(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 diffusionuntil the activator's range is exceeded and then switch orientationto the perpendicular direction. Similar results occur if the activatorand inhibitor species have different preferred directions of diffu-sion (Shoji et al., 2002). An anisotropic model should also facil-itate modeling the serrated appearance of migraine fortificationsand the finding that fans and concentric circles are easier toproduce and more salient than spirals (Billock & Tsou, 2007),whereas isotropic models show no such preference. Bressloff et al. 's (2001, 2002) neural pattern-formation model illustrates the power of stimulus-specific networks for understand-ing complicated hybrid hallucinations, like textures superimposedon a geometric pattern (see, e. g., Figures 1F and 12, which showa finer lattice texturing a coarser spiral structure). Cortical area V1is functionally subdivided into an array of hypercolumns, whichcontain neurons with various preferences in feature orientation. In Bressloff et al. 's models, the neurons within a column interactwithout respect to orientation preference, but connections betweencortical columns are between neurons of common orientationpreference. These models behave almost as if there is a separate Ermentrout-Cowan-like pattern-forming network for each orientedpopulation that competes for the right to fire; the patterns that arisecan appear textured if the winning neurons are perceptually labeledfor orientation (see also Golubitsky et al., 2003). 6The orientation of the resulting cortical stripe patterns determines the overall geometry of the illusion, while the orientation preferences of theactivated neurons determine the texture attached to the geometry. Moreover, the resulting individual stripes of activity need not havethe same orientation labels; richly complicated percepts, like her-ringbone textures, can result. Bressloff et al. 's work drasticallyexpands the range of hallucinatory percepts that can be modeledand points the way to still more powerful treatments; one canimagine Bressloff et al.-like networks with neurons perceptuallylabeled for contrast, depth, color, and motion. A vast array ofperceptual phenomena could be modeled by such networks. Bakerand Cowan (2009) created a vector-based model that expands thenumber of perceptual attributes that such models can pin to spon-taneous hallucinatory pattern formation. In another interestingvariation, Golubitsky et al. (2003) introduced a weak level ofanistrophy into the model by making connection strength vary withorientation preference (but any stimulus preference should apply). Curiously, this also creates time-varying pattern formation, withrotating and pulsating hallucinations, and tunnel-like percepts withstructure seeming to emerge from or disappear into the opening ofthe tunnel. Stochastic Resonance in Neural/Perceptual Pattern Formation Stochastic resonance—a qualitative change in a system's behav-ior when noise is added—comes in three forms: (a) paradoxicalincrease in signal-to-noise ratio, (b) induced multistability, and (c)autonomous spatiotemporal pattern formation. The first two phe-nomena are often enhanced by using random fractal (1/f n) noise in place of the usual white noise. However, pattern formation via6This explanation is an oversimplification that does not do complete justice to Bressloff et al. 's (2001, 2002) approach. For example, their model not only forms orientation-specific textures but also sharpens theorientation specificity of the network, relative to the specificity of theindividual cortical neurons. Figure 12. Neural connectivity affects perceptual pattern formation. Top: In Bressloff, Cowan, Golubitsky, Thomas, and Wiener's (2001, 2002)models, these connections can be specific to sensory neural preferences,like feature orientation. Here, local connections ( /H110211 mm) within a cortical orientation hypercolumn are indiscriminate, while connections betweenhypercolumns are dependent on orientation selectivity. Bottom: Bressloffet al. 's model leads to textured hallucinations that resemble some complexreported hallucinations, in this case Siegel and Jarvik's (1975) THC-induced hallucination (see Figure 1F). From “Geometric Visual Halluci-nations, Euclidean Symmetry and the Functional Architecture of Striate Cortex,” by P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas, &M. C. Wiener, 2001, Philosophical Transactions of the Royal Society of London: Series B. Biological Sciences, 356, pp. 306 & 325. Copyright 2001 by the Royal Society. Reprinted with permission. 762 BILLOCK AND TSOU