## Abstract

Previous optical imaging studies used the vector-summation (VS) method for calculating direction and orientation preference maps. However, for direction maps it often resulted in direction vectors which showed a steep angle to that of orientation vectors violating the ‘aperture rule’. The present report provides a simple procedure for calculating direction preference maps using the ‘electro- physiologist's ear’ approach. This approach takes into account the strongest directional response component (vector-maximum, VM) in each pixel of the optical image, reminiscent of how electro- physiologists determine direction preference by audio-monitoring of the firing rate of neurons. The major advantage of this method is that the orthogonal relationship between orientation and direction preference vectors is preserved and that for most image pixels direction preference can be faithfully described by a single vector parameter. Here we used the VM method for calculating direction and the VS method for calculating orientation preference maps and quantified their spatial relationship. The results showed that, typically, an iso-orientation domain contained a pair of patches that preferred opposite directions orthogonal to the orientation. Rate-of-change maps for direction revealed that virtually all direction discontinuity lines linked orientation centres. Close to orientation centres, direction discontinuity lines ran chiefly parallel with iso-orientation lines, whereas more remotely they had either parallel or perpendicular courses.

## Introduction

Optical imaging of intrinsic signals has been used to reveal the spatial distribution of direction selectivity across large visual cortical regions (Shmuel and Grinvald, 1996; Weliky et al., 1996; Roerig and Kao, 1999). Commonly, direction maps, as orientation maps, are calculated using the vector-summation (VS) algorithm. Accordingly, activity images obtained for a full set of stimulus conditions are vectorially summed on a pixel-by-pixel basis. Although for orientation maps the VS method has been shown to be reliable (Blasdel and Salama, 1986; Grinvald et al., 1986), its validity in computing direction maps has not been established yet. In fact, close inspection of published direction maps, calculated with this method, revealed that at several locations the direction vectors ran in parallel with the corresponding orientation vectors. Such an angular relationship between direction and orientation vectors goes against the so-called ‘aperture rule’, according to which the perceived direction of movement must be perpendicular to the orientation of the stimulus, provided that the stimulus exceeds the size of the receptive field (Henry et al., 1974; Movshon et al., 1985). In the present study, we show that this conflict stems from the fact that direction tuning of optical signals is most often bimodal, in contrast to orientation tuning that is unimodal. Thus when the preferred and non-preferred direction vectors are nearly equal in magnitude, the VS method yields an erroneous estimate of direction preference. Consequently, the VS method should not be used for calculating direction maps. Here, we propose an alternative procedure that preserves the orthogonal relationship between direction and orientation vectors. In addition to this, we provide a quantitative description of the relationship between orientation and direction preference maps in the cat primary visual cortex.

## Materials and methods

For the present investigation, optical images were obtained from four adult cats (8–14 months old). All surgical procedures were in accordance with institutional and federal guidelines for the care and use of laboratory animals (Arnsberg, Germany).

### Surgical Preparations

The animals were prepared for surgery as previously described (Buzás et al., 1998; Yousef et al., 1999). Briefly, after initial anaesthesia (ketamin, 7 mg/kg, Ketanest, Parke-Davis, Berlin, Germany and xylazin, 1 mg/kg, Rompun, Bayer Belgium, Sint-Truiden, Belgium; i.m.) prolonged anaesthesia was maintained using artificial ventilation of 0.4–0.6% halothane (Halothan Eurim, Eurim-Pharm, Piding, Germany) in a 1:2 mixture of O2 and N2O. For muscle relaxation, alcuronium chloride (0.15 mg/kg/h, Alloferin, Hoffman-La Roche, Grenzach-Whylen, Germany) was infused with glucose (24 mg/kg/h, Glucosteril, Fresenius, Bad-Homburg, Germany) and Ringer solution (Ringerlösung Fresenius, Fresenius, Bad-Homburg, Germany) i.a. via a femoral catheter. End-tidal CO2 (3–4%), blood pressure (100–140 mmHg) and body temperature (38–39°C) were monitored continuously. A large craniotomy was made, exposing both hemispheres between Horsley–Clarke coordinates AP (–)4 and (+)9, and LM (+)0.5 and (+)6.5. Then, a round copper chamber was mounted over the exposed cortical region, filled with silicone oil (50 cSt viscosity, Aldrich, Milwaukee, WI, USA) and sealed with a round cover-glass.

### Optical Imaging of Intrinsic Signals

Optical imaging of intrinsic signals was carried out using the Imager 2001 imaging system (Optical Imaging, Germantown, NY, USA) and VDAQ data acquisition software (version 2.18k, Optical Imaging, Germantown, NY, USA). During data acquisition of intrinsic signals, the camera was focused 650–750 μm below the cortical surface (Fig. 1A) and the cortex was illuminated through the cover glass with 609 ± 5 nm light from a circular fibre-optic slit lamp (Schott, Mainz, Germany) surrounding the camera optics, which consisted of two SMC Pentax lenses, 1:1.2, f = 50 mm, arranged in a ‘tandem’ manner (Ratzlaff and Grinvald, 1991). Visual stimuli were presented on a video screen (Sony, Pencoed, UK) in 120 Hz non-interlaced mode 28.5 cm in front of the cats' eyes, covering 40–60° of their visual field. Refraction of the eyes was corrected with contact lenses. High-contrast, square-wave gratings were generated at four or eight orientations (spaced equally at 45° and 22.5°, respectively) at optimal spatial (0.1–0.2 cycle/degree) and temporal frequencies (1–2 Hz) using the Vision Works (Vision Research Graphics, Durham, NH, USA) or VSG Series Three (Cambridge Research Systems, Rochester, UK) stimulus generator systems. For obtaining direction and orientation preference maps, a single stimulus trial consisted of eight (16) stimulus conditions: gratings of four (eight) orientations varied in a pseudo-random sequence by multiples of 45° (22.5°) and drifted unidirectionally with optimal velocity along the orthogonal axis of the orientation. Each data acquisition period (during which the stimulus grating moved) was preceded by an ‘interstimulus interval’ of 10 s, when the animals viewed a stationary image of the grating to be presented during the next data acquisition period. Video frames were acquired for 4.5 s, commencing 1 s after the stimulus grating began to move.

### Analysis of the Optical Images

For data analysis, single condition maps (SCMs) were calculated by summing the images associated with a particular stimulus attribute (i.e. orientation and direction) using TVMIX software (Optical Imaging, Germantown, NY, USA). For orientation selectivity, the acquired activity images were summed for the same orientation and opposite directions (Fig. 1B). Orientation SCMs were normalized to the sum of images recorded for all stimulus conditions, called the ‘cocktail blank’ (Bonhoeffer and Grinvald, 1993). For calculating direction selectivity maps, two kinds of SCM were generated for each stimulus direction. First, activity maps were divided by the ‘cocktail-blank’ (Fig. 1C,D). Second, differential direction SCMs (dSCMs) were produced by dividing activity maps of opposite directions (Fig. 1E,F). The grey value distribution of the image pixels of each SCM was ‘clipped’ by discarding extreme grey values which were outside the range defined as ±1.5–3× the SD around the average grey value. Then the SCMs were re-scaled to the range between 0 and 255. Saturation of pixel values (0 or 255) was allowed only in regions containing reflection artefacts outside the exposed cortical tissue. Subsequent calculations were performed using a custom-made program written in the IDL language (Research Systems, Boulder, CO, USA). The SCMs were filtered in two steps. First, high-pass filtering was applied using a Laplace filter of 1064 μm (50 pixels) kernel size to remove low- frequency noise due to uneven illumination. Then the images were smoothed using a spatial filter (boxcar filter) averaging over 106–234 μm (5–11 pixels). Orientation angle maps were computed using pixel- by-pixel VS of SCMs corresponding to four (eight) orientations (Blasdel and Salama, 1986; Bonhoeffer and Grinvald, 1991). Direction angle maps were calculated from SCMs representing eight (16) directions using either the usual VS or VM methods described in this paper. Gradient maps (rate of change, ROC) calculated for each pixel of the orientation and direction angle maps comprised vectors whose length was proportional with the ROC per pixel and whose angle denoted the axis of highest ROC on a scale from 0 to 180°. Histograms of the pixel values of ROC maps were based on regions which excluded large surface blood vessels and bone.

## Results

A direction preference map is a two-dimensional vector field that shows the preferred direction at each pixel position of the optically imaged cortical region. Direction preference vectors are calculated from SCMs, each of which represents the distribution of activity to a particular stimulus direction. One of the essential prerequisites of direction vectors is that they must be orthogonal to the corresponding orientation vectors (Henry et al., 1974; Adelson and Movshon, 1982; Movshon et al., 1985; Scannell et al., 1996). However, none of the calculation methods used so far — vector-summation and Fourier transform (Shmuel and Grinvald, 1996) — fulfilled this requirement. The necessity of this orthogonal relationship is based on the fact that the receptive field of an individual neuron is much smaller than the size of the stimulus grating, implying that the perceived direction of movement of the stimulus must be perpendicular to its orientation — the ‘aperture rule’. In this case, the activity of neurons contributing to the optical responses must abide by the ‘aperture rule’.

Previous optical imaging studies have relied on the VS method in calculating direction maps (Malonek et al., 1994; Shmuel and Grinvald, 1996; Weliky et al., 1996; Roerig and Kao, 1999). This method generates a single direction vector at each pixel position by adding corresponding image pixels of all dSCMs as vectors. The angles of the vectors represent stimulus directions and their lengths correspond to the grey-scale values of the pixels. We noticed that using the VS method, after summation of the dSCMs, many of the generated direction vectors (black arrows in Fig. 2A) were not perpendicular at all or were even parallel with the preferred orientation axis (black bars in Fig. 2A) of the same locations. At these image locations, the VS results clearly violated the ‘aperture rule’. Our aim was to explore the reasons that caused the discrepancy between the expected and the calculated angles. Therefore, we conducted a step-by-step analysis of the data processing used in calculating direction maps.

### Direction Vectors Calculated with the VS Method

$\mathit{DI}\ {=}\ {[}\mathit{R}(\mathit{p})\ {\mbox{--}}\ \mathit{R}(\mathit{n}){]}/\mathit{R}(\mathit{p})$
where R(p) is the optical response (grey value) in the direction producing a maximal response (preferred direction) and R(n) is the optical response (grey values) in the opposite (non-preferred) direction. Then the pixels were grouped in three directional categories according to their DI values using the criteria of Albus (Albus, 1980): bidirectional, direction asymmetric and direction selective. We found that 67.2% of the pixels were bidirectional (DI < 0.20), showing equal or nearly equal values at the preferred and non-preferred directions, 32.8% fell in the directional-asymmetric category (0.20 ≤ DI < 1), while no direction-selective (DI = 1) pixels were found. The lack of strictly direction selective locations is not surprising, since the optical images are based on spatial and temporal signal averaging (see Discussion).

### Normalization of the Direction Vectors Obtained with the VM Method

It needs to be emphasized that the VM method relies on activity images which did not undergo response normalization to the opposite direction of motion. Hence, one can argue that the lengths of the resulting direction vectors contain orientation- related response components in addition to those of direction. In order to normalize the strongest response component of an image pixel as chosen by the VM method, the opposite response components must be subtracted. Importantly, such a post- normalization process results in a vector whose direction is identical with that of the strongest response component before normalization and whose length is proportional to the strength of direction selectivity at that location.

### Quantitative Comparison between the VS and VM Methods

As we have shown above, the VS method is rather sensitive to situations where the directionally opposite and strongest responses can cancel out. This leads to wrong estimates of direction preference. A quantitative description of this effect is shown by the scatter plot in Figure 4. Here we plotted the ‘error’ of the VS method as expressed by the angle differences between direction vectors calculated with the VS and VM methods against the length (magnitude) of the VS direction vector. Statistically, the data points are distributed according to a negative correlation (r = –0.26), implying that the smaller the direction vector calculated with the VS method, the larger the angle difference from its VM counterpart.

## Discussion

The VM method described here offers a simple, reliable procedure for generating direction vectors which preserve the required orthogonal relationship with orientation vectors. In principle, the VM method utilizes the strongest components of the optical responses to direction stimuli. In this regard, it resembles how direction preference is determined in in vivo electrophysiological experiments by taking the highest amplitude (loudest) of stimulus-evoked responses. One of the advantages of the VM method lies in the fact that it returns a single vector to each pixel position of the direction map; this method is, therefore, readily applicable for quantitative purposes, e.g. in determining direction preferences (functional topography) of horizontal connections.

### Previous Attempts to Estimate Direction Preference Maps

Former optical imaging studies relied on the VS method for calculating direction selectivity maps and, in this regard, they adopted the same algorithm applied for calculating orientation maps (Bonhoeffer and Grinvald, 1991, 1993). We have shown here that the VS method is inappropriate because it is sensitive to situations where the strongest and opposite direction components of the optical responses have equal or near equal magnitudes. Mathematically, this situation is equivalent to the problem of finding a single parameter for every image pixel to describe the bimodal tuning function of direction selectivity. In fact, for bimodal, circular variables such a calculation of direction using the mean angle by VS has little or no meaning (Batschelet, 1981). Apart from the VS method, the only function that has been applied so far for optically imaged orientation and direction data (Malonek et al., 1994; Shmuel and Grinvald, 1996), is the cosine-based fast Fourier transform (FFT) on the responses in the polar domain, a method proposed by Wörgötter and Eysel (Wörgötter and Eysel, 1987). These two methods reportedly gave nearly identical results. Indeed, it was recently demonstrated that the VS and Fourier methods are equivalent, at least for calculating orientation and direction preferences (Swindale, 1998).

### Considerations on the Use of Grating and Random Dot Stimuli to Obtain Direction Maps

Commonly, optical imaging studies apply luminance grating stimuli for revealing direction maps. Grating stimuli, however, contain orientational information in addition to directional information since they are composed of oriented edges. While this is a general problem with which all studies of direction selectivity using oriented stimuli are confronted, there have been attempts to extract the directional component from the activity images by producing so-called ‘differential images’. These are generated by subtracting images obtained to the same orientation but opposite directions (Malonek et al., 1994). As described above, such a normalization is compatible with the VM method. Importantly, however, the normalization step should be restricted to the strongest response component at every image pixel.

A more direct approach is to use random dot stimuli (texture) which contain directional but no orientational information. However, one must consider that most primary visual cortical neurons elicit strong axial responses to a dot-like stimulus moving parallel to the axis of the preferred orientation across the receptive field centre (Henry et al., 1974; Wörgötter and Eysel, 1989). Indeed, electrophysiological experiments revealed that the direction tuning of cortical cells for random dots can be different from the direction tuning for bars or gratings (Hammond, 1978; Skottun et al., 1988). Concomitantly, random dot stimuli provide direction maps whose activity distribution differs from that obtained with oriented stimuli (grating). While texture stimuli have not been exploited in optical imaging experiments so far, data from studies using dot-like stimuli have lent some support to this argument (Malonek et al., 1994; Shoham and Grinvald, 1996). For example, in figure 2A and C of Malonek et al. (Malonek et al., 1994), the direction maps generated to moving random dot-like patterns differ at many sites from the direction maps obtained with grating stimuli of the same region.

### Should the VM Method be Used for Calculating Orientation Maps?

The short answer is no. Since most visual cortical neurons display a unimodal orientation preference, the VS method yields a reliable estimate of it. It should be noted, however, that some cells show preferences to more than one orientations. The so-called ‘cross-detectors’ are known to possess two orientation preferences, chiefly at orthogonal orientations (Shevelev et al., 1994). For these neurons, the VM method might be a better choice in describing the preferred orientation than the VS method. Our calculations of orientation maps using the VS and VM methods revealed no major differences in the fine structure of orientation maps, suggesting that the majority of optical responses derive from cells with unimodal orientation tuning, at least under the conditions used.

### Limitations of the VM Method

The angle of the direction vector defined by the VM method always equals with one of the stimulus directions. Consequently, the angle resolution of the VM method is inherently poorer than that of the VS method. The data presented here were accumulated from experiments using 8–16 stimulus directions corresponding to 22.5–45° resolution. Interestingly, for the VM direction vectors such as those in Figure 3, the eight stimulus conditions (directions) were sufficient to reach a considerably smaller deviation from the expected orthogonality to the orientation vectors than for those calculated with the VS method. It is possible, however, that in experiments where the majority of direction responses are unimodal, the error of the VS method decreases, for example in cortical areas where neurons are organized in strictly direction-selective columns. Even in these cases, the VM method can be successfully used, although it may be necessary to increase the angle resolution of the stimulus set.

### Implications of the Columnar Distribution of Direction Selectivity on Optical Responses

#### Differences in the Strength of the Optical Signals Obtained with Orientation and Direction Stimuli

It has been shown that stimulus gratings moving in opposite directions produce activity images which differ only slightly from each other [Fig. 1C,D; see also figure 1 in Shmuel and Grinvald (Shmuel and Grinvald, 1996)]. In general, the amplitude of the direction SCMs is 3–5 times weaker than that of the orientation SCMs (adult cat) (Shmuel and Grinvald, 1996). A plausible interpretation of this phenomenon is that the columnar organization of the two attributes differs strongly. Indeed, in electrode penetrations which were perpendicular to the cortical surface, orientation selectivity was found rather constant throughout the different layers (Hubel and Wiesel, 1962; Murphy and Sillito, 1984). Contrary to this, direction preference changes abruptly by 180°, at least once, along vertical electrode penetrations, most commonly in the granular and infragranular layers (Berman et al., 1987). These findings suggest that while an optimally oriented stimulus that moves back and forth along the axis perpendicular to its orientation can activate the entire orientation column, the same oriented stimulus moving in one of the two opposite directions can activate only a subset of cells of the same cortical column. As a result of this, the average strength of the optical signal in the latter case must be weaker than in the former.

Another explanation for the signal strength difference between orientation and direction stimuli could be the high occurrence of bidirectionally tuned cells. In such a situation, the stimulus moving in one direction would elicit roughly half the number of spikes as the same stimulus moving in the same then in the opposite directions. This reasoning is supported by electrophysiological assessments indicating that 35–55% of units in area 18 subserving the central 0–15° of vision can respond to opposite direction stimuli (non-direction selective and direction tuned cells) (Orban et al., 1981). Importantly, many of these cells having complex receptive field types are found in the superficial layers. Thus, neurons with strongly bimodal direction tuning characteristics are likely to be encountered within the effective depth range of optical recording, providing a plausible explanation for the weaker optical signals of direction stimuli compared with those of orientation stimuli.

#### Bidirectional Locations in the Direction Maps

Bidirectional locations were quite frequent in our direction maps. As we have mentioned above, one of the possible explanations for their occurrence is that these sites contain a high number of bidirectional cells. None the less, an equally plausible explanation can be based on the coexistence of cell clusters with opposite direction preferences within the same cortical column (Berman et al., 1987). In this case, both clusters of cells of opposite direction preferences can contribute to the direction tuning of the same pixel in the optical map. The above considerations raise the question of from what cortical depth the optical signals derive. Based on the wavelength of the light (609 nm) used for illuminating the cortex and the optical index of refraction of the living brain tissue (nbrain = 1.38–1.41) (Bolin et al., 1989), it is estimated that not only the superficial layers but also layer 4 must have contributed to the measured activity pattern. Thus, it cannot be ruled out that the strongly bidirectional (bimodal) signals detected at a number of cortical locations represent the sum activity of two or more clusters of cells in the same column which show opposite direction preferences to each other.

### Comparison of the Organization between Orientation and Direction Maps

Former optical imaging studies concluded, although without compelling evidence, that direction preference vectors have a strong tendency to be orthogonal to orientation preference vectors (Shmuel and Grinvald, 1996; Weliky et al., 1996). Here we used a simple method with which this orthogonality could be readily demonstrated and applied to study the relationship between orientation and direction preference maps. While our method inherently resulted in discrete direction angles depending on the number of stimulus directions used, a generally smooth representation of direction preferences along the cortical surface was seen. The only exceptions were the locations of direction fracture lines where direction preferences had up to 180° reversals. The overall pattern and lateral spacing of fracture lines were in agreement with previous findings (Swindale et al., 1987; Shmuel and Grinvald, 1996; Weliky et al., 1996). Accordingly, they originate chiefly from orientation centres, but endpoints of fracture lines were also detected in orientation domains. Using a statistical evaluation, we showed that orientation centre regions contain, indeed, a high density of direction fracture lines (see Fig. 6D,E). With regard to the course of these fracture lines, they did not necessarily link orientation centres via the shortest possible route. Occasionally, two or three fracture lines were found to converge on the same orientation centre zone. On the basis of the present data, all orientation centres emit one or more direction fractures and all direction fractures are connected to at least one orientation centre. We also showed that orientation centres of the same or different polarity can be linked by direction fracture lines.

#### Direction Patches

The present analysis revealed that that iso-direction patches are, typically, 300–400 μm in diameter, about half the size of an orientation domain — see, for example, Figure 6A (Berman et al., 1987). Intriguingly, the fracture lines which divided direction patches of opposite signs seldom ran across the centre (ridge-like zone) of orientation domains. In this respect, an idealized relationship would be that each orientation domain is symmetrically divided into two equally sized direction patches so that an equal representation of all direction preferences could be achieved. For the majority of orientation domains studied here, no such a symmetric occurrence of opposite direction patches was found. We do not know whether the observed asymmetry represents a genuine organizational feature of direction or stems from limitations of the optical imaging technique.

#### Implications for Lateral Connections

It has been generally assumed that one of the main driving forces in establishing long-range lateral connections is functional similarity (Bosking et al., 1997; Schmidt et al., 1997). In this regard, previous studies have shown that lateral connections tend to link similar orientations (Gilbert and Wiesel, 1989; Bosking et al., 1997) and this tendency is somewhat stronger for excitatory connections than for inhibitory ones (Kisvárday et al., 1997). It is tempting to speculate that similarity in direction preference is also a driving force just as is orientation preference. Obviously, scheme (A) in Figure 7 fits better with this assumption than scheme (B), because in the former the same axon travelling along an iso-orientation line (thin lines) can readily establish contacts at iso-directions, whereas in the other situation (Fig. 7B) it has to make large jumps across patches of opposite directions (divided by direction fracture lines) in order to contact iso-directions. It remains a challenging task to explore how exactly the intracortical microcircuitry conforms to the above specific spatial constraints of orientation and direction preference maps.

## Notes

Authors would like to thank Ms Éva Tóth for her excellent technical assistance and Mr Vesselin Tzvetkov for his computer expertise. We wish to thank Drs David Fitzpatrick and Alan Humphrey for helpful discussions on direction selectivity. This work was supported by a grant of the Deutsche Forschungsgemeinschaft (SFB509 TP/A6 to ZFK).

Figure 1.

Optical images used in the present study. All activity images were taken by focusing the camera optics 650 μm below the cortical surface shown in (A). For each activity image, dark zones represent higher activity than light zones. Orientation SCMs, such as the one shown for a vertical stimulus grating in (B), were generated by summing activity maps representing the same orientation but opposite directions (C,D). dSCMs were computed by dividing activity maps produced by the same orientation but opposite directions. Examples of dSCMs to a vertical grating drifting to the right and left are shown in (E) and (F), respectively. For comparison, the same cortical regions are framed. Notice that the orientation domain overlapping with the framed area in (B) is composed of similar (although not identical) activity domains (C,D) corresponding to opposite stimulus directions. Normalization of the latter activity images to the opposite directions reveals distinct direction domains (E,F). L = lateral, A = anterior, scale bar (AF) = 1 mm.

Figure 1.

Optical images used in the present study. All activity images were taken by focusing the camera optics 650 μm below the cortical surface shown in (A). For each activity image, dark zones represent higher activity than light zones. Orientation SCMs, such as the one shown for a vertical stimulus grating in (B), were generated by summing activity maps representing the same orientation but opposite directions (C,D). dSCMs were computed by dividing activity maps produced by the same orientation but opposite directions. Examples of dSCMs to a vertical grating drifting to the right and left are shown in (E) and (F), respectively. For comparison, the same cortical regions are framed. Notice that the orientation domain overlapping with the framed area in (B) is composed of similar (although not identical) activity domains (C,D) corresponding to opposite stimulus directions. Normalization of the latter activity images to the opposite directions reveals distinct direction domains (E,F). L = lateral, A = anterior, scale bar (AF) = 1 mm.

Figure 3.

Quantitative comparison between the angle differences of orientation and direction vectors according to four calculation paradigms (64 000 pixels, data pooled from four hemispheres). (A) Differential direction SCMs followed by VS. (B) Differential direction SCMs followed by VM. (C) SCMs normalized to the ‘cocktail blank’ followed by VS. (D) SCMs normalized to the ‘cocktail blank’ followed by VM. The mean of each of the four distributions is close to 90°. However, only the VM method in conjunction with the ‘cocktail blank’ normalization gave rise to a small deviation of individual direction vectors from the expected 90° difference.

Figure 3.

Quantitative comparison between the angle differences of orientation and direction vectors according to four calculation paradigms (64 000 pixels, data pooled from four hemispheres). (A) Differential direction SCMs followed by VS. (B) Differential direction SCMs followed by VM. (C) SCMs normalized to the ‘cocktail blank’ followed by VS. (D) SCMs normalized to the ‘cocktail blank’ followed by VM. The mean of each of the four distributions is close to 90°. However, only the VM method in conjunction with the ‘cocktail blank’ normalization gave rise to a small deviation of individual direction vectors from the expected 90° difference.

Figure 4.

Scatter plot showing the angle differences between the direction vectors generated with the VS and VM methods plotted against the magnitude (length) of the direction vector calculated with the VS method. Each data point represents a different pixel of the same image (n = 44695). It is noticeable that the shorter the direction vector calculated with the VS method, the larger the angle difference. This recognition is supported by a statistically significant negative correlation (r = –0.26, P < 0.0001, Pearson correlation).

Figure 4.

Scatter plot showing the angle differences between the direction vectors generated with the VS and VM methods plotted against the magnitude (length) of the direction vector calculated with the VS method. Each data point represents a different pixel of the same image (n = 44695). It is noticeable that the shorter the direction vector calculated with the VS method, the larger the angle difference. This recognition is supported by a statistically significant negative correlation (r = –0.26, P < 0.0001, Pearson correlation).

Figure 5.

Demonstration of angular differences (C,G) for orientation (AD) and direction maps (EH) calculated with the VS (A,E) and VM (B,F) methods of an experiment using eight stimulus orientations and 16 directions. For orientation preferences, the VS (A) and VM (B) methods yield the same map, except at orientation centres where a few pixels difference can cause considerable change in the preferred orientation [light spots in (C)]. For direction preferences, the VS (E) and VM (F) methods resulted in rather different maps. Light-grey to white areas differ up to 180° between the VS and VM maps (G). (D,H) Quantitative evaluation of the differences between maps calculated with the VS and VM methods. The majority (90%) of pixels differed by up to 22° for orientation maps and 97° for direction maps. Color schemes: bars to (A,B), palette to (E,F). Grey-scale schemes: 0–180° to (C), 0–360° to (G). L = lateral, A = anterior, scale bar (AG) = 1 mm.

Figure 5.

Demonstration of angular differences (C,G) for orientation (AD) and direction maps (EH) calculated with the VS (A,E) and VM (B,F) methods of an experiment using eight stimulus orientations and 16 directions. For orientation preferences, the VS (A) and VM (B) methods yield the same map, except at orientation centres where a few pixels difference can cause considerable change in the preferred orientation [light spots in (C)]. For direction preferences, the VS (E) and VM (F) methods resulted in rather different maps. Light-grey to white areas differ up to 180° between the VS and VM maps (G). (D,H) Quantitative evaluation of the differences between maps calculated with the VS and VM methods. The majority (90%) of pixels differed by up to 22° for orientation maps and 97° for direction maps. Color schemes: bars to (A,B), palette to (E,F). Grey-scale schemes: 0–180° to (C), 0–360° to (G). L = lateral, A = anterior, scale bar (AG) = 1 mm.

Figure 6.

(A) Relationship between iso-orientation domains and direction patches. Dark patches represent regions which responded best to 135° stimulus grating. Arrows indicate regions which responded best to stimulus movements perpendicular to 135° (lower left scheme). These vectors were calculated using the VM method. Their length is proportional with the difference between the optimal and the opposite direction response components. Notice that two of the three iso-orientation domains are subdivided into two almost equal parts by opposite direction patches. At other locations, iso-orientation domains can be devoted largely to a single direction patch as shown on the left of the image. (B) Superimposition of direction ROC map [scheme below (B)] and iso-orientation contour lines (in black). The direction ROC map was calculated from a direction map using the VM method. Direction fractures representing high ROC appear in dark red. Light red lines indicate small changes of direction preferences between patches of slightly different direction response maximum. Notice that almost all direction fractures originate from orientation centres. In many cases, more than one direction fracture is associated with the same orientation centre. Close to the orientation centres, direction fractures run parallel with the iso-orientation lines, while at other locations they can intersect each other at right angles (arrows). (C) Quantitative comparison of the angle differences between rate of change values of orientation and direction maps (abscissa) showed that for the most part of the map, the direction fractures run parallel with the iso-orientation lines. This histogram is based on genuine fractures, including only those pixels of the map whose direction preference differed by 180° from its neighbour's. (D) Superimposition of orientation ROC map (blue shades) with direction ROC map [scheme below (B)]. Positive and negative symbols mark the polarity of orientation centres around which orientation preference changes, respectively, clockwise and counter-clockwise. Interestingly, direction fracture lines can link orientation centres of any polarity. (E) Histogram showing the frequency distribution of direction rate of change values around (eight pixel radius) all the 25 orientation centres (red bars) shown in (D) and at 25 randomly selected regions of the same size (yellow bars). Clearly, near to orientation centres direction fractures (high ROC of direction preference) are more likely to be encountered than elsewhere (Mann–Whitney U-test, P < 0.0001). Scale bar (A) = 1 mm.

Figure 6.

(A) Relationship between iso-orientation domains and direction patches. Dark patches represent regions which responded best to 135° stimulus grating. Arrows indicate regions which responded best to stimulus movements perpendicular to 135° (lower left scheme). These vectors were calculated using the VM method. Their length is proportional with the difference between the optimal and the opposite direction response components. Notice that two of the three iso-orientation domains are subdivided into two almost equal parts by opposite direction patches. At other locations, iso-orientation domains can be devoted largely to a single direction patch as shown on the left of the image. (B) Superimposition of direction ROC map [scheme below (B)] and iso-orientation contour lines (in black). The direction ROC map was calculated from a direction map using the VM method. Direction fractures representing high ROC appear in dark red. Light red lines indicate small changes of direction preferences between patches of slightly different direction response maximum. Notice that almost all direction fractures originate from orientation centres. In many cases, more than one direction fracture is associated with the same orientation centre. Close to the orientation centres, direction fractures run parallel with the iso-orientation lines, while at other locations they can intersect each other at right angles (arrows). (C) Quantitative comparison of the angle differences between rate of change values of orientation and direction maps (abscissa) showed that for the most part of the map, the direction fractures run parallel with the iso-orientation lines. This histogram is based on genuine fractures, including only those pixels of the map whose direction preference differed by 180° from its neighbour's. (D) Superimposition of orientation ROC map (blue shades) with direction ROC map [scheme below (B)]. Positive and negative symbols mark the polarity of orientation centres around which orientation preference changes, respectively, clockwise and counter-clockwise. Interestingly, direction fracture lines can link orientation centres of any polarity. (E) Histogram showing the frequency distribution of direction rate of change values around (eight pixel radius) all the 25 orientation centres (red bars) shown in (D) and at 25 randomly selected regions of the same size (yellow bars). Clearly, near to orientation centres direction fractures (high ROC of direction preference) are more likely to be encountered than elsewhere (Mann–Whitney U-test, P < 0.0001). Scale bar (A) = 1 mm.

Figure 7.

Spatial relationship between iso-orientation lines (thin lines) and direction fracture lines (thick lines). Two extreme cases can be observed in optical maps. (A) In most of the cases, the direction fracture line runs parallel to the iso-orientation lines and splits the orientation domain into patches of opposite direction preferences. The flanking orientation domains contain chiefly the same direction. (B) The fracture can split multiple orientation domains by running orthogonal to the iso-orientation lines. As a result of this, the ratio of areas devoted to opposite directions is equal in a series of orientation domains. Interestingly, this situation is less frequent than that in (A).

Figure 7.

Spatial relationship between iso-orientation lines (thin lines) and direction fracture lines (thick lines). Two extreme cases can be observed in optical maps. (A) In most of the cases, the direction fracture line runs parallel to the iso-orientation lines and splits the orientation domain into patches of opposite direction preferences. The flanking orientation domains contain chiefly the same direction. (B) The fracture can split multiple orientation domains by running orthogonal to the iso-orientation lines. As a result of this, the ratio of areas devoted to opposite directions is equal in a series of orientation domains. Interestingly, this situation is less frequent than that in (A).

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