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Article  |   January 2015
Evidence for tilt normalization can be explained by anisotropic orientation sensitivity
Author Affiliations
  • Katherine R. Storrs
    School of Psychology, The University of Queensland, St. Lucia, Queensland, Australia
    k.storrs@uq.edu.au
  • Derek H. Arnold
    School of Psychology, The University of Queensland, St. Lucia, Queensland, Australia
    d.arnold@psy.uq.edu.au
Journal of Vision January 2015, Vol.15, 26. doi:10.1167/15.1.26
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      Katherine R. Storrs, Derek H. Arnold; Evidence for tilt normalization can be explained by anisotropic orientation sensitivity. Journal of Vision 2015;15(1):26. doi: 10.1167/15.1.26.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Some data have been taken as evidence that after prolonged viewing, near-vertical orientations “normalize” to appear more vertical than they did previously. After almost a century of research, the existence of tilt normalization remains controversial. The most recent evidence for tilt normalization comes from data suggesting a measurable “perceptual drift” of near-vertical adaptors toward vertical, which can be nulled by a slight physical rotation away from vertical (Müller, Schillinger, Do, & Leopold, 2009). We argue that biases in estimates of perceptual stasis could, however, result from the anisotropic organization of orientation-selective neurons in V1, with vertically-selective cells being more narrowly tuned than obliquely-selective cells. We describe a neurophysiologically plausible model that predicts greater sensitivity to orientation displacements toward than away from vertical. We demonstrate the predicted asymmetric pattern of sensitivity in human observers by determining threshold speeds for detecting rotation direction (Experiment 1), and by determining orientation discrimination thresholds for brief static stimuli (Experiment 2). Results imply that data suggesting a perceptual drift toward vertical instead result from greater discrimination sensitivity around cardinal than oblique orientations (the oblique effect), and thus do not constitute evidence for tilt normalization.

Introduction
Prolonged exposure to a visual stimulus can alter perception in two ways—by changing the appearance of the stimulus itself (e.g., fading color saturation; see Webster, 1996), or by changing the appearance of other subsequent stimuli (e.g., the direction aftereffect; see Anstis, Verstraten & Mather, 1998). Patterns of adaptation-induced changes for a particular perceptual dimension can give insight into how that dimension is neurally encoded (see, e.g., Kohn, 2007). 
It is well documented that after prolonged exposure to one orientation, the orientation of subsequent stimuli can seem repelled away from the adapted orientation—the tilt aftereffect (Gibson, 1933). Controversially, it has also been argued that vertical and horizontal (the cardinal orientations) constitute ‘norms' for orientation perception (e.g., Gibson, 1933; Howard, 1982; Müller et al., 2009), and that over time near-cardinally oriented stimuli can undergo a perceptual change, being drawn toward the nearest cardinal axis—tilt normalization (Day & Wade, 1969; Gibson & Radner, 1937; Held, 1963; Müller et al., 2009; Prentice & Beardslee, 1950; Vaitkevicius et al., 2009; Vernon, 1934). 
Tilt aftereffects are often explained via models in which perceived orientation is determined by activity across a population of orientation-tuned neurons, and can be biased by adaptation-induced changes in the distribution of that activity (Barlow & Hill, 1963; Clifford, Ma Wyatt, Arnold, Smith, & Wenderoth, 2001; Clifford, Wenderoth, & Spehar, 2000; Day, 1962; Girshick, Landy, & Simoncelli, 2011). However, population models of orientation coding do not strongly predict normalization (and can even predict the opposite; see below). Some have suggested there might therefore be separate neural processes underlying tilt normalization and tilt aftereffects (Coltheart, 1970; Morant & Harris, 1965; Morant & Mistovich, 1960; Müller et al., 2009). An alternative possibility is that data thought to demonstrate tilt normalization have been misinterpreted. 
The earliest reports of normalization were based on the finding that after adapting to a tilted line, lines adjusted to appear vertical tend to be displaced toward the adapting orientation (Day & Wade, 1969; Gibson & Radner, 1937; Vernon, 1934). This can be considered as evidence only for a standard tilt aftereffect. A more direct method to measure normalization is to adapt observers for a period in one part of their visual field, then briefly present a second stimulus elsewhere, estimating the orientation at which the second stimulus appears parallel with the persistent adaptor. Data from such paradigms suggest that a persistent near-vertical standard is seen as less tilted than a brief comparison to which it is either physically (Prentice & Beardslee, 1950; Vaitkevicius et al., 2009) or perceptually (Held, 1963) matched prior to adaptation (although at least one study failed to replicate this effect: Heinemann & Marill, 1954). 
While there is some evidence that persistent and transient stimuli differ in apparent orientation, it is unclear whether this mismatch is due to normalization of the prolonged stimulus or to misperception of the brief stimulus. As in the studies described above, Andrews (1965, 1967) estimated the orientation at which a brief test appeared parallel to a prolonged standard and found that when the standard was near vertical or horizontal, physically matched tests were judged as more tilted. However, Andrews (1967) showed that the magnitude of this mismatch depended on both the presentation time and spatial extent of the comparison. Larger mismatches were found for shorter lines, and the bias reduced by around two-thirds within the first second of comparison exposure. This dependence on the properties of the comparison suggests the mismatch between brief and prolonged near-cardinal orientations might be a measurement of misperceptions in briefly-presented stimuli (Howard, 1982, p. 151; for a similar argument regarding putative curvature normalization, see Coren & Festiniger, 1967). Data from experiments in which the orientation of a persistent “adaptor” is compared to brief comparators (e.g., Andrews, 1965, 1967; Heinemann & Marill, 1954; Held, 1963; Prentice & Beardslee, 1950; Vaitkevicius et al., 2009) are therefore ambiguous. 
Recently, Müller et al. (2009) presented a novel method by which to measure normalization that eliminates the need for a comparison. They suggested that tilt normalization involves a detectable “perceptual drift” of tilted stimuli toward vertical or horizontal, and that this can be quantified by measuring the rate of physical rotation away from a cardinal axis required to null the perceptual rotation. The experimenters used an adaptive procedure to estimate points of subjective stasis. For stimuli with an initial orientation of 0°, 90°, or ±45° subjective stasis estimates coincided with physical stasis. For all other orientations, subjective stasis estimates corresponded with a physical rotation away from the nearest cardinal axis, with the “perceptual drift rate” peaking at ±15° from cardinal axes. The magnitude of perceptual drift measured in this manner was smaller than that of the tilt aftereffect, followed a different pattern as a function of adapting orientation, and magnitudes of the two effects were not correlated across observers. These data therefore seemed to present conclusive evidence both that normalization occurs, and that it arises from a different process than the tilt aftereffect. However, we propose that the measured “drift” can be more parsimoniously explained as an artefact arising from the oblique effect (i.e., lower detection and discrimination thresholds for orientations near cardinal than oblique axes; Appelle, 1972). 
Modeling
Here we will show that a neurophysiologically plausible model of orientation coding, which predicts the oblique effect because of anisotropies in the bandwidths of orientation-selective neurons (Girshick, Landy, & Simoncelli, 2011; Li, Peterson, & Freeman, 2003; Rose & Blakemore, 1974), also predicts asymmetric just-noticeable-differences (JNDs) for orientation changes from near-vertical and near-horizontal standard orientations. While this type of orientation coding model is well-established, this predictive feature of such models has not previously been appreciated or formally expressed. 
Figure 1a shows an idealized population of 601 V1 orientation-selective neurons. Each neuron's tuning curve is represented by a von Mises distribution (Swindale, 1998) drawn on a circular continuum spanning 0 to 180° using the Circular Statistics Toolbox for Matlab (Berens, 2009). For each tuning curve, the distribution mean indicates the preferred orientation of the simulated neuron, and the distribution variance indicates the bandwidth of the tuning curve (e.g., Clifford et al., 2000, 2001; Pouget, Dayan, & Zemel, 2000). The value of a neuron's tuning curve at any given orientation represents its average firing rate (as a proportion of its maximum firing rate) when presented with that orientation. 
Figure 1
 
(a) The tuning curves of 60 orientation-selective neurons simulated using von Mises distributions, with tuning bandwidths ∼1.4 times wider near the oblique than cardinal axes (two cardinal and two oblique tuning curves are shown in bold to highlight these differences). (b) The response to any presented orientation can be decoded by taking the average (in bold) of the vector responses to that stimulus across the population. (c) The physical orientation difference between two inputs required to reach an arbitrary just-noticeable-difference (JND) of 5° between decoded orientations varies along the orientation continuum. This produces the oblique effect, wherein discrimination sensitivity is higher for cardinal orientations and lower for oblique ones. Thresholds are shown separately for clockwise (blue) and counter-clockwise (red) deviations from each orientation. (d) Clockwise thresholds minus counterclockwise thresholds. Sensitivity to clockwise and counterclockwise displacements is equal at the cardinal and oblique axes, but asymmetric elsewhere. In this model, these asymmetries peak at approximately ± 20° away from the cardinal axes, where thresholds are lower for detecting displacements toward than away from the nearest cardinal orientation.
Figure 1
 
(a) The tuning curves of 60 orientation-selective neurons simulated using von Mises distributions, with tuning bandwidths ∼1.4 times wider near the oblique than cardinal axes (two cardinal and two oblique tuning curves are shown in bold to highlight these differences). (b) The response to any presented orientation can be decoded by taking the average (in bold) of the vector responses to that stimulus across the population. (c) The physical orientation difference between two inputs required to reach an arbitrary just-noticeable-difference (JND) of 5° between decoded orientations varies along the orientation continuum. This produces the oblique effect, wherein discrimination sensitivity is higher for cardinal orientations and lower for oblique ones. Thresholds are shown separately for clockwise (blue) and counter-clockwise (red) deviations from each orientation. (d) Clockwise thresholds minus counterclockwise thresholds. Sensitivity to clockwise and counterclockwise displacements is equal at the cardinal and oblique axes, but asymmetric elsewhere. In this model, these asymmetries peak at approximately ± 20° away from the cardinal axes, where thresholds are lower for detecting displacements toward than away from the nearest cardinal orientation.
The bandwidth of tuning curves across the population of neurons varies as a rectified sinusoidal function of preferred orientation. Consistent with recorded properties of cat V1 cells (Li et al., 2003; Orban & Kennedy, 1981), the full-width half-height of tuning curves reaches a maximum of approximately 39° at the oblique axes and a minimum of 28° at cardinal axes. Any oriented input elicits a response from a number of neurons, some reaching close to their maximum firing rate, others responding only slightly. Each neuron's activity can be described by a vector, as illustrated in Figure 1b, where the vector's angle represents the neuron's preferred orientation (the orientation that it is “voting for”), and the vector's length represents the neuron's firing rate (the “weight of evidence” provided by the neuron's activity). One simple method to decode the presented orientation from the population activity is by taking the average of all response vectors (Pouget et al., 2000). 
One can calculate the model's threshold for discriminating any two oriented stimuli by declaring an arbitrary just noticeable difference (JND) value, by which two decoded orientations must differ in order to be discriminable—say 5°. For each input orientation one can then estimate the amount by which a second input would need to differ in order for the population to signal one JND. Thresholds calculated in this manner are depicted in Figure 1c for both clockwise and counterclockwise differences from each orientation. Because bandwidths are anisotropic across the orientation continuum, a JND between decoded orientations will in some cases be reached when the difference between physical inputs is less than the declared JND (e.g., rotations away from 0°), and in other cases a JND will only be reached when the difference between physical inputs is greater than the declared JND (e.g., rotations away from 45°). As depicted in Figure 1c, the model predicts a clear oblique effect for discrimination thresholds, i.e., sensitivity is greater about cardinal orientations and lower about oblique orientations. 
As well as predicting the well-known oblique effect, the pattern of anisotropies captured in this model predicts asymmetric sensitivity to clockwise vs counterclockwise changes from certain orientations. In Figure 1d, the discrimination thresholds for counterclockwise differences have been subtracted from the thresholds for clockwise differences, to reveal orientations at which sensitivity is asymmetric. In this model, threshold asymmetries peak at approximately ±20° from cardinal orientations, with greater sensitivity predicted for orientation changes toward the cardinal axes than away. 
We believe this asymmetric sensitivity can explain data that have been taken as evidence for tilt normalization. For instance, Müller et al. (2009) propose that tilt normalization involves a measurable “perceptual drift” of tilted stimuli toward vertical or horizontal, which can be nulled by a slight physical rotation away from the cardinal axis. Rotation speeds required to null this putative drift were estimated by an adaptive procedure (a single staircase, see Cornsweet, 1962). For 0°, 90°, and ±45° initial orientations, the procedure converged on physical stasis, but for other orientations converged on a slight rotation away from the nearest cardinal axis. 
Müller et al.'s (2009) method of estimating perceptual stasis assumes a single point of stasis lying midpoint between the speeds of rotation that result in chance performance when judging whether a stimulus is rotating clockwise or counterclockwise. Judging direction of rotation involves two perceptual thresholds, one for detecting that stimuli are rotating clockwise, another for detecting that stimuli are rotating counter-clockwise. If these two thresholds are equal in magnitude, and physically static inputs are seen as static, an adaptive procedure searching for a single estimate of perceptual stasis should converge on a physically static stimulus. If, however, thresholds are asymmetric, with observers being more sensitive to orientation changes toward, rather than away from, cardinal axes, then an adaptive procedure will likely converge on a stimulus that is slowly rotating away from a cardinal orientation, even if physically static inputs are seen as static. 
In Experiment 1 we seek to replicate the effect reported by Müller et al. (2009) by measuring the apparent rotation direction of oriented stimuli, to which varying amounts of physical rotation have been applied via adaptive staircase procedures. Unlike in the previous report, we determine independent clockwise and counterclockwise rotation speed thresholds for tests originating at a range of orientations. 
Experiment 1
Method
Participants
There were 12 participants, comprising the first author and 11 experienced psychophysical observers who were naïve to the hypotheses of the experiment. All experimental procedures were approved by the School of Psychology's Ethical Review Board at the University of Queensland. 
Stimuli and apparatus
Stimuli were presented on a gamma-corrected 19-inch Trinitron Multiscan CPD-G520 monitor (Sony, Tokyo, Japan), with a resolution of 1024 × 768 pixels and a refresh rate of 120Hz. Matlab software (MathWorks, Natick, MA) was used to drive a VSG 2/3 stimulus generator from Cambridge Research Systems (Kent, UK). Participants viewed stimuli from a distance of 52 cm, with their head restrained by a chin rest. Stimuli were viewed through a black cylindrical cardboard tube 16 cm in diameter and 52 cm in length (see Figure 2a). The surface of the monitor could be viewed through a circular aperture cut into a piece of black cardboard attached to the end of the viewing tube, which subtended 17.5 degrees of visual angle (dva). This minimized the influence of potential reference orientations, such as those provided by the edge of the monitor screen or by peripheral objects in the room. 
Figure 2
 
(a) Apparatus used in Experiments 1 and 2. The participant viewed stimuli through a black cardboard tube, with a circular aperture cut into a sheet of black cardboard covering the screen. (b) Trial sequence in Experiment 1. After a 1-s fixation, a single Gabor appeared, rotating for 6 s at a speed and direction determined by an adaptive procedure. At the end of this period, the participant reports if the stimulus had rotated clockwise or counterclockwise. They were subsequently given auditory feedback.
Figure 2
 
(a) Apparatus used in Experiments 1 and 2. The participant viewed stimuli through a black cardboard tube, with a circular aperture cut into a sheet of black cardboard covering the screen. (b) Trial sequence in Experiment 1. After a 1-s fixation, a single Gabor appeared, rotating for 6 s at a speed and direction determined by an adaptive procedure. At the end of this period, the participant reports if the stimulus had rotated clockwise or counterclockwise. They were subsequently given auditory feedback.
On each trial a single Gabor was presented (see Figure 2b), with a red 1-pixel fixation dot at its center (approximately 0.03 dva). The spatial Gaussian envelope for Gabors subtended 9.9 dva, with a standard deviation of 1.7 dva. The Michelson luminance contrast of Gabors was 0.50 and they had a spatial frequency of 5 cycles/dva. The phase of the Gabor waveform was randomized on a trial-by-trial basis, and the display background was gray (CIE chromaticity coordinates x = 0.27, y = 0.30, Y = 45). The average luminance of test stimuli was matched to the background. 
Procedure
On each trial the test Gabor rotated at a speed determined by an adaptive procedure for a period of 6 s (see below). After the test presentation, the display was filled with static white noise (individual elements subtended 0.03dva × 0.03dva), which persisted until the participant indicated in which direction they felt the Gabor had rotated, by pressing one of two mouse buttons. Feedback was then provided in the form of a high-pitched (correct) or low-pitched (incorrect) beep. 
Speed thresholds for detecting the direction of rotational motion were measured at eight orientations. These consisted of the two cardinal orientations (0° and 90°), the two obliques (45° and 135°), and four orientations located ± 15° to either side of the cardinal orientations (i.e., 15°, 75°, 105°, and 165°). Here and throughout, orientations are labeled according to the geometric convention, with 0° indicating a horizontal line and 90° a vertical line. 
Adaptive “staircase” procedures (Cornsweet, 1962) were used to determine speed thresholds for detecting rotation direction, independently for clockwise and counterclockwise rotations from stimuli initialized at each of eight test orientations. Each of the 16 staircase procedures began by presenting a clearly rotating stimulus, at a speed of 0.8 angular degrees/s. Test speeds were then adjusted according to a “two-down, one-up” procedure, to derive an estimate of the 71% threshold (i.e., if the participant got one response wrong, speed was increased for that staircase, and if they got two successive responses correct within a staircase procedure, speed was decreased; see Levitt, 1970). Whenever the direction of adjustment (increasing vs. decreasing speed) differed from the direction of the previous adjustment within a staircase, a “reversal” was recorded for that staircase. 
As all staircases began with clearly rotating stimuli, stimulus speed was decreased by 0.16°/s on each trial until the first reversal was recorded within a given staircase procedure (i.e., until the participant's first incorrect response), after which speed was adjusted by 0.04°/s for that staircase. If such an adjustment would generate a negative speed for that staircase, stimulus speed was set to 0°/s, ensuring that staircase procedures searching for clockwise rotation thresholds did not contain any counterclockwise rotating stimuli, and vice versa. Each staircase terminated after six reversals, and the mean of the stimulus speeds at the last three reversals was taken as an estimate of the threshold for identifying directional rotation from the initial test orientation. 
To prevent fatigue, the 16 staircase procedures were completed in four blocks of trials, each taking approximately 12 min to complete. Within a block of trials, individual trials were drawn randomly from one of four interleaved staircases. In block A, staircases were conducted to determined speed thresholds for clockwise and counterclockwise rotations from initial test orientations of 0° and 90°. In block B initial test orientations were 75° and 165°, in block C initial test orientations were 45° and 135°, and in block D initial test orientations were 15° and 105°. Half of the participants completed blocks in the order A, B, C, D, and the other half completed blocks in the order D, C, A, B. 
Results
We determined thresholds for each of the eight test orientations, independent of rotation direction, by averaging clockwise and counterclockwise threshold estimates for each test orientation for each participant. This revealed a clear oblique effect (see Figure 3a), with highest thresholds for oblique test orientations (45° and 135°) and lowest thresholds for cardinal orientations (0° and 90°). 
Figure 3
 
(a) Rotation speed thresholds (averaged across clockwise and counterclockwise conditions) for detecting rotation direction in a stimulus displayed for 6 s, starting at various initial test orientations. The oblique effect is evidenced by lower thresholds for detecting rotation direction from cardinal orientations, and higher thresholds for oblique orientations. (b) Rotation speed thresholds for detecting rotation direction for tests with an initial orientation ± 15° from a cardinal axis. These data were grouped, for each participant, by whether the stimulus rotated toward or away from the nearest cardinal axis. All data show threshold estimates averaged across participants. Error bars show ± 1 SEM.
Figure 3
 
(a) Rotation speed thresholds (averaged across clockwise and counterclockwise conditions) for detecting rotation direction in a stimulus displayed for 6 s, starting at various initial test orientations. The oblique effect is evidenced by lower thresholds for detecting rotation direction from cardinal orientations, and higher thresholds for oblique orientations. (b) Rotation speed thresholds for detecting rotation direction for tests with an initial orientation ± 15° from a cardinal axis. These data were grouped, for each participant, by whether the stimulus rotated toward or away from the nearest cardinal axis. All data show threshold estimates averaged across participants. Error bars show ± 1 SEM.
To estimate thresholds for test stimuli rotating toward the nearest cardinal axis we averaged individual data from the 15° clockwise, 75° counterclockwise, 105° clockwise, and 165° counterclockwise conditions. To estimate thresholds for test stimuli rotating away from the nearest cardinal axis we averaged individual data from the 15° counterclockwise, 75° clockwise, 105° counterclockwise, and 165° clockwise conditions. Analyses revealed that speed thresholds for detecting rotation toward the nearest cardinal axis (Mean 0.22 ± SEM 0.03 °/s) were lower than thresholds for detecting rotation away from the nearest cardinal axis (0.33 ± 0.03; two-tailed paired t11 = −2.58, p = .026; see Figure 3b). One interpretation of this result is that sensitivity is symmetric, but that the point of subjective stasis (PSS) deviates from physical stasis. Assuming symmetric sensitivity, the PSS could be estimated by taking the signed average of the toward- and away-from-cardinal axis thresholds for each participant, yielding a mean PSS of a 0.05°/s (± 0.02) away from the nearest cardinal axis. These data are broadly consistent with Müller et al.'s (2009) key results (see their Figure 3b), which were taken as evidence for a perceptual drift of near-cardinal orientations. 
However, we do not believe our data provide unambiguous evidence that an observer's PSS differs from physical stasis. Another interpretation is that subjective stasis corresponds with physical stasis, but sensitivity is higher for toward-cardinal than away-from-cardinal axis rotations. This is consistent with the predictions of the population model of orientation coding outlined in the previous section. If sensitivity to rotational motion of an oriented stimulus is limited by orientation coding sensitivity, or if observers rely on judging absolute orientation at the end of a test presentation (as opposed to actually detecting movement), participants would be more sensitive to rotations that change orientation toward a cardinal axis, as opposed to rotations that change orientation toward an oblique axis. A third possibility is that asymmetric thresholds measured using a method of single stimuli result from a systematic response bias, for example a tendency to report that stimuli are rotating away from the nearest cardinal axis when in doubt (Morgan, Dillenburger, Raphael, & Solomon, 2012). In Experiment 2 we disambiguate these possibilities by using brief static tests, and a bias-free measure of sensitivity. 
Experiment 2
If asymmetric thresholds for detecting rotations toward and away from cardinal orientations measured in Experiment 1 result from perceptual drift, one would expect that a prolonged stimulus exposure would be necessary to measure this evidence. If, on the other hand, orientation sensitivity differences due to anisotropic orientation tuning (Li et al., 2003) are responsible for asymmetric thresholds, we should find asymmetric sensitivity when people try to detect orientation differences among brief static tests. In Experiment 2 we tested this proposition using a bias-free measure of orientation sensitivity—a three-interval forced-choice “odd one out” task. 
Method
Details were as for Experiment 1, with the following exceptions. 
Participants
There were six participants, comprising the first author and five experienced observers who were naïve to experimental hypotheses. 
Stimuli and apparatus
Each trial involved three sequential presentations of static Gabors, with a spatial frequency of three cycles/dva, a randomized waveform phase and a Michelson contrast of 1.0 (see Figure 4). These were shown at fixation for 300 ms, each separated by 200 ms intervals filled by dynamic noise (individual elements subtending 0.03 dva × 0.03 dva, updated at the monitor refresh rate). Two of the Gabors were of a standard orientation, and the orientation of the third “oddball” was varied on a trial-by-trial basis according to adaptive staircase procedures (see below). Order of presentation was randomized on a trial-by-trial basis. After the final dynamic noise sequence, a static white noise field was displayed until the participant indicated, by pressing one of three mouse buttons, if the first, second, or third Gabor had had a different orientation from the other two. Auditory feedback was given, and the next trial began after 1 s. 
Figure 4
 
Example trial sequence in Experiment 2. Three static Gabors are shown sequentially for 300 ms each, preceded and followed by 200-ms bursts of dynamic white noise. Two of the Gabors had a standard orientation, and one deviated by an amount determined by an adaptive procedure (here the oddball is presented in the second interval). At the end of this sequence, participants reported in which interval the oddball had appeared, followed by auditory feedback.
Figure 4
 
Example trial sequence in Experiment 2. Three static Gabors are shown sequentially for 300 ms each, preceded and followed by 200-ms bursts of dynamic white noise. Two of the Gabors had a standard orientation, and one deviated by an amount determined by an adaptive procedure (here the oddball is presented in the second interval). At the end of this sequence, participants reported in which interval the oddball had appeared, followed by auditory feedback.
Orientation difference thresholds for correctly detecting the “oddball” were measured at the same eight orientation points tested in Experiment 1, using four separate staircase procedures for each point. Two of the staircases sampled oddballs tilted clockwise from standards, one initiated at a large orientation difference (20°, a “shrinking” staircase) and another at no orientation difference (0°, a “growing” staircase). A complementary pair of staircases sampled oddballs tilted counter-clockwise from standards. Oddball orientations were adjusted by 1° within each staircase procedure until the first six reversals were recorded, after which orientation was adjusted in steps of 0.5° for that staircase. Staircase values had a set minima of 0°, ensuring that staircase procedures searching for clockwise thresholds did not contain any counterclockwise-tilted stimuli, and vice versa. Each staircase terminated after 12 reversals. 
Orientation values corresponding to the last six reversals in each of the paired shrinking and growing staircases were averaged to estimate independent orientation difference thresholds for each participant for oddballs tilted clockwise and counterclockwise from each of the eight standard orientations. 
Participants completed two blocks of trials, each of which sampled tests drawn randomly from one of 16 interleaved staircase procedures. One block of trials tested thresholds around cardinal and oblique orientations (i.e., four staircases centered around each of 0°, 45°, 90°, and 135°), while the other tested thresholds around near-cardinal orientations (i.e., 15°, 75°, 105°, and 165°). These groupings ensured that participants were not adapting to an average orientation during any block of trials. Half of the participants completed the block containing cardinal and oblique tests first, the other half completed the block containing near-cardinal orientations first. 
Results
Oddball detection thresholds, expressed as a function of standard orientation (see Figure 5a), display a clear oblique effect, with largest orientation difference thresholds for oblique standards (45° or 135°), and smallest for cardinal standards (0° or 90°). 
Figure 5
 
(a) Mean thresholds for detecting oddballs tilted relative to various standard orientations. Note the oblique effect. (b) Average thresholds at near-cardinal test points, for detecting an oddball tilted either toward or away from the nearest cardinal axis. Error bars show ± 1 SEM.
Figure 5
 
(a) Mean thresholds for detecting oddballs tilted relative to various standard orientations. Note the oblique effect. (b) Average thresholds at near-cardinal test points, for detecting an oddball tilted either toward or away from the nearest cardinal axis. Error bars show ± 1 SEM.
Discrimination thresholds for near-cardinal standards (15°, 75°, 105°, and 165°), grouped and averaged according to whether oddballs were tilted toward or away from the nearest cardinal axis, revealed that larger physical differences were required to detect a brief static oddball tilted away from a cardinal axis (10.1 ± 1.2°) than toward a cardinal axis (6.3 ± 0.5°; two-tailed paired t5 = −4.81, p = .005; see Figure 5b). 
General discussion
Our data demonstrate greater sensitivity to changes in orientation toward than away from a nearby cardinal axis. This was true both when changes in orientation were gradual over time (rotation-direction detection thresholds in Experiment 1) and when changes were between brief static stimuli (Experiment 2). An anisotropic model of orientation encoding, in which neurons selective for cardinal orientations have narrower tuning curves than those selective for oblique orientations, predicts these asymmetric discrimination thresholds (see Modeling section), in addition to the well-known oblique effect (Girshick et al., 2011; Li et al., 2003; Rose & Blakemore, 1974; see also Cohen & Zaidi, 2007). We hold that asymmetric sensitivity to changes toward versus away from cardinal axes provides a more parsimonious explanation for Müller et al.'s (2009) data than does perceptual drift caused by tilt normalization. 
Müller et al. (2009) reported that the magnitude of the perceptual drift suggested by their data was smaller, by a factor of five, than the magnitude of the tilt aftereffect. They also reported that there was no correlation between the magnitudes of these two effects. Our account explains this dissociation, as it posits that the tilt aftereffect arises from adaptation-induced changes in the responsiveness of orientation-selective neurons, whereas the bias measured when nulling the rotational motion of oriented stimuli results from asymmetric sensitivity to orientation changes which is present before adaptation. 
We have not yet addressed Müller et al.'s (2009) curvature data, but suspect they are amenable to a similar explanation. Greater sensitivity to decrements than to increments in curvature would produce data consistent with the “perceptual uncurling” of slightly curved stimuli posited by Müller et al. (2009). One possible reason for such asymmetric sensitivity is that when a slightly-curved near-vertical line segment (as in Müller et al., 2009) is straightened, the local orientation changes are toward vertical, whereas when the segment is further curved the local orientation changes are toward oblique axes. Enhanced sensitivity for toward-cardinal compared to toward-oblique changes would therefore predict enhanced sensitivity to decrements compared to increments in curvature for such stimuli. This would explain why Müller et al.'s (2009) “perceptual uncurling” rate estimates are tied to the aspect ratio of the stimulus rather than to absolute curvature, as the local orientations comprising a stimulus do not change with stimulus size. If this explanation is correct, a reversed pattern of results should ensue for obliquely-oriented curved line segments, which, when straightened, would contain local orientations further from the cardinal axes. 
While the predictions of our model agree qualitatively with our data both in terms of the shape of the oblique effect (compare Figure 1c to Figures 3a and 5a), and the direction of the near-cardinal discrimination threshold asymmetries, our model makes no firm predictions regarding the magnitudes of these effects. The threshold values predicted by the model depend on an arbitrarily declared average JND (5°), and the size of any asymmetry predicted would vary substantially for different JND values. Further, we have not made any assumptions regarding the temporal evolution or integration of neural orientation signals. In Experiment 1 participants were able to detect a change in orientation from vertical of approximately 1.2° during 6-s presentations. In Experiment 2 they were able to detect a difference in orientation of approximately 3.5° between successive static inputs, each presented for just 300 ms. This discrepancy suggests an evolution in signal to noise, possibly due to reiterative processes. Since there is insufficient background information to justify assumptions about these processes, our modeling makes no quantitative predictions for JND values in different tasks. 
If it did happen, how could tilt normalization arise?
Although evidence for tilt normalization remains underwhelming, our data do not rule it out as a possibility. There are various ways in which normalization could be predicted. Vaitkevicius et al. (2009) recently proposed a population-coding model to account for both tilt aftereffects and tilt normalization. In their model narrowband orientation selectivity in cortex is derived from subcortical mechanisms that have broad selectivity for the two cardinal orientations, and normalization occurs as a flow-on effect of adaptation within these cardinal channels. However, this model seems to fail on empirical grounds, erroneously predicting that adaptation to 0°, 90° and ±45° should not induce a tilt aftereffect (but see Templeton, Howard, & Easting, 1965), and that the appearance of stimuli oriented ±22.5° and ±67.5° from vertical should not be affected by adaptation (but see, e.g., Mitchell & Muir, 1976). 
Models similar to that described here could be modified to predict either normalization or anti-normalization, depending the on average channel bandwidth and the degree of bandwidth variance. Given that there are no firm estimates from investigations of human visual cortex to constrain assumptions regarding these parameters, the fact that any particular instantiation of a model might, or might not, predict normalization cannot be regarded as firm evidence for the existence or absence of this process. Another possibility is that tilt normalization happens because a population code, like that described here, is pre-adapted to an environment containing a preponderance of cardinal, compared to oblique, orientations. Arguably, this could be expected from adaptation to natural scenes (Hansen & Essock, 2004; Ganguli & Simoncelli, 2010; Girshick et al., 2011). Data pertaining to blur normalization has been similarly explained, by assuming pre-adaptation to a 1/f spatial frequency distribution (Elliott, Georgeson, & Webster, 2011). It remains to be seen, however, if such explanations are necessary to explain tilt normalization, as it is unclear if tilt normalization exists; our data suggests it does not. 
A general methodological point
We believe our data highlight an important methodological issue. Sensitivity is nonuniform along most, if not all, perceptual continua. For magnitude continua (luminance, weight, loudness, etc), Weber's law dictates that sensitivity to a decrement will be superior to an increment of the same magnitude. Along other continua, sensitivity is often maximal around null points, such as zero binocular disparity (e.g., Stevenson, Cormack, Schor, & Tyler, 1992) a white/gray hue (Webster, 1996), or at subjective boundaries, such as those between phonemes, colors (Goldstone & Hendrickson, 2009) and complex object categories (Beale & Keil, 1995; Newell & Bülthoff, 2002). Most methods for establishing points of subjective equality will systematically err in all of these circumstances. Staircase procedures converge on the center of a region of uncertainty, which will coincide with the point of subjective equality only when discrimination thresholds are symmetric. Symmetric thresholds are also assumed when fitting data with symmetric psychometric functions. If this assumption is violated, the estimate of central response tendency will deviate from the point of subjective equality. As uneven sensitivity to stimulus changes is common, procedures and analyses should generally be preferred that avoid the assumptions of symmetric sensitivity and or symmetric criteria placement (see Yarrow, Jahn, Durant, & Arnold, 2011). 
Conclusion
The existence of tilt normalization remains under question. Recent data taken as evidence for tilt normalization might instead result from anisotropic sensitivity to orientation differences—the oblique effect. 
Acknowledgments
This research was supported by an Australian Research Council Discovery project grant to DHA (DP0878140) and an Australian Postgraduate Award to KRS. The authors have no competing financial interests. 
Commercial relationships: none. 
Corresponding author: Katherine R. Storrs. 
Email: k.storrs@uq.edu.au. 
Address: School of Psychology, The University of Queensland, St. Lucia, Queensland, Australia. 
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Footnotes
1  The values predicted by the model for discrimination thresholds (Figure 1c and d) are near-identical across a wide range of numbers of simulated neurons (at least from 20 to 80). We chose the figure of 60 for the sake of consistency with a similar model (Girshick, Landy, & Simoncelli, 2011).
Figure 1
 
(a) The tuning curves of 60 orientation-selective neurons simulated using von Mises distributions, with tuning bandwidths ∼1.4 times wider near the oblique than cardinal axes (two cardinal and two oblique tuning curves are shown in bold to highlight these differences). (b) The response to any presented orientation can be decoded by taking the average (in bold) of the vector responses to that stimulus across the population. (c) The physical orientation difference between two inputs required to reach an arbitrary just-noticeable-difference (JND) of 5° between decoded orientations varies along the orientation continuum. This produces the oblique effect, wherein discrimination sensitivity is higher for cardinal orientations and lower for oblique ones. Thresholds are shown separately for clockwise (blue) and counter-clockwise (red) deviations from each orientation. (d) Clockwise thresholds minus counterclockwise thresholds. Sensitivity to clockwise and counterclockwise displacements is equal at the cardinal and oblique axes, but asymmetric elsewhere. In this model, these asymmetries peak at approximately ± 20° away from the cardinal axes, where thresholds are lower for detecting displacements toward than away from the nearest cardinal orientation.
Figure 1
 
(a) The tuning curves of 60 orientation-selective neurons simulated using von Mises distributions, with tuning bandwidths ∼1.4 times wider near the oblique than cardinal axes (two cardinal and two oblique tuning curves are shown in bold to highlight these differences). (b) The response to any presented orientation can be decoded by taking the average (in bold) of the vector responses to that stimulus across the population. (c) The physical orientation difference between two inputs required to reach an arbitrary just-noticeable-difference (JND) of 5° between decoded orientations varies along the orientation continuum. This produces the oblique effect, wherein discrimination sensitivity is higher for cardinal orientations and lower for oblique ones. Thresholds are shown separately for clockwise (blue) and counter-clockwise (red) deviations from each orientation. (d) Clockwise thresholds minus counterclockwise thresholds. Sensitivity to clockwise and counterclockwise displacements is equal at the cardinal and oblique axes, but asymmetric elsewhere. In this model, these asymmetries peak at approximately ± 20° away from the cardinal axes, where thresholds are lower for detecting displacements toward than away from the nearest cardinal orientation.
Figure 2
 
(a) Apparatus used in Experiments 1 and 2. The participant viewed stimuli through a black cardboard tube, with a circular aperture cut into a sheet of black cardboard covering the screen. (b) Trial sequence in Experiment 1. After a 1-s fixation, a single Gabor appeared, rotating for 6 s at a speed and direction determined by an adaptive procedure. At the end of this period, the participant reports if the stimulus had rotated clockwise or counterclockwise. They were subsequently given auditory feedback.
Figure 2
 
(a) Apparatus used in Experiments 1 and 2. The participant viewed stimuli through a black cardboard tube, with a circular aperture cut into a sheet of black cardboard covering the screen. (b) Trial sequence in Experiment 1. After a 1-s fixation, a single Gabor appeared, rotating for 6 s at a speed and direction determined by an adaptive procedure. At the end of this period, the participant reports if the stimulus had rotated clockwise or counterclockwise. They were subsequently given auditory feedback.
Figure 3
 
(a) Rotation speed thresholds (averaged across clockwise and counterclockwise conditions) for detecting rotation direction in a stimulus displayed for 6 s, starting at various initial test orientations. The oblique effect is evidenced by lower thresholds for detecting rotation direction from cardinal orientations, and higher thresholds for oblique orientations. (b) Rotation speed thresholds for detecting rotation direction for tests with an initial orientation ± 15° from a cardinal axis. These data were grouped, for each participant, by whether the stimulus rotated toward or away from the nearest cardinal axis. All data show threshold estimates averaged across participants. Error bars show ± 1 SEM.
Figure 3
 
(a) Rotation speed thresholds (averaged across clockwise and counterclockwise conditions) for detecting rotation direction in a stimulus displayed for 6 s, starting at various initial test orientations. The oblique effect is evidenced by lower thresholds for detecting rotation direction from cardinal orientations, and higher thresholds for oblique orientations. (b) Rotation speed thresholds for detecting rotation direction for tests with an initial orientation ± 15° from a cardinal axis. These data were grouped, for each participant, by whether the stimulus rotated toward or away from the nearest cardinal axis. All data show threshold estimates averaged across participants. Error bars show ± 1 SEM.
Figure 4
 
Example trial sequence in Experiment 2. Three static Gabors are shown sequentially for 300 ms each, preceded and followed by 200-ms bursts of dynamic white noise. Two of the Gabors had a standard orientation, and one deviated by an amount determined by an adaptive procedure (here the oddball is presented in the second interval). At the end of this sequence, participants reported in which interval the oddball had appeared, followed by auditory feedback.
Figure 4
 
Example trial sequence in Experiment 2. Three static Gabors are shown sequentially for 300 ms each, preceded and followed by 200-ms bursts of dynamic white noise. Two of the Gabors had a standard orientation, and one deviated by an amount determined by an adaptive procedure (here the oddball is presented in the second interval). At the end of this sequence, participants reported in which interval the oddball had appeared, followed by auditory feedback.
Figure 5
 
(a) Mean thresholds for detecting oddballs tilted relative to various standard orientations. Note the oblique effect. (b) Average thresholds at near-cardinal test points, for detecting an oddball tilted either toward or away from the nearest cardinal axis. Error bars show ± 1 SEM.
Figure 5
 
(a) Mean thresholds for detecting oddballs tilted relative to various standard orientations. Note the oblique effect. (b) Average thresholds at near-cardinal test points, for detecting an oddball tilted either toward or away from the nearest cardinal axis. Error bars show ± 1 SEM.
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