December 2018
Volume 18, Issue 13
Open Access
Article  |   December 2018
Spatial scaling of illusory motion perceived in a static figure
Author Affiliations
  • Rumi Hisakata
    School of Engineering, Tokyo Institute of Technology, Kanagawa, Japan
    Department of Life Sciences, the University of Tokyo, Tokyo, Japan
    Japan Society for the Promotion of Science
    https://rumihisakata.wordpress.com/
    rumi.hisakata@icloud.com
  • Ikuya Murakami
    Department of Life Sciences, the University of Tokyo, Tokyo, Japan
    Department of Psychology, the University of Tokyo, Tokyo, Japan
    ikuya@l.u-tokyo.ac.jp
Journal of Vision December 2018, Vol.18, 15. doi:10.1167/18.13.15
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      Rumi Hisakata, Ikuya Murakami; Spatial scaling of illusory motion perceived in a static figure. Journal of Vision 2018;18(13):15. doi: 10.1167/18.13.15.

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Abstract

In a phenomenon known as the Rotating Snakes illusion (Kitaoka & Ashida, 2003), illusory motion is perceived in a static figure with a specially designed luminance profile. It is known that the strength of this illusion increases with eccentricity, suggesting that the underlying mechanism of the illusion has a spatial property that changes with eccentricity. If a change in receptive-field size of responsible neurons causes the eccentricity dependence of the illusion, its strength should be spatially scalable using a scaling factor that increases with eccentricity, because the receptive field size of neurons in visual areas with retinotopy generally obeys quantitative dependence on eccentricity. For the luminance micropatterns comprising the figure for the Rotating Snakes illusion, we varied eccentricity from 9 to 15 deg and spatial frequency from 0.25 to 1.6 cycles/deg, and measured illusion strength. Illusion strength was found to increase with decreasing spatial frequency and with increasing eccentricity. Furthermore, the profiles of illusion strength at different eccentricities were spatially scalable into a single parabola as a function of the spatially scaled visual angle. The estimated scaling factors linearly increased with eccentricity with a slope similar to the eccentricity dependence of the receptive field size of V1 neurons, suggesting the involvement of early visual areas in the generation of the illusion.

Introduction
Motion illusions perceived in static figures have received sustained attention among both vision scientists and nonspecialists, and a number of striking demonstrations have been devised (e.g., Fraser & Wilcox, 1979; Ouchi, 1977; Pinna & Brelstaff, 2000). One of the most powerful of these illusions is “The Rotating Snakes illusion” (Ashida & Kitaoka, 2003; Kitaoka & Ashida, 2003). The figure for this illusion is actually composed of luminance micropatterns, each of which is characterized by a luminance array ordered as white, light gray, black, and dark gray (Figure 1a). Such micropatterns are repeated circularly, and illusory motion is perceived in the direction of this luminance order (Figure 1b). This illusion is dramatic and powerful, but its underlying mechanism is not fully understood. 
Figure 1
 
Structure of a figure for the Rotating Snakes illusion. (a) A gray-scale figure for the illusion (reproduced, with permission, from the website of Akiyoshi Kitaoka, who originally created this figure http://www.ritsumei.ac.jp/∼akitaoka/index-e.html). (b) A simplified figure for the Rotating Snakes illusion. Luminance micropatterns, each comprised of white, light gray, black, and dark gray, are repeated circularly. In this example, a rotary illusory motion is perceived in the clockwise direction.
Figure 1
 
Structure of a figure for the Rotating Snakes illusion. (a) A gray-scale figure for the illusion (reproduced, with permission, from the website of Akiyoshi Kitaoka, who originally created this figure http://www.ritsumei.ac.jp/∼akitaoka/index-e.html). (b) A simplified figure for the Rotating Snakes illusion. Luminance micropatterns, each comprised of white, light gray, black, and dark gray, are repeated circularly. In this example, a rotary illusory motion is perceived in the clockwise direction.
Several models and hypotheses have been proposed, though a unified conclusion remains elusive. Backus and Oruç (2005) proposed that adaptation to luminance and contrast could explain the long-lasting slow illusory motion. Conway, Kitaoka, Yazdanbakhsh, Pack, and Livingstone (2005) suggested that differential latencies of neural responses for high and low contrasts would generate the illusory motion in a cortically registered stimulus pattern. Murakami, Kitaoka, and Ashida (2006) found a positive intersubject correlation between illusion strength and fixation instability, and proposed a model in which motion detectors that are sensitive to image oscillations due to fixational eye movements cause the illusory motion; according to this model, a transient temporal property in the visual system plays an important role in generating the illusion. Fermüller, Ji, and Kitaoka (2010) proposed a model in which motion signals detected with a combination of symmetric spatial filters and asymmetric temporal filters are misestimated when an image has locally asymmetric intensity signals visible through certain spatial frequency channels. 
Some studies have investigated the illusion using brain-imaging experiments. By using functional magnetic resonance imaging (fMRI), Kuriki, Ashida, Murakami, and Kitaoka (2008) investigated brain activity corresponding to illusion occurrence and observed a larger blood oxygenation level dependent (BOLD) signal in hMT+, during observation of a static figure for this motion illusion, than during observation of a control figure producing no illusion. Using the technique of fMRI adaptation and structural equation modeling, Ashida, Kuriki, Murakami, Hisakata, and Kitaoka (2012) estimated the cortical network most likely responsible for the occurrence of the Rotating Snakes Illusion: The network involved a dorsal pathway from area V1 to hMT+. 
We previously reported that strength of illusion decreases with decreasing retinal illuminance and increases with increasing eccentricity (Hisakata & Murakami, 2008). In this study, a small ring-shaped stimulus designed for the Rotating Snakes Illusion was placed at various eccentricities, and the velocity of the physical rotation nulling the illusory motion was measured. The first finding, retinal-illuminance dependence, supported Murakami et al.'s (2006) model because a sufficient decrease in retinal illuminance is known to change the temporal property of the visual system from transient-dominant to sustained-dominant (Burr & Morrone, 1993; Hess & Snowden, 1992; Kelly, 1962; Snowden, Hess, & Waugh, 1995; Takeuchi & DeValois, 1997). Indeed, each subject's temporal property was measured at various retinal illuminances and was confirmed to change as noted above. Therefore, the first finding demonstrated that the strength of illusion depends on the transient channel of the visual system. 
The second finding, i.e., eccentricity dependence, we discuss here in more detail. The increase in illusory motion in the periphery had previously only been reported phenomenologically (Kitaoka & Ashida, 2003); thus, we psychophysically measured illusion strength at various eccentricities in order to obtain quantitative data (Hisakata & Murakami, 2008). Our results that the illusion became stronger with increasing eccentricity support anecdotal observations and are consistent with another study on illusory motions in static figures (Naor-Raz & Sekuler, 2000). However, the underlying mechanism of this dependence remains unclear. One possibility is that a spatial resolution in the visual system is involved in the increase in illusion strength. Because the receptive-field sizes of cells, or of functional computational units, in visual areas with retinotopic organization expand with eccentricity, as such, spatial resolution in the visual system gets worse in the periphery (e.g., Albright & Desimone, 1987; Amano, Wandell & Dumoulin, 2009; Dougherty et al., 2003; Dow, Snyder, Vautin, & Bauer, 1981). Differences in resolution across locations in the visual field affect a wide variety of visual performance including contrast sensitivity, visual acuity, critical flicker frequency, color perception, and differential motion among others (Abramov, Gordon, & Chan, 1992; Millodot, Johnson, Lamont, & Leibowitz, 1975; Murakami & Shimojo, 1995; Rovamo & Virsu, 1979; Virsu & Rovamo, 1979). These visual characteristics share one certain feature: They are spatially scalable by factors such as cortical magnification, receptive-field size, and other physiological and/or anatomical structures that vary across locations in the visual field (Curcio, Sloan, Kalina, & Hendrickson, 1990; Dow et al., 1981). 
Spatial scaling is a psychophysical method in which stimuli are presented at a size systematically varied according to their eccentricities, with the goal of equating stimulus size relative to the receptive field size of the relevant cortical units. As the receptive-field size of cortical cells, or their functional equivalents in the visual system, tends to increase with eccentricity, a physically identical stimulus size diminishes its relative size with respect to the average receptive-field size. Conversely, differences in performance across eccentricities may disappear if the physical stimulus size is enlarged in proportion to receptive-field size. The logic underlying the method of spatial scaling is that if performance can be equated at different eccentricities by simply scaling a stimulus, the effect of eccentricity likely depends on quantitative parameters of the visual system such as receptive field size, rather than qualitative differences between central and peripheral locations. If the eccentricity dependence of the strength of the Rotating Snakes illusion originates from a mere difference in receptive-field size, the effect should be spatially scalable. Nevertheless, it was difficult to quantify the Rotating Snakes illusion at only a few degrees from the fovea at which the illusion is not always vigorous (Hisakata & Murakami, 2008). Possible reasons for suboptimal appearance of the illusion around the fovea would include the decreases in amplitude and rate of small eye movements, which are deemed important for the illusion to occur (Murakami et al, 2006; Otero-Millan, Macknik, & Martinez-Conde, 2012), in the presence of fine-grained stimuli around the fovea, and higher sensitivity for position signals there. Therefore, our research question, i.e., whether experimental manipulation of size and eccentricity would affect the illusion strength of the Rotating Snakes illusion in an orderly manner, was examined within the eccentricity range from 9 to 15 deg within which the illusion is known to occur vigorously (Hisakata & Murakami, 2008). 
To address this issue, we applied a spatial scaling approach to the Rotating Snakes illusion. As we had no prior knowledge regarding the scaling factor that can be applied to eccentricity dependence of illusion strength, we used an assumption-free method of spatial scaling (Murakami & Cavanagh, 2001; Murakami & Shimojo, 1995, 1996; Whitaker, Rovamo, Macveigh, & Mäkelä, 1992). According to this method, scaling factors can be empirically estimated from psychophysical data obtained at various eccentricities and with various stimulus sizes. Fermüller et al. (2010) proposed that the size of luminance micropatterns phenomenologically affects the strength of the Rotating Snakes illusion. Thus, we manipulated the size of luminance micropatterns (each comprised of white, light gray, black, and dark gray) as an independent variable related to stimulus size at each eccentricity. Manipulation of eccentricity was achieved by altering the overall size of a ring-shaped stimulus, centered on a central fixation point (Figure 2). In sum, we examined the stimulus size dependence of the Rotating Snakes illusion at several eccentricities and asked whether the increases in illusion strength at larger eccentricities, which have been confirmed in our previous study (Hisakata & Murakami, 2008), could be explained in terms of spatially scaled stimulus size relative to receptive field size. 
Figure 2
 
Stimulus configuration. A fixation point was always presented at the center of the display, and the background was filled with static random dots.
Figure 2
 
Stimulus configuration. A fixation point was always presented at the center of the display, and the background was filled with static random dots.
Methods
Subjects
Eight subjects who were naive to the experimental purpose and one of the authors (RH) participated. All had normal or corrected-to-normal visual acuity. All subjects gave written informed consent prior to the participation. This study followed the Declaration of Helsinki guidelines for ethical treatments of human volunteers and had been approved by the Ethics Committee of the Graduate School of Arts and Sciences, the University of Tokyo, and was carried out in accordance with the approved guidelines. 
Apparatus
Stimuli were generated by a computer (Apple PowerMac G5), and displayed on a 22-in. CRT monitor (Mitsubishi Electric RDF223H, 1600 × 1200 pixels, refresh rate 75 Hz, 0.025 deg/pixel, driven by a videoboard with 10-bit depth). The mean luminance was 48 cd/m2. The viewing distance was 57 cm. Participants had their head movements restricted using a chin rest; subjects observed the stimuli with both eyes in a dark room. 
Stimuli
We used a ring-shaped stimulus composed of micropatterns, each of which consisted of four stepwise luminance levels, white, light gray, black, and dark gray; they had luminance values of 1, 2/3, 0, and 1/3, respectively, multiplied by the maximum luminance of 96 cd/m2. The stimulus was blurred with a spatial low-pass filter (Gaussian distribution with a sigma of 4/253 cycles of micropatterns) to enable subpixel animation (Hisakata & Murakami, 2008). The width of the ring was always 2 deg, following previous spatial-scaling studies that used a constant window size across different eccentricities (e.g., Johnston, 1987; Virsu & Rovamo, 1979). A central fixation point was always presented at the center of the stimulus ring. The midpoint between the inner and outer radii of the ring was taken as representing the eccentricity of the stimulus (Figure 2). Four eccentricities, namely 9, 11, 13, and 15 deg, were tested. Note that the notation “1 deg” is consistently used to refer to one degree of visual angle, not angle of rotation. The number of micropatterns occupying 1 deg along a tangential line was called the fundamental spatial frequency, so that the stimulus would be said to have 1 cycle/deg when the four luminance levels comprising a single micropattern spatially occupied just one degree of visual angle. To manipulate fundamental spatial frequency, the number of micropatterns of the ring stimulus was varied in five steps, namely 24, 40, 56, 72, and 88. Accordingly, the variety of the fundamental spatial frequencies differed across stimuli at different eccentricities. 
There were two types of stimuli: a clockwise stimulus (CW) and a counterclockwise stimulus (CCW). The CW stimulus was composed of the abovementioned luminance micropatterns arranged in a clockwise direction and appeared to rotate clockwise when it was physically stationary. The CCW stimulus was the mirror reversal of the CW stimulus and appeared to rotate counterclockwise when it was physically stationary. The background was filled with static random dots made by rendering every pixel either in black or white with equal probabilities. 
Procedure
Our aim was to quantify illusion strength at each fundamental spatial frequency and each eccentricity. To this end, the ring was physically rotated about the fixation point, and the stimulus velocity that gave the point of subjective stationarity was measured with the method of constant stimuli. In all conditions, the stimulus was rotated either clockwise or counterclockwise at a velocity chosen from {±0.27, ±0.18, ±0.09, 0} rpm for the condition of 24 cycles, {±0.16, ±0.107, ±0.053, 0} rpm for 40 cycles, {±0.12, ±0.08, ±0.04, 0} rpm for 56 cycles, {±0.09, ±0.06, ±0.03, 0} rpm for 72 cycles, and {±0.07, ±0.047, ±0.023, 0} rpm for 88 cycles. The stimulus was presented for 0.5 s. The subject made a two-alternative judgment as to whether the stimulus ring appeared to rotate clockwise or counterclockwise. For each velocity, 30 responses were collected. Each psychometric curve was fitted with cumulative Gaussian function. The combination of velocity, fundamental spatial frequency, and stimulus type (CW or CCW) was chosen in random order within each session. From session to session, eccentricity was changed in a random order. All eccentricities were tested once before being repeated in successive sessions, and there were five sessions in total for each eccentricity. 
Results
In Figure 3, we show examples of psychometric functions for a representative subject, obtained at the four eccentricities in the condition of 88 cycles. With the cancellation method used to quantify the illusion strength, the function for CW stimulus shifted to the counterclockwise direction. This means that if the subject observed a static version of the stimulus, it more likely appeared to rotate clockwise than counterclockwise. Therefore, if the illusion occurred as we intended, the two psychometric functions for the CW and CCW stimuli in each panel should separate from each other, with the CW and CCW stimuli shifting rightward and leftward, respectively, which was actually the case. As an index of illusion strength, we defined the cancellation velocity as the half the distance between the points of subjective stationarity for the two functions. 
Figure 3
 
Examples of psychometric functions for subject AI. Each panel shows results under the condition of 88 cycles at each eccentricity. The black dots and solid curves indicate the data for the CCW stimulus whereas the open dots and dashed curves indicate the data for the CW stimulus. The positive values of velocity indicate that the stimulus rotated counterclockwise.
Figure 3
 
Examples of psychometric functions for subject AI. Each panel shows results under the condition of 88 cycles at each eccentricity. The black dots and solid curves indicate the data for the CCW stimulus whereas the open dots and dashed curves indicate the data for the CW stimulus. The positive values of velocity indicate that the stimulus rotated counterclockwise.
For the intersubject average of illusion strength, a set of one-sample t tests with Bonferroni correction confirmed that the Rotating Snakes illusion significantly occurred at all conditions except at 24 cycles at 15°eccentricity. Also, a repeated-measures analysis of variance (ANOVA) revealed significant main effects of eccentricity, F (3, 24) = 5.39, p = 0.0056, and of fundamental spatial frequency, F (4, 32) = 6.85, p = 0.0004. Overall, the illusion was stronger at larger eccentricities, replicating our previous report (Hisakata & Murakami, 2008). At all eccentricities, illusion strength was initially reduced with increasing spatial frequency, reached a trough at a particular frequency, and was slightly elevated beyond it. For example, the illusion strength at 9 deg eccentricity reached the trough at 0.05 cycles/deg. The trough point was found at different frequencies across different eccentricities, but the U-shaped profiles of frequency dependence were more or less similar. With increasing eccentricity, the profile systematically shifted to the left and up, keeping its shape (Figure 4). 
Figure 4
 
(a) Intersubject average of the strength of the Rotating Snakes illusion plotted against fundamental spatial frequency, with eccentricity as a parameter. Error bars indicate standard error. Both axes are in log scale. (b) Results of spatial scaling. The gray curve indicates the best-fit parabola. FSF = fundamental spatial frequency. Other conventions are identical to those in (a).
Figure 4
 
(a) Intersubject average of the strength of the Rotating Snakes illusion plotted against fundamental spatial frequency, with eccentricity as a parameter. Error bars indicate standard error. Both axes are in log scale. (b) Results of spatial scaling. The gray curve indicates the best-fit parabola. FSF = fundamental spatial frequency. Other conventions are identical to those in (a).
This relationship led us to hypothesize that the profiles of illusion strength at different eccentricities might overlap, if they were appropriately shifted in an oblique direction. Scaling of visual angle at an eccentricity in question results in translation of a profile in a log-log plot. For example, consider a hypothetical case in which the stimulus occupying 1 [deg] of visual angle at x [deg] eccentricity makes the same impact as the stimulus occupying 2 [deg] at 13 [deg] eccentricity. This relationship can be written as 1 [degx] = 2 [deg13], where “degx” denotes the unit “deg” at x deg eccentricity. Equivalently, 1 [deg13] = 0.5 [degx], and this value, 0.5, is called the scaling factor, f, for x deg eccentricity. Accordingly, a stimulus size z [degx] will be converted to z [1/0.5 deg13] = 2z [deg13] (more generally, Display Formula\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\({f^{ - 1}}z\) [deg13]), and the log plot will be shifted rightward by one log2 unit after scaling. Similarly, a spatial frequency of q [cycles/degx] will be converted to q [cycles/{1/0.5 deg13}] = 0.5q [cycles/deg13], and the log plot will be shifted left by one log2 unit, after scaling. Therefore, spatial scaling of fundamental spatial frequency is equivalent to a horizontal shift of the plot. Illusion strength, m, is similarly altered by spatial scaling. For rotation speed s [rpm] at eccentricity x [deg], angular change per second θ = Display Formula\({\pi \over {30}}s\) [rad]; arc length per second m = x tan (θ) [degx]. By the same logic as above, illusion strength as represented by m [degx] will be converted to m [1/0.5 deg13] = 2m [deg13], and be converted back again to rotation speed, Display Formula\({{30} \over \pi }{\tan ^{ - 1}}{{2m} \over x}\) [rpm]. Since tan (θ) ≈ [rad], where k ≈ 0.0175, for small θ, the above unit conversion through scaling is reduced to linear scaling: Display Formula\({\tan ^{ - 1}}\left( {{f^{ - 1}}\tan \theta } \right) \approx {{{f^{ - 1}}k\theta } \over k} = {f^{ - 1}}\theta \). Therefore, spatial scaling of illusion strength is equivalent to a vertical shift of the plot. We assumed that a common scaling factor should be used to scale the abscissa and ordinate, and thus the translation of each profile was constrained in the diagonal direction with a slope of −1 in the log-log plot. Factors greater than unity would shift the profile diagonally to the right and down, whereas factors less than unity would shift it to the left and up. What scaling factor would best characterize the behavior at each eccentricity? To answer this question, we applied the assumption-free method for spatial scaling (Whitaker et al., 1992). Graphically, the procedure was equivalent to shifting the profiles for different eccentricities along the diagonal line so that all profiles collapsed into a single function. More specifically, we stabilized the profile at 13 deg eccentricity as standard and attempted to find the scaling factors for the three other eccentricities that would deliver all the profiles over the profile at 13 deg eccentricity, and that would minimize the residual error of curve fitting applied to the scaled dataset. The model curve we chose was the parabola, y = a (xc)2 + b, the simplest formulation depicting U-shaped nonlinearity. 
The degrees of freedom in the curve fitting comprised the three variables in the parabola and three scaling factors for 9, 11, and 15 deg eccentricities, but to prevent overfitting, we freed only four parameters; three variables in the parabola (a, b, and c), and the slope of scaling factor against eccentricity, on the linearity assumption that Display Formula\(f = 1 + g\left( {x - 13} \right)\), where g was the fourth free parameter. This approximation was consistent with a number of previous studies demonstrating that scaling factor is indeed characterized by a linear increasing function of eccentricity (e.g., Levi, Klein, & Aitsebaomo, 1985; Watson, 1987; Whitaker et al., 1992). The results of the spatial scaling for the illusion can be seen in Figure 4b: The scaling was more or less successful, and the data at different eccentricities seemingly obeyed a single parabola (see figure inset for the best-fit equation, adjusted R2 = 0.92, p = 1.614 × 10−11). Estimated scaling factors were 0.76, 0.88, and 1.12, for eccentricity 9, 11, and 15 deg respectively. The success in applying scaling factors as a linear function of eccentricity (Figure 5) strengthens the validity of our approach, applying spatial scaling to the Rotating Snakes illusion. Also, this implies that differences in illusion strength at different eccentricities are derived not from qualitative differences in processing across different eccentricities, but from systematically scaled cortical architecture such as the difference in receptive-field size of a functional unit underlying the illusion. 
Figure 5
 
Comparison among psychophysical as well as physiological scaling factors: Scaling factors obtained in this study are shown as open circles. Each linear function indicates the normalized scaling factors against eccentricity in previous studies, summarized by Murakami and Shimojo (1996). Detection threshold from Levi et al. (1985): scaling factor = 1 + 1.05−1E. RF size in the macaque monkey V1 neurons from Dow et al. (1981): Log (60RF) = 1.1438 + 0.1920x + 0.0712x2 + 0.0619x3, where x = log (60E), but was approximated by a linear function here. Contrast sensitivity from Watson (1987): scaling factor = 1 + 0.24E. Vernier threshold from Whitaker et al. (1992): Scaling factor = 1 + 1.66(E − 0.267). Detection thresholds for relative motion from Levi et al. (1985): Scaling factor = 1+5.99−1E. RF size in monkey MT neurons from Albright and Desimone (1987): RF = 1.04 + 0.61E.
Figure 5
 
Comparison among psychophysical as well as physiological scaling factors: Scaling factors obtained in this study are shown as open circles. Each linear function indicates the normalized scaling factors against eccentricity in previous studies, summarized by Murakami and Shimojo (1996). Detection threshold from Levi et al. (1985): scaling factor = 1 + 1.05−1E. RF size in the macaque monkey V1 neurons from Dow et al. (1981): Log (60RF) = 1.1438 + 0.1920x + 0.0712x2 + 0.0619x3, where x = log (60E), but was approximated by a linear function here. Contrast sensitivity from Watson (1987): scaling factor = 1 + 0.24E. Vernier threshold from Whitaker et al. (1992): Scaling factor = 1 + 1.66(E − 0.267). Detection thresholds for relative motion from Levi et al. (1985): Scaling factor = 1+5.99−1E. RF size in monkey MT neurons from Albright and Desimone (1987): RF = 1.04 + 0.61E.
Discussion
In this study, we show that the Rotating Snakes illusion is spatially scalable. It is known that the strength of illusion increases with eccentricity (e.g., Hisakata & Murakami, 2008), and we confirmed similar eccentricity dependence in our stimulus configuration. To spatially scale the illusion, we determined four free parameters, namely three variables of the model function, i.e., the parabola, fitted to illusion strength against fundamental spatial frequency (in cycles/deg13), and one variable governing the linear increase of scaling factor with eccentricity. As the result of spatial scaling, illusion strength at all tested eccentricities overlapped and a single parabola could summarize the entire dataset. 
Our spatial scaling analysis would indicate that the increase of the illusion strength basically reflected the quantitative and orderly increase of the space constants, such as the receptive-field size, of functional units underlying the illusion, not qualitative differences in visual functions across eccentricities. Previous studies have determined scaling factors across eccentricities in relation to visual functions as well as receptive-field sizes of neurons in several visual areas in the cortex. Here, we compared the eccentricity dependence of the scaling factors estimated for the Rotating Snakes illusion to some of those previously reported that may have a potential relationship to visual motion and/or spatial perception; these include the detection thresholds for absolute motion and relative motion, vernier acuity, contrast detection threshold, and receptive-field size of neurons in V1 and MT of the macaque (Albright & Desimone, 1987; Dow et al., 1981; Levi et al., 1985; Watson, 1987; Whitaker et al., 1992; Figure 5). All scaling factors and receptive-field sizes as a function of eccentricity were normalized so as to pass through the point (13, 1), enabling comparison with respect to slope, within the eccentricity range tested in the present study. The scaling factors for 9, 11, 13, and 15 deg eccentricities obtained in the present study were plotted with circles (Figure 5). 
In this study, we found that illusion strength decreases with increasing fundamental spatial frequency. The detection thresholds for absolute motion and gratings have a similar dependence on spatial frequency (Johnston, 1987; Kelly, 1979; Mckee & Nakayama, 1984; Robson, 1966; Rovamo & Virsu, 1979; Rovamo, Virsu, & Näsänen, 1978; Watson & Eckert, 1994; Wright, 1987). Wright (1987) measured the minimum phase difference of a grating for motion detection while manipulating spatial frequency and eccentricity; the displacement of grating at detection threshold increased with increasing spatial frequency and the profiles of spatial frequency dependence systematically shifted from high to low spatial frequencies with increasing eccentricity. This indicates that displacement sensitivity decreases with both spatial frequency and eccentricity. Johnston (1987) measured the contrast sensitivity at several spatial frequencies and at several eccentricities and found that it was also lowered at high spatial frequencies. The observed spatial property of the Rotating Snakes illusion, namely that the illusion is weaker at higher fundamental spatial frequencies, may reflect spatial processing that causes the lowering of sensitivity for higher spatial frequencies. From the scaling results, it could be argued that the basic mechanisms involved in the generation of illusory motion may be present and located in the stages of early visual processing, such as V1, although it is not possible to unequivocally identify the neural correlates of our psychophysically estimated scaling factors. 
In early visual processing, conventional psychophysical investigations led to the development of a consensus that there are two functional systems, a transient system preferring lower spatial frequencies and higher temporal frequencies, and a sustained system preferring higher spatial frequencies and lower temporal frequencies (Hess & Snowden, 1992; Snowden & Hess, 1992; Watson & Robson, 1981). In our previous study, we showed that the transient system in visual processing plays an important role in producing the Rotating Snakes illusion, and argued that having a biphasic temporal impulse response function, which corresponds to a band-pass filter with respect to temporal frequency, is a necessary condition (Hisakata & Murakami, 2008). The results of the present study indicating a preference for lower fundamental spatial frequencies are consistent with our previous suggestion regarding major involvement of the transient channel. This suggestion is also in accordance with physiological results showing that neurons activated by first-order motion in the primary visual cortex are also activated by the luminance pattern in the Rotating Snakes (Conway et al., 2005). 
One peculiar finding in the current study is that illusion strength reached minimum at a particular spatial frequency and partially recovered beyond it. The same pattern of spatial frequency tuning was repeated at all tested eccentricities, which was not only helpful in offering anchor points that strongly constrained spatial scaling but also indicative of a general rule of the Rotating Snakes illusion in a concentric stimulus pattern centered at the fixation point. In a previous physiological study of the responses of cat visual neurons, Sestokas and Lehmkuhle (1986) showed that the latency of X-cells was shortest for gratings around 0.75 cycles/deg and longer for lower and higher spatial frequencies. Conway et al. (2005) suggested that if a difference of the latencies of neurons is crucial for the illusion, long latencies of a subpopulation of neurons for low and high spatial frequencies might give rise to latency differences that are large enough to elicit the illusion. Another possibility is that the sustained system might interfere with the generation of illusory motion. The contrast sensitivity to static gratings has a band-pass shape with the peak around 3∼5 cycles/deg at 7.5∼14 deg eccentricity (Johnston, 1986; Rovamo et al., 1978). In the current study, each cycle of our fundamental spatial frequency corresponds to a micropattern containing four steps of luminance (black, dark gray, white, and light gray), meaning that the stimulus had rich harmonics, some of which may have overlapped the contrast sensitivity peak at the trough of illusion strength against fundamental spatial frequency. The sustained system presumably governs detection of correlation structures such as collinearity and relative position, and reliable detection of positional relationship may have overshadowed potential contribution from the transient system. 
In an fMRI study, Kuriki et al. (2008) examined BOLD signal changes in response to a figure for the Rotating Snakes illusion but the significant change was observed in hMT+, not in V1. The message from this study was that the illusion involves cortical stages in which local motions are integrated into a global solution. It is therefore possible that the cortical stage where motion representations corresponding to illusory motion are generated in local resolution is distinct from the stage where a motion integration process binds such local signals altogether into a representation of a single rigid object that appears to move coherently. Ashida, Kuriki, Murakami, Hisakata and Kitaoka (2012) attempted to see direction-specific BOLD signal changes in response to the figure for the Rotating Snakes illusion by changing the fMRI protocol from a conventional block design to an adaptation paradigm, whereby the effect of direction-specific adaptation was revealed in many visual areas including V1, strengthening the notion that early visual processing registers local motion directions that are ultimately brought upon perception as the Rotating Snakes illusion via higher order processes. 
We have illustrated a potential two-stage model for the Rotating Snakes illusion (Figure 6). In the first stage, seeming motion information is locally extracted from each luminance micropattern. In the second, such local and small spurious motion information is spatially integrated into global rotation, which is eventually brought upon the observer's consciousness as illusory motion in a static figure. This study demonstrated that the spatial scaling by fundamental spatial frequency of micropatterns worked well, implying that the scaling factor reflects the spatial property of the first stage, which is sensitive to how well the stimulus size of input micropatterns fits with the space constant of processing units, rather than properties of the second stage, which is dealing with vectorial information and presumably insensitive to local luminance structure. Our finding that the estimated scaling factor exhibited an eccentricity dependence similar to those of the receptive-field sizes of neurons in lower visual areas, such as V1, suggests that the first stage of the two-stage model could be located early in the visual cortex. 
Figure 6
 
Illustration of the two-stage model for the Rotating Snakes illusion. Local and small spurious motion information is extracted from each micropattern, and subsequently, the local motions are spatially integrated to establish a globally consistent motion at a higher visual stage.
Figure 6
 
Illustration of the two-stage model for the Rotating Snakes illusion. Local and small spurious motion information is extracted from each micropattern, and subsequently, the local motions are spatially integrated to establish a globally consistent motion at a higher visual stage.
Precisely which mechanism is responsible for the extraction of seeming motion information from local micropatterns is still open to discussion. Recently Atala-Gérard and Bach (2017) reported that the perceived direction of the Rotating Snakes illusion reversed when the two middle gray areas of each micropattern had high luminance. They argued that the models proposed in three previous studies (Conway et al., 2005; Fermüller et al, 2010; Murakami et al, 2006) could not explain such a behavior, whereas only the model proposed by Backus and Oruç (2005) would predict the directional reversal of illusory motion if a different adaptation curve was applied to their model. The present study has no particular suggestion for or against any of the models. Future investigations will shed more light on the underlying mechanism. 
Conclusion
In summary, we show that the Rotating Snakes illusion is spatially scalable, and from the scaling results we propose that the generation of the illusory motion involves the transient system preferring low spatial and high temporal frequencies in an early visual processing stage. 
Acknowledgments
We thank Hiroshi Ashida, Ichiro Kuriki and Akiyoshi Kitaoka for their constructive comments. RH was supported by a Grant-in-Aid for JSPS fellows from the Japan Society for the Promotion of Science. IM was supported by the JSPS Funding Program for Next Generation World-Leading Researchers (NEXT Program LZ004) and by JSPS KAKENHI Grant Number 18H01099. 
Commercial relationships: none. 
Corresponding author: Rumi Hisakata. 
Address: School of Engineering, Tokyo Institute of Technology, Kanagawa, Japan. 
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Figure 1
 
Structure of a figure for the Rotating Snakes illusion. (a) A gray-scale figure for the illusion (reproduced, with permission, from the website of Akiyoshi Kitaoka, who originally created this figure http://www.ritsumei.ac.jp/∼akitaoka/index-e.html). (b) A simplified figure for the Rotating Snakes illusion. Luminance micropatterns, each comprised of white, light gray, black, and dark gray, are repeated circularly. In this example, a rotary illusory motion is perceived in the clockwise direction.
Figure 1
 
Structure of a figure for the Rotating Snakes illusion. (a) A gray-scale figure for the illusion (reproduced, with permission, from the website of Akiyoshi Kitaoka, who originally created this figure http://www.ritsumei.ac.jp/∼akitaoka/index-e.html). (b) A simplified figure for the Rotating Snakes illusion. Luminance micropatterns, each comprised of white, light gray, black, and dark gray, are repeated circularly. In this example, a rotary illusory motion is perceived in the clockwise direction.
Figure 2
 
Stimulus configuration. A fixation point was always presented at the center of the display, and the background was filled with static random dots.
Figure 2
 
Stimulus configuration. A fixation point was always presented at the center of the display, and the background was filled with static random dots.
Figure 3
 
Examples of psychometric functions for subject AI. Each panel shows results under the condition of 88 cycles at each eccentricity. The black dots and solid curves indicate the data for the CCW stimulus whereas the open dots and dashed curves indicate the data for the CW stimulus. The positive values of velocity indicate that the stimulus rotated counterclockwise.
Figure 3
 
Examples of psychometric functions for subject AI. Each panel shows results under the condition of 88 cycles at each eccentricity. The black dots and solid curves indicate the data for the CCW stimulus whereas the open dots and dashed curves indicate the data for the CW stimulus. The positive values of velocity indicate that the stimulus rotated counterclockwise.
Figure 4
 
(a) Intersubject average of the strength of the Rotating Snakes illusion plotted against fundamental spatial frequency, with eccentricity as a parameter. Error bars indicate standard error. Both axes are in log scale. (b) Results of spatial scaling. The gray curve indicates the best-fit parabola. FSF = fundamental spatial frequency. Other conventions are identical to those in (a).
Figure 4
 
(a) Intersubject average of the strength of the Rotating Snakes illusion plotted against fundamental spatial frequency, with eccentricity as a parameter. Error bars indicate standard error. Both axes are in log scale. (b) Results of spatial scaling. The gray curve indicates the best-fit parabola. FSF = fundamental spatial frequency. Other conventions are identical to those in (a).
Figure 5
 
Comparison among psychophysical as well as physiological scaling factors: Scaling factors obtained in this study are shown as open circles. Each linear function indicates the normalized scaling factors against eccentricity in previous studies, summarized by Murakami and Shimojo (1996). Detection threshold from Levi et al. (1985): scaling factor = 1 + 1.05−1E. RF size in the macaque monkey V1 neurons from Dow et al. (1981): Log (60RF) = 1.1438 + 0.1920x + 0.0712x2 + 0.0619x3, where x = log (60E), but was approximated by a linear function here. Contrast sensitivity from Watson (1987): scaling factor = 1 + 0.24E. Vernier threshold from Whitaker et al. (1992): Scaling factor = 1 + 1.66(E − 0.267). Detection thresholds for relative motion from Levi et al. (1985): Scaling factor = 1+5.99−1E. RF size in monkey MT neurons from Albright and Desimone (1987): RF = 1.04 + 0.61E.
Figure 5
 
Comparison among psychophysical as well as physiological scaling factors: Scaling factors obtained in this study are shown as open circles. Each linear function indicates the normalized scaling factors against eccentricity in previous studies, summarized by Murakami and Shimojo (1996). Detection threshold from Levi et al. (1985): scaling factor = 1 + 1.05−1E. RF size in the macaque monkey V1 neurons from Dow et al. (1981): Log (60RF) = 1.1438 + 0.1920x + 0.0712x2 + 0.0619x3, where x = log (60E), but was approximated by a linear function here. Contrast sensitivity from Watson (1987): scaling factor = 1 + 0.24E. Vernier threshold from Whitaker et al. (1992): Scaling factor = 1 + 1.66(E − 0.267). Detection thresholds for relative motion from Levi et al. (1985): Scaling factor = 1+5.99−1E. RF size in monkey MT neurons from Albright and Desimone (1987): RF = 1.04 + 0.61E.
Figure 6
 
Illustration of the two-stage model for the Rotating Snakes illusion. Local and small spurious motion information is extracted from each micropattern, and subsequently, the local motions are spatially integrated to establish a globally consistent motion at a higher visual stage.
Figure 6
 
Illustration of the two-stage model for the Rotating Snakes illusion. Local and small spurious motion information is extracted from each micropattern, and subsequently, the local motions are spatially integrated to establish a globally consistent motion at a higher visual stage.
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