation in the supragranular layers (Malach et al., 1993;
Bosking et al., 1997
). Assuming that feed-forward and other sources of input endowed all neurons in the map with the same initial degree of selectivity, these connections would have the effect of smoothing the tuning curves of individual neurons in much the same way as was done here in our control calculations, thereby inducing a positive correlation between tuning width and orientation gradient.
Evidence that the variations in tuning width are not completely the result of either optical or hemodynamic blurring of signals comes from tetrode recording experiments (Hetherington and Swindale, 1999) which showed a strong positive correlation between orientation tuning width and variation of orientation preference among groups of nearby neurons. This is consistent with the correlation between orientation gradient and tuning width observed here (Fig. 3b
). Further evidence for physiological variations in tuning width across the map and around singularities comes from a study that combined intracellular recording with optical imaging of intrinsic signals (Schummers et al., 2001
). The orientation tuning width of membrane potentials of neurons in the vicinity of singularities was found to be wider on average than elsewhere. This suggests that the tuning width of a neuron is a function of the distribution of orientation preferences of neurons in its immediate vicinity. Since synaptic activity provides a major contribution to the hemodynamic response (Logothetis et al., 2001
), Schummers et al
.s findings suggest that the spatial variations of tuning width (Figs 4c and 5c
) represent similar variations in the tuning of the underlying electrical activity.
Although control calculations show that isotropic Gaussian blurring can reproduce some of the observed variation in tuning widths, it cannot account for all of it (Fig. 10, Table 5
). Thus non-isotropic or non-Gaussian local network interactions (Bosking et al., 1997;
Sincich and Blasdel, 2001
) may need to be invoked to explain the correlation between orientation gradient and tuning width.
The Direction Fractures
A topological argument (Swindale et al., 1987) shows that, given half-rotation orientation singularities and an orthogonal relationship between direction and orientation preferences, at least one or an odd number of direction fracture lines must connect with each orientation singularity. This was clearly observed in the present study. Similar direction fracture regions, bearing the same relationship to orientation singularities, have been described before (Swindale et al., 1987;
Malonek et al., 1994;
Shmuel and Grinvald, 1996;
Weliky et al., 1996;
Kim et al., 1999;
Kisvardáy et al., 2001
). However, with the exception of the last two of these studies, which did not use vector averaging, the relationship between regions of high direction gradient and the singularities was not as clear-cut as seen here. This is almost certainly the result of deviations from orthogonality produced by the vector method, which tend to be greatest near orientation singularities and direction fractures. As a result, the direction fractures often terminate near a singularity, rather than in the singularity itself.
Direction fracture lines appear to have a tendency to equidistant spacing, and it is hard to find circular regions with a diameter larger than 0.75 mm within which one or more fracture lines do not occur. This is true even in regions that lack orientation singularities (e.g. Fig. 5f
). Such behaviors might be expected on the grounds of coverage uniformity (Swindale, 1991
), i.e. that the cortex attempts to represent all possible directions of motion as uniformly as possible across the map, as seems to be true for other parameters (Swindale et al., 2000
). Having the fracture lines divide iso-orientation domains into equally sized sub-regions preferring opposite directions of motion might be one way to do this. Fractures also have a tendency to run in straight lines before suddenly changing direction. While this behavior tends to minimize their length, there must be other constraints on their arrangement. It would certainly be possible to connect the singularities (e.g. in Fig. 4f
) by shorter routes than the ones actually present. However, such an arrangement, although it would give a more continuous direction map, might give poorer coverage. It remains to be seen whether or not the arrangement of direction fractures that we observe represents an optimal trade-off between smoothness and coverage constraints.
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Footnotes |
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References |
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