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Perception of spatiotemporal random fractals: an extension of colorimetric methods to the study of dynamic texture Vincent A. Billock, Douglas W. Cunningham, and Paul R. Havig Logicon, Inc., U. S. Air Force Research Laboratory, P. O. Box 317258, Dayton, Ohio 45437-7258 Brian H. Tsou U. S. Air Force Research Laboratory, Building 146, Wright-Patterson Air Force Base, Ohio 45433 Received October 23, 2000; accepted April 4, 2001; revised manuscript received April 16, 2001 Recent work establishes that static and dynamic natural images have fractal-like 1/ faspatiotemporal spectra. Artiﬁcal textures, with randomized phase spectra, and 1/ faamplitude spectra are also used in studies of tex-ture and noise perception. Inﬂuenced by colorimetric principles and motivated by the ubiquity of 1/ faspatial and temporal image spectra, we treat the spatial and temporal frequency exponents as the dimensions char-acterizing a dynamic texture space, and we characterize two key attributes of this space, the spatiotemporalappearance map and the spatiotemporal discrimination function (a map of Mac Adam-like just-noticeable-difference contours). © 2001 Optical Society of America OCIS codes: 330. 1730, 330. 5510, 330. 6100, 330. 6110, 330. 6790, 100. 6740. 1. GENERAL INTRODUCTION Early color science struggled with problems of quantify-ing, representing, and predicting color appearance. Thedevelopment of colorimetry solved many of these prob-lems with the creation of color spaces that quantify colors,relate them in appearance, and make predictions aboutdiscriminability. 1,2Texture perception suffers from simi-lar difﬁculties and seems in need of similar solutions. Ina pioneering study, Richards showed that colorimetrictexture and ﬂicker matches can be made to arbitrary spa-tial and temporal stimuli by mixing several spatial ortemporal ''primaries. '' 3,4This approach ﬁts well with both the multiscale structure of natural images and withthe multichannel visual processing of spatial and tempo-ral information, but the dimensionality of a spatiotempo-ral match would be cumbersomely large. This dimen-sionality can be reduced in some situations where there isa lawful relationship between the contributions at variousscales. Such an approach is suggested by fractal math-ematics and by a plethora of recent studies on fractal-likeproperties of natural images. Fractional Brownian tex-tures (favorite objects of study in recent textureexperiments 5-7), natural images,8-11and ''fractal forger-ies'' of natural images12-14all share 1/ fbspatial frequency spectra. Similarly, temporal sequences of naturalimages 15-18and the electronics used in imaging devices19,20produce 1/ fatemporal spectra. This suggests that a reasonable model for the amplitude spectra ofmany spatiotemporal textures is A ~fs,ft!5Kfs2bft2a. (1) where fsand ftare spatial and temporal frequency, re-spectively. Throughout this paper we will refer to dy-namic textures—with randomized phase spectra and power law [Eq. (1)] Fourier amplitude spectra—as spa-tiotemporal random fractals. 21We mapped the percep-tual space of such dynamic fractal textures, quantifyingappearance and discriminability. Speciﬁcally, we soughtto create an appearance map of the a,bdynamic texture space, analogous to a CIE space with labeled color bound-aries. We also sought to characterize discriminability ofdynamic textures, by measuring just noticable differences(JNDs) for the four cardinal directions in this space,analogous to the measurement of Mac Adam's ellipses incolorimetry. 2. EXPERIMENT 1: QUANTIFICATION OF APPEARANCE SPACE A. Introduction Every color researcher has a fairly clear notion of the ap-pearance of a speciﬁed aperture color based on its CIE co-ordinates. This expectation is based on a general aware-ness of the approximate CIE coordinates of many spectraland desaturated colors. Most CIE charts plot the spec-tral locus and the blackbody locus. Some reference CIEcharts are even helpfully colored over their entire surface. One can consider such a chart an ''appearance map'' of thecolorimetric space (see Fig. 1 for an example). 22-24We set out to produce an appearance map for spatiotemporalrandom-fractal space by varying the aandbparameters of Eq. (1) and asking several observers to characterize theappearance of these dynamic fractals. B. Methods 1. Participants The four observers (PH, DC, SF, and VB) were all myopes wearing their refractive prescriptions, yielding correc-2404 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al. 0740-3232/2001/102404-10$15. 00 © 2001 Optical Society of America	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
tions to at least 6/6 binocular acuity. All participants are published psychophysicists and highly experienced ob-servers, with prior work in the psychophysics of ''white''and ''colored'' (spectrally nonuniform) spatial noise. Three of the observers were also authors of this study; oneadditional observer (SF) was naı ¨ve to the purpose of the experiment. 2. Apparatus All stimuli were generated and presented on a Silicon Graphics O2 at 30 Hz. The lookup table was modiﬁed tolinearize the display. All stimuli were viewed binocu-larly with natural pupils in a well-lit room (the luminanceof a diffuse reﬂector in the plane of the monitor was 3. 5cd/m 2). Subjects were comfortably ﬁxed in place by a chin rest at a viewing distance of 40 cm from the monitor. 3. Stimuli The stimuli were 70 spatiotemporal, gray-scale, random-phase fractal-like image sequences. For each stimulusimage, the average luminance was constant at 8. 57 cd/m 2 Fig. 1. Appearance map of CIE 1931 color space. Reproduced from Ref. 22 by permission of the Optical Society of America. Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2405	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
(not including the diffuse room illumination). Owing to computational constraints, each fractal was limited to64364 pixels (18 318 mm) in size and 64 frames in du-ration. Thus each stimulus subtended 2. 58 arc deg em-bedded in a 43. 9 (H) by 36. 4 (V) deg dark surround andlasted 2. 133 s. Each 64-frame stimulus sequence wascreated by ﬁrst producing a series of random white noiseimages. Gray-scale values are computed digitally withuse of 15-bit ﬂoating-point numbers and are displayedwith 8 bits (256-level gray scale). Each image in the se-ries was Fourier transformed, and the amplitudes of allspatial frequencies were equalized to ensure that thenoise was uniformly white. The resulting frequency dis-tributions were then ﬁltered so that the amplitude spec-trum varied over time following the power law relation-ship 1/ f taand over space (isotrophically over both Cartesian coordinates) following the power law relation-ship 1/ f sb@s5(x21y2)1/2#. Ten spatial exponents ( b Fig. 2. Appearance map of spatiotemporal fractal space. The appearance map is based on the responses of four observers. To help render the chart more readable to a diverse audience, the real-world descriptions provided by the subjects (which were highly individu-alistic and culturally based) have been replaced where possible with stimulus-based descriptions provided by the other subjects. 2406 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al.	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
50. 4, 0. 6, 0. 8, 1. 0, 1. 2, 1. 4, 1. 6, 1. 8, 2. 0, and 2. 2) were factorially combined with seven temporal exponents( a50. 2, 0. 4, 0. 6, 0. 8, 1. 0, 1. 2, and 1. 4) to create the 70 dynamic image sequences we employed. The resultingﬁltered spectra were inverse Fourier transformed to gray-scale levels for display. For noise stimuli the traditionalmeasures of contrast have proved inadequate. A usefulmetric of noise contrast is root mean square (RMS)contrast, 25which we express here in percentage terms as the standard deviation of the gray levels in the display di-vided by the mean. The standard deviation and mean ofthe gray-scale distribution in each stimulus were mea-sured. The gray levels in the ﬁnal image were thenrenormalized to a standard deviation of 28 and a meanlevel of 127, corresponding to a RMS contrast of 22%. 4. Procedure Observers were asked to provide a written description of the phenomenal appearance of each fractal sequence andwere free to use any terms they chose. They were givenan unlimited amount of time to respond. Each fractal se-quence was shown in a continuous loop until the observerindicated readiness to proceed to the next display. Thefractals were presented one at a time in order, starting with the lowest spatial and temporal exponents. All tem-poral exponents were shown for a given spatial exponentbefore the spatial exponent was increased. C. Results and Discussion There was considerable regularity across the four observ-ers' descriptions, so the responses were combined to pro-duce a consensus map of the appearance of the spatiotem-poral texture space (see Fig. 2). Very idiosyncratic termsprovided by some of the subjects were replaced by moregeneral terms consistent with that description. As ex-pected from previous work, 5-8,26,27the primary effect of increasing the spatial exponent bwas to increase the ap-parent coarseness of the fractal texture (see Fig. 3 for anillustration of the effect of bon static fractal textures). The primary effect of increasing the temporal exponent a was to decrease the apparent speed and jitter of the dy-namic textures. Perceptually, for aexponents of 0. 2 and 0. 4, the motion appeared very jittery; e. g., motion to theright was more often than not followed by motion to theleft. In the theory of fractional (biased) Brownian mo-tions, such behavior is called antipersistent and is differ-entiated from random (Brownian) motions and persistentmotions. 28-31Interestingly, when the spatial and tempo-ral exponents were both within the range of natural im-ages, the subjects tended to provide real-world descrip-tions of the stimuli. There were strong interactions between the spatial and temporal dimensions for the appearance of dynamic tex-ture. There are three primary examples of this interac-tion. First, the emergent texture regions were oftenmore apparent, larger, and more diffuse at higher tempo-ral exponents (slower speeds). Second, when the tempo-ral exponent was between 0. 6 and 1. 0, the texture oftenappeared to rotate, but only for smaller spatial exponents. Finally, when the temporal exponent was between 0. 4 and0. 8 and the spatial exponent was small, there was a ten-dency to see two transparent textures, with one moving infront of the other. 3. EXPERIMENT 2: DISCRIMINATION OF DYNAMIC TEXTURES A. Introduction In addition to being able to anticipate the appearance of an aperture color, given its CIE coordinates, color re-searchers are also able to gauge how tolerant subjectswould be to small perturbations in the CIE coordinates. Several studies, especially Wright's and Mac Adam's pio-neering work, 32-35have measured the just noticeable dif-ferences for perturbations of CIE coordinates. Becausethe CIE space is usually depicted as a plane, the locus of JNDs around a single point is usually an ellipse. A mapof such ''Mac Adam ellipses'' in the CIE space (Fig. 4) is auseful tool, which we wished to characterize for the spa-tiotemporal random-fractal space. So far as we areaware, prior fractal discrimination studies have been lim-ited to spatial discrimination. 26,27 Fig. 3. Static, random phase, fractal-like textures produced by 1/fbspatial-frequency ﬁltering of random white noise. Static snapshots of ﬁve of the ten fractal exponents (for one seed valueof the random number generator) are shown here. From top tobottom, the values of the exponent bare 0. 4, 0. 8, 1. 2, 1. 6, 2. 0. Note that as the spatial exponent increases, the apparent coarse-ness of the texture also increases. Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2407	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
B. Methods 1. Participants and Apparatus These were the same as in Experiment 1. 2. Stimuli The stimuli were identical to those used in Experiment 1, with the sole exception that static fractals were also em-ployed. 3. Procedure The smallest detectable increase (Above JND) and de-crease (Below JND) in the spatial exponent band in the temporal exponent awere separately measured with an adaptive staircase procedure. For each of the four typesof discrimination, a reference and a comparison imagewere presented side by side (1. 1 mm apart) for 2. 133 s oruntil the observer responded, whichever came ﬁrst. Ineach case, the subject was given instructions that wereequivalent to identifying the reference stimulus. The lo-cation of the reference stimulus (left versus right) wasrandomized across trials. Immediately after the ob-server responded (using a computer keyboard), feedbackwas provided on the accuracy of this response. If the ob-server did not respond within the display time, the screenwas blanked and the observer was prompted for an an-swer. If the observer accurately identiﬁed the referencestimulus three times in a row (79% correct criteria), the difference between the exponents of the comparison andreference images was decreased by 10%. If the observerresponded incorrectly, the difference between the expo-nents was increased by 10%. Each staircase continuedfor eight reversals, and the mean of the last six reversalswas used as a measure of the threshold. When spatial JND's were measured, the reference and the comparison fractals were assigned the same temporalexponent and the spatial exponent of the comparisonstimulus was adaptively varied. The observers wereasked to identify the reference image. That is, they wereto ﬁnd the more ''ﬁne-grained'' texture for the Above dis-criminations and the less ﬁne-grained texture for the Be-low discriminations. When temporal JND's were measured, the spatial ex-ponent was held constant while the temporal exponentwas adaptively varied. The observers were again askedto identify the reference image. In other words, theywere to ﬁnd the faster and more jittery texture for the Above discriminations and the slower and less jittery tex-ture for Below discriminations. For all subjects, the 80 Above spatial discriminations were presented ﬁrst, followed by the 80 Below spatial dis-criminations, then the 70 Above temporal discrimina-tions, and ﬁnally the 70 Below temporal discriminations. Fig. 4. Mac Adam's ellipses. 33Discrimination thresholds (JNDs) for various directions in CIE 1931 color space. Ellipses are for Mac-Adam's observer PGN and are shown at ten times their actual size. Reproduced by permission of the Optical Society of America. 2408 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al.	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
Within each type of discrimination, the order of presenta-tion of the fractal exponents was randomized, with eachobserver receiving a different random order. We alsorandomized the ''seed'' values of the random number gen-erator used to generate the white noise before ﬁltering. The same seed values were used within each staircase butnot for subsequent staircases. Thus the data in Figs. 5-9, which are collapsed across subjects or conditions, represent discrimination for an ensemble of randomphase spectra rather than any particular pattern of ran-dom phases. Each subject required roughly 50 h to com-plete the measurements. C. Results and Analysis Figure 5 plots just-noticeable-difference contours (analo-gous to Mac Adam ellipses) for the dynamic fractals. It isinteresting to note that the ellipse size is fairly constantover much of the space. In color, much attention hasbeen devoted to obtaining colorimetric spaces in whichdiscrimination thresholds are relatively constant: the socalled Uniform Color Spaces (e. g., CIE 1976 Uniform Color Space). 1,2The spatial and temporal discrimina-tions will be discussed separately in more detail below. 1. Spatial Just-Noticeable Differences Overall, spatial discriminations were easiest when bwas between 1. 4 and 1. 8. This is consistent with previous re-search on static fractal discrimination. 26,27Spatial JND's are plotted in Fig. 6 as a function of spatial expo-nent. The data for both Above and Below JND's are re-markably similar across subjects. 2. Effect of Temporal Characteristics on Spatial Discriminations If the spatial and temporal dimensions are independent in dynamic fractals, as is often assumed, similar discrimi-nation thresholds should be found for dynamic fractals re-gardless of temporal exponent. Figure 7 helps to illus-trate the point that the spatial JND's did not vary muchas a function of the temporal nature of the stimuli. Al-though there is a slight suggestion of lower spatial JNDsfor the lowest temporal exponents, it seems that spatialdiscriminations are largely independent of the temporalcharacteristics in the stimuli. 3. Temporal Just-Noticeable Differences Figure 8 plots the temporal JND's as a function of tempo-ral exponent. For each subject, temporal discriminationswere easiest when awas between 0. 8 and 1. 0, a typical range of exponents for natural stimuli. 15-18The pattern of discriminations was remarkably consistent across sub-jects. Of particular interest is that the antipersistentstimuli were extremely difﬁcult to discriminate. More-over, for three of the four subjects, the Below JND's forthe antipersistent stimuli tended to be elevated forsmaller temporal exponents. Notice that for these three Fig. 5. Wright-Mac Adam-like contours for discrimination thresholds in four directions in a spatiotemporal fractal space. Average of four observers. The contours are simple quarter-circles ﬁtted to the JNDs that span their quadrants. These contours are not theo-retically motivated and are used to reduce the confusion that occurs for overlapping JND crosses. Note that over a wide range of thespace the discrimination thresholds vary only slightly; e. g., the space approaches the ideal of a uniform colorimetric space (such as the CIE 1976 uniform chromaticity space). 1,2Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2409	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
observers, the JND at a50. 2 was greater than 0. 2; i. e., the exponent a2Dabecomes negative, and the slope of the 1/ fa2Daspectra for the comparison image is slightlypositive. This can occur only for the Below discrimina-tions and is the most likely explanation of the asymmetrybetween Above and Below thresholds. Fig. 6. Spatial JND's collapsed across temporal exponents. Here the spatial discriminations are plotted as a function of spatial expo-nent. Each subﬁgure is for an observer named by his initials. Fig. 7. Spatial JND's collapsed across spatial exponents. Here the spatial discriminations are plotted as a function of temporal expo-nent. The temporal exponent labeled ''0'' is a place holder for the static condition. Observers are identiﬁed by initials. 2410 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al.	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
4. Effect of Spatial Characteristics on Temporal Discriminations Figure 9 plots the temporal JND's as a function of spatial exponent. For all four observers, thresholds increased asthe spatial exponent increased. In other words, discrimi-nating the temporal nature of the stimuli became more difﬁcult as the texture became coarser. This makes in-tuitive sense, as it would be easier to see a 1-mm object Fig. 8. Temporal JND's collapsed across spatial exponents. Here the temporal discriminations are plotted as a function of temporal exponent. Each subﬁgure is for an observer named by his initials. Fig. 9. Temporal JND's collapsed across temporal exponents. Here the temporal discriminations are plotted as a function of spatialexponent. Each subﬁgure is for an observer named by his initials. Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2411	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
move 2 mm (200% of its size) than it would be to see a 100-m object move the same distance (0. 002% of its size). 4. GENERAL DISCUSSION The fractal space described here was chosen not only forits multiscale elegance but for its potential applicabilityto real-world images and image sequences. Fractals—mathematical entities that are self-similar over manyspatial or temporal scales—are ideal tools for represent-ing natural phenomena that take place on many scalesand for studying vision, which contains multiple channelstuned on the spatial and temporal dimensions. Randomfractal textures (sometimes called fractional Browniantextures) have been widely studied because their 1/ f b spectra simulate the spectra of natural images without being confounded by phase information. If bis in the range of a typical natural object ( b50. 9-1. 2),11subjects will often spontaneously report its appearance12: a psy-chophysical version of the venerable children's game ofnaming the shapes in clouds. If phase is manipulated,fractal textures can take on the appearance of naturalscenes: Fractal forgeries of landscapes, coastlines andclouds are extremely compelling. 13,14In the temporal di-mension, sequences of events with 1/ faspectra are ubiq-uitous and fractal forgeries are especially striking; 1/ f acoustic sequences sound musical. 36In vision, 1/ faam-plitude spectra are found for dynamic images, where ais indicative of the character of the movement. 18,28-31The use of a common mathematical framework for character-izing both dynamic noise and dynamic images may alsofacilitate the study of masking of images by noise. ACKNOWLEDGMENTS We thank Jer Sen Chen, Steven Fullenkamp, and Eric Heft for technical support. Douglas Cunningham's current address is Max Planck Institute for Biological Cybernetics, Spemannstrasse 38,Tubingen 72076, Germany. REFERENCES AND NOTES 1. D. B. Judd and G. Wyszecki, Color in Business, Science and Industry, 2nd ed. (Wiley, New York, 1963). 2. G. Wyzecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982). 3. W. Richards, ''Quantifying sensory channels: generalizing colorimetry to spatiotemporal texture, touch and tones,''Sens. Processes 3, 207-229 (1979). In general, Richards found that arbitrary textures could be matched by mixturesof four separated texture primaries and that temporalmodulations could be matched by mixtures of three sepa-rated ﬂicker primaries. 4. The analogy to colorimetry cannot be pushed too far here. Brill, in an interesting gedanken experiment, showed thatunder some circumstances Grassman's additivity law is vio-lated for some textures: M. H. Brill, ''Formalizing Grass-man's laws in a generalized colorimetry,'' Sens. Processes 3, 370-372 (1979). 5. J. E. Cutting and J. J. Garvin, ''Fractal curves and complex-ity,'' Percept. Psychophys. 42, 365-370 (1987). 6. T. Kumar, P. Zhou, and D. A. Glaser, ''Comparison of human performance with algorithms for estimating fractal dimen-sion of fractional Brownian statistics,'' J. Opt. Soc. Am. A 10,1 1 3 6-1146 (1993). 7. A. Pentland, ''Fractal-based description of surfaces,'' in Natural Computation, W. Richards, ed. (MIT Press, Cam-bridge, Mass., 1988), pp. 279-299. 8. A. P. Pentland, ''Fractal-based description of natural scenes,'' IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661-674 (1984). 9. D. J. Field, ''Relations between the statistics of natural im-ages and the responses of cortical cells,'' J. Opt. Soc. Am. A4, 2379-2394 (1987). 10. D. J. Tolhurst, Y. Tadmor, and T. Chou, ''The amplitude spectrum of natural images,'' Ophthalmic Physiol. Opt. 12, 229-232 (1992). 11. V. A. Billock, ''Neural acclimation to 1/ fspatial frequency spectra in natural images transduced by the human visualsystem,'' Physica D 137, 379-391 (2000). 12. B. E. Rogowitz and R. F. Voss, ''Shape perception and low-dimensional fractal boundaries,'' Human Vision and Elec-tronic Imaging: Models, Methods, and Applications,J. P. Allebach and B. E. Rogowitz, eds., Proc. SPIE 1249, 387-394 (1990). 13. R. F. Voss, ''Random fractal forgeries,'' in Fundamental Al-gorithms for Computer Graphics, R. A. Earnshaw, ed. (Springer, Berlin, 1985), pp. 805-835. 14. B. B. Mandelbrot, The Fractal Geometry of Nature (Free-man, New York, 1983). 15. M. P. Eckert, G. Buchsbaum, and A. B. Watson, ''Separabil-ity of spatiotemporal spectra of image sequences,'' IEEETrans. Pattern Anal. Mach. Intell. 14, 1210-1213 (1992). 16. D. W. Dong and J. J. Atick, ''Statistics of time varying im-ages,'' Network Comput. Neural Syst. 6, 345-358 (1995). 17. J. H. van Hateren, ''Processing of natural time series by the blowﬂy visual system,'' Vision Res. 37, 3407-3416 (1997). 18. V. A. Billock, G. C. De Guzman, and J. A. S. Kelso, ''Fractal time and 1/ fspectra in dynamic images and human vision,'' Physica D 148, 136-146 (2001). 19. M. Savilli, G. Lecoy, and J. P. Nougier, Noise in Physical Systems and 1/f Noise (Elsevier, New York, 1983). 20. M. S. Keshner, ''1/ fnoise,'' Proc. IEEE 70, 212-218 (1982). 21. Technically, fractals have inﬁnite spectral bandwidths. Purists would designate textures physically obtainable ondisplays as prefractals or pseudofractals. 22. K. L. Kelly, ''Color designations for lights,'' J. Opt. Soc. Am. 33,6 2 7-632 (1943). 23. P. Keller, ''1976-UCS chromaticity diagram with color boundaries,'' Proc. Soc. Inf. Disp. 24, 317-321 (1983). 24. Color appearance maps of Kelly's CIE 1931 and Keller's 1976 spaces are available from Photo Research, 3000 N. Hollywood Way, Burbank, California 91505. 25. B. Moulden, F. Kingdom, and L. F. Gatley, ''The standard deviation of luminance as a metric for contrast in random-dot images,'' Perception 19,7 9-101 (1990). 26. D. C. Knill, D. Field, and D. Kersten, ''Human discrimina-tion of fractal textures,'' J. Opt. Soc. Am. A 7, 1113-1123 (1990). 27. Y. Tadmor and D. J. Tolhurst, ''Discrimination of changes in the second-order statistics of natural and synthetic images. ''Vision Res. 34, 541-554 (1994). 28. In the theory of fractional (biased) Brownian motion, the motion bias is quantiﬁed by the distance that a biasedwalker moves from the origin in unit time. A Brownian(random) walker's distance over unbroken ground is pro-portional to Atand has an amplitude spectrum exponent of 1. 0. Greater distance than this is covered if the motion ispersistently biased (if bias is perfect, distance is propor-tional to time), less distance is covered if the bias is anti-persistent. B. B. Mandelbrot and J. W. van Ness, ''Frac-tional Brownian motions, fractional noises andapplications,'' SIAM Rev. 10,4 2 2-437 (1968). 29. Over broken ground, a random walker covers less distance from the origin. The case closest to our stimuli that hasbeen studied in statistical physics is Brownian motion on afractal fractured surface. The temporal frequency spec-trum exponent afor this case is 0. 5. W. Lehr, J. Machta,2412 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al.	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
and M. Nelkin, ''Current noise and long time tails in biased disordered random walks,'' J. Stat. Phys. 36,1 5-29 (1984). This is in accord with our ﬁnding that stimuli with expo-nents below 0. 5 appear to move in a jittery fashion, whereasexponents above 0. 5 move smoothly. 30. In an interesting experiment Snippe and Koenderink stud-ied human ability to perceive correlations between mem-bers of a row of light sources. Positive correlations be-tween two lights were detected as apparent motion, but thecorresponding anticorrelation between the same two lightswas undetectable as motion. H. P. Snipe and J. J. Koen-derink, ''Detection of noise-like luminance functions,'' Per-cept. Psychophys. 55,2 8-41 (1994). 31. One oddity of random motion is that the visual system does not seem to be able to compensate for motion blur forstimuli undergoing Brownian motion. S. N. J. Watama-niuk, ''Visual persistence is reduced by ﬁxed-trajectory mo-tion but not random motion,'' Perception 21, 791-802 (1992). No one has yet studied deblurring for anticorre-lated (antipersistent) motion. 32. W. D. Wright, ''The graphical representation of small color differences,'' J. Opt. Soc. Am. 33, 632-636 (1943). 33. D. L. Mac Adam, ''Visual sensitivity to color differences in daylight,'' J. Opt. Soc. Am. 32,2 4 7-274 (1942). 34. L. Silberstein and D. L. Mac Adam, ''The distribution of color matchings around a color center,'' J. Opt. Soc. Am. 35, 32-39 (1945). 35. W. R. J. Brown and D. L. Mac Adam, ''Visual sensitivities to combined chromaticity and luminance differences,'' J. Opt. Soc. Am. 39, 808-834 (1949). 36. R. F. Voss and J. Clarke, ''1/ fnoise in music: music from 1/fnoise,'' J. Acoust. Soc. Am. 63, 258-263 (1978). Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2413	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
APPROVED FOR PUBLIC RELEASEPresented at the Human Factors Issues in Combat Identification Workshop, Gold Canyon, Arizona, May 13, 2008. WHAT VISUAL DISCRIMINATION OF FRACTAL TEXTURES CAN TELL US ABOUT DISCRIMINATION OF CAMOUFLAGED TARGETS Vincent A. Billock General Dynamics Advanced Information Systems Douglas W. Cunningham University of Tübingen Brian H. Tsou U. S. Air Force Research Laboratory Abstract Most natural images have 1/fȕ Fourier image statistics, a signature which is mimicked by fractals and which forms the basis for recent applications of fractals to camouflage. To distinguish a fractal camouflaged target (with 1/fȕ* statistics) from a 1/fȕ natural background (or another target), the exponents of target and background (or other target) must differ by a critical amount (dȕ ȕȕ*), which varies depending on experimental circumstances. The same constraint applies for discriminating between friendly and enemy camouflaged targets. Here, we present data for discrimination of both static and dynamic fractal images, and data on how discrimination varies as a function of experimental methods and circumstances. The discrimination function has a shallow minimum near ȕ=1. 6, which typifies images with less high spatial frequency content than the vast majority of natural images ( ȕ near 1. 1). This implies that discrimination between fractal camouflaged objects is somewhat more difficult when the camouflaged objects are sufficiently similar in statistics to the statistics of natural images (as any sensible camouflage scheme should be), compared to the less natural ȕ value of 1. 6. This applies regardless of the ȕ value of the background, which has implications for fratricide; friendlies and hostiles will be somewhat harder to tell apart for naturalistically camouflaged images, even when friendlies and hostiles are both visible against their backgrounds. The situation is even more perverse for “active camouflage”. Because of perceptual system nonlinearities (stochastic resonance), addition of dynamic noise to targets can actually enhance target detection and identification under some conditions.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 2 APPROVED FOR PUBLIC RELEASEIntroduction Discrimination tasks in combat target identification are legion. For example, operators need to discriminate a target against a background and to discriminate a set of similar targets from one another. The first task is a necessary, but not sufficient condition for targeting, while the second task is essential to solve decoy and friendly-fire problems. Both tasks are complicated by camouflage. If it were necessary to consider the set of all possible targets, backgrounds and camouflage, the combinatorial problem would be disheartening. However, a consideration of visual psychophysics, image science and fractal mathematics suggests that a particularly simple optical signature provides a low-dimensional solution. Figure 1. An Expeditionary Fighting Vehicle (General Dynamics, Inc. ) concealed against foliage with two different camouflage schemes. The rear of the vehicle is in standard single-scale NATO camouflage and pops-out from the foliage background. The front of the vehicle (see figure bottom) is concealed by a two-scale MARPAT camouflage pattern and is less conspicuous (O'Neill et al., 2004). If the number of scales increases, the perception of fractal-like camouflage is less distance dependent. Courtesy of the United States Marine Corps Systems Command. Background: Perceptual Popout, Fractals and Camouflage It is well known that humans effortlessly (and preattentively) discriminate images which differ significantly in their second-order statistics (the so-called "pop-out" phenomenon), while images	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 3 APPROVED FOR PUBLIC RELEASEthat have similar second-order statistics must usually be compared on a more laborious point-by-point basis (Julesz & Caelli, 1979; Caelli, 1981). Most natural (and many artificial) images have surprisingly regular 1/fȕ Fourier spatial amplitude spectra (Table 1; see Field & Brady, 1997 and Billock, 2000 for review), a signature which is mimicked by random fractals and which forms the basis for fractal forgeries (Voss, 1985) and digital camouflage. The exponent ȕ (the slope of the Fourier spectra when plotted on log-log coordinates) is a second-order statistic. A growing body of evidence suggests that humans are adapted to this statistical regularity in the environment and that this evolutionary/developmental adaptation forms the basis for neural image enhancement and debluring (Billock, 2000; Billock et al., 2001a,b; Campbell et al., 1978; Hammett & Bex, 1996). A hallmark of random fractal images is the presence of statistically similar features at every spatial scale. The lawful relationship between spatial scales is termed self-similarity and is one of the properties of natural images that give rise to 1/fȕ spectra. This property is what enables random fractals to mimic natural images and backgrounds. For example, a tree branch gives rise to several smaller branches, which give rise to many twigs-a random fractal that distributes and scales its features similarly can emulate foliage and act as camouflage. The ȕ value (or equivalently, fractal dimension) is often used as a mathematical measure of image texture and its perceptual correlates (Cutting & Garvin, 1987; Kumar et al., 1993; Pentland, 1988; Rogowitz & Voss, 1990; Taylor et al., 2005). It follows that some aspects of fractal image discrimination can emulate natural image discrimination (Hansen & Hess, 2006; Thomson & Foster, 1997; Parraga & Tolhurst, 2000; Tolhurst & Tadmor, 2000). Table 1. Second-order statistics of natural images Study Number of images ȕ±1sd Burton & Morehead (1987) 19 1. 05±. 12 Field & Brady (1997) 20 1. 10±0. 14 Parraga (1998) 29 1. 11±0. 13 Ruderman (1994) 45 0. 905 Webster & Miyahara (1997) 48 1. 13 Thomson & Foster (1997) 82 1. 19 Field (1993) 85 1. 10 van Hateren (1992) 117 1. 065±. 18 Tolhurst et al. (1992) 135 1. 20±. 13 Schaaf & Hateren (1996) 276 0. 94±0. 21 Dong & Atick (1995) 320 1. 15 Weighted average 1176 1. 08 If natural backgrounds are fractal-like, camouflage should be designed along similar principles. Newer camouflage schemes like MARPAT (U. S Marines) and CADPAT (Canadian Armed Forces) use a two-scale scheme which is noticeably better at blending into terrain and foliage than the older single-scale schemes. For example, detection times for MARPAT (Fig. 1) camouflaged targets are about 2. 5 times longer than detection of NATO single-scale camouflage	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 4 APPROVED FOR PUBLIC RELEASEand recognition times following detection increase by an additional 20% (O'Neill et al., 2004). Some newer camouflage schemes-inspired by fractals-have more than two scales. (True fractal camouflage would be defined by statistical similarity at every visible spatial scale, but limited size and printing resolution result in a restricted range of scales. ) More complicated schemes are possible, including the use of multi-fractals which mimic blends of particular textures that occur in natural images (e. g., plant growth on fractured rock). Here, we study human abilities to discriminate images based on small differences in the ȕ signature and place the results in context with camouflage and with earlier texture discrimination studies. Figure 2. Fractal textures like those used in the experiments. Each fractal in this figure has 1/fβ amplitude spectrum and identical phase spectra, and is synthesized by spatial frequency filtering the same set of random gray levels. The lack of weight in the higher spatial frequencies can easily be seen in the coarseness of the images as the exponent β increases.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 5 APPROVED FOR PUBLIC RELEASEMethods Participants The four observers were all myopes corrected to at least 20/20 binocular acuity. All are professional psychophysicists and highly experienced observers, with prior work in the psychophysics of "white" and "colored" (spatiotemporally non-uniform) visual noise. One subject (SF) was naïve to the purpose of the experiment. Another subject (VB) has a diagnosed, mild, congenital visual condition-optic nerve hypoplasia (a low density of neurons in the optic nerve). Although his vision is considered normal by standard clinical measures (including acuity), his contrast sensitivity is slightly depressed (about 1 sd) at all spatial frequencies relative to a large sample of age-matched-normals; this depression in sensitivity is worse for higher spatial frequencies. His vision is relevant here because it provides us a gauge of the effects of spatial under-sampling in an otherwise intact visual system. Apparatus All stimuli were generated and presented on a Silicon Graphics O 2 graphics workstation with a linearized 30 Hz display. Stimuli were viewed binocularly with natural pupils in a well-lit room (ambient luminance in the plane of the monitor was 3. 5 cd/m2). Subjects were comfortably fixed in place by a chin rest at two viewing distances, 40 cm and 100 cm. The far distance was a limiting case (e. g., each pixel subtends 0. 016 deg at 100 cm, matching the subjects' 1 arc min spatial resolution). The stimuli consisted of static, grayscale, random-phase fractals (e. g., Fig. 2) whose Fourier amplitude spectra were described by A(fs) = kf sȕ Eq. 1 Where k is a constant and f s is spatial frequency. (In visual psychophysics, amplitude rather than power spectra are used, because amplitude is proportional to perceptual contrast for each spatial frequency component. ) For each stimulus, the average luminance was constant at 8. 57 cd/m2and the Root Mean Square Contrast (a good measure of perceptual contrast for noise-like textures; Moulden et al., 1990; Peli, 1990, 1997) was 10. 98%. For consistency with another study, each fractal contained 64x64 pixels (18x18mm). Thus, at 40 cm, the stimuli subtended 2. 58O embedded in a 43. 9O horizontal by 36. 4O vertical dark surround. At 100 cm, each stimulus subtended 1. 03O embedded in a 21. 1O horizontal by 16. 4O vertical dark surround. Both a reference and a comparison image were generated for each trial. The images were created by filling a 64X64 array with random white noise (256 gray-levels). This white-noise image was Fourier-transformed and the amplitudes of all spatial frequencies were equalized to ensure that the noise was uniformly flat. The resulting amplitude spectra were filtered so that they followed a power law relationship (Eq. 1), and then inverse-Fourier transformed to produce the stimuli. Procedure Just noticeable discrimination thresholds (79% correct criterion) for fractal spatial exponents were measured using a two-alternative forced-choice adaptive staircase procedure with a 1 db step size (Mac Millan & Creelman, 1991). Ten β exponents were used (0. 4, 0. 6, 0. 8, 1. 0, 1. 2, 1. 4, 1. 6, 1. 8, 2. 0, and 2. 2) for the reference images. For the comparison image, the fractal exponent was equal to the exponent, β, of its reference image plus a small increment, Δβ. Observers were asked to identify the image with the lower spectral exponent, and were provided with immediate	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 6 APPROVED FOR PUBLIC RELEASEfeedback on the accuracy of their response. If the observer correctly identified the reference image three times in a row, the difference between the two images' exponents Δβ was decreased. In contrast, Δβ was increased after each incorrect response. Each staircase continued for 8 reversals, with the mean of the last 6 reversals being used as a measure of the threshold. Two presentation conditions (Sequential and Simultaneous) were used. In the Simultaneous condition, the reference and comparison stimuli were presented side by side (1. 1 mm apart) for 2. 133 seconds. The location of the reference image (left versus right) was randomized across trials. In the Sequential condition, the two stimulus images were sequentially presented in the center of the screen for 2. 133 seconds each. The screen was blanked for 500 ms between the two images to prevent masking effects. The order of presentation of the reference and comparison images was randomized across trials. Combining two viewing distances with two presentation modes yielded four experimental conditions. Two observers were presented with the Near conditions first, and two with the Far conditions first. For all observers, the presentation style (simultaneous vs. sequential) alternated after each threshold. The order of presentation of the 10 exponents was randomized for each of the 4 conditions. Each threshold was measured 3 times, with the thresholds in all 4 conditions being completed once before being re-measured. This required approximately 20 hours of data collection per subject, which was generally completed in 2 one-hour sessions each day, over a two-week period. Figure 3. Group averages for all 4 conditions. Results and Analysis General Findings Discrimination thresholds (d ȕ) are generally in the range of 0. 05-0. 20 for ȕ values of 0. 4-2. 2 (see Figs. 3, 4). The discrimination function is not flat; it has higher (worse) discrimination thresholds for both low and high values of ȕ, and lower (better) discrimination thresholds for in-between	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 7 APPROVED FOR PUBLIC RELEASEvalues of B. The minimum is near ȕ=1. 6, which typifies images with less high spatial frequency content than the vast majority of natural images ( ȕ near 1. 1). This implies that discrimination between fractal camouflaged objects is somewhat more difficult when the statistics of camouflaged objects are sufficiently similar to the statistics of natural images (as any sensible camouflage scheme should be), compared to the less natural B value of 1. 6. This applies regardless of the background's ȕ value, which has implications for fratricide; friendlies and hostiles will be somewhat harder to tell apart for naturalistically camouflaged images, even when friendlies and hostiles are both visible against their backgrounds. Figure 4. Individual data from all 4 observers for all 4 conditions.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 8 APPROVED FOR PUBLIC RELEASEEffect of Viewing Distance For ideal 1/f images, there should be little effect of viewing distance, because increasing viewing distance would simply shift a lower spatial frequency component into a higher spatial frequency, but the relationship between the spatial frequencies would be preserved. However, all physically obtainable fractals are limited to a range of spatial scales set at the lower end by the size of the image and at the upper end by the size of the pixels. Shifting the viewing distance from 40 to 100 cm therefore shifts the spatial frequency range of the fractal image by a factor of 2. 5, but no information is lost because the stimuli were designed so that the individual pixels were resolvable at the far viewing distance by all observers. Accordingly, for three of the observers (DC, PH, and SF), viewing distance had little effect, although there is a slight trend suggesting lower thresholds in the nearer viewing distances (see Figure 4a-c). For VB, however, the Far thresholds (and their variability) were noticeably elevated compared to either his Near thresholds or to the other observers' Far thresholds (see Fig. 4). VB's anomalous results may be due to sampling problems induced by a mild congenital defect-a developmental paucity of retinal ganglion cells (optic nerve hypoplasia). Electrophysiological studies in VB and other hypoplastic subjects and post-mortem histology in other hypoplastics indicate that both retinal pre-processing and cortical post-processing seem to be normal (Billock et al., 1994) and point to reduction in retinal ganglion cell numbers as the sole cause of abnormal vision in hypoplasia. In the case of VB, perimetric thresholds are flattened relative to normals, suggesting the subject did not gain a full measure of the elevated density of foveal ganglion cells that develops in normals. Since pixel size and stimulus size are fixed, any sampling problems would more likely manifest as a threshold elevation at the far viewing conditions. Moreover, if the reduced sampling is not homogeneous, then this could increase variability (because, from trial to trial, filtered noise features would fall on neighboring retinal locations with different retinal sampling densities). Effect of Presentation Style Simultaneous viewing simulates the task of making a side-by-side comparison of fractal camouflaged targets, while sequential viewing simulates the task of comparing a target to one that is in memory; in theory and experiment the two paradigms can lead to somewhat different results (Garcia-Perez et al., 2005; Hansen & Hess, 2006). The discrimination function is similar for both conditions (Figs. 3 and 4) but there is a small advantage for simultaneously-viewed images, relative to sequentially-viewed ones, especially for small values of ȕ. This tendency can be clearly seen when the data from the 4 subjects are pooled (Fig. 3). This is contrary to Hansen & Hess (2006), who found an advantage for sequential viewing, and attributed differences between the two conditions to differences in the portions of retina they cover. However, our near-sequential and far-simultaneous stimuli covered very similar regions of central retina (2. 6o and 2. 3o respectively), and yet simultaneous viewing yielded lower thresholds for nine of ten exponents (and tied for the tenth). This suggests that for our experiment, the memory demands of sequential viewing were disadvantageous, a design consideration for combat target displays. Discussion Comparison to Related Studies: Static Fractals Some prior studies of fractal discrimination overlap our work. Our discrimination functions resemble those of Knill et al. (1990), particularly their low-contrast near condition (17. 5% RMS	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 9 APPROVED FOR PUBLIC RELEASEcontrast at 1 meter with 64x64 pixel images), which is similar to our Far Sequential condition. While the average exponent of natural scenes is around 1. 1 (Table 1), the greatest sensitivity to changes in a fractal image's exponent are consistently found to be around 1. 6 across a wide range of conditions. We see no evidence for a second minimum at low ȕ (circa 0. 6) reported by Tadmor and Tolhurst, even when we used a simultaneous viewing condition similar to Tadmor and Tolhurst (1994). Nor is the discrepancy due to the angular size of the image, as all three studies had stimuli that were similar in size. Hansen & Hess (2006) note that the spatial presentation task uses two different parafoveal patches of retina, while the temporal task uses the same patch of central fovea; they find that fovea and parafovea yield somewhat different patterns of discriminability as a function of ȕ, but none of their data show a second minimum at low ȕ (rather, they find a maximum at ȕ=0. 8, with better thresholds on either side, similar to our findings). Another possible source of this difference in discriminability functions may be the specific nature of Tadmor and Tolhurst's task. In both the present study and Knill et al. (1990), standard two-Alternative Forced-Choice psychophysical procedures were used. In contrast, Tadmor and Tolhurst (1994) used an odd-one-out task (i. e., three images were presented simultaneously, two of which had identical exponents-the task was to choose the image that was different from the other two). In other words, Tadmor and Tolhurst's task was one of simple discrimination, while our task (and Knill's) requires discrimination and some form of identification (once the two images could be told apart, the subjects had to decide which had a lower exponent). These tasks coincide in difficulty only if all information required for identification is present at the discrimination threshold, which will most often take place when a single channel mediates performance of the task. Indeed, Tolhurst and Tadmor (1997) have shown that simple discrimination data is often consistent with a single channel mediating discrimination. However, since a comparison of channel outputs is required to estimate the spectral exponent of an image, discrimination plus identification would likely require a comparison of channel outputs, perhaps raising the JND for β near 0. 4 sufficiently to eliminate the second minimum that Tadmor and Tolhurst (1994) found. Figure 5. Camouflage can be dynamic in several ways, including simple movement. A Jordanian F-16 painted in Hyper Stealth Biotechnology Corp. 's fractal-like camouflage.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 10 APPROVED FOR PUBLIC RELEASEComparison to Related Studies: Dynamic Fractals So far we have discussed only perception of static fractals. But camouflaged images may move against their backgrounds and camouflage may be dynamic in other ways. In general, the effect of motion on such fractals is described by f t= v*f s (where v is velocity and f t is temporal frequency). Thus, if the spatial spectrum is a 1/f distribution, then for simple movement the temporal frequency distribution is a linear transform of spatial frequency, which is no more interesting than the viewing distance condition. There are however more interesting dynamic manipulations of fractals that are worth study. For example, it is possible to extend our study of discrimination to spatiotemporal fractals-fractals whose individual pixel intensities vary over time in a manner described by fractional Brownian motion. Such images have Fourier amplitude spectra of A(ft,fs) = kf tĮfsȕ Eq. 2 In general, as Į becomes larger, the motion of the texture becomes more coherent and can be used to mimic various biological motions (Billock et al., 2001a). Figure 6 shows the human perceptual discrimination space for spatiotemporal fractals (dynamic textures). Figure 6. Discrimination contours for spatiotemporal fractal textures (Eq. 2) using the same observers in Figures 3 and 4. The contours are estimated using JNDs for discrimination in four directions in the perceptual space and are fit using simple quarter-circles of no theoretical significance.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 11 APPROVED FOR PUBLIC RELEASETo define two-dimensional JNDs we measured discrimination in four directions: both increments and decrements for both spatial ( ȕ) and temporal ( Į) exponents. Not surprisingly, the JNDs in this space are ellipsoidal (they resemble color discrimination JNDs). Interestingly, the interior portion of the resulting two-dimensional discrimination space is remarkably flat, a feature that some psychophysicists have gone to great lengths to obtain in nonlinear mappings of other perceptual systems (e. g., color discrimination). Implications for Future Work It is worth noting that humans can become proficient at naming the spectral exponents of images (or equivalently, fractal dimension, which is a linear transform of the exponent; Cutting & Garvin, 1987; Kumar et al., 1993; Pentland, 1988). A neural ability to estimate the spectral drop-off and exploit it has been speculated on and deserves additional attention (Billock, 2000; Billock et al., 2001b; Campbell et al., 1978; Hammett & Bex, 1996; Rogowitz & Voss, 1990). Taken together with the natural image regularities and perceptual pop-out findings discussed earlier, this suggests that ȕ is a key signature, both for images and for the visual systems that evolved to transduce images. Of particular interest is the finding that, under some conditions (nonlinear systems near threshold), adding noise can facilitate detection and identification of some signals, including images (Repperger et al., 2001; Simonotto et al., 1997; Yang, 1998)-an example of stochastic resonance as an image enhancement mechanism. Dynamic noise is more effective than static (Simonotto et al, 1997). Because other studies of stochastic resonance show that 1/fȕ noise can be more efficient than white noise in inducing stochastic resonance effects (Billock & Tsou, 2007; Hangi et al., 1993; Nozaki et al., 1999), further studies of discrimination in spatiotemporal fractal noise (at various contrast levels) would be warranted and might uncover some practical applications. Implications of Fractal Discrimination for Camouflage and Combat ID Based on this and other work we can enumerate some implications for camouflage and combat ID: (i) Natural images have 1/fȕ spatial amplitude spectra. The most reasonable value of ȕ for general purpose camouflage is around 1. 1. Particular environments will vary in this statistic and in coloration. (ii) Keeping the difference between the ȕtarget and ȕbackground less than 0. 2 generally avoids preattentive popout, but discrimination will still be possible using a point-by-point search. (iii) Using many spatial scales makes camouflage effectiveness almost independent of distance. (iv) For IFF purposes, friendly camouflage schemes should have different ȕs than the unfriendly camouflage patterns, but this may conflict with concealment goals. The best outcome would be for hostile and friendly camouflage statistics to be on opposite sides of the ȕbackground value, with the friendly scheme not easily discriminable from background but discriminable from the hostile. (v) For identification purposes, side-by-side viewing of sensor and reference images is preferable. Sensor operators should be screened for spatial sampling problems (sub-clinical amblyopia) by measuring their contrast sensitivity functions. (vi) It may be possible to break many camouflage schemes by adding filtered noise to the sensor images. This seemingly perverse aspect of stochastic resonance should be exploited if possible. Since stochastic resonance's effectiveness is often dependent on the Fourier spectral qualities of the noise, fractal camouflage may be particularly vulnerable (because the spectral qualities of simple fractals are easily matched by varying one noise parameter). Multi-fractals may be less vulnerable in this regard. It would be ironic if the beautiful mathematical attributes of fractals (which give it so	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 12 APPROVED FOR PUBLIC RELEASEmuch utility in describing the natural environment and make it such an elegant solution to the problem of designing camouflage) also prove to be its Achilles' heel. Author Note and Acknowledgements Vincent A. Billock,General Dynamics, Inc., Suite 200, 5200 Springfield Pike, Dayton, OH 45431 ; vince. billock@gd-ais. com. Douglas W. Cunningham, University of Tübingen, Tübingen, Germany ;douglas. cunningham@gris. uni-tuebingen. de. Brian H. Tsou, AFRL/RHCI, Wright-Patterson Air Force Base, OH 45433 ;brian. tsou@afrl. af. mil. We thank Jer Sen Chen, Steven Fullenkamp, Paul Havig and Eric Heft for technical support.	Fractal_BCT20BILLOCK.pdf
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Billock, Cunningham & Tsou (2008) 14 APPROVED FOR PUBLIC RELEASEJulesz, B. & Caelli, T. (1979). On the limits of Fourier decompositions in visual texture perception. Perception, 8, 69-73. Knill, D. C., Field, D., & Kersten, D. (1990). Human discrimination of fractal images. Journal of the Optical Society of America A, 7, 1113-1123. Kumar, T., Zhou, P., & Glaser, D. A. (1993). A comparison of human performance with algorithms for estimating fractal dimension of fractional Brownian statistics. Journal of the Optical Society of America A, 10, 1136-1146. Mac Millan, N. A., & Creelman, C. D. (1991). Detection Theory: A User's Guide. Cambridge: Cambridge University Press. Moulden, B., Kingdom, F., & Gatley, L. F. (1990). The standard deviation of luminance as a metric for contrast in random-dot images. Perception, 19, 79-101. Nozaki, D., Collins, J. J., & Yamamota, Y. (1999). Mechanism of stochastic resonance enhancement in neuronal models driven by 1/f noise. Physical Review E, 60, 4637-4644. O'Neill, T., Matthews, M., & Swiergosz, M. (2004). Marine Corps innovative camouflage. Midyear meeting of the American Psychological Association, Divisions 19 & 21. Supplementary data at http://www. hyperstealth. com/digital-design/index. htm Parraga, C. A., Brelstaff, G., Troscianko, T., & Morehead, I. R. (1998). Color and luminance information in natural images. Journal of the Optical Society of America A, 15, 563-569. Parraga, C. A., & Tolhurst, D. J. (2000). The effect of contrast randomization on the discrimination of changes in the slopes of the amplitude spectra of natural scenes. Perception, 29, 1101-1116. Peli, E. (1990). Contrast in complex images. Journal of the Optical Society of America A, 7, 2032-2040. Peli, E. (1997). In search of a contrast metric: matching the perceived contrast of Gabor patches at different phases and bandwidths. Vision Research, 23, 3217-3224. Pentland, A. (1988). Fractal-based descriptions of surfaces, in W. Richards (Ed. ), Natural Computation (pp. 279-299). Cambridge: MIT Press. Repperger, D. W., Phyllips, C. A., Neidhard, A., & Haas, M. (2001). Designing human machine interfaces using principles of stochastic resonance. AFRL Technical Report ARRL-HE-WP-TR-2002-0187. DTIC# ADA412330. Rogowitz, B. E., & Voss, R. F. (1990). Shape perception and low-dimension fractal boundaries. Proceedings of the SPIE, 1249, 387-394. Ruderman, D. L. (1994). The statistics of natural images. Network: Computation in Neural Systems, 5, 517-548. van der Schaaf, A., & van Hateren, J. H. (1996). Modeling the power spectra of natural images: statistics and information. Vision Research, 36, 2759-2770. Simonotto, E., Riani, M., Seife, C., Roberts, M., Twitty, J., & Moss, F. (1997). Visual perception of stochastic resonance. Physical Review Letters, 78, 1186-1189. Tadmor, Y., & Tolhurst, D. J. (1994). Discrimination of changes in the second-order statistics of natural and synthetic images. Vision Research, 34, 541-554. Taylor, R. P., Spahar, B., Wise, J. A., Clifford, C. W. G., Newell, B. R., & Martin, T. P. (2005). Perceptual and physiological responses to the visual complexity of fractal patterns. Nonlinear Dynamics in Psychology and Life Sciences, 9, 89-114. Thomson, M. G. A., & Foster, D. H. (1997). Role of second-and third-order statistics in the discriminability of natural images. Journal of the Optical Society of America A, 14, 2081- 2090.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 15 APPROVED FOR PUBLIC RELEASETolhurst, D. J., & Tadmor, Y. (2000). Discrimination of spectrally blended natural images: Optimization of the human visual system for encoding natural images. Perception, 29, 1087-1100. Tolhurst, D. J., & Tadmor, Y. (1997). Band-limited contrast in natural images explains the detectability of changes in the amplitude spectra. Vision Research, 23, 3203-3215. Tolhurst, D. J., Tadmor, Y., & Chao, T. (1992). The amplitude spectra of natural images. Ophthalmic and Physiological Optics, 12, 229-232. Webster, M. A., & Miyahara, E. (1997). Contrast adaptation and the spatial structure of natural images. Journal of the Optical Society of America A, 14, 2355-2366. Voss, R. F. (1985). Random fractal forgeries, in R. A. Earnshaw, Ed., Fundamental Algorithms for Computer Graphics (pp. 805-835). Berlin: Springer. Yang, T. (1998). Adaptively optimizing stochastic resonance in visual system. Physics Letters A, 245, 79-86.	Fractal_BCT20BILLOCK.pdf
What do catastrophic visual binding failures look like? Vincent A. Billock1and Brian H. Tsou2 1General Dynamics, Suite 200, 5200 Springﬁeld Pike, Dayton, OH 45431, USA 2US Air Force Research Laboratory, Wright-Patterson Air Force Base, OH 45433, USA Ordinary vision is considered a binding success: all the pieces and aspects of an image are bound together,despite being processed by many different neurons in several different cortical areas. How this is accom-plished is a key problem in visual neuroscience. The study of visual binding might be facilitated if we had ways to induce binding failures. A particularly interest-ing failure would involve a loss of the physical integrity of the image. Here, we identify conditions that induce such perceptual failures (e. g. the melting together ofequiluminant colored images and the fragmentation of retinally stabilized images) and we suggest that these should studied using electrophysiological measuresof binding. In visual cortex, many neurons are specialized for detecting different visual attributes such as form, color, motion and depth. Often these cells are physically separated-in some cases, they seem to aggregate by stimulus preference in different cortical areas or at different spots of the same cortical area. So how does thevisual system combine these responses to create uniﬁed and veridical percepts? A leading theory suggests that the stimulus-based responses of widely spaced neurons can bebound together by the temporal synchronization of their gamma band (30-90 Hz) activity. Although controversial, an impressive body of theoretical and electrophysiologicalevidence for this theory has accumulated [1-12]. Some of the most compelling binding studies compare perceptual and electrophysiological measures in humans [10-12]. T o further such efforts, it is necessary to understand what perceptually constitutes a binding success or failure, so that its electrophysiological correlates can be sought. There are two especially interesting aspects of the 'binding problem': feature-binding and part-binding. Feature-bind-ing correctly associates various features of imagestogether, for example so that we see a yellow school bus moving towards us, rather than yellowness, curves and angles, and movement, as in a dynamic abstract painting. Part-binding refers to correct construction of spatially coherent percepts, which need to be extracted (segmented)from the noisy and often ambiguous retinal image. To study feature-binding, Treisman introduced the notion of feature-binding failures called illusory conjunctions-under some circumstances, especially in peripheral vision,two features such as shape and color can be incorrectly associated with one another-and this has been very useful for psychophysical studies of feature-binding [1]. However, there have been no comparable studies of part-binding failures (except in brain lesions [8]). For electro-physiological studies it would be useful to be able to produce part-binding failures in normal humans using reversible and noninvasive methods. So, what would a really spectacular part-binding failure look like? Has anyone ever seen one? Do they occur in normal (brain-undamaged) observers? In addressing these questions, we realized that we have seen candidates for such bind-ing failures. Recently, we studied the perception of retinally stabil-ized colored images [13]. We coupled a motorized mirror to an optical eye-tracker (which measures eye movements using infrared reﬂections from the cornea and lens). Subjects view the stimulus as a reﬂection in the mirror. By deﬂecting the mirror to compensate for eye movements, the reﬂected image remains stationary on the retina [13,14]. The textbook account of these stabilized images is that they fade (because of transient temporal responses of some retinal mechanisms), and fade they do. However, they also become unstable and lose their spatial coherence: their internal structure can ﬂow about like ﬁgures made from melting wax, or their internal borders can fade away altogether so that the colors of the image diffuse into oneanother. These and similar bizarre phenomena are not explicable in terms of peripheral neural mechanisms. But what central mechanism could exhibit such behaviors? It occurred to us that some of the behaviors that we (and many earlier investigators of equiluminant and stabilized images) found were good examples of part-binding fail-ures. Because the vast majority of these studies were made in an era innocent of binding theory, we have a rare opportunity to harness several decades of misunderstood research in the goal of advancing the current research in binding. Expected failure modes of part-binding There is an extensive literature of algorithms for part-binding. In general, these models need to bind things that belong together and segment them from backgrounds and other objects. At this level, one could imagine binding failures that result from an inability to link parts into a whole (such a binding failure will be discussed in a later section of this article). Additional possibilities can be Corresponding author: Vincent A. Billock (vince. billock@wpafb. af. mil). Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 www. sciencedirect. com 0166-2236/$-see front matter. Published by Elsevier Ltd. doi:10. 1016/j. tins. 2003. 12. 003	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
derived from segmentation models that rely on balancing cooperation and competition [15-18]. Competitive net-works deﬁne the locations of borders and edges; coopera-tive networks reinforce mechanisms that agree about thepresence of a border, and ﬁll-in areas between borders. Ifthe balance between these mechanisms breaks down, part-binding should fail in distinctive ways. If cooperativemechanisms are too strong, then the network will settleon a common value for all points in an image. If this occurs slowly enough in an already segmented image, the image would appear to fade, or gray-out, or melt. Conversely, if competition is too strong or cooperation istoo weak, the image should appear to fragment. Thisoccurs for some neural networks that use nearest neighborconnectivity [17,18]. Binding-like failures in retinal stabilization: fading and blackout Images can be mechanically stabilized from eye move-ments, either by mechanical coupling to a contact lens, orvia an eye-tracker [14,19]. When an image is stabilized, the most common perceptual effect is a gradual fading ofthe image. Similar effects can be obtained using steadyﬁxation, afterimages or ganzfelds. If the image starts outas a rather textureless surface (a ganzfeld), then gradually all aspects of the percept are lost and the subject perceives a uniform, depthless gray (eigengrau) even if the ganzfeldwas initially colored. Fading is usually attributed to thetransient properties of certain retinogeniculate units-thetransient M cells. This explanation is inadequate on atleast two grounds: (i),80% of retinogeniculate units belong to the P cell pathway, which has a sustained response to both achromatic form and color stimuli [20], and (ii) most visual mechanisms (including achromaticform, color and depth) receive inputs from P cells and canbe driven by P cells alone, if M cells are silenced [21]. And this information is utilized by higher mechanisms; forexample, the depth and multistability of a Necker cubeare just as salient when stabilized, so long as it remains visible [22]. Although P cells signal the attributes of stabilized images to cortex (a necessary condition for corticalprocessing), this is not always a sufﬁcient condition forveridical perception. Consider some properties of stabil-ized images, paradoxical to P cell function, implicatingcentral mechanisms in stabilized-image fading. For example, P cells have sustained responses for high spatial frequencies and high contrasts (unlike M cells, whichbecome more transient at high contrasts [23]); yet Purkinje's tree (the high-contrast pattern of blood vesselshadows on the retina) fades unusually rapidly, and fadesfastest (as fast as 80 ms) in the parafovea, where P cells arepresent in high densities [24]. Similarly, although P cells have strong sustained responses to chromatic stimuli, stabilized color images fade faster and more completelythan achromatic stimuli [25,26]. Actually, many attributes of visual fading are more consistent with central than peripheral mechanisms. Forexample, the fading of a stabilized image in one eye can beslowed or reversed by modulating the stimulation of theother eye [27]. Conversely, fading of a stabilized image inone eye makes images viewed by the other eye less visible. Remarkably, even if the stabilized image in one eye has completely disappeared, it can still be binocularly fused with a duplicate moving image in the other eye to yield aparadoxical percept of motion in depth [14]. Finally, fading of images can be slowed or reversed by stimulatingattention or stimulating the other senses [19]. Blackout is a related, albeit more dramatic effect. Sometimes imagestabilization, even if only in one eye, induces an extremely abrupt, binocularly simultaneous visual blackout. Recov-ery can be initiated by a blink. Descriptions of this 'blackﬁeld' as 'more black than black' do not do it justice [19]. Our subjects describe it as 'like someone cut my optic nerves,and I don't have a visual system anymore' [13]. Ditchburn attributes the black-ﬁeld effect to a fading mechanismcombined with a cortical inhibitory feedback loop that runs out of control [19]. Binding-like failures in retinal stabilization: image fragmentation From a binding failure perspective, the most interestingretinal stabilization effect is fragmentation [28-30] (Figures 1,2 ). In general, complex patterns are more likely to fragment than simple ones, angular patterns fragmentmore than round ones, and parallel lines tend to disappear and reappear together [31,32]. Sometimes the fragmenta-tion takes the form of a series of breaks in the image(Figure 1 ), whereas at other times, entire portions of the ﬁgure disappear and reappear ( Figure 2 ). The probability of fragmentation versus unitary fading grows with linelength and reaches equal probability for lines subtending45 0, suggesting a range for the neural processes involved [32]. Interestingly, if a stabilized image with gaps in its structure fades and then reappears, the gaps in the imageare often ﬁlled-in [19]. Moreover, ﬁlling-in can occur even as the overall image is fading [33]. When images fragment, they tend to fade and revive in clusters that are cognitivelymeaningful, and follow Gestalt-like rules of association(Figure 2 ). These processes act on the central image; fragmentation can eliminate contours present only in a binocular image and not in its retinal precursors [34]. Similar fragmentation can occur during migraine attacks[35] and in strabismic amblyopia (a cortical visual defect induced by poor coordination of the eyes during visualdevelopment), especially for high spatial frequency stimuli[36](Figure 1 ). The connection of fragmentation phenom-ena to amblyopia is especially interesting because, in cat, cortical cells driven by the amblyopic eye are poorlysynchronized compared with cell populations driven by thenormal eye-especially for high spatial frequency stimuli[37]. No studies of synchronization during stabilization have been made. However, some early electroencephalo-gram (EEG) studies report that alpha-rhythm (9-10 Hz) power increased,0. 7-1. 0 s before disappearances; higher frequencies (including the gamma band, which is nowimplicated in binding) were suppressed until,1 s before spontaneous reappearances [38,39]. Oddities of equiluminant images Like stabilized images, pictures in which hues have equalluminance are less perceptually salient than natural Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 85 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
images. Luminance differs slightly from brightness and is usually deﬁned operationally; for example, if two colors are shown in the same place in rapid sequence (a procedure known as heterochromatic ﬂicker photometry), the sen-sation of ﬂicker is drastically reduced for a certain ratio of the radiances of the colors-this is deﬁned as equilumi-nance. Slightly harder to use in practice, but of morerelevance to this discussion, is the minimum border criterion: at equiluminance, the apparent border contrast between differently colored image parts is minimized. Similarly, perception of form, depth, motion and color are all degraded for equiluminant stimuli [40-42]. Sometimes these phenomena are attributed to a lack of chromaticinput to these mechanisms, but several studies show thatthese systems can be driven by P cells in general and by hue-speciﬁc mechanisms in particular [43]. Indeed, even some putatively chromatic functions (e. g. chromaticrivalries) are disrupted by equiluminance [41]. Othershave attributed some effects to the lower acuity of chromatic mechanisms. This can explain some effects[42]but the most interesting effects are out of proportion to the acuity loss [43], or are seemingly unrelated to it [40,41]. From a binding point of view, the most signiﬁcant of these are linking failures and image melting. Binding-like failures at equiluminance: linking failures and perceptual melting A variety of symptoms suggest that there are part-binding failures in equiluminance. Included in these are failures tolink pieces of a surface. Consider random dot stereopsis. Itis possible to build a stereogram in which the two images contain only random dots. The two images are near duplicates; one image has a group of dots shifted overseveral pixels. Binocular fusion of the two images yields apercept of a group of random dots ﬂoating above or belowthe random dot background. In an ordinary random dot Figure 1. Examples of fragmentation percepts. (a)A square (left) breaks up during retinal stabilization, with fragmentary losses and fading starting at the corners. Repro-duced, with permission, from Ref. [29]. (b)The effects of stabilizing a long bar (left). The bar fragments in various places and the fragments fade. Reproduced from Ref. [19] by permission of Oxford University Press. The probability of such fragmentation increases with bar length [28]and some studies show a loss of high-frequency electro-encephalogram power during such stabilization-induced fragmentation and fading [38,39]. (c)Mosaic vision. An odd fragmentation phenomenon reported by a small min-ority of patents during epileptic seizures and migraine. According to Sacks, 'mosaic vision denotes the fracture of the visual image into irregular, crystalline, polygonal facets, dovetailed together as a mosaic. The size of the facets may vary greatly. If they are extremely ﬁne, the visual world presents an appearance of c rystalline iridescence or “graininess”, reminiscent of a pointillist painting... If the facets become larger, the visual image takes on the appearance of a classical mosaic... o r even a “cubist” appearance. ' Reproduced, with permission, from Ref. [35]. (d)A possibly related phenomenon, described by subjects with strabismic amblyopia. Vision is normal for low spatial frequency gratings but distorted at high spatial frequencies, as if the features were improperly bound by location. At the highest spatial fr equencies, breaks appear in the features. The phenomenon is not due to an inability to transduce the features because contrast sensitivity is normal (graph). 'c/d' (cycles per degree) refers to the periodicity of the coarse and ﬁne gratings (e. g. 1 c/d mean that one black and one white bar ﬁt into one degree of visual angle, which is roughly the angle s ubtended by the width of the thumb held at arms length; the best human acuity-20/10-is equivalent to being able to see a 60 c/d grating). Reproduced, with permission, f rom Ref. [36]. Normal Pointilliste Mosaic Cubist Vertical Horizontal Contrast sensitivity 110100300 0. 1 1 10 100 Spatial frequency (c/d)1c/d 4c/d 10c/d 15 c/d4o(a) (b) (c) (d)Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 86 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
stereogram, when a group of dots is segmented from the background, the pattern of dots deﬁnes a textured surfaceof common depth. Near equiluminance, depth is seen for the individual colored pixels but, unlike ordinary random dot stereopsis, the pixels seem to be separate rather thanpart of the same textured surface [41]. This is not a curiosity of stereopsis: 2D surfaces are not linked together when deﬁned only by hue. An image composed ofequiluminous patches does not appear uniﬁed but, rather, appears as patches of one color ﬂoating on the other [41,42]. In other cases, all sense of object and surface can be lost and the target is perceived as a 'jumble of lines' [41] or 'extremely confusing and hard to describe' [40]. More-over, the contrast of equiluminant images can seemunstable-Gregory [40]describes such images as 'jazzy'. In 1927, Liebmann (translated in Ref. [42]) reported that there is a 'critical zone [where] everything ﬂows... glim-mers... most everything is soft, jelly-like, colloidal. Often... parts which belong together in the normal ﬁgure now have nothing to do with one another. [It is] a world without ﬁrm things, without solidity. ' Such effects are mostsevere when the borders are deﬁned by tritan (S-cone) modulation: the borders of the images can appear to melt like hot wax and distort like pulled taffy (reminiscentof Liebmann's comment), and the colors diffuse into oneanother in the same way as spilled inks, blending into continuous color gradients [13,44]. There are interesting local boundary effects: fading is sometimes localized to particular features or regions [40]and when equiluminant features or edges fade away, they are sometimes replacedwith an illusory achromatic border [40,41] ). Cross-modal reinforcement: what is lost in stabilized and equiluminant images?We have discussed how a balance of cooperation and competition is required to produce reliable segmentation networks, and how these networks sometimes fail,especially for nearest-neighbor connectivity [17,18]. One way to avoid this is to allow the activity of a secondnetwork to reinforce the ﬁrst-in effect, the secondnetwork can act as a second set of nearest neighbors. Inthe segmentation literature, this reinforcement between different sensory modalities-called 'cross-modal con-struction' [45]-improves the precision and likelihood of both correct segmentation and feature binding [45-47]. I f part-binding mechanisms evolved to exploit this reinforce-ment, then its loss could compromise part-binding. Theevidence described in this article is consistent withreliance on cross-modal reinforcement of color segmenta-tion mechanisms by achromatic form mechanisms, and reliance on reinforcement of achromatic form segmenta-tion systems by mechanisms sensitive to motion ortemporal modulation. It takes two: combining equiluminance and stabilization makes everything worse If this cross-modal construction hypothesis is correct, thenwe would expect that compromising more feature systems would further worsen part-binding. This seems to be the case. A familiar example is the minimum border effect-the weakened border formed by an equiluminant bipartiteﬁeld can disappear completely when eye ﬁxation is steady[44]. A more dramatic example is our own experiment using stabilized opponent colors [13]; unlike the equilu-minant experiments already discussed, binding failures occur for any color combination. Perhaps the most estab-lished law in experimental psychology is that certain colorcombinations such as reddish green and bluish yellow arenever allowed; this observation formed the basis for Hering's theory of color opponency and much of moderncolor science. However, it had been reported that somesubjects saw these forbidden colors when colored bipartite ﬁelds such as those shown in Figure 3 were stabilized [14]. When we attempted to replicate these results, we foundthat the effect depended on the combination of equilumi-nance and stabilization. When we stabilized non-equilu-minant red-green or blue-yellow borders, we saw avariety of part-binding failures, which included effectswhere the border would disappear in places; the unbound colors both diffuse and compete, resulting in illusory textures such as red glitter on a ﬁeld of green or bluestreaks on yellow [13]. Sometimes, the two colored sides of the bipartite ﬁeld would appear to switch places or evenrotate 90 8(an effect reminiscent of Treisman's illusory conjunctions). Only when we made the two sides of thebipartite ﬁeld equiluminant did subjects report forbidden Figure 2. Some gestalt-like attributes of image fragmentation in retinal stabilization that implicate central mechanisms. (a-d) Fragmentation of monocular stabilized images. Reproduced, with permission, from Ref. [29]. Pictures on the left show unstabilized images and pictures to the right show stabilized percepts. Note that images tend to fragment and fade, such that the visible fragments follow rules. In (a) and (b), meaningful fragments are preserved. In (c), parallel fragments are pre-served. In (d), gestalt grouping rules are mostly followed. (e)Effects of stabilizing the monocular parts (left-eye image and right-eye image) of a binocular (fused) percept. Note that disappearance of image fragments is characteristic of the fused image, not the monocular forms, ruling out peripheral explanations. Reproduced,with permission, from Ref. [34]. (a) (b) (c) (d) (e) Left-eye image Right-eye image Fused image Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 87 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
colors. The border collapsed completely and the two colored sides appeared to run together, like diffusing inks or melting waxes, creating new color combinationsundreamt of in Hering's philosophy. In some cases, thesemanifest as a gradient which ran from red on one side togreen on the other, with every possible shade of reddishgreen in between. In other cases, subjects report seeing aspatially homogeneous reddish green, the red and greencomponents of which were as compelling as the red andblue components of a purple. We modeled this opponencyloss in terms of a cortical competition model whosecompetition (between color-labeled neurons) is disabled[13]. Why this should occur is not clear, but there is a model of cortical competition, whose competition is gated bygamma-band oscillations [48]; this is the frequency range that is implicated in many binding studies and thefrequency range in EEG that is lost during stabilization-induced fading and fragmentation [38,39]. Electrophysiological testing of part-binding failures We argue that phenomena such as border disappearance, fragmentation, diffusive color spreading, and linkingfailures of equiluminous and stabilized images are examples of part-binding failures, and that these particu-lar failure conditions are clues that part-binding relies onachromatic mechanisms reinforcing segmentation bycolor, and on motion-driven mechanisms reinforcingsegmentation by achromatic form. Testing these ideaswill require some notion of what the electrophysiologicalcorrelates of binding are and how they could be detectednon-invasively in intact responding humans. Becausegamma-band activity is implicated in binding processes,the loss of high-frequency EEG power that accompaniesstabilized-image fragmentation [38,39] is suggestive but not deﬁnitive. One potential problem is that some sourcesof gamma-band power in EEGs might be unrelated tobinding. Another potential problem is that our bindingfailures might actually be incorrect bindings and thereforenot distinguishable by EEG. Recent experiments have castlight on both questions, by using fragmented and camou-ﬂaged images in which subjects could eventually learn to see coherent embedded objects [10,12]. (These studies are especially relevant here because the fragmentation-to-order shift is the complement to the order-to-fragmenta-tion phenomena we studied. ) There was gamma-bandactivity whether subjects saw the coherent picture or not,but there were two differences in EEG when gestaltperception was obtained: the temporal output of widelyspaced electrodes became more correlated (consistent withsynchronization of widely spaced neural mechanisms [12]) and a second kind of gamma-band response, the 'inducedgamma response', was recorded. On every physicallyidentical stimulus presentation there is a gamma-bandresponse that is phase-locked to the camouﬂaged orfragmented stimulus, but only on trials where the gestaltpercept is seen does a second signature emerge thatconsists of gamma-band bursts in variable phase to theactual stimulus [10,12]. We therefore predict that EEG signatures of binding, such as the induced gamma response, will be reduced during binding-like failures induced by retinal stabilization or equiluminance, andthat these signatures will be further minimized oreliminated during the more dramatic perceptual effectscreated by combining stabilization and equiluminance[13]. A sophisticated version of this experiment could be done using the methods of Billock et al. [13], but the simpler methods of Buck et al. [44]might well sufﬁce. Acknowledgements We thank Kenneth Blum, Scott Kelso, Oliver Sacks, Wolf Singer and Anne Treisman for particularly helpful suggestions and remarks. References 1 Treisman, A. (1998) Feature binding, attention and object perception. Philos. Trans. R. Soc. Lond. B Biol. Sci. 353, 1295-1306 2 von der Malsburg, C. (1999) The what and why of binding: the modeler's perspective. Neuron 24, 95-104 3 Horn, D. and Opher, I. (2000) Temporal segmentation and binding in oscillatory neural systems. In Oscillations in Neural Systems (Levine, D. S. et al., eds), pp. 201-216, Erlbaum Figure 3. An image used in the stabilized-image studies of Billock et al. [13]. The image was stabilized on the retina and viewed through an unstabilized aperture. This arrangement keeps the whole image from fading [14]but does not protect the border between red and green ﬁelds. Typically, portions of the border melt or fade and the opponent colors tend to ﬂow and compete, creating complex illusory textures. Sometimes the two sides will switch colors, or even rotate 90 8. This phenomenon resembles an illusory conjunction of location and color (a feature-binding failure). If the two sides are made equal in luminance, the border disap-pears and the colors ﬂow together to form mixture colors of reddish-green, inviolation of Hering's law of color opponency. Sometimes these colors are uniform, other times they form a smooth gradient ranging from reddish green to greenish red. Similarly, stabilizing an equiluminant blue-yellow bipartite ﬁeld results inperception of bluish-yellow forbidden colors. These phenomena can be modeled by a breakdown of competitive interactions between some color-labeled cortical units [13]. 1 4 8and 24 8are degrees of visual angle for which the stabilized stimulus subtended horizontally and vertically, respectively. Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 88 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
4 Castelo-Branco, M. et al. (2000) Neural synchrony correlates with surface segregation rules. Nature 405, 685-689 5 Singer, W. (2001) Consciousness and the binding problem. Ann. N. Y. Acad. Sci. 929, 123-146 6 Singer, W. (2003) Synchronization, binding and expectancy. In The Handbook of Brain Theory and Neural Networks (Arbib, M. A., ed. ), pp. 1136-1143, MIT Press 7 Robertson, L. C. (2003) Binding, spatial attention and perceptual awareness. Nat. Rev. Neurosci. 4, 93-102 8 Humphreys, G. W. (2003) Conscious visual representations built from multiple binding processes: evidence from neurophysiology. Prog. Brain Res. 142, 243-255 9 Ursino, M. et al. (2003) Binding and segmentation of multiple objects through neural oscillators inhibited by contour information. Biol. Cybern. 89, 56-70 10 Bertrand, O. and Tallon-Baudry, C. (2000) Oscillatory gamma activity in humans: a possible role for object representation. Int. J. Psychophysiol. 38, 21 1-223 11 Muller, M. M. et al. (2000) Modulation of induced gamma band activity in the human EEG by attention and visual information processing. Int. J. Psychophysiol. 38, 283-299 12 Gruber, T. et al. (2002) Modulation of induced gamma band responses in a perceptual learning task in the human EEG. J. Cogn. Neurosci. 14, 732-744 13 Billock, V. A. et al. (2001) Perception of forbidden colors in retinally stabilized equiluminant images: an indication of softwired cortical color opponency? J. Opt. Soc. Am. A. Opt. Image Sci. Vis. 18, 2398-2403 14 Crane, H. D. (1994) The Purkinje image eyetracker, image stabiliz-ation, and related forms of stimulus manipulation. In Visual Science and Engineering (Kelly, D. H., ed. ), pp. 15-89, Dekker 15 Grossberg, S. and Wyse, L. (1992) Figure-ground separation of connected scenic ﬁgures: boundaries, ﬁlling-in and opponent proces-sing. In Neural Networks for Vision and Image Processing (Carpenter, G. A. and Grossberg, S., eds), pp. 161-194, MIT Press 16 Levine, D. S. (2000) Introduction to Neural and Cognitive Modeling, Erlbaum 17 Terman, D. and Wang, D. L. (1995) Global competition and local cooperation in a network of neural oscillators. Physica D. 81, 148-176 18 Marr, D. et al. (1978) Analysis of a cooperative stereo algorithm. Biol. Cybern. 28, 223-239 19 Ditchburn, R. W. (1973) Eye movements and Visual Perception, Clarendon 20 Ingling, C. R. Jr and Martinez-Uriegas, E. (1985) The spatiotemporal properties of the r-g X-cell channel. Vision Res. 25, 33-38 21 Merigan, W. H. and Maunsell, J. H. R. (1993) How parallel are the primate visual pathways? Annu. Rev. Neurosci. 16, 369-402 22 Ingling, C. R. Jr and Grigsby, S. S. (1990) Perceptual correlates of magnocellular and parvocellular channels: seeing form and depthin afterimages. Vision Res. 30, 823-828 23 Benardete, E. A. et al. (1992) Contrast gain control in the primate retina: P cells are not X-like, some M-cells are. Vis. Neurosci. 8, 483-486 24 Coppola, D. and Purves, D. (1996) The extraordinarily rapid disappearance of entopic images. Proc. Natl. Acad. Sci. U. S. A. 93, 8001-800425 Weintraub, D. J. (1964) Successive contrast involving luminance and purity alterations of the Ganzfeld. J. Exp. Psychol. 68, 555-562 26 Kelly, D. H. (1981) Disappearance of stabilized chromatic gratings. Science 214, 1257-1258 27 Cohen, H. B. (1961) The effect of contralateral visual stimulation on visibility with stabilized retinal images. Can. J. Psychol. 15, 212-219 28 Pritchard, R. M. et al. (1960) Visual perception approached by the method of stabilized images. Can. J. Psychol. 14, 67-77 29 Pritchard, R. M. (1961) Stabilized images on the retina. Sci. Am. 204, 72-78 30 Davies, P. (1973) The role of central processes in the perception of visual after-images. Br. J. Psychol. 64, 325-338 31 Evans, C. R. (1965) Some studies of pattern perception using a stabilized retinal image. Br. J. Psychol. 56, 121-133 32 Evans, C. R. (1967) Further studies of pattern perception and a stabilized retinal image. Br. J. Psychol. 58, 315-327 33 Cardu, B. et al. (1971) The inﬂuence of peripheral and central factors on the way that stabilized images disappeared. Vision Res. 11, 1337-1343 34 Evans, C. R. and Wells, A. M. (1967) Fragmentation phenomena associated with binocular stabilization. Br. J. Physiol. Opt. 24, 45-50 35 Sacks, O. (1999) Migraine, Vintage Books 36 Hess, R. F. et al. (1990) The puzzle of amblyopia. In Vision: Coding and Efﬁciency (Blakemore, C., ed. ), pp. 267-280, Cambridge University Press 37 Roelfsema, P. R. et al. (1994) Reduced synchronization in the visual cortex of cats with strabismic amblyopia. Eur. J. Neurosci. 6, 1645-1655 38 Lehmann, D. et al. (1965) Changes in patterns of the human electroencephalogram during ﬂuctuations of perception of stabilizedretinal images. Electroencephalogr. Clin. Neurophysiol. 19, 336-343 39 Keesey, U. T. and Nichols, D. J. (1967) Fluctuations in target visibility as related to the occurrence of the alpha component of theelectroencephalogram. Vision Res. 7, 959-977 40 Gregory, R. L. (1977) Vision with isoluminant colour contrast. Perception 6, 113-119 41 Livingstone, M. S. and Hubel, D. H. (1987) Psychophysical evidence for separate channels for perception of form, color, movement and depth. J. Neurosci. 7, 3416-3468 42 Cavanagh, P. (1991) Vision at equiluminance. In Limits of Vision (Kulikowski, J. J. et al., eds), pp. 234-250, CRC Press 43 Mullen, K. T. and Kingdom, F. A. A. (1991) Colour contrast in form perception. In The Perception of Colour (Gouras, P., ed. ), pp. 198-217, CRC Press 44 Buck, S. L. et al. (1977) Initial distinctness and subsequent fading of minimally distinct borders. J. Opt. Soc. Am. 67, 1126-1128 45 Finkel, L. H. and Edelman, G. M. (1989) Integration of distributed cortical systems by reentry: a computer simulation of interactivefunctionally segregated visual areas. J. Neurosci. 9, 3188-3208 46 Schillen, T. B. and Ko ¨nig, P. (1994) Binding by temporal structure in multiple feature domains of an oscillatory neuronal network. Biol. Cybern. 70, 397-405 47 Poggio, T. et al. (1988) Parallel integration in visual modules. Science 242, 436-440 48 Niebur, E. et al. (1993) An oscillation-based model for the neuronal basis of attention. Vision Res. 33, 2789-2802Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 89 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
A Role for Cortical Crosstalk in the Binding Problem: Stimulus-driven Correlations that Link Color, Form, and Motion Vincent A. Billock1and Brian H. Tsou2 Abstract &The putative independence of cortical mechanisms for color, form, and motion raises the binding problem—how isneural activity coordinated to create unified and correctly segmented percepts? Binding could be guided by stimulus-driven correlations between mechanisms, but the nature ofthese correlations is largely unexplored and no one has(intentionally) studied effects on binding if this joint informa-tion is compromised. Here, we develop a theoretical frame-work which: (1) describes crosstalk-generated correlationsbetween cortical mechanisms for color, achromatic form, and motion, which arise from retinogeniculate encoding; (2) showshow these correlations can facilitate synchronization, segmen-tation, and binding; (3) provides a basis for understanding perceptual oddities and binding failures that occur forequiluminant and stabilized images. These ideas can be testedby measuring both perceptual events and neural activity while achromatic border contrast or stabilized image velocity is manipulated. & INTRODUCTION How does the brain coordinate the activity of visual mechanisms to create unified percepts? There are twoparticularly important aspects to this binding problem(see Treisman, 1996 for a review of all aspects)—thebinding of different parts of the same image and thebinding of different image features (e. g., form, color, and motion). Part-and feature-binding are closely related because each feature class contributes to the segmenta-tion of images from backgrounds (Regan, 2000). Thisarticle is not a theory of binding, nor is it a theory ofmotion, color, and luminance defined form—on whichvast literatures already exist. Rather, we create a frame-work in which binding and its perceptual attributes maybe better understood. Towards this end, we tackle some key questions that are too seldom addressed in binding studies: (1) Just how independent are putatively inde-pendent cortical mechanisms for color, achromatic form,and motion, and what is the nature of any crosstalkbetween them? (2) How could crosstalk between thesemechanisms contribute to binding? (3) If ordinary visionis a resounding binding success, what would a catastroph-ic part-binding failure look like (feature-binding failures are called illusory conjunctions; Treisman, 1996), and under what conditions would binding failures occur?Background on Parallel Perceptual Mechanisms and Binding Theory Ample physiological and psychophysical evidence sug-gest that human vision is mediated by multiple mecha-nisms, each responding to selective combinations ofimage attributes (color, spatial frequency, orientation,motion, etc. ; for reviews see Regan, 2000; Zeki, 1993). This parallel processing—although crucial to under-standing visual detection and appearance—raises a bind-ing problem: How are unified and correctly segmented percepts created from the activity of putatively indepen-dent, spatially separated c ortical mechanisms? One theory is that binding stems from coordination of mech-anisms responding to an image (von der Malsburg, 1981,1995). Electrophysiological studies show coordinated g-band (generally 30-90 Hz) oscillations between cells inthe same and different orientation columns of the same cortical area, in different cortical areas, and in different hemispheres (for review and debates, see Gray, 1999;Shadlen & Movshon, 1999; Singer, 1999; Singer & Gray,1995). Recently, a series of compelling studies have tiedspecific g-band EEG activity in humans to specific per-cepts (Gruber, Mu ¨ller, & Keil, 2002; Tallon-Baudry & Bertrand, 1999). This impressive corpus of work onbinding and neural synchronization compels us to ad-dress a key question: What information governs binding? In principle, a sufficien tly potent coupling could synchronize any two neurons' activity, but indiscrimi-1General Dynamics,2U. S. Air Force Research Laboratory D2004 Massachusetts Institute of Technology Journal of Cognitive Neuroscience 16:6, pp. 1036-1048	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
nant perceptual binding is undesirable. There are two key kinds of information available to bind sensorymechanisms: location coding and correlated activity. Some studies suggest that binding exploits locationcoding; for example, two cells in different cortical areas,each responding to a different image feature (e. g., colorand orientation), are likely to synchronize if both areactive and both are coded for the same retinotopic location (for review, see Finkel & Edelman, 1989). This is plausible because of the substantial retinotopicallyspecific connections between cortical areas. We do notdenigrate the role of retinotopic mapping in binding,but because large receptive fields in the extrastriatecortex degrade localization, it is useful to have anindependent supplemental source of information forbinding. Moreover, supplementing location coding should speed and improve binding (time is at a premi-um when segmentation/binding takes only 50-200 msec;von der Malsburg, 1999). We suggest that a type ofstimulus-specific common neural input may provideinformation useful for binding. THEORETICAL BUILDING BLOCKS A Role for Stimulus-induced Correlations in Oscillatory Binding A tenet of binding theory is that binding is driven by (and makes manifest) correlations between mechanismsresponding to common image features. In principle, aninitial weak correlation between oscillatory mechanisms is amplified into mutual synchronization (von der Mals-burg, 1981). However, what guarantees that synchroni-zation will be correctly linked to real stimulus features?To avoid binding everything, competitive networkscould select the strongest correlations to be synchro-nized. Correlated activity in neural mechanisms is in-duced by causal relations between temporal events (e. g.,by perception-action interactions) and by common neu-ral inputs (Feng, 2000; von der Malsburg, 1999). One particularly promising source of correlated activity—crosstalk—is both stimulus-driven and induced bycommon neural inputs. Crosstalk is the unintendedportion of a selectively filtered signal. For example, Pcells carry both hue and luminance information (De Valois & De Valois, 1988). A cortical cell that filters the P-cell input to extract luminance may also extract a portion of the chromatic signal as well (Billock, 1995). This crosstalk—a nuisance in visual parallel processingtheory—has been neglected in binding theory. Theclosest exception is Horn, Sagi, and Usher's (1991)segmentation/binding network, which uses correlatednoise to synchronize neural oscillators responding todifferent visual modalities. Consider two such systems(E 1,E2) coupled through a common set of inhibitory (I) neurons. d E1=dt¼/C0E1þ Ft ða E1/C0b I/C0/C18E/C0c R1þ INPUT 1þ Q1;2Þ d R1=dt¼ð1=d/C01ÞR1þ E1 d E2=dt¼/C0E2þ Ft ða E2/C0b I/C0/C18E/C0c R2þ INPUT 2þ Q1;2Þ d R2=dt¼ð1=d/C01ÞR2þ E2 d I=dt¼/C0Iþ Fðf E1þg E2/C0h I/C0/C18IÞ Here E1,E2are firing rates driven by different visual features (INPUT 1, INPUT 2), which in this article could be achromatic contrast, color, or motion. R1,R2are dy-namic thresholds; a,b,c,d,g,u,tare constants; and Ft(x) = 1/(1+ e/C0x/t). Key attributes of Equation 1 are that E1and E2oscillate (fire) in response to their sensory inputs and can synchronize those oscillations. The Q1,2 term is an input common to both systems which speeds and strengthens synchronization—a result likely genericto coupled systems. In Horn et al. 's (1991) model, it re-presents correlated noise. Because Horn et al. considerbinding of color and form, they speculate that somenoisy ''early mixed representation of shape and color information exists in the input layer. '' Another model uses a cell class tuned along two stimulus dimensionsand couples it to a cell class that has a common stimulusdimension (Roelfsema, Engel, Ko ¨nig, & Singer, 1996; e. g., oriented cells for chromatic edges coupled to similarlyoriented cells for achromatic contrast). The correlatednoise and common dimension approaches are moresimilar than they seem. An ''early mixed representation'' is an apt description of retinogeniculate multiplexing. Attempts to create cortical cells tuned along (and labeledfor) one stimulus dimension inevitably induce stimulus-specific crosstalk from other stimulus dimensions (e. g.,Billock, 1995). Below, we quantify this crosstalk for twospecific sensory interactions. Fragmentation and Fading—Characteristic Failure Modes of Segmentation Networks Generally, within a sensory modality, segmentation in-volves cooperation between mechanisms that agree andcompetition between mechanisms that disagree. Thesemethods are generic and apply whether an image is to be segmented on the basis of achromatic form, color, motion, depth, and so forth (Levine, 2000; Wang &Terman, 1997; Grossberg & Wyse, 1992; von der Mals-berg & Buhmann, 1992). Competitive networks definethe location of edges and borders. Cooperative net-works fill in areas between borders and reinforce theresponse of retinotopically neighboring cells signaling(1) Billock and Tsou 1037	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
an edge. Such segmentation systems have characteristic failure modes: If cooperation runs amok, then all mech-anisms agree and the image is grayed-out, rather thansegmented. If such a failure occurs in an already seg-mented image, the image would appear to melt or fade. If competition is too strong, or cooperation too weak,then the image may fragment. For example, Terman and Wang's (1995) neural network (using nearest neighbor connectivity) tends to fragment images. Similarly, stere-opsis segmentation networks fragment images if thresh-olds for keeping cells active and/or entraining adjacentinactive cells are set too high (Marr, Palm, & Poggio,1978). Below, we examine situations (equiluminance andretinal stabilization) where percepts fade, melt, and/orfragment and argue these phenomena are understand-able in a segmentation/part-binding framework. These fading/fragmentation problems may be alleviated if mul-tiple segmentation networks share information aboutlocation of borders. Reinforcement between parallelnetworks—called ''cross-modal construction'' (Finkel& Edelman, 1989)—has the potential to reinforce andrefine estimates of image features (Roelfsema et al.,1996; Schillen & Ko ¨nig, 1994; Horn et al., 1991; Poggio, Gamble, & Little, 1988). Cross-modal construction is intuitive: Realistic segmentation networks have re-sponse thresholds; weak cooperative interactions be-tween segmentation networks responding to differentfeatures of the same image (e. g., chromatic and achro-matic form) allow the more active network to give theother network's units a boost above threshold, but donot entrain units not driven by sensory input, if theirthresholds are chosen appropriately (Horn et al., 1991). We argue that cross-modal construction depends on a stimulus-linked variant of the correlated noise that Horn used for synchronization. ORIGIN AND NATURE OF CROSSTALK-BASED CORRELATIONS BETWEEN ACHROMATIC AND CHROMATIC CORTICAL FORM MECHANISMS Retinogeniculate Origins of Correlations between Chromatic and Achromatic Mechanisms There is much evidence for chromatic/achromatic inter-actions in human vision (Mullen & Kingdom, 1991). Some interactions arise early in the visual pathway. About 80% of LGN cells have Type I receptive fields(center of different spectral sensitivity than its spatiallyopponent surround; Wiesel & Hubel, 1966); these P cellsrespond to both chromatic and achromatic stimuli (De Valois & Pease, 1971). P cells account for both the threshold chromatic and achromatic spatio-temporal contrast sensitivity functions (except for the achromaticlow spatial, high temporal frequency corner; Merigan &Maunsell, 1993; Kelly, 1983). Although the P cell's mixedsignal seems ambiguous, Ingling and Martinez's (1983,1985) algebraic identity factors the P cell's sensitivity into psychophysically meaningful terms. The expansion forr+g/C0cells (similar identities obtain for other cell types and for mixed cone surrounds; Billock, 1996) is: Prþg/C0¼RSe Te/C0GSi Ti ¼ð Rþ GÞfð Se/C0Si Þð Teþ Si Þþð Seþ Si Þð Te/C0Ti Þg=4 (Achromatic, Spatio-temporally bandpass) þð R/C0GÞfð Se/C0Si Þð Te/C0Ti Þþð Seþ Si Þð Teþ Ti Þg=4 (Chromatic, Spatio-temporally lowpass Þ Here, Rand Gare absorptions in L-and M-cones, Se,Si are center (excitatory) and surround (inhibitory) spatial weighting functions, and Te,Tiare center and surround temporal impulse response functions. Se,Si,Te,Tiare all lowpass functions of spatial/temporal frequency, so their sums are lowpass and differences are bandpass. A variation of Equation 2 models both the chromaticand achromatic contrast sensitivity functions (Burbeck& Kelly, 1980). For our current purposes, this formula-tion is unnecessarily complex; we develop some basicideas using a simplification and then return to thefull version when necessary. Temporarily neglectingtime gives Prþg/C0¼RCenter /C0GSurround ¼ð Rþ GÞð Se/C0Si Þ=2 (achromatic, bandpass tuning) þð R/C0GÞð Seþ Si Þ=2 (chromatic, lowpass tuning)ð3Þ For simplicity, we restrict space to one dimension ( x) and define the stimulus in terms of L and M coneabsorptions: Z(x)={ R(x),G(x)}. The response of the P cell to the stimulus is Prþg/C0ðx Þ/C10Zðx Þ¼0:5f Rðx Þþ Gðx Þg /C10 f Seðx Þ/C0Siðx Þg (Achromatic response) þ0:5f Rðx Þ/C0Gðx Þg /C10 f Seðx Þþ Siðx Þg (Chromatic response) where the convolution of two functions A(x)/C10B(x)=R A(t)B(x/C0t)dtand tis a dummy variable of in-tegration. Equations 3 and 4 reveal a subtle encoding of chromatic and achromatic information that is deci-pherable by cortical decoders. To see how, let theexcitatory center ( S e) and inhibitory surround ( Si) spatial weighting functions of this Type I cell be representedby Gaussians. The achromatic term is a difference of Gaussians that closely approximates a second spatialderivative of a Gaussian. The chromatic term is the sum of these Gaussians and is fit by a Gaussian with a space constant ( s) about 1. 83 times that of the achromatic term. If some simple assumptions (Billock, 1995) hold,(2) (4) 1038 Journal of Cognitive Neuroscience Volume 16, Number 6	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
then the spatial tuning of the P cell is the Fourier transform ( F[P]) of Equation 3. F1/2Prþg/C0/C138¼0:5ð Rþ GÞa2f2 s Hsðfs Þ(Achromatic tuning) þð R/C0GÞH1:83sðfs Þ(Chromatic tuning) ð5Þ where Hs(fs) is exp[ /C02(psfs)2] (the Fourier transform of the center Gaussian), a=2p, and fsis spatial frequency. The achromatic term is a spatial bandpassfilter; the chromatic term is lowpass, multiplexing thechromatic and achromatic information into different frequency ranges. If the multiplexing filters had no overlap, two well-chosen cortical filters could perfectlyseparate (demultiplex) the chromatic and achromaticinformation (Kingdom & Mullen, 1995; Billock, 1991,1995). For P cells, the encoding filters overlap, inducingcrosstalk in the cortical decoders, especially at moderatespatial frequencies where the overlap is greatest. Consider the extraction of achromatic information from P cells by the simplest spatial bandpass filter—spatial differentiation (Billock, 1995). Taking the nth local derivative of the P-cell array is equivalent to differ-entiating Equations 3 or 5, with respect to a spatialdimension; in neural terms it corresponds to buildingcortical dual-opponent cells (Thorell, De Valois, &Albrecht, 1984) by lateral inhibition between afferentgeniculate cells (for details, see Billock, 1995). In the spatial domain, such derivative operators look like simple cells; 1-D differentiation produces oriented cellsand the order of differentiation determines the numberof alternating excitatory and inhibitory lobes; Equation 6describes a cell that has n+ 3 receptive field lobes when probed with achromatic spots, but n+ 1 lobes for equiluminant hue probes. The spatial frequency re-sponse ( A(f s)) of such an achromatic mechanism to a stimulus (defined in terms of L-and M-cone absorp-tions) Z(fs)={ R(fs),G(fs)} is Aðfs Þ¼Zðfs Þð F1/2Dn Pðfs Þ/C138Þ ¼0. 5[R(fs)+G(fs)]anþ2fnþ2 s HS(fs) (achromatic response) þ1/2Rðfs Þ/C0Gðfs Þ/C138anfn s H1:83sðfs Þ (chromatic crosstalk)ð6Þ This bandpass filtering model correctly predicts the spatial tuning of cortical cells sensitive to both achro-matic and chromatic contrast (but presumably labeledonly for achromatic contrast; Billock, 1995). Increasingnshifts the tuning to higher spatial frequencies and narrower bandwidths. The typical cortical cell has anachromatic bandwidth of 1. 4 octaves (De Valois & De Valois, 1988) matching the achromatic term of Equation 6 if n= 2 (i. e., matched filtering for lumi-nance signals in P cells). Similarly, color signals can beextracted by matched lowpass filtering (second-order integration), resulting in achromatic crosstalk at all butthe lowest spatial frequencies. Cðfs Þ¼Zðfs Þð F1/2D/C02Pðfs Þ/C138Þ ¼0:51/2Rðfs Þþ Gðfs Þ/C138H1:83sðfs Þ (luminance crosstalk) þ[R(fs)--G(fs)]H3:35s(fs) (chromatic response) An engineer would find this odd—this kind of cross-talk is generally undesired and avoided in communica-tions systems. Yet, as discussed below, crosstalk is useful information for binding. Estimation of Stimulus-induced Correlations between Spatial and Chromatic Mechanisms The discussion of crosstalk above was deliberately over-simplified to illustrate general principles. To estimatecrosstalk-based correlations, we generalize Equations 6and 7, to include the temporal response. Selective two-dimensional ( x,t) spatio-temporal matched filtering of Equation 2 yields Aðfs;ft Þ¼[R(fs,ft)+G(fs,ft)][Se(fs)Te(ft)--Si(fs)Ti(ft)]2 f Achromatic response g þ1/2Rðfs;ft Þ/C0Gðfs;ft Þ/C1381/2Seðfs ÞTeðft Þ þ Siðfs ÞTiðft Þ/C1381/2Seðfs ÞTeðft Þ/C0Siðfs ÞTiðft Þ/C138 f Chromatic Crosstalk g Cðfx;ft Þ¼[R(fs,ft)--G(fs,ft)][Se(fs)Te(ft)+Si(fs)Ti(ft)]2 f Chromatic Response g þ1/2Rðfs;ft Þþ Gðfs;ft Þ/C1381/2Seðfs ÞTeðft Þ þ Siðfs ÞTiðft Þ/C1381/2Seðfs ÞTeðft Þ/C0Siðfs ÞTiðft Þ/C138 f Achromatic Crosstalk g Although not as elegant as Equations 6 and 7, Equa-tions 8 and 9 are computable if the Se,Si,Te,Tifunctions can be estimated (see Kelly, 1989; Burbeck & Kelly,1980; for methods and estimated functions). We could use Equations 8 and 9 as INPUT 1, INPUT 2in Equation 1 and ignore Q1,2; the correlated input is implicit. Howev-er, it is enlightening to use only the achromatic andchromatic response terms (the terms in bold type) as INPUT 1, INPUT 2and to use Equations 8 and 9 (with the crosstalk terms) to estimate the stimulus-driven correla-tions ( Q 1,2in Equation 1) between spatial mechanisms that extract information from P cells about achromatic form and other mechanisms that extract information about color or chromatic form. We define spectralcorrelation as the integrated overlap of the Fourierspectra of two functions (an analog of Signal Detection(7) (8) (9) Billock and Tsou 1039	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
Theory's cross-ambiguity function). The spectral corre-lation between A(fs,ft) and C(fs,ft)i s Q1;2¼Aðfs;ft Þ2Cðfs;ft Þ ¼Z Aða;b ÞCðfsþa;ftþb Þdadb ð10Þ where a,bare dummy variables of integration. For an equiluminant stimulus, the achromatic signal to theachromatic mechanism is zero and the correlatedcrosstalk consists of the correlation between the chromatic signal in the chromatic pathway [the bold term in the C(f s) equation] and the chromatic crosstalk extracted by the achromatic pathway [the plain typeterm in the A(f s) equation]. Thus, at equiluminance, the visual system attempts to bind something to nothing,based on the correlation between something andbandpass filtered noise. This situation corresponds tosome very odd perceptual effects. Effects of Equiluminance on Perception At equiluminance perception of form, depth and mo-tion are degraded (Cavanagh, 1991; Livingstone & Hubel, 1987; Gregory, 1977). Equiluminance can alsodisrupt segmentation and binding. Segmentation bybinocular disparity is severely degraded at equilumi-nance for random dot stereograms (Lu & Fender,1972). Surfaces are not linked together if their featuresare defined only by color; an image made of equilu-minant colors appears as patches of those colors, not as a unified whole (Cavanagh, 1991; Livingstone & Hubel, 1987). Moreover, perception of equiluminousforms can be unstable. Gregory (1977) found equilu-minant images ''looked uns table in contrast and 'jazzy'. '' This instability is sometimes attributed to aninability of ocular accommodation to use color infor-mation, but this is contradicted by Kotulak, Morse, and Billock (1995). Liebmann (1927) found that there is a critical luminance zone within which ''everything flows... glimmers... everything is soft, jelly-like, colloidal. Often... parts which belong together in the normal figure now have nothing to do with each other. (It is) aworld without firm things, without solidity'' (translatedin Cavanagh, 1991). Studies using luminance minimizedborders show these effects are particularly severe forborders defined by S-cone (tritan) modulation; the borders collapse and the color fields blend into a continuous color gradient (Buck, Frome, & Boynton,1977). No Evidence for Explanations Based on Lack of Parvo Inputs to Central Pathways Livingstone and Hubel (1987) ascribe the detrimental effects of equiluminance to a lack of parvo inputs tomotion, form, and depth channels, but evidence con-tradicts this (Merigan & M aunsell, 1993; Ingling & Grigsby, 1990; Schiller, Logothesis, & Charles, 1990). Moreover, there are chromatic mechanisms sensitive toform, motion, and depth (Regan, 2000; Cavanagh,1991; Mullen & Kingdom, 1991). Indeed, paradoxically,some chromatic phenomena are adversely affected byequiluminance, including color discrimination, color contingent aftereffects, and color rivalries (for review, see Mullen & Kingdom, 1991; Livingstone & Hubel,1987), suggesting a more subtle origin for the effectsof equiluminance. Several possibilities are consideredbelow. Evidence Contradicting Explanations Based on Low Chromatic Acuity Although the low acuity of chromatic mechanisms af-fects some visual phenomena (Cavanagh, 1991), it does not explain the poor and unstable contrast of some equiluminous borders. Border sharpness is not strictly ahigh spatial frequency phenomenon; a blurred edgelooks sharp if the missing harmonics of its Fourier seriesare below detection threshold (Campbell, Hopwell, &Johnstone, 1978). Similarly, reducing acuity by dimmingillumination has little effect on contrast over a largerange and edges modulated at 15 Hz appear sharp, even though acuity is reduced by a factor of 2. 5 (Livingstone & Hubel, 1987). Finally, as Mullen and Kingdom (1991)put it, ''it would be surprising if the lower borderdistinctness rated for S cone mechanisms compared to M-L cone ones was due to their differences in acuitysince a greater difference in acuity occurs betweenluminance and M-L chromatic mechanisms with no lossof border distinctness. '' Luminance as a Master Signal? Gregory (1977) posits that luminance is a master signal necessary for demarcating borders. This fits luminancecaptures color phenomena but does not explain whytritan equiluminous borders suffer excessively relative toequiluminous borders that stimulate the red/green sys-tem. Nor can the vulnerability of tritanopic borders bedue to a lack of S-cone driven color contrast mecha-nisms; double opponent b-y cells (Livingstone & Hubel, 1984), are well suited for transducing S-cone modulatedchromatic contrast and multiple S-cone driven spatialfrequency channels are found psychophysically (Human-ski & Wilson, 1993). A Binding-level Explanation? A possible explanation for the insalience of tritanopic borders stems from the origins of the r/C0g,r+g (luminance) and b-y signals. The major input of r/C0g 1040 Journal of Cognitive Neuroscience Volume 16, Number 6	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
signal to the cortex comes from spatially opponent P cells that multiplex luminance and color signals, induc-ing crosstalk between cortical luminance and r/C0gcolor mechanisms. Conversely, it appears that a majorsource of retinogeniculate y-b signal is from cells withspatially nonopponent Type II receptive fields (de Monasterio & Gouras, 1975; Wiesel & Hubel, 1966). These cells carry no multiplexed luminance signal and therefore do not induce correlations between cortical luminance and color detectors. The paucity of S-coneinputs to luminance (for review, see Cavanagh, 1991)removes another source of correlations between y-band luminance mechanisms. If correlations betweenmechanisms responding to borders are exploited forbinding, perhaps the y-b system, deprived of correla-tions to luminance borders, evolved only a weak input to the border contrast system. We suggest that binding allows achromatic mechanisms to reinforce r/C0gmech-anisms involved in segmentation and border formation(the luminance-driven form mechanism may need lessreinforcement from the chromatic system because—asdiscussed later—it receives reinforcement from motionmechanisms for unstabilized stimuli). CORRELATED CORTICAL SPATIAL/TEMPORAL MECHANISMS Why Form and Motion (or Space and Time) Are Not Independent There is evidence for separate mechanisms mediating detection of spatial and temporal variation (for review,see Zeki, 1993). There is also evidence for interactions between spatially and temporally tuned mechanisms (Burt, 1987; Breitmeyer & Ganz, 1976). Such interac-tions—as unavoidable as chromatic/achromatic interac-tions—are due to the physical entanglement of spatialand temporal information, and to spatio-temporal mul-tiplexing in retinogeniculate neurons. The physicalentanglement of motion and form is obvious; thevelocity (deg/sec) of a moving grating is V=f t (cycles/sec)/ fs(cycles/deg), where fs,ftare spatial and temporal frequency (each spatial and temporal fre-quency in the stimulus's Fourier representation canbe so treated). The neural entanglement of spatialand temporal information is more complicated, butcan be analyzed analogously to chromatic/achromaticinteractions. Consider the psychophysical achromaticspatio-temporal contrast sensitivity function. In princi-ple, if the function is separable (decomposable into the product of spatial and temporal functions), then inde-pendent spatial and temporal information can be ex-tracted by a homomorphic filter (a log transformfollowed by a matched filter). Unfortunately, contrastsensitivity is inseparable. However, Burbeck and Kelly(1980) showed that an Ingling-Martinez identity couldachieve a limited separati on, decomposing contrast sensitivity into the sum of two separable surfaces. Fora stimulus Z(f x,ft) the frequency response (assuming equal integrated excitatory/inhibitory sensitivity) is Xðfx;ft Þ¼0:5Zðfx;ft Þ1/2Seðfx Þ/C0Siðfx Þ/C1381/2Teðft Þþ Tiðft Þ/C138 Sustained (spatially tuned) response þ0:5Zðfx;ft Þ1/2Seðfx Þþ Siðfx Þ/C1381/2Teðft Þ/C0Tiðft Þ/C138 Transient (temporally tuned) response ð11Þ where Se,Te,Si,Tiare the excitatory and inhibitory spatial and temporal response functions defined in Equation 2. In Equation 11, the terms are labeledsustained and transient. The sustained term hasbandpass tuning for spatial frequencies and lowpasstuning for temporal frequencies. The transient term has lowpass tuning for spatial frequency and bandpass tuning for temporal frequency. In the psychophysicalliterature, the terms sustained and transient usually referto separate retinogeniculate spatial and temporalprocessing pathways. Yet, Burbeck and Kelly's analysisshows that both kinds of responses are embedded in allretinogeniculate cells with center/surround receptivefields, and are well modeled by Equation 11 for cells that obey superposition (P cells and X-like M cells). More-over, although the sustained and transient signals can beseparated algebraically, there is no plausible spatio-temporal filter that enables a strict spatial and temporalseparation by physiological means; the spectral contentof the terms overlap and attempts to extract one signalwill extract a small crosstalk signal as well. Quantifying Stimulus-induced Spatial and Temporal Mechanism Correlations Here, we estimate the stimulus-driven correlation be-tween temporally and spatially tuned pathways, createdby cortical filtering of the spatio-temporal signals carriedby LGN afferents. To extract the maximum signal fromthe spatially tuned (sustained) component of an array of active X-cells, we apply a filter matched to the sustained component, yielding a frequency response SPðfx;ft Þ¼0. 5Z( fx,ft)[Se(fx)--Si(fx)]2[Te(ft)+Ti(ft)]2 (Spatially tuned term) þ0:5Zðfx;ft Þ1/2Seðfx Þ2/C0Siðfx Þ2/C1381/2Teðft Þ2/C0Tiðft Þ2/C138 (Common information term) Similarly, a matched filter can be used to attempt to extract the temporally tuned (transient) component ofthe X-cell signals, yielding a frequency response TEðfx;ft Þ¼0:5Zðfx;ft Þ1/2Seðfx Þ2/C0Siðfx Þ2/C1381/2Teðft Þ2/C0Tiðft Þ2/C138 (common information term) þ0. 5Z(fx,ft)[Se(fx)+Si(fx)]2[Te(ft)--Ti(ft)]2 (temporally tuned term)(12) (13) Billock and Tsou 1041	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
Note that the matched filters are only partially suc-cessful in extracting signals tuned along one stimulusdimension. In both cases, the undesired portions of thesignal—tuned along both space and time—are identical,and so spectral correlation between the mechanisms isdominated by the crosstalk. As before, we use the boldterms of Equations 12 and 13 as input signals INPUT 1, INPUT 2in Equation 1 and estimate correlated crosstalk (Q1,2in Equation 1) by computing the spectral correla-tion between Equations 12 and 13 (neglecting unknownsources of correlated noise; e. g., Horn et al., 1991) Q 1;2¼SPðfs;ft Þ2TEðfs;ft Þ ¼ZZ SPða;b ÞTEðfsþa;ftþb Þdadb ð14Þ where aand bare dummy variables of integration. These spatio-temporal interactions present two possi-bilities for crosstalk-mediated segmentation: (1) Mutualfeedback between correlated form and motion mecha-nisms both responding to moving edges (the cross-modal construction model discussed above for form/color interactions); (2) Motion-correlated segmentationwithin the form pathway; that is, stimulus motion givesrise to two responses in the form pathway: the sustained form signal (the first term in Equation 12) and the trans-ient crosstalk signal (the second term in Equation 12). Because these crosstalk signals arise only in formdetectors stimulated by the target's motion, they are apotential cue for binding by correlated motion. A Binding Failure—Perception of Stabilized Images During normal vision images are in constant motion; microtremors of the eye make volitional stabilizationalmost impossible. However, images can be stabilized mechanically or by producing afterimages on the retina. These stabilized images can fade away rather quickly, aneffect often attributed to the temporal response prop-erties of retinogeniculate neurons. We argue that thisexplanation is grossly inadequate and that a failure of cor-tical segmentation mechanisms is indicated. However,the cortex can only operate on what afferent mecha-nisms send it, so next we analyze what early and cortical mechanisms contribute to stabilized image perception. What Early Mechanisms Do to Perception of Stabilized Images Some stabilized image percepts are consistent with retinogeniculate cell properties. Low contrast, low spa-tial frequency stimuli (favoring transduction by transientmagno [M] cells) usually fade faster than high contrast,high spatial frequency stimuli (favoring transduction bysustained parvo [P] cells; Ingling & Grigsby, 1990). Moreover, nonbleaching chromatic afterimages elevateboth achromatic and chromatic thresholds, suggesting a common P cell-like pathway (Kelly & Martinez-Uriegas,1993). There are some sustained M cells that respond tostatic images for a few seconds, but their temporalresponse becomes transient for high contrast stimuli(Benardete, Kaplan, & Knight, 1992), leaving only P cellsto mediate perception of afterimages. Moreover, King-Smith, Rosten, and Alvarez (1980) describe a subject who (on psychophysical grounds) appears to be missing the parvo system; this subject was also unable to per-ceive afterimages. Hence, transduction by P cells is anecessary condition for perception of most stabilizedimages. However, retinogeniculate properties cannotaccount for at least seven lines of evidence that centralmechanisms are responsible for some binding-failure-like oddities of stabilized images. Three Properties of Stabilized Image Perception Paradoxical to P-and M-Cell Properties (1) Anomalously Rapid Fading of Images that Should Tap High Acuity Sustained Channels Recall that Equation 2 shows that P cells have a sustained response for high spatial frequencies. Near the fovea,these cells have midget receptive fields capable oftransducing spatial frequencies up to 60 c/deg (conesampling limit). Additionally, these P cells (unlike M cells) are sensitive to high contrasts (Benardete et al., 1992). So, it is odd that the stabilized pattern of highcontrast shadows cast by blood vessels on the retinafades faster than other stabilized images (Coppola &Purves, 1996). Moreover, the closer the blood vessels areto the fovea, the faster they fade (as fast as 80 msec forthe highest spatial frequency components), in contra-diction to the response characteristics and retinal distri-bution of P cells. (2) Anomalously Rapid Fading for Stimuli that Tap Chromatic Sustained Mechanisms P cells have a chromatic response that is lowpass in both space and time (Equation 2 shows a bandpass chromaticcomponent as well, but this merely adds to the responseat moderate spatial and temporal frequencies without driving down the response to low frequencies). Al-though Equation 2 must hold if P cells obey linearsuperposition, it does not model psychophysical chro-matic contrast sensitivity for very low temporal frequen-cies (<0. 2 Hz), where the temporal CSF slope isconsistent with a first-order temporal differentiation(Kelly, 1981); this extra temporal derivative is probablya cortical process acting on P-cell inputs. A perceptual manifestation of this is the dramatic elevation of stabi-lized chromatic grating detection thresholds (a factor ofat least 45 greater than for chromatic unstabilized gra-tings and much higher than the elevation for achromatic 1042 Journal of Cognitive Neuroscience Volume 16, Number 6	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
gratings; Kelly, 1983). To reemphasize, unlike achromat-ic information, which is also transmitted by magno units,color is carried by P cells with sustained temporalproperties. If retinogeniculate mechanisms are solelyresponsible for fading of stabilized images, one wouldexpect color to be affected less by stabilization relative tostabilized luminance signals, not more. (3) Fading of Dynamic Images is Paradoxical to a Peripheral Mechanism Explanation Empty figures defined by a twinkling random dot back-ground fade if fixated steadily (Spillman & Kurtenbach,1992). Dynamic image fading cannot be due to transientperipheral mechanisms; fading is actually faster fordynamic stimuli than for static noise. This suggests a central mechanism that requires coherent modulation to extract kinetic edges. Four Lines of Evidence that Central Mechanisms Mediate Stabilized Image Phenomena (1) Effects of Binocular and Other Sensory Interactions(a) Stabilized monocular image fading can be reversed by similarly patterned stimulation of the other eye(Cohen, 1961), especially if placed on correspondingpoints of the stabilized and unstabilized retinae (effec-tiveness drops off monotonically with retinal disparity). In dichoptic presentation, modulating the background eye slows fading of the image in the stabilized eye(Gerling & Spillmann, 1987). (b) Conversely, fading ofa stabilized image makes unstabilized stimuli in theother eye less visible (Krauskopf & Riggs, 1959). (c)Even after a stabilized image has disappeared in one eye,it can be combined with an unstabilized duplicatemoving image in the other eye, to yield a sensation of motion in depth (Crane, 1994). (d) Some stabilized images induce an abrupt absolute blindness (Billock,Gleason, & Tsou, 2001; Ditchburn, 1973). Both blind-ness and recovery are binocularly simultaneous, imply-ing a central mechanism. (e) Visibility of stabilizedimages can be maintained by listening to auditorystimuli; the effect decays to baseline in about 15 min ifthe stimulation is not varied, suggesting a role for attention (Ditchburn, 1973). (2) Fragmentation Davies (1973), Evans (1965), and Pritchard, Heron, and Hebb (1960) find complex stabilized images dis/reap-pear, not as wholes but in fragmented forms (forillustrations of fragmentation percepts, see Billock & Tsou, 2004). This image fragmentation is not explained by the properties of peripheral neurons (e. g., fragmen-tation can eliminate contours that are present only in abinocular image, but not in the separate retinal images;Evans & Wells, 1967). Moreover, image fragments that wax and wane together obey Gestalt-like rules (Evans,1965, 1967): (a) Short lines appear and disappear as aunit. Fragmentation probability increases with linelength; for foveal vision fragmentation is likely for linelengths of roughly 45 arc min (an order of magnitudelarger than foveal retinogeniculate receptive fields),suggesting a range for the cortical mechanisms in-volved. (b) Disappearance and reappearance of parallel lines are correlated. (c) Random patterns are moreunstable than meaningful ones. (d) Angular patternsare more fragmented and perceptually unstable thanrounded patterns. (e) Fragmentation and fading in oneregion of a field is strongly affected by activity inneighboring regions. (f) Complex patterns are morelikely to fragment than simple ones. Simulations show that Gestalt-like grouping rules are emergent properties of cooperative segmentation and binding networks(Horn & Opher, 2000; Wang & Terman, 1997; Sporns,Tononi, & Edelman, 1991). Interestingly, some modelsof synchronized segmentation tend to fragment imagesif the spatial properties of the model do not pool over alarge enough set of units to discount noisy stimulusinputs (Terman & Wang, 1995). Similar fragmentation is reported by some subjects with migraine (mosaic vi-sion; Sacks, 1995) and amblyopia (especially for highspatial frequency stimuli; Hess, Field, & Watt, 1990); alink between amblyopia and neural synchronization issuggested by Roelfsema, Ko ¨nig, Engel, Sireteanu, and Singer (1994) finding that cortical cells are normal inspatial frequency selectivity, but cells driven by theamblyopic eye do not synchronize well, especially for high spatial frequency stimuli. (3) Filling-in of Stabilized Images If a broken figure is stabilized and fades—then regener-ates—the reappearance of the figure is often marked bycompletion of the break (Ditchburn, 1973). Moreover,Cardu, Gilbert, and Stabel (1971) find that objects made up of dashed lines often (45% of the time) exhibit completion just before fading. This effect resemblesthe completion of images broken by scotomas and iscompatible with binding algorithms that incorporatecooperative interactions. (4) Stabilization Effects on EEG Correlated to Perceptual Phenomena Although no stabilization experiment has looked for synchronization of g-band activity, several studies find that the relative level of a-rhythm activity increases when stabilized images disappear or when perceptualblanks occur during Ganzfeld viewing (Keesey & Nich-ols, 1967, 1969; Lehmann, Beeler, & Fender, 1967,Lehmann, Beeler, & Fender, 1965; Evans & Smith, 1964). Following stabilization, power in the EEG a-band Billock and Tsou 1043	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
(9-10 Hz) rises about 0. 7-1. 0 sec before target disap-pearance and higher frequencies are suppressed untilabout 1 sec prior to reappearance of target structures(Keesey & Nichols, 1967; Lehmann et al., 1965). Overview of Stabilized Image Studies: Only a Central Explanation Will Do The textbook account of stabilized image perception is that fading is due to the transient response propertiesof some retinogeniculate units. That analysis cannotaccount for counterintuitively rapid fading of stimulithat favor highly sustained parvocellular mechanisms. Nor can it account for any of the explicitly corticaleffects discussed above. Of these, image fading andfragmentation are consistent with known failure modes of segmentation networks that require a balance of competitive and cooperative interactions; fading canstem from relative weakness of competitive interac-tions; fragmentation can stem from weakness of coop-erative interactions. The Gestalt properties shown instabilized image studies correspond to emergent prop-erties of cooperative/competitive segmentation net-works (Horn & Opher, 2000; Wang & Terman, 1997; Sporns et al., 1991). As early as 1960, Pritchard et al. argued that fragmentation and partial reappearance offaded stabilized images is evidence for the dynamicformation of Hebbian cell assemblies—essentially thesame argument advanced in modern binding theory. Here, we argue that binding of units responding tononstabilized images is facilitated by the common in-formation in each form and motion mechanism induced by eye movements perturbing the retinal image, allow-ing motion and form-based segmentation networks toreinforce each other. DISCUSSION Summary of Reasoning At this point, it is useful to summarize our reasoning and the evidence that supports it so that we can lay out its strengths and weaknesses, and point out where futurework needs to be done: 1. Feng (2000) and Horn et al. (1991) showed that a preexisting correlation between mechanisms facilitates synchronizing those mechanisms' activity, a form offeature binding. In Horn et al. 's and Feng's work, theseinitial correlations stem from unspecified noise. What weadd here is an explicit physiological source for suchsignals (crosstalk induced by cortical demultiplexing of LGN signals), which results in stimulus-driven correla-tions between cortical mechanisms for achromatic form and color, and between achromatic form and motion. 2. Several investigators point out that feature-binding between mechanisms responding to common parts ofan image can improve part-binding/segmentation (Schil-len & Ko ¨nig, 1994; Finkel & Edelman, 1989; Poggio et al., 1988), a process called crossmodal reinforcement. 3. Other investigators point out that some part-binding (segmentation) models can fail in interestingways, for example, fragmentation of images (Terman &Wang, 1995; Marr et al., 1978). What we add to this is adiscussion of various ways a segmentation networkcould fail (depending on whether cooperative or com-petitive interactions were inadequate). 4. We point out that some odd percepts (e. g., fragmentation of stabilized images and melting ofequiluminant images) closely resemble these expectedfailure modes of segmentation models. We suggestthat these part-binding failures may be due to loss ofcrossmodal reinforcement (Point 2) that under normalcircumstances is available and relatively easy to bind because of stimulus-driven correlations (Point 1). Modeling—What Remains To Be Done Although each point in the above analysis is supported by experiments or mathematical models, it would bedesirable to supplement this analysis with a formalintegrated simulation and with a set of experimentalpredictions to be tested. We provide some such predic-tions in the sections below. A formal integrated simula-tion lies beyond the scope of this article, but we can lay out what would be required to implement one. For simplicity, we discuss form and color binding here—the steps required for form and motion binding arequite similar. (a) Start with a segmentation model, like Terman and Wang's (1995) that fails in one of theinteresting ways described in Point 4, and manipulatethe balance of competition and cooperation in themodel until it so fails. Two copies of this segmentation network are created, although they need not be sym-metrical (only failure in the color segmentation system isbeing studied in this example). (b) Create two sets ofinput mechanisms to drive the networks in (a)—an arrayof color units obeying Equation 9 and an array ofachromatic form units obeying Equation 8. To parame-terize these equations, the S e,Te,Si, and Tifunctions describing the spatial and temporal excitatory and in-hibitory responses of P cells need to be specified using psychophysical or physiological data. Suitable functions(derived psychophysically) can be found in Kelly (1989)and Burbeck and Kelly (1980). Under ordinary (non-equiluminant) conditions, there should be a correlationbetween the color inputs described by Equation 9 andthe achromatic units described by Equation 8. Thiscorrelation is implicit in the crosstalk terms of these equations, but can be made explicit using Equation 10. Now run the segmentation networks so that they syn-chronize for normal stimuli (stimuli that contain bothhue and luminance information). (c) Set the stimuli tobe equiluminant. The response of the achromatic form-1044 Journal of Cognitive Neuroscience Volume 16, Number 6	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
driven network should become disorganized (because it is being driven by noise), and the chromatic networkshould desynchronize from it. Segmentation mistakes inthe unreinforced chromatic network should now occur. Combining Equiluminance and Stabilization— Effects and Predictions Many of the binding failures discussed above seem connected. Equiluminant border melting has obvioussimilarities to border fading in image stabilization. Sim-ilarly, the ''jazziness'' of equiluminant borders may be afast time-scale analog to the fragmentation and dis/reappearance of some stabilized images. The comple-tion of gaps in reappearances of stabilized imagesresembles filling-in phenomena seen for some equilumi-nous images. In general, nulling the signal to a reinforc-ing channel should be detrimental to the channel itreinforced: The reinforcement is lost and lost informa-tion is replaced with mere noise. Significantly, for sub-jects who have one of the color, motion, and achromaticform systems disrupted, interrupting another systemresults in a more dramatic effect. For example, Sacksand Wasserman (1987) find that for an achromatopsic observer, boundaries between areas dissolved during periods of steady fixation (see below for analogousfindings in color normals). Similarly, stimuli that stimu-late only the b-y pathway (tritanopic stimuli) are partic-ularly susceptible to dissolution of contours (Cavanagh,1991). If the visual system requires two cooperatingsystems to achieve segmentation, then we would predictthat combining equiluminance and stabilization should be especially devastating. In fact, steadily fixated colored stimuli do fade more readily when equiluminant to theirsurroundings (Livingstone & Hubel, 1987; Buck et al.,1977; Krauskopf, 1967). Still more dramatic effects areobtained by combining retinal stabilization and equilu-minance. Billock et al. (2001) and Crane and Piantanida(1983) report that if a stabilized red/green or blue/yellowbipartite field is viewed through an unstabilized aper-ture, the colors flow and mix across the faded border. Some observers perceive uniform reddish greens oryellowish blues, in violation of color opponency. Otherobservers report formation of fine color textures orunstable islands of one color continuously forming anddissolving in a sea of the other color. Billock et al. foundthis spurious segmentation occurs only when there is aluminance difference between the two stabilized colored fields. Moreover, while melting together of adjacent equiluminant fields is normally seen only for tritanopicpairs, Billock et al. found that the phenomenon occursfor any stabilized color pair. (Related and extreme casesof binding failures are found in some neurologicaldisorders; Friedman-Hill, Robertson, & Treisman, 1995;Sacks, 1995; Zeki, 1993; Critchley, 1965). Given our reasoning and these preliminary results, a test of our reinforcement hypothesis would be tomeasure both perceptual segmentation and electro-physiological measures of binding while equiluminanceand stabilization are manipulated. The psychophysics isstraightforward. For experiments on equiluminance, animage is made of at least two colors, and radianceratios of the colors are varied through the range inwhich the equiluminance ratio must lie. Whether lossof neural binding signatures occurred at equiluminance can be determined by measuring figure and back-ground luminosities using standard psychophysicaltechniques (Cavanagh, 1991; Wyszecki & Stiles, 1982). Specific information should be gathered with respect toperceptual state (e. g., border instability, border melting,surface delinking). Similarly, image stabilization isstraightforward, albeit technically challenging and ex-pensive (Crane, 1994); useful results can often be had with afterimages or steady fixation (Ditchburn, 1973). Electrophysiological states should be correlated withthe specific perceptual state elicited (e. g., fragmenta-tion, fading, etc. ). If a stabilization system is available,t h ed e g r e eo fs t a b i l i z a t i o nc a nb em a n i p u l a t e db ymoving the image independently of eye movementsor by manipulating the gain of the eye-movement $ image-movement feedback loop (Crane, 1994). Finding what neural signatures to use (and how to test for them noninvasively in responding humans) is more challeng-ing. Since g-band activity is implicated in binding, the loss of higher-frequency EEG power during perceptuallosses in image stabilization is suggestive (Keesey &Nichols, 1967; Lehmann et al., 1965). There are twolimitations with this approach: Some sources of g-band power may be unrelated to binding and some binding failures may represent incorrect bindings and therefore not be identifiable by EEG. Recent experiments offerhelp on both counts, by studying gestalt perception(pop-out) of fragmented images hidden in camouflage. This fragmentation-to-order shift is complementary tothe order-to-fragmentation effects we wish to study. Subjects had g-band activity whether they saw a coher-ent image pop-out or not, but for coherent percepts the outputs of widely spaced electrodes became more correlated (as would be expected for synchronization;Gruber et al., 2002) and a different kind of g-band activity ''the induced gamma response'' was recorded. On every physically identical trial, there was a g-band response phase-locked to the stimulus, but only ontrials where the gestalt percept was obtained was thereanother EEG signature, consisting of g-band bursts in variable phase to the stimulus (Gruber et al., 2002; Tallon-Baudry & Bertrand, 1999; Tallon-Baudry, Ber-trand, Delpuech, & Pernier, 1996). We therefore predictthat EEG signatures like ''induced gamma'' will bereduced during binding failures induced by retinalstabilization (especially fragmentation) and equilumi-nance (e. g., loss of Gestalt symmetry in Glass patterns;Cavanagh, 1991). Moreover, we posit that these signa-tures should be further reduced by combining equilumi-Billock and Tsou 1045	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
nance with stabilization, and should be monitored when color boundaries collapse and forbidden colorsare perceived (Billock et al., 2001). An easier variation ofthis experiment could be done with steady fixation on aminimized border (Buck et al., 1977); under these con-ditions, some color boundaries collapse, creating uniformcolor mixtures. Acknowledgments We thank Randolph Blake, Angela Brown, Patrick Cavanagh, Viktor Jirsa, Daniel Levine, Lynn Olzak, Wolf Singer, De Liang Wang, and Scott Watamaniuk for critical readings of the manuscript. Special thanks to J. A. Scott Kelso for suggesting this problem. Reprint requests should be sent to V. A. Billock, General Dynamics, Suite 200, 5200 Springfield Pike, Dayton, OH 45431,USA, or via e-mail: Vince. Billock@wpafb. af. mil. REFERENCES Benardete, E. A., Kaplan, E., & Knight, B. W. (1992). Contrast gain control in the primate retina: P cells are not X-like, some M-cells are. Visual Neuroscience, 8, 483-486. Billock, V. A. (1996). 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RESEARCH Pattern forming mechanisms of color vision Zily Burstein1, David D. Reid1, Peter J. Thomas2, and Jack D. Cowan3 1Department of Physics, University of Chicago, Chicago, IL, USA 2Department of Mathematics, Applied Mathematics, and Statistics; Department of Biology; Department of Cognitive Science, Case Western Reserve University, Cleveland, OH, USA 3Department of Mathematics, University of Chicago, Chicago, IL, USA Keywords: Color vision, V1, Pattern formation, Turing mechanism ABSTRACT While our understanding of the way single neurons process chromatic stimuli in the early visual pathway has advanced significantly in recent years, we do not yet know how these cellsinteract to form stable representations of hue. Drawing on physiological studies, we offer adynamical model of how the primary visual cortex tunes for color, hinged on intracortical interactions and emergent network effects. After detailing the evolution of network activity through analytical and numerical approaches, we discuss the effects of the model 's cortical parameters on the selectivity of the tuning curves. In particular, we explore the role of the model 's thresholding nonlinearity in enhancing hue selectivity by expanding the region of stability, allowing for the precise encoding of chromatic stimuli in early vision. Finally, in theabsence of a stimulus, the model is capable of explaining hallucinatory color perception via a Turing-like mechanism of biological pattern formation. AUTHOR SUMMARY We present a model of color processing in which intracortical neuronal dynamics within thevisual cortex serve as the substrate for hue perception. Our analytical and numerical treatments of the emergent behavior seek to characterize the population dynamics underlying chromaticprocessing within the visual cortex, as well the roles of the various cortical parameters in determining the selectivity of the steady-state network response. We show that the system is self-organizing, capable of encoding stable representations of hue regardless of the stimulusstrength, and generating spontaneous color hallucinations in the absence of any input. INTRODUCTION Our experience of color begins in the early visual pathway, where, from the moment lightstrikes the retina, cone-specific neuronal responses set off the mechanisms by which the pho-tons'chromatic information is converted to the hues we ultimately see. While color vision scientists agree that the single-cell processing of chromatic stimuli occurs along the two inde-pendent cone-opponent L-M and S-(L+M) pathways ( Conway, Eskew, Martin, & Stockman, 2018 ;Kaiser & Boynton, 1996 ), there is yet no consensus as to how and where the divergent signals are synthesized to encode hue. To complicate matters, cone-opponency, observed in electrophysiological recordings of single neurons ( Shapley & Hawken, 2011 ), is often con-founded with hue-opponency, a phenomenon first theorized by Ewald Hering in the 19than open access journal Citation: Burstein, Z., Reid, D. D., Thomas, P. J., & Cowan, J. D. (2023). Pattern forming mechanisms of color vision. Network Neuroscience,7(2), 679-711. https://doi. org/10. 1162/netn_a _00294 DOI: https://doi. org/10. 1162/netn_a_00294 Received: 19 August 2022 Accepted: 17 November 2022 Competing Interests: The authors have declared that no competing interests exist. Corresponding Author: Zily Burstein ceburst@gmail. com Handling Editor: Gustavo Deco Copyright: © 2023 Massachusetts Institute of Technology Published under a Creative Commons Attribution 4. 0 International (CC BY 4. 0) license The MIT Press Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
century and later mapped out in clinical studies by Jameson and Hurvich ( De Valois, Cottaris, Elfar, Mahon, & Wilson, 2000 ;Jameson & Hurvich, 1955 ;Shevell & Martin, 2017 ). Best depicted in the Derrington-Krauskopf-Lennie (DKL) stimulus space (Figure 1 ), cone-opponency predicts that neurons tuned to either the L-Mo r S-(L+M) pathway will not respond to light whose wavelengths isolate the other ( Derrington, Krauskopf, & Lennie, 1984 ). It is tempting to equate these null responses to the four unique hues of color-opponent theory, in which unique blue, for example, is observed when the “redness ”and“greenness ”of a per-ceived color exactly cancel. But the wavelengths of the unique hues specified by perceptualstudies ( Jameson & Hurvich, 1955 ) only roughly match the wavelengths isolating either cone-opponent pathway ( Wool et al., 2015 ;Wuerger, Atkinson, & Cropper, 2005 ;Xiao, 2014 ), and, more fundamentally, we do not yet understand the mechanisms behind the processing that theanalogy implies ( Mollon & Jordan, 1997 ;Stoughton & Conway, 2008 ;Valberg, 2001 ). That is, how do we get from the single neurons 'chromatic responses to our perception of color? The necessary processing has often been attributed to higher-level brain function ( De Valois & De Valois, 1993 ;Lennie, Krauskopf, & Sclar, 1990 ;M. Li, Liu, Juusola, & Tang, 2014 ; Mehrani, Mouraviev, & Tsotsos, 2020 ;Zaidi & Conway, 2019 ) or yet unidentified higher order mechanisms ( Valberg, 2001 ;Wuerger et al., 2005 ). A central question of color vision research is whether these mechanisms rely on paralle l or modular processing to encode stimulus chromaticity ( Conway, 2009 ;Garg, Li, Rashid, & Callaway, 2019 ;Liu et al., 2020 ;Nauhaus, Nielsen, Disney, & Callaway, 2012 ;Schluppeck & Engel, 2002 ;Shapley & Hawken, 2011 ). If signaling about chromaticity is transmitted w ith information about other visual features, such as brightness, orientation, and spatial frequency, how do these features get teased apart? If not, where is the purported color center of the brain ( Conway, Moeller, & Tsao, 2007 ; Gegenfurtner, 2003 )? Several authors have addressed these questions through combinatorial models that param-eterize the weights of the L, M, and S cones contributing to successive stages of processing(De Valois & De Valois, 1993 ;Gegenfurtner & Ennis, 2015 ;Judd, 1949 ;Mehrani et al., 2020 ; Stockman & Brainard, 2010 ). Though differing in their assumptions of modularity, the theories share a mechanistic framework for the transitio n of single-cell receptive field properties Figure 1. The DKL space maps chromatic stimuli onto a circle with two “cardinal ”axes represent-ing the L-Ma n d S-(L+M) pathways. The excitatory or inhi bitory effect of a stimulus on cone-opponent cells tuned to either pathway can be thought of as a projection of its location in DKLspace onto the relevant axis. Stimuli isolating the two pathways correspond roughly to wavelengthsassociated with the red, green, blue, and yellow unique hues of color-opponent theory, leading tothe common, but mistaken, conflation of chromatic stimulus and color perception. Stimulus space: A geometrical construct in whichchromatic stimuli are represented bythe relative cone responses they yield. Color-opponent theory: Theory of color appearance thatpostulates that the four unique hues — red, green, blue, and yellow —are perceived antagonistically. That is,there is no such thing as a reddishgreen or a bluish yellow. Cone-opponency: Receptive field property of neuronsin the early visual pathway, by whichchromatic stimuli are processedthrough the comparison of therelative L, M, and S cone responses. Network Neuroscience 680Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
(Brown, 2014 ). Starting with cells in the retina and lateral geniculate nucleus (LGN) known to be tuned broadly to the cone-opponent axes, these proposed mechanisms build up to cells invarious cortical areas more narrowly tuned to divergent (and debated) chromatic directions in DKL space. While parsimonious, this formalism comes at the cost of tuning the cone weights arbitrarily, disregarding specific properties of real neurons 'receptive fields ( Eskew, 2009 ; Kaiser & Boynton, 1996 ;Stockman & Brainard, 2010 ). Furthermore, the linear combinatorial mechanism is not, on its own, able to account for the variety of color cells observed in the visual cortex ( Garg et al., 2019 ;Johnson, Hawken, & Shapley, 2001 ;Shapley & Hawken, 2011 ). In addition to the forward flow of chromatic information through the successive stages ofprocessing, the encoding of color reflects t he neuronal dynamics within each. Modelers agree that the next forays into a mechanistic theory of color vision should consider these intracortical circuits, but di sagree about where such interactions first become important (De Valois & De Valois, 1993 ;Hanazawa, Komatsu, & Murakami, 2000 ;Liu et al., 2020 ; Wachtler, Sejnowski, & Albright, 2003 ). Electrophysiological studies of macaque visual cortex have shed some light on this ques-tion, showing that the processing of individual hues previously associated with higher level mechanisms has its origins in the primary visual cortex (V1) ( Garg et al., 2019 ;Gegenfurtner, 2003 ;Hanazawa et al., 2000 ;Li et al., 2022 ;Wachtler et al., 2003 ;Xiao, 2014 ;Xiao, Casti, Xiao, & Kaplan, 2007 ). These experiments have identified the emergence of neurons in V1 tuned to the gamut of hues in DKL space, as well as to the role of processing nonlinearitiesin determining their tuning curves ( De Valois et al., 2000 ;Hanazawa et al., 2000 ;Lennie et al., 1990 ;Wachtler et al., 2003 ). Puzzlingly, these cells mainly inhabit the so-called CO “blobs, ” patchy regions rich in cytochrome oxidase that display a sensitivity to stimuli modulating either of the cone-opponent axes rather than the full set of hues ( Landisman & Ts 'o, 2002b ; Li et al., 2022 ;Livingstone & Hubel, 1984 ;Salzmann, Bartels, Logothetis, & Schüz, 2012 ). Some have speculated that this colocaliza tion stems from a mixing of cell populations encoding the two cardinal pathways (Li et al., 2022 ;Xiao, 2014 ) while others indicate a distinct population of hue-sensitive neurons in the “interblob ”regions, more conclusively associated with orientation tuning ( Garg et al., 2019 ;Landisman & Ts 'o, 2002a ). As a whole, however, these studies point to the need for a p opulation theory of chromatic processing remarkably early in the visual pathway. In this article, we present a model of color pr ocessing in which intracortical neuronal dynamics within V1 serve as the substrate for hue perception. Drawing on the canonical Wilson-Cowan neural field equations and the ring model of orientation tuning, we show that this population approach allows us to account for cells responsive to the full rangeof DKL directions without the need to fine-tune input parameters ( Ben-Yishai, Bar-Or, & Sompolinsky, 1995 ;Burstein, 2022 ;Hansel & Sompolinksy, 1998 ;Wilson & Cowan, 1972,1973 ). The threshholding we employ bears in mind the input-response nonlinearities of previous combinatorial models, but zooms out of the single-cell, feedforward interpreta-tion of input as the stimulus-driven LGN afferents to individual neurons. Rather, we model input as the total synaptic current into a population of cells, taking into account both thecone-opponent LGN afferents as well as the hue-dependent connectivity between distinct neuronal populations. The resulting demarcation between the cone-opponent and the hue-selective mechanisms in the same population of cells points to the importance of V1 in the transition from chromatic stimulus to color perception. To characterize this role, we study the effects of the model 's con-nectivity parameters and processing nonlinearities on the narrowness and stability of the hue tuning curves. In the final part of the paper, we show that the model is able to explain color Cardinal pathways: In DKL space, the two orthogonalaxes representing stimuli isolating the L-M and S-(L+M) cone-opponent pathways. Wilson-Cowan neural field equations: Coupled set of partial differentialequations describing the networkdynamics of excitatory and inhibitoryneural populations. Network Neuroscience 681Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
responses in the absence of LGN input, evoking color hallucinations via a Turing-like mech-anism of spontaneous pattern formation in DKL space. MODEL In light of the patchy distribution of color-sensitive cells reported in Landisman and Ts 'o (2002b), Li et al. (2022),Livingstone and Hubel (1984), and Salzmann et al. (2012), we model the color map of V1 as a set of neuronal networks, each encoding the chromaticity of its corresponding region of the visual field. This organization brings to mind the hypercolumnar structure of ori-entation preference within V1 ( Hubel & Wiesel, 1974 ), which, on the basis of its feature-based connectivity properties, allows for the representation of network activity as a function of a local-ized feature space. Here, we assume a mean hue-dependent activity a(θ,t) where θrepresents a direction in the DKL stimulus space, a strictly physiological conception of “hue”from the hues categorizing color perception, as explained above. In drawing this distinction, and in agreementwith Wool et al. (2015) and Li et al. (2022), we give no special status to V1 cells tuned to the DKL directions associated with the unique hues of color-opponent theory, while simultaneously emphasizing the cone-opponent nature of feedforward afferents from the LGN. The resulting activity a(θ,t) of a network of hue-preferring cells, expressed as a firing rate in units of spikes/second, is dominated by the membrane properties of its constituent cells, whose potential variations occur on the order of the membrane time constant τ 0, taken to be 10 msec (Ben-Yishai et al., 1995 ;Carandini & Ringach, 1997 ;Izhikevich, 2010 ). In the vein of previous neural mean field models of feature detection ( Bressloff & Cowan, 2002,2003b ;Bressloff, Cowan, Golubitsky, Thomas, & Wiener, 2001 ;Dayan, Abbott, & Labahn, 2001 ;Ermentrout, 1998 ;Gutkin, Pinto, & Ermentrout, 2003 ), and in close analogy to the ring model of orien-tation tuning ( Ben-Yishai et al., 1995 ;Hansel & Sompolinksy, 1998 ), we let a(θ,t) evolve according to the single-population firing-rate formulation of the Wilson-Cowan equations: τ0daθ;tð Þ dt¼-aθ;tð Þ þ ghθ;tð Þ1/2/C138 ; (1) where h(θ,t), the synaptic input, takes into account both excitatory and inhibitory afferents into a population of cells preferring hue θ, and g(h) is an activation function, as described below. To analyze the relationships between feedforward and recurrent processing and to distin-guish between their respective effects on a(θ,t), we write h(θ,t) as a sum of the stimulus-driven synaptic input from the LGN and the intracortical input resulting from the hue-dependentnetwork connectivity within V1: hθ;tð Þ ¼ h ctxθ;tð Þ þ hlgnθð Þ: (2) We express the input as the average effect of the net synaptic current on the membrane potential of a cell, following the conventions of Ermentrout (1998) and Carandini and Ringach (1997). Thus, h(θ,t) has units of m V and can take on both positive and negative values, chosen here so that a(θ,t) typically ranges from 0 to 60 spikes/sec, consistent with electrophysiological experiments penetrating individual color-responsive cells ( Conway, 2001 ;Johnson et al., 2001 ;Landisman & Ts 'o, 2002a ;Wachtler et al., 2003 ). The input is converted to a firing rate according to the nonlinear activation function ghð Þ ¼ β⋅h-T ð Þ ⋅Hh-Tð Þ ; (3) where H(x) is the Heaviside step function defined as H(x) = 1 for x> 0 and zero for x≤0. Note that in the context of machine learning, this form of activation is also known as the rectified Spontaneous pattern formation: A system 's ability to self-generate new symmetries in the absence ofexternal input. Activation function: A function mapping the afferentinput into a population of neurons(expressed as a current or membranepotential) to the population 's firing rate or probability of firing. Network Neuroscience 682Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
linear unit function, or Re LU for short. By constraining the network activity to levels below 60 spikes/sec, we ignore the effects of neuronal saturation commonly implemented in modelsofg(h)(Ben-Yishai et al., 1995 ;Ermentrout, 1998 ). Here, Tis the threshold potential of a neuron, below which the synaptic input has no effect on the mean firing rate of the network. Interestingly, as a processing feature, this thresholding nonlinearity has been speculated toaccount for the chromatic responses of individual neurons in V1 ( Hanazawa et al., 2000 ). The amplification of these responses, and thus the mean network response, is modulated by β, the neural gain measured in spikes · sec-1/m V. We assume that βis determined by far-ranging internal and external influences, from attentional mechanisms to hallucinogenicinput ( Ferguson & Cardin, 2020 ;Michaiel, Parker, & Niell, 2019 ). Feedforward Input To parameterize the input, prior work has relied on the direct relationship between cortical feature preferences and properties of the visual stimulus ( Ben-Yishai et al., 1995 ;Bressloff & Cowan, 2003b ). Cells in the cortex labeled, for instance, by their spatial frequency preferences can be mapped directly onto a visual space parameterized by the same variable. Thus, theactivity of each neuronal population is no longer labeled purely by its position on the corticalsheet, but also by its preferred stimulus in an analogous feature space. The corresponding network topology may be modeled on the cortical histology, such as the orientation map of Bosking, Zhang, Schofield, and Fitzpatrick (1997) or spatial frequency maps addressed in Bressloff and Cowan (2002),Bressloff and Cowan (2003a), and Bressloff and Cowan (2003b). Conversely, it may be based entirely on functional considerations, as, for instance, in the orientation tuning model of Sompolinksy et al. ( Ben-Yishai et al., 1995 ; Hansel & Sompolinksy, 1998 ), also known as the “ring model, ”which posits a topology based on the experimentally motivated assumption that populations with similar orientation prefer-ences are maximally connected ( Ben-Yishai et al., 1995 ) and on the argument that the impor-tant features of such a connectivity are captured by its first-order Fourier components ( Hansel & Sompolinksy, 1998 ). Our model deviates in this regard by emphasizing that the stimulus 's chromatic information is first discretized along the two cone-opponent pathways. We incorporate this aspect of early processing by projecting the stimulus 's DKL space position θ/C22onto the two cardinal axes: l¼cosθ/C22 s¼sinθ/C22:(4) The magnitudes of landsare thus taken to represent the normalized strengths of the L-M and S-(L+M) cone-opponent signals respectively. The feedforward input is then given by h lgn¼clcosθþssinθ ð Þ ; (5) where cis the signal strength, or contrast, expressed as the mean postsynaptic coarse mem-brane potential (in units of m V) of the target hue population generated by the presynaptic LGN neurons ( Carandini & Ferster, 2000 ). Formulated in this way, the input captures the colocali-zation of cone-opponency and hue selectivity in the activity of V1 cells as observed in Li et al. (2022) and Xiao et al. (2007). The hue tuning networks, parameterized by θ, are not only responsive to the individual cone-opponent stimulus signals, land s, but also implement the combinatorial mechanisms by which they are first mixed ( De Valois et al., 2000 ). Substitut-ing the expressions for land sinto Equation 5, we obtain hlgn¼ccosθ-θ/C22ð Þ : (6) Network Neuroscience 683Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
With this form, we point out the similarity of our combinatorial scheme to that of Mehrani et al. (2020), in which the input from cone-opponent V2 cells into hue tuning V4 cells is weighted as a function of the difference in their preferred hue angles. Most evidently, we differ from this model by first combining the cone-opponent signals in V1 rather than V4, in accordance with the above-mentioned studies. But beyond pointing to V1 as the origin of mixing, these exper-iments indicate that the combinatorial feedforward scheme is not sufficient to account forthe variability of neuronal hue preferences. Li et al. (2022) showed, for instance, that the con-tribution of signals isolating the S-(L+M) pathway is too small to explain the shifting of hue preferences away from the L-M axis by purely combinatorial means. As put forward by Shapley and Hawken (2011),Wachtler et al. (2003),a n d Lehky and Sejnowski (1999),a more complete understanding of neuronal hue encoding within V1 requires us to consider the nonlinear population dynamics therein. Recurrent Interactions We begin by characterizing the connectivity of the target hue tuning populations with a trans-lation invariant cortical connectivity function w(\|x-x0\|), such that the interactions between neurons in a single CO blob (length scale ∼0. 5 mm) depend only on the cortical distance between them ( Bullmore & Sporns, 2012 ;Salzmann et al., 2012 ). The network 's connectivity comprises the interactions of both its excitatory and inhibitory populations, wx-x0jjð Þ ¼ wexcþwinh; (7) each of which we model as a sum of an isotropic and distance-dependent term: wexc¼E0þ E1cos x-x0jjð Þ winh¼-I0-I1cos x-x0jjð Þ :(8) We set E0≥E1> 0 and I0≥I1> 0 so that wexcandwinhare purely excitatory and inhibitory, respectively, in accordance with Dale 's law ( Ben-Yishai et al., 1995 ;Dayan et al., 2001 ). Next, we map the weighting function onto hue space, drawing from the hue tuning micro-architecture revealed by the imaging studies of Liu et al. (2020) and Xiao et al. (2007). These studies point to a linear relationship between distance and hue angle difference, which min-imizes the wiring length of cells tuned to similar hues ( Liu et al., 2020 ). The hue-preferring cells inhabit the so-called “color regions, ”defined as such for their activation by red-green grating stimuli ( Liu et al., 2020 ). These regions predominantly overlap with the V1 CO blobs (Landisman & Ts 'o, 2002b ;Li et al., 2022 ) and are responsive to the full range of hues, much like the patchy distribution of orientation maps within the V1 hypercolumns. Thus, in a similarmanner to the local feature processing models of Bressloff and Cowan (2003b) and Ben-Yishai et al. (1995), we model the CO blob as a single color-processing unit consisting of Nneurons labeled by the continuous hue preference variable θ2[-π,π](Bressloff & Cowan, 2003b ). Figure 2 shows the distribution of hue-responsive neurons within a typical color region (Figure 2A ) as well as a more coarse-grained demarcation of peak activity within several of these regions ( Figure 2B ). To describe the spatial organization of their hue preference data, Xiao et al. (2007) and Liu et al. (2020) applied a linear fit to the cortical distance between two cell populations as a function of the difference in their preferred hue stimuli Δθ≡\|θ-θ 0\| apart in DKL space. Note, this implies a discontinuity between θ= 0 and θ=2π, allowing for the 2 πperiodicity of the hue preference label. Liu et al. (2020) report that the linear fit was able to capture the micro-organization of 42% of their tested hue maps, and a regression per-formed by Xiao et al. (2007) on an individual hue map gave a squared correlation coefficient of R2= 0. 91. Network Neuroscience 684Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
In agreement with these findings, we let \| x-x0\|=\|θ-θ0\|, absorbing the regression param-eters into the connectivity strength values E0,E1,I0,a n d I1in Equation 8. Substituting this change of variables and setting J0=E0-I0,J1=E1-I1(measured in m V/spikes · sec-1) gives wθ-θ0ð Þ ¼ J0þ J1cosθ-θ0ð Þ : (9) As detailed in Figure 3, for J1> 0, this functional form captures the local excitation and lateral inhibition connectivity ansatz typically assu med in neural field models as an analogy to diffusion-driven pattern formation ( Amari, 1977 ;Bressloff, 2003 ;Hoyle, 2006 ;Kim, Rouault, Druckmann, & Jayaraman, 2017 ;Turing, 1952 ). Notably, neurons in close proximity in both cortical and hue space maximally excite each other, and those separated by Δθ=πmaximally inhibit each other, evoking the hue-opponency of perception on a cellular level. We empha-size, however, that this choice of metric is guided by our physiological definition of hue and does not associate a perceived color difference to measurements in hue space. Here, it is also important to distinguish between the connectivity function and the center-surround receptive fields of single-and double-opponent color cells ( Shapley & Hawken, 2011 ). While the structures of both can be approximated by the same functional form, the resemblance is superficial: the former characterizes the interactions between different neuro-nal populations, and the latter is a property of single cells, often adapted for computer vision algorithms ( Somers, Nelson, & Sur, 1995 ;Turner, Schwartz, & Rieke, 2018 ). Finally, we weigh the influence of the presynaptic cells by convolving the connectivity function with the network activity, arriving at the cortical input to the target hue population at time t: h ctxθ;tð Þ ¼Zπ-πwθ-θ0ð Þ aθ0;tð Þ dθ0: (10) The recurrent input is thus a continuous function in θ, derived from the population-level inter-actions. As put forward by the above-mentioned imaging studies, these interactions are colo-calized with the cone-opponent feedforward input, h lgn, within the same CO blob regions of Figure 2. (A) Hue map of individual hue-selective cells obtained by 2-photon calcium imaging of neuronal responsiveness to seven test hues. Scale bar: 200 μm. (B) Regions of peak response to test hues (solid contours). The dashed white lines demarcate the color-preferring regions, colocalized with the CO blobs. Scale bar: 0. 5 mm. Modified with permission from Liu et al. (2020). Network Neuroscience 685Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf

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