Exp Brain Res (2014) 232:2317–2324 DOI 10.1007/s00221-014-3928-7
Research Article
Two independent sources of anisotropy in the visual representation of direction in 2‑D space Nikolaos Smyrnis · Asimakis Mantas · Ioannis Evdokimidis
Received: 8 January 2014 / Accepted: 20 March 2014 / Published online: 3 April 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract It is known that visual direction representation is more accurate for cardinal directions compared to oblique, a phenomenon named the “oblique effect”. It has been hypothesized that there are two sources of oblique effect, a low level one confined to vision and a high level one extending to different modalities and corresponding to higher cognitive processes. In this study directional error (DE) was measured when normal individuals tried to align the direction of an arrow presented in the center of a computer monitor to the direction of a peripheral target located in one of 32 directions equally spaced on an imaginary circle of 60 mm radius. Task difficulty was manipulated by varying arrow length (15, 30, 45 and 60 mm). By measuring mean DE and its variance we identified two independent sources of the oblique effect. A low level oblique effect was manifested in higher accuracy or equivalently lower variance of DE in the alignment for cardinal orientations compared to oblique. A second oblique effect was manifested measuring mean DE resulting in space expansion in the vicinity of cardinal directions and space contraction in the vicinity of oblique directions. Only this latter source of oblique effect was modulated by arrow length as predicted from a theoretical model postulating that this oblique
N. Smyrnis (*) Laboratory of Sensorimotor Control, University Mental Health Research Institute, Soranou Efesiou 2, 11527 Athens, Greece e-mail: [email protected] N. Smyrnis · A. Mantas · I. Evdokimidis Neurology Department, Eginition Hospital, National and Kapodistrian University of Athens, Athens, Greece N. Smyrnis Psychiatry Department, Eginition Hospital, National and Kapodistrian University of Athens, Athens, Greece
effect is produced by a cognitive process of 2-D space categorization. Keywords Oblique effect · Space categorization · Space representation · Arrow pointing · Mean directional error · Variable directional error
Introduction In a series of studies we investigated the directional accuracy of planar pointing movements to visually presented targets using a memory delay paradigm (Smyrnis et al. 2000; Gourtzelidis et al. 2001; Smyrnis et al. 2007). We showed that when subjects pointed to the location of previously seen targets, a systematic directional error (DE) was observed, that varied with target direction. This systematic DE reflected a bias for movement endpoints to cluster towards the oblique directions between the cardinal axes. After excluding more trivial explanations for this error, relating to the mechanical properties of the arm (Smyrnis et al. 2000), we sought to explain this phenomenon as an effect of spatial working memory (Gourtzelidis et al. 2001). The same pattern of systematic DE has been observed in a series of studies of spatial working memory where subjects had to memorize the location of a dot within a circle and then use a pen to draw the dot in an empty circle (Huttenlocher et al. 1991). A category-adjustment theoretical model was developed to account for the pattern of errors in both direction and amplitude, proposing that these errors emerged from a strategy of subjects to categorize space, in order to help them memorize the spatial location of the targets (Huttenlocher et al. 2004). These systematic errors though are not confined to memory. In a study measuring DE in slow pointing movements
13
2318
it was found that the initial movement direction consistently deviated from the target direction and the pattern of systematic DE that emerged, was surprisingly identical to the one we observed in fast pointing movements performed in memory conditions (de Graaf et al. 1991). We also observed the same pattern of DE in the initial direction of fast pointing movements (Mantas et al. 2008). The same pattern of systematic DE were observed when subjects used a pointer to point in the direction of a target in 2-D space, suggesting that these errors originated in perception (de Graaf et al. 1991). In a follow up study de Graaf et al. (1994) showed that the same systematic DE were observed when targets were presented using kinesthetic instead of visual input. Also in a study of pointing movements, where the movement endpoints where defined by the passive positioning of the arm in a location in 2-D space by a robot arm, the same pattern of systematic DE emerged (Baud-Bovy and Viviani 2004). A modification of the category-adjustment model was used to provide a theoretical explanation of these results (BaudBovy and Gentaz 2012). The same pattern of systematic DE was also observed in the direction of smooth eye pursuit movements (Krukowski and Stone 2005). In that study the authors measured the gain, that is the change of mean DE in smooth eye pursuit between neighboring directions, and showed that the pattern of systematic DE in smooth eye pursuit reflected a space expansion in the vicinity of the cardinal directions (horizontal and vertical) and a space compression in the vicinity of the oblique directions. They then linked this spatial anisotropy to a known phenomenon in perception called the “oblique effect”. The oblique effect refers to the superiority of discrimination of direction or orientation for cardinal directions or orientations compared to oblique (Appelle 1972). We reached the same conclusion in our previous study (Smyrnis et al. 2007) showing that the systematic error pattern in pointing movements was linked to an oblique effect in visual direction discrimination. In conclusion then studies in different modalities had confirmed the presence of a directional anisotropy in 2-D space that corresponds to an oblique effect in the perception of direction. This oblique effect is caused by space expansion in the vicinity of cardinal directions and space contraction in the vicinity of oblique directions and has been explained by a theoretical model suggesting that spatial representation of direction involves a categorization of space in which oblique directions (Huttenlocher et al. 2004) or cardinal and oblique directions (Baud-Bovy and Gentaz 2012) serve as categories. Theoretically the oblique effect in the representation of direction in 2-D space could result from two independent sources. The first source is related to the systematic distortion of space that we previously described. The second source is related to differences in the accuracy of
13
Exp Brain Res (2014) 232:2317–2324
representation of direction. Studies that used reproduction of visual orientation showed that the variance in the reproduction of cardinal orientations is significantly smaller compared to the variance in the reproduction of oblique orientations (Appelle 1972). Maffei and Campbell (1970) showed that the amplitude of the evoked potential for visually presented vertical and horizontal gratings was larger than that for oblique gratings. In an fMRI study it was also observed that the magnitude of the BOLD response in area V1 for horizontal and vertical lines was larger than that for oblique lines (Furmanski and Engel 2000). These findings suggested that the oblique effect in vision was the result of a larger number of detectors for cardinal directions in primary visual cortex leading to a larger accuracy for representing these directions. A theoretical framework by Essock (1980) proposed the existence of two classes of oblique effect, a purely visual one (class 1) related to low level visual processing that is observed using measures of visual spatial resolution (acuity) and contrast sensitivity and a higher level oblique effect that is evoked in several sensory modalities (visual, vestibular, kinesthetic, haptic), consistent with the idea that this effect stems from a bias of “higher-level” representation, memory, and/or recall of orientation information (class 2). In subsequent studies Essock et al. (1992, 1997) showed that low-level sensitivity on the human finger ventral surface shows a proximal–distal bias where proximal– distal (“vertical”) stimuli were detected best and medial– lateral (“horizontal”) stimuli were the least-well detected. They concluded that for the visual modality, both the lowlevel (class 1) and high-level (class 2) directional biases show an “oblique-effect” anisotropic pattern. However, depending upon the ability measured and the location on the body tested, there are somatosensory low level (class 1) anisotropies that show other patterns. In our previous study (Smyrnis et al. 2007) we did not observe significant differences in the variance of DE for the arm pointing task. Thus the oblique effect observed for movement endpoints was based only on the systematic DE pattern. The same was true for the oblique effect in smooth eye pursuit (Krukowski and Stone 2005) and the haptic oblique effect (Gentaz et al. 2008). Based on these results one could argue that the oblique effect seen in other modalities such as the haptic oblique effect (Gentaz et al. 2008) or the oblique effect in movement endpoints (Smyrnis et al. 2007) is a class 2 oblique effect that is based upon the systematic DE while the oblique effect in vision according to the theoretical scheme of Esscok and colleagues is a combination of class 1 and class 2 oblique effect reflected accordingly in the accuracy (variable DE) and the systematic DE. In this study we used the arrow alignment task similar to that of de Graaf et al. (1991) in which the subject has to align the direction of an arrow to point to a visual target
Exp Brain Res (2014) 232:2317–2324
at different directions in 2-D space and manipulated the amount of available directional information and thus the difficulty of the task by introducing four arrow lengths. The hypothesis investigated was that in this purely visual task we would observe two independent sources of oblique effect. The first would be caused by better accuracy in the representation of cardinal directions compared to oblique. This effect would manifest itself by measuring the variance of the DE. Our prediction was that this low level oblique effect would not be affected by the amount of available directional information (arrow length), a manipulation affecting task difficulty. The second oblique effect would be caused by the systematic DE introduced by the cognitive process of spatial categorization (Huttenlocher et al. 2004; Baud-Bovy and Gentaz 2012) and would manifest itself by measuring the mean DE and the gain (change of mean DE with target direction) in the arrow alignment task. Our prediction was that this oblique effect would increase with decreasing amount of directional information (decreasing arrow length) reflecting an increased reliance on spatial categories with increasing task difficulty.
Methods Subjects Five healthy adults performed the arrow alignment experiment (age span: 25–42 years, 2 men). All participants were naïve to the purposes of this study and gave written informed consent for participation in the study after a detailed explanation to them, of the experimental procedures. The experimental protocol was approved by the Eginition Hospital Scientific and Ethics Committee. All participants were right handed and performed the tasks using their preferred right hand.
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Fig. 1 Graphic representation of the arrow pointing task. On each trial a single arrow of one length appeared and the subject was instructed to point the white target by moving the arrow clockwise or counterclockwise from its initial position. For presentation reasons all four arrow lengths are presented on the figure
of an imaginary circle of 6 cm radius in one of 32 directions (11.25° intervals) and a yellow arrow appeared originating at the center target. The yellow arrow pointed 45° away from the target clockwise or counterclockwise chosen at random and the subject was instructed to align the arrow to the direction of the target by moving it in the opposite direction using the arrow buttons of the keyboard (Fig. 1). Each button press moved the arrow by 1.4°. The arrow length varied among four values (15, 30, 45 and 60 mm). Each subject performed one experimental session every day. In each session the subject performed four repetitions for every arrow length, for every target direction in a randomized sequence, for a total of 512 trials (4 repetitions × 4 arrow lengths × 32 target directions) presented in a random order. Each subject performed four sessions in 4 days (total number of trials for each subject was 4 × 512 = 3,072).
Set up and procedure Data analysis Subjects sat comfortably in front of a computer monitor (HITACHI CM630ET, 32.8 cm horizontal × 24.5 cm vertical) at a distance of approximately 60 cm. The amplitude of the target stimuli was 6 cm from the center target, thus the degrees of visual angle for all stimuli were 5.7°. It should be mentioned that this is an approximate value since the head was not constrained and the subjects were not instructed to fixate at the center during the experiment. The subjects used two fingers of their right hand to press the left or right arrow key on the computer keyboard. Each trial started when a filled red disk (5 mm diameter) appeared at the center of the screen (center target). After a variable period of 1–2 s, a second white filled disk (5 mm diameter), the peripheral target, appeared at the circumference
The DE was the polar angular difference in degrees of the direction of the arrow minus the direction of the peripheral target. A counter-clockwise deviation from the peripheral target was defined as positive DE. We excluded trials where the latency of the first arrow key press was 2,000 ms and DE > 8.4° or DE 1. Finally in the last case (open triangles and dashed line) the mean direction of arrow endpoints for n − 1 is larger than n − 1 and for n + 1 is smaller than n + 1 thus the gain is smaller than 1
arrow length data were excluded, since performance at this arrow length was almost perfect (ceiling effect). This was expected since this arrow length was equal to the amplitude of the target. This condition was used to make sure that subjects were concentrated on task performance and were trying to perform as accurately as possible. Gain is a measure of the rate with which the direction of the arrow varies for different target directions and shows whether the directional space is expanded or contracted with respect to the target directional space (see also Krukowski and Stone 2005). Figure 2 presents how this measure is derived. The figure plots the mean direction of the arrow on the Y axis (output) versus the direction of the target on the X axis (input). We called gain for target direction n, the slope of the best fitting regression line for mean arrow directions at the two neighbor targets n − 1, n + 1 and target n. This line is given by the equation:
Predicted mean arrow direction = constant + gain ∗ target direction
Oblique_sd = (Mean SD oblique − Mean SD cardinal)/ (Mean SD cardinal + Mean SD oblique) (2) The second index represents the oblique effect caused by gain modulation: Oblique_gain = (Mean gain cardinal − Mean gain oblique)/ (Mean gain cardinal + Mean gain oblique) (3)
A one way ANOVA was performed to test the effect of arrow length on each of the oblique effect indexes. For all statistical analyses we used the STATISTICA 7.0 software (StatSoft, Inc. 194-2001).
(1)
In the hypothetical case of no anisotropy between target and arrow direction (filled squares in Fig. 2), the gain for target n equals 1 (solid line). Thus in this case the process of aligning the arrow direction results in no distortion of directional space in the vicinity of target neither expansion nor contraction. In the case where the mean arrow directions for neighbor target directions are shifted away from
13
target n (open circles in Fig. 2), the gain will be >1 (dotted line in Fig. 2). Thus the process of aligning the arrow results in the expansion of directional space in the vicinity of target n (larger output difference for the same input difference). Finally, in the case where the mean directions of the arrow for the neighbor target directions are shifted towards n (filled triangles in Fig. 2), the gain will be 1) then it is obligatory that directional space will be contracted in another neighboring region (gain
Research Article
Two independent sources of anisotropy in the visual representation of direction in 2‑D space Nikolaos Smyrnis · Asimakis Mantas · Ioannis Evdokimidis
Received: 8 January 2014 / Accepted: 20 March 2014 / Published online: 3 April 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract It is known that visual direction representation is more accurate for cardinal directions compared to oblique, a phenomenon named the “oblique effect”. It has been hypothesized that there are two sources of oblique effect, a low level one confined to vision and a high level one extending to different modalities and corresponding to higher cognitive processes. In this study directional error (DE) was measured when normal individuals tried to align the direction of an arrow presented in the center of a computer monitor to the direction of a peripheral target located in one of 32 directions equally spaced on an imaginary circle of 60 mm radius. Task difficulty was manipulated by varying arrow length (15, 30, 45 and 60 mm). By measuring mean DE and its variance we identified two independent sources of the oblique effect. A low level oblique effect was manifested in higher accuracy or equivalently lower variance of DE in the alignment for cardinal orientations compared to oblique. A second oblique effect was manifested measuring mean DE resulting in space expansion in the vicinity of cardinal directions and space contraction in the vicinity of oblique directions. Only this latter source of oblique effect was modulated by arrow length as predicted from a theoretical model postulating that this oblique
N. Smyrnis (*) Laboratory of Sensorimotor Control, University Mental Health Research Institute, Soranou Efesiou 2, 11527 Athens, Greece e-mail: [email protected] N. Smyrnis · A. Mantas · I. Evdokimidis Neurology Department, Eginition Hospital, National and Kapodistrian University of Athens, Athens, Greece N. Smyrnis Psychiatry Department, Eginition Hospital, National and Kapodistrian University of Athens, Athens, Greece
effect is produced by a cognitive process of 2-D space categorization. Keywords Oblique effect · Space categorization · Space representation · Arrow pointing · Mean directional error · Variable directional error
Introduction In a series of studies we investigated the directional accuracy of planar pointing movements to visually presented targets using a memory delay paradigm (Smyrnis et al. 2000; Gourtzelidis et al. 2001; Smyrnis et al. 2007). We showed that when subjects pointed to the location of previously seen targets, a systematic directional error (DE) was observed, that varied with target direction. This systematic DE reflected a bias for movement endpoints to cluster towards the oblique directions between the cardinal axes. After excluding more trivial explanations for this error, relating to the mechanical properties of the arm (Smyrnis et al. 2000), we sought to explain this phenomenon as an effect of spatial working memory (Gourtzelidis et al. 2001). The same pattern of systematic DE has been observed in a series of studies of spatial working memory where subjects had to memorize the location of a dot within a circle and then use a pen to draw the dot in an empty circle (Huttenlocher et al. 1991). A category-adjustment theoretical model was developed to account for the pattern of errors in both direction and amplitude, proposing that these errors emerged from a strategy of subjects to categorize space, in order to help them memorize the spatial location of the targets (Huttenlocher et al. 2004). These systematic errors though are not confined to memory. In a study measuring DE in slow pointing movements
13
2318
it was found that the initial movement direction consistently deviated from the target direction and the pattern of systematic DE that emerged, was surprisingly identical to the one we observed in fast pointing movements performed in memory conditions (de Graaf et al. 1991). We also observed the same pattern of DE in the initial direction of fast pointing movements (Mantas et al. 2008). The same pattern of systematic DE were observed when subjects used a pointer to point in the direction of a target in 2-D space, suggesting that these errors originated in perception (de Graaf et al. 1991). In a follow up study de Graaf et al. (1994) showed that the same systematic DE were observed when targets were presented using kinesthetic instead of visual input. Also in a study of pointing movements, where the movement endpoints where defined by the passive positioning of the arm in a location in 2-D space by a robot arm, the same pattern of systematic DE emerged (Baud-Bovy and Viviani 2004). A modification of the category-adjustment model was used to provide a theoretical explanation of these results (BaudBovy and Gentaz 2012). The same pattern of systematic DE was also observed in the direction of smooth eye pursuit movements (Krukowski and Stone 2005). In that study the authors measured the gain, that is the change of mean DE in smooth eye pursuit between neighboring directions, and showed that the pattern of systematic DE in smooth eye pursuit reflected a space expansion in the vicinity of the cardinal directions (horizontal and vertical) and a space compression in the vicinity of the oblique directions. They then linked this spatial anisotropy to a known phenomenon in perception called the “oblique effect”. The oblique effect refers to the superiority of discrimination of direction or orientation for cardinal directions or orientations compared to oblique (Appelle 1972). We reached the same conclusion in our previous study (Smyrnis et al. 2007) showing that the systematic error pattern in pointing movements was linked to an oblique effect in visual direction discrimination. In conclusion then studies in different modalities had confirmed the presence of a directional anisotropy in 2-D space that corresponds to an oblique effect in the perception of direction. This oblique effect is caused by space expansion in the vicinity of cardinal directions and space contraction in the vicinity of oblique directions and has been explained by a theoretical model suggesting that spatial representation of direction involves a categorization of space in which oblique directions (Huttenlocher et al. 2004) or cardinal and oblique directions (Baud-Bovy and Gentaz 2012) serve as categories. Theoretically the oblique effect in the representation of direction in 2-D space could result from two independent sources. The first source is related to the systematic distortion of space that we previously described. The second source is related to differences in the accuracy of
13
Exp Brain Res (2014) 232:2317–2324
representation of direction. Studies that used reproduction of visual orientation showed that the variance in the reproduction of cardinal orientations is significantly smaller compared to the variance in the reproduction of oblique orientations (Appelle 1972). Maffei and Campbell (1970) showed that the amplitude of the evoked potential for visually presented vertical and horizontal gratings was larger than that for oblique gratings. In an fMRI study it was also observed that the magnitude of the BOLD response in area V1 for horizontal and vertical lines was larger than that for oblique lines (Furmanski and Engel 2000). These findings suggested that the oblique effect in vision was the result of a larger number of detectors for cardinal directions in primary visual cortex leading to a larger accuracy for representing these directions. A theoretical framework by Essock (1980) proposed the existence of two classes of oblique effect, a purely visual one (class 1) related to low level visual processing that is observed using measures of visual spatial resolution (acuity) and contrast sensitivity and a higher level oblique effect that is evoked in several sensory modalities (visual, vestibular, kinesthetic, haptic), consistent with the idea that this effect stems from a bias of “higher-level” representation, memory, and/or recall of orientation information (class 2). In subsequent studies Essock et al. (1992, 1997) showed that low-level sensitivity on the human finger ventral surface shows a proximal–distal bias where proximal– distal (“vertical”) stimuli were detected best and medial– lateral (“horizontal”) stimuli were the least-well detected. They concluded that for the visual modality, both the lowlevel (class 1) and high-level (class 2) directional biases show an “oblique-effect” anisotropic pattern. However, depending upon the ability measured and the location on the body tested, there are somatosensory low level (class 1) anisotropies that show other patterns. In our previous study (Smyrnis et al. 2007) we did not observe significant differences in the variance of DE for the arm pointing task. Thus the oblique effect observed for movement endpoints was based only on the systematic DE pattern. The same was true for the oblique effect in smooth eye pursuit (Krukowski and Stone 2005) and the haptic oblique effect (Gentaz et al. 2008). Based on these results one could argue that the oblique effect seen in other modalities such as the haptic oblique effect (Gentaz et al. 2008) or the oblique effect in movement endpoints (Smyrnis et al. 2007) is a class 2 oblique effect that is based upon the systematic DE while the oblique effect in vision according to the theoretical scheme of Esscok and colleagues is a combination of class 1 and class 2 oblique effect reflected accordingly in the accuracy (variable DE) and the systematic DE. In this study we used the arrow alignment task similar to that of de Graaf et al. (1991) in which the subject has to align the direction of an arrow to point to a visual target
Exp Brain Res (2014) 232:2317–2324
at different directions in 2-D space and manipulated the amount of available directional information and thus the difficulty of the task by introducing four arrow lengths. The hypothesis investigated was that in this purely visual task we would observe two independent sources of oblique effect. The first would be caused by better accuracy in the representation of cardinal directions compared to oblique. This effect would manifest itself by measuring the variance of the DE. Our prediction was that this low level oblique effect would not be affected by the amount of available directional information (arrow length), a manipulation affecting task difficulty. The second oblique effect would be caused by the systematic DE introduced by the cognitive process of spatial categorization (Huttenlocher et al. 2004; Baud-Bovy and Gentaz 2012) and would manifest itself by measuring the mean DE and the gain (change of mean DE with target direction) in the arrow alignment task. Our prediction was that this oblique effect would increase with decreasing amount of directional information (decreasing arrow length) reflecting an increased reliance on spatial categories with increasing task difficulty.
Methods Subjects Five healthy adults performed the arrow alignment experiment (age span: 25–42 years, 2 men). All participants were naïve to the purposes of this study and gave written informed consent for participation in the study after a detailed explanation to them, of the experimental procedures. The experimental protocol was approved by the Eginition Hospital Scientific and Ethics Committee. All participants were right handed and performed the tasks using their preferred right hand.
2319
Fig. 1 Graphic representation of the arrow pointing task. On each trial a single arrow of one length appeared and the subject was instructed to point the white target by moving the arrow clockwise or counterclockwise from its initial position. For presentation reasons all four arrow lengths are presented on the figure
of an imaginary circle of 6 cm radius in one of 32 directions (11.25° intervals) and a yellow arrow appeared originating at the center target. The yellow arrow pointed 45° away from the target clockwise or counterclockwise chosen at random and the subject was instructed to align the arrow to the direction of the target by moving it in the opposite direction using the arrow buttons of the keyboard (Fig. 1). Each button press moved the arrow by 1.4°. The arrow length varied among four values (15, 30, 45 and 60 mm). Each subject performed one experimental session every day. In each session the subject performed four repetitions for every arrow length, for every target direction in a randomized sequence, for a total of 512 trials (4 repetitions × 4 arrow lengths × 32 target directions) presented in a random order. Each subject performed four sessions in 4 days (total number of trials for each subject was 4 × 512 = 3,072).
Set up and procedure Data analysis Subjects sat comfortably in front of a computer monitor (HITACHI CM630ET, 32.8 cm horizontal × 24.5 cm vertical) at a distance of approximately 60 cm. The amplitude of the target stimuli was 6 cm from the center target, thus the degrees of visual angle for all stimuli were 5.7°. It should be mentioned that this is an approximate value since the head was not constrained and the subjects were not instructed to fixate at the center during the experiment. The subjects used two fingers of their right hand to press the left or right arrow key on the computer keyboard. Each trial started when a filled red disk (5 mm diameter) appeared at the center of the screen (center target). After a variable period of 1–2 s, a second white filled disk (5 mm diameter), the peripheral target, appeared at the circumference
The DE was the polar angular difference in degrees of the direction of the arrow minus the direction of the peripheral target. A counter-clockwise deviation from the peripheral target was defined as positive DE. We excluded trials where the latency of the first arrow key press was 2,000 ms and DE > 8.4° or DE 1. Finally in the last case (open triangles and dashed line) the mean direction of arrow endpoints for n − 1 is larger than n − 1 and for n + 1 is smaller than n + 1 thus the gain is smaller than 1
arrow length data were excluded, since performance at this arrow length was almost perfect (ceiling effect). This was expected since this arrow length was equal to the amplitude of the target. This condition was used to make sure that subjects were concentrated on task performance and were trying to perform as accurately as possible. Gain is a measure of the rate with which the direction of the arrow varies for different target directions and shows whether the directional space is expanded or contracted with respect to the target directional space (see also Krukowski and Stone 2005). Figure 2 presents how this measure is derived. The figure plots the mean direction of the arrow on the Y axis (output) versus the direction of the target on the X axis (input). We called gain for target direction n, the slope of the best fitting regression line for mean arrow directions at the two neighbor targets n − 1, n + 1 and target n. This line is given by the equation:
Predicted mean arrow direction = constant + gain ∗ target direction
Oblique_sd = (Mean SD oblique − Mean SD cardinal)/ (Mean SD cardinal + Mean SD oblique) (2) The second index represents the oblique effect caused by gain modulation: Oblique_gain = (Mean gain cardinal − Mean gain oblique)/ (Mean gain cardinal + Mean gain oblique) (3)
A one way ANOVA was performed to test the effect of arrow length on each of the oblique effect indexes. For all statistical analyses we used the STATISTICA 7.0 software (StatSoft, Inc. 194-2001).
(1)
In the hypothetical case of no anisotropy between target and arrow direction (filled squares in Fig. 2), the gain for target n equals 1 (solid line). Thus in this case the process of aligning the arrow direction results in no distortion of directional space in the vicinity of target neither expansion nor contraction. In the case where the mean arrow directions for neighbor target directions are shifted away from
13
target n (open circles in Fig. 2), the gain will be >1 (dotted line in Fig. 2). Thus the process of aligning the arrow results in the expansion of directional space in the vicinity of target n (larger output difference for the same input difference). Finally, in the case where the mean directions of the arrow for the neighbor target directions are shifted towards n (filled triangles in Fig. 2), the gain will be 1) then it is obligatory that directional space will be contracted in another neighboring region (gain
