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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,... | 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 ¨... | 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 sequenc... | 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. ... | 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 expon... | 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... | 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. ... | 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 beca... | 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 entit... | 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 K... | 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. Cunni... | 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 suf... | 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 (Ta... | 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 visibl... | 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... | 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 s... | 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 ... | 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 ... | 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 t... | 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 fra... | 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 JND... | 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... | Fractal_BCT20BILLOCK.pdf |
Billock, Cunningham & Tsou (2008) 13 APPROVED FOR PUBLIC RELEASEReferences Billock, V. A. (2000). Neural acclimation to 1/f spatial frequency spectra in natural images and human vision. Physica D, 137, 379-391. Billock, V. A., Cunningham, D. W., Havig, P., & Tsou, B. H. (2001a). Perception of spatiotemporal random frac... | Fractal_BCT20BILLOCK.pdf |
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, ... | 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 natura... | 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 as... | 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 break... | 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-th... | 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... | 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 o... | 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 Network... | 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... | 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., ... | 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 ... | 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... | 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 consist... | 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 M... | 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 bo... | 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... | 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 im... | 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 a... | 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 ... | 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 |
Hess, R. F., Field, D. J., & Watt, R. J. (1990). The puzzle of amblyopia. In C. Blakemore (Ed. ), Vision: coding and efficiency (pp. 267-280). Cambridge: Cambridge University Press. Horn, D., & Opher, I. (2000). Temporal segmentation and binding in oscillatory neural systems. In D. S. Levine, V. R. Brown, & V. T. Shire... | 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 |
Tallon-Baudry, C., & Bertrand, O. (1999). Oscillatory gamma activity and its role in object representation. Trends in Cognitive Sciences, 3, 151-162. Tallon-Baudry, C., Bertrand, O., Delpuech, C. & Pernier, J. (1996). Stimulus specificity of phase-locked and non-phase-locked 40 hz visual responses in human. Journal of ... | 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 |
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 R... | 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... | 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 form... | 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. (... | 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 n... | 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 co... | 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 F... | netn_a_00294.pdf |
V1. Collectively, our formulation of h(θ,t) implements the mixing rules posited by these experiments, without requiring us to arbitrarily fine-tune the relative weights of the afferentsignals. RESULTS Evolution of Network Activity We start by observing that by virtue of the invariance of w(θ-θ0) under translations of θ... | netn_a_00294.pdf |
we make the change of variables θ-θ0=ϕ, so that the left-hand side of Equation 12 can be rewritten as-Zθ-π θþπwϕð Þe-iμϕ 1ffiffiffiffiffiffi 2πp eiμθdϕ¼Zπ-πwϕð Þe-iμϕ 1ffiffiffiffiffiffi 2πp eiμθdϕ: (13) The eigenvalues are thus: λμ¼Zπ-πwϕð Þe-iμϕdϕ: (14) Next, we assume a(θ,t) is separable in tand θand bounded on [-π,π] so that we may expand... | netn_a_00294.pdf |
To determine δ1and δ2, we reformulate the Heaviside as a function of θ. Given that the input h(θ,t) is a real-valued function, c02ℝand c1=c/C3-1. For mathematical convenience, we then rewrite Equation 17 in terms of c0, Re( c-1)≡c R-1, and Im( c-1)≡c I-1as hθ;tð Þ ¼λ0c0tð Þffiffiffiffiffiffi 2πp þclþffiffiffi 2 πr λ-1c R-1tð Þ ! cosθð Þ... | netn_a_00294.pdf |
With this reformulation, the system of equations for the evolution of the coupled cν (Equation 19) takes the more explicit form: τ0dcνtð Þ dt¼-cνtð Þ þ βZδ2 δ1q0tð Þ þ chtð Þcosϕþγtð Þ 1/2/C138 1/2/C138 ^e/C3 νϕð Þdϕ: (24) Note that, for all cν, the integrand of Equation 24 is a function of q0(t),ch(t), and γ(t) and th... | netn_a_00294.pdf |
state value. Within each time step (typically chosen to be 1 msec), we coarse-grain the net-work into n= 501 populations with hue preferences separated evenly across the DKL angle domain [-π,π]. The choice of an odd nallows us to numerically integrate Equation 10 using the Composite Simpson 's Rule, whereupon we rectif... | netn_a_00294.pdf |
It is also important to note here the difference between a network tuning curve and a single-neuron tuning curve. The former is a coarse-grained representation of the CO blob response,with the horizontal axis representing the gamut of hue preferences within a single network. A relatively large tuning width would theref... | netn_a_00294.pdf |
Figure 8. Effect of Ton the tuning curve properties. θ/C22=0, β=1, J0=-3,J1= 2, and c= 10. Figure 7. Effect of con the tuning curve for varying values of Twith β=1, J0=-1,J1= 0. 2, and θ/C22= 0. Note that the small network response rates are due to the low values of cchosen here. (A) T=-5. (B) T=-1. (C) T=0. ( D ) T= 0... | netn_a_00294.pdf |
comparable magnitudes of the stimulus strength and threshold, | c|∼|T|, we see a transition in which Talso begins to sharpen the tuning curve and continues to do so until the threshold surpasses h(θ,t)f o ra l l θ(i. e., for δ⋆ 1=δ⋆ 2= 0). Accordingly, for higher stimulus strengths, the thresholding nonlinearity plays ... | netn_a_00294.pdf |
relatively large value of J1does not restrict the growth of the network response with increasing stimulus strength. Thus, the anisotropic tuning introduced by the external input and the recur-rent interactions act cooperatively to raise the network 's response to the stimulus hue, and competitively to tune its selectiv... | netn_a_00294.pdf |
We thus conclude that the emergent hue curves in V1 are both inherited from the LGN and built on the recurrent interactions. The competition between J1andcpoints to a continuum of regimes in which either hlgnorhctxdominates. However, in all regimes, J0works cooperatively with cto narrow the curves, and all the parame t... | netn_a_00294.pdf |
In contrast, our model does not apportion separate regions of the parameter space to exter-nal and recurrent mechanisms. Rather, in both the analytical and extended regimes, the rolesofcand J 1exist on a spectrum, where the effect of each parameter is suppressed by larger values of the other. Of course, this suppressio... | netn_a_00294.pdf |
and, consequently, the analytical intractability of the associated stability analysis. We therefore set up the Jacobian matrix for a numerical analysis of the local stability. We begin by adding a small perturbation of the form δaθ;tð Þ ¼X μDμtð Þ^eμθð Þ (29) and substituting the resulting activity into Equation 1. The... | netn_a_00294.pdf |
The entries of the corresponding Jacobian matrix consist of the bracketed prefactors, and may equally be obtained from the general system of equations for the global network dynamics, asfollows: J¼∂f1 ∂c0∂f1 ∂c R-1∂f1 ∂c I-1 ∂f2 ∂c0∂f2 ∂c R-1∂f2 ∂c I-1 ∂f3 ∂c0∂f3 ∂c R-1∂f3 ∂c I-12 66666666643 7777777775 c⋆ 0;c R⋆-1;c I... | netn_a_00294.pdf |
of T. The stability conditions on J0and J1are thus uniquely determined by βalone. Further-more, for the general diagram (i. e., with βfixed and c,Tunfixed), each point of the analytical regime permits linear solutions, in addition to the ones that implement thresholding. Accord-ingly, the uniqueness of the bifurcation ... | netn_a_00294.pdf |
never deviates far from the underlying homogeneous steady state, the two dynamical state equations for their concentrations, Xand Y, take the linear form d Xθ;tð Þ dt¼a Xθ;tð Þ þ b Yθ;tð Þ þ DX∇2Xθ;tð Þ d Yθ;tð Þ dt¼c Xθ;tð Þ þ d Yθ;tð Þ þ DY∇2Yθ;tð Þ ;(34) where a,b,c, and drepresent the chemical reaction rates, and D... | netn_a_00294.pdf |
of an embryo along various directions from an original spherical state. A hallmark of each of these examples is that there is no input into the system, so the emergent patterns reflect amechanism of spontaneous symmetry breaking, onset by a perturbation of “some influences unspecified ”(Turing, 1952 ). In light of this... | netn_a_00294.pdf |
Figure 15. Spontaneous pattern formation in the absence of input ( c=0 ). β=1. ( A-B):J0=-2,J1= 0. 1 (A) T= 0 (B) For T< 0, the homo-geneous steady-state value increases. Here, T=-10. (C-D): Pattern formation in the extended regime for J0=-2,J1= 0. 4. (C) No hue tuning curve emerges for T≥0. Here, T=0. ( D ) T=-10. A h... | netn_a_00294.pdf |
We observe that within the analytical regime, the system generates a stable homogeneous steady-state solution a∞(θ) = const ≥0 for all values of the parameters β,T,J0,a n d J1(Figure 15A-B). As such, from the closed-form linear steady-state solution ( Methods :Linear Solution ), we obtain a∞θð Þ ¼-βT 1-2πβJ0for T≤0 0 f... | netn_a_00294.pdf |
thereby transforms the discrete cone-opponent signals to a continuous representation of chromatic information. Such mixing mechanisms have been implemented by previous com-binatorial models of color processing, though through a largely feedforward approach or at the level of the single neuron. Our theory bears in mind ... | netn_a_00294.pdf |
the visual cortex is an early stop in the visual pathway, we point to the need for further field theory approaches to our understanding of color perception. METHODS Linear Solution We assume in the linear case that the net input h(θ,t) is above threshold throughout the dynamics such that the activity profile is never c... | netn_a_00294.pdf |
Evolution of Peak Angle We first assume that upon receiving a stimulus θ/C22at time t= 0, the network has a random spon-taneous firing rate a(θ, 0). Using Equation 15, with c02ℝandcμ=c/C3-μ, we expand the activity profile in terms of the initial values of the corresponding coefficients cμ(0): aθ;0ð Þ ¼X μcμ0ð Þ^eμθð Þ ... | netn_a_00294.pdf |
As before, we note that the evolution of cμ(t), and therefore of Fμ(t),∀μ2ℤdepends only on the first-order coefficients c|μ|≤1(t). Therefore, the steady-state values of the higher order coefficients c R⋆-μ¼βffiffiffiffiffiffi 2πp F⋆ μcosμγ⋆ð Þ c I⋆-μ¼-βffiffiffiffiffiffi 2πp F⋆ μsinμγ⋆ð Þ(48) and the corresponding ϕμ, that is, tanϕ⋆ μ/C16/C17... | netn_a_00294.pdf |
where δh(θ,t) is a perturbation to the input due to δa(θ,t). Taylor expanding the right-hand side of Equation 54 in h(θ,t)≡h∞(θ)+δh(θ,t) about h(θ,t)=h∞(θ) then yields τ0dδaθ;tð Þ dt¼-a∞θð Þþδaθ;tð Þ ð Þ þβh∞θð Þ-T ð Þ H h∞θð Þ-T ð Þ þ δhθ;tð Þ H h∞θð Þ-T ð Þ þ Oδh2/C0/C1 /C8/C9 : (55) For small perturbations, the high... | netn_a_00294.pdf |
REFERENCES Amari, S.-I. (1977). Dynamics of pattern formation in lateral-inhibition type neural fields. Biological Cybernetics,27(2), 77-87. https://doi. org/10. 1007/BF00337259,P u b M e d : 911931 Ben-Yishai, R., Bar-Or, R. L., & Sompolinsky, H. (1995). Theory of orientation tuning in visual cortex. Proceedings of th... | netn_a_00294.pdf |
Cambridge, UK: Cambridge University Press. https://doi. org/10. 1017/CBO9781107337930. 005 Gross, T. (2021). Not one, but many critical states: A dynamical sys-tems perspective. Frontiers in Neural Circuits,15, 614268. https://doi. org/10. 3389/fncir. 2021. 614268, Pub Med: 33737868 Gutkin, B., Pinto, D., & Ermentrout,... | netn_a_00294.pdf |
Salzmann, M. F. V., Bartels, A., Logothetis, N. K., & Schüz, A. (2012). Color blobs in cortical areas V1 and V2 of the new worldmonkey Callithrix jacchus, revealed by non-differential opticalimaging. Journal of Neuroscience,32(23), 7881-7894. https:// doi. org/10. 1523/JNEUROSCI. 4832-11. 2012, Pub Med: 22674264 Schlup... | netn_a_00294.pdf |
Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms Vincent A. Billock and Brian H. Tsou U. S. Air Force Research Laboratory An extraordinary variety of experimental (e. g., flicker, magnetic fields) and clinical (epilepsy, migraine) conditions give rise to a surprisingly commo... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
“windows on the visual brain. ” Second, elementary hallucinations illustrate an important general principle in cognitive science. It isnow well recognized that complex systems, such as the humanbrain, have collective properties that are not inherent in the indi-vidual neural elements; it has become common to ascribe so... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
model describes the overall changes in arc shape and apparent speed as it propagates across cortex; this behavior is generic forweakly excitable media and can be mimicked by reaction-diffusionmodels (Dahlem & Hadjikhani, 2009). A temporary scotoma(blind region) is left in the wake of the fortification arc's move-ment. ... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
spreading depression. One discrepancy is that the symptoms of migraine normally do not spread as far as the cortical spreadingwave, suggesting that the strength of the wave falls below anactivation threshold in the unaffected region (Dahlem & Had-jikhani, 2009). Since the first neural effect of cortical spreading depre... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
mation stretches the horizontal cell layer, depolarizing on-center bipolar cells and hyperpolarizing off-center cells. Continued deepocular pressure (for more than about 40 s) results in a temporaryblindness via ischemia (like applying a choke hold directly to theretina)—a useful technique often employed by psychophysi... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
dermaier (1976) found that it was easier to electrically stimulate geometric hallucinations at high altitudes, presumably as a sideeffect of hypoxia. This is interesting because hypoxia alsofacilitates triggering the cortical spreading depression phenom-enon linked to migraine (Dahlem & Mu ¨ller, 2004; Grafstein, 1963;... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
ular vision, thresholds were higher, and the function's minimum was (for two thirds of observers) shifted to higher frequencies. Slightly lower thresholds for binocular vision are expected oninformation theoretic grounds (by a factor of √2), which con-trasts with the near absolute requirement of binocular pressurefor o... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
propeller shape rotating in time with the illusory movements of the flashing icons (see Figure 3). Mac Kay effects. Donald Mac Kay (1957a, 1957b, 1965, 1978) described a series of phenomena that have become known as Mac Kay effects, without much separate consideration of theirdiverse nature. Most of these effects invol... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
also show activity in the posterior fusiform gyrus, which is known to contain concentrations of color sensitive neurons. Bexton et al. (1954) recognized from the beginning that the hallucinations re-ported for sensory deprivation were probably related to CBS(although they did not use the term); they cited similar cases... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
mechanism for spreading their influences. Both can manifest in subtle ways. For the cortex, the conceptually simplest model is aset of coupled integrodifferential equations, each pair of whichrepresents dynamic neural interactions in one particular corticallocation. 3 /H11509E /H11509t/H11005/H11002E/H11001SE/H20853a W... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
mation in the spirit of “... and then a miracle occurs...,” like Aphrodite sprung from the sea foam. To better understand self-organized pattern formation in neural systems, it is useful to examineother pattern-forming systems. Early work on hallucinatory modelingused fluid dynamics as an analogy: A pan of liquid, heat... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
Figure 7. Reaction-diffusion (RD) simulations of pattern formation compared to actual biological patterns. A: Some stable states that RD systems can generate. B: Some two-dimensional simulations produced by a simple Turing model. C top: Actual and simulated shell patterns produced by Meinhardt's (2003) RD model. C bott... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
/H11509A//H11509t/H11005F/H20849A,I/H20850/H11001DAƒ2A;/H11509I//H11509t/H11005G/H20849A,I/H20850/H11001DIƒ2I, (2) where DAand DIare diffusion rates for the activator ( A) and inhibitor ( I) chemicals, respectively, and ƒ2is the second derivative (Laplacian) operator /H115092x//H11509x2/H11001/H115092y//H11509y2, which... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
/H11509G//H11509t /H11005 a G/H11002 b GM /H11001 c G2/H11001 DGƒ2G prey population growth births violent deaths G-cooperation diffusion, /H11509M//H11509t /H11005 e M/H11001 f GM /H11002 g S2/H11001 DMƒ2M predator population growth births feeding M-competition diffusion. (4) For some parameterizations, this kind of mo... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
hallucinations. Because each location in perceptual space is rep-resented by a cell that responds on every other cycle, this createsstanding wave patterns on cortex, with the two populations re-sponding both temporally and spatially out of phase with oneanother. The overall effect should resemble the directionally mul-... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
Intriguing Synergies Between Hallucinatory Conditions The interplay between increasing excitation via the coupling constants (e. g., a, c) and via the Input term in Equation 1 suggests that condition pairings that drive both should interact synergisti-cally. Five lines of evidence support this: (a) Subjects given asubh... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
hot and falling cold fluid, which from above appear like the stripes in the Ermentrout-Cowan model. As in the Ermentrout-Cowanmodel, there is no preferred orientation to these stripes. However,even a single line of injected fluid can dictate the final orientationof the entire system (Bestehorn & Haken, 1991). Billock a... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
shaped hallucinations are more salient than induced circular pat-terns, which in turn are more salient than induced spirals. Thispattern preference resembles Kenet et al. 's (2003) finding thatsome spontaneous cortical patterns recur much more often thanothers. In general, biased hallucinations and their interactions w... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
(mathematically similar to Ermentrout-Cowan neural networks) produce similar stripe-orientation opponencies; for example, inpattern formation on fish skins, if diffusion is easier in one direc-tion than another and the inhibitor species has greater range thanthe activator, then stripes form parallel to the direction of... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
stochastic resonance—first predicted for mathematical systems (Ditzinger, Ning, & Hu, 1994)—is not yet well documented inneural systems; there are only a few relevant models and onerelated psychophysical study. For example, in a simulation of200/H11003200 coupled neural oscillators, 1/f noise was far more efficient tha... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
Other Complications: Alternate Pattern-Formation Mechanisms and Neural Loci Complicating this picture are alternative mechanisms that could lead to hallucinatory geometric patterns. For example, considerrotating spirals and pinwheels: In the Ermentrout-Cowan model,these correspond to a pattern of oblique stripes of neu... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
sometimes appear to be more a continuum of states than a set of discrete percepts. For example, in TMS of cortex, single-anddouble-wedge (bow-tie/butterfly) phosphenes are common. Kammer et al. (2005) found that single wedges are seen at lowerstimulus intensities and that butterfly wedges extend into bothvisual fields ... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
some theoreticians use the mapping from geometric patterns on retina to stripes on V1 to argue that these mappings evolved toallow the cortex to exploit certain regularities and invariancesin image processing (Caelli, 1977; Dodwell, 1991; Schwartz,1977, 1980); if other sensory cortices exploit similar ap-proaches, it s... | Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf |
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