/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 is used to make the diffusion flowdown the Aor Iconcentration gradient. F( A,I) and G( A,I) are generally quadratic functions of chemical concentration but are specific to thesystem being modeled; for example, in Gierer and Meinhardt (1972), F/H20849A,I/H20850/H11005k 1/H11002k2A/H11001k3A2/I;G/H20849A,I/H20850/H11005k4A2/H11002k5I. (3) Turing was interested in the spot-and-stripe patterns of animal coats, such as leopards and zebras, and showed that a necessary condition for thediffusion rate constants is that D A/HS11005DI. Because of the asymmetry between substance A's and substance I's actions ( Iinhibits A's action, but not vice versa), the diffusion can drive an instability. This was a revolu-tionary development—diffusion had previously been thought an agent ofstability—but the combination of the two Turing asymmetries creates the Turing instability. Murray (1988, 1989) made some compelling simula-tions of animal coats using Turing-like reaction-diffusion models (see,e. g., Figure 6). Shoji, Iwasa, Mochizuki, and Kondo, (2002) and Kondoand Miura (2010) extended Turing models to complicated patterns foundon fish skins and other surfaces (see Figure 7). The wave of corticalspreading depression—believed to be the physiological basis of mi-graine—is an example of a reaction-diffusion process in an excitablemedium. It is therefore not surprising that several investigators havecreated reaction-diffusion models of the spread of the migraine fortifica-tion arc and its accompanying scotoma. Reggia and Montgomery's(1996) reaction-diffusion model acting on orientation specific neuronscan result in a moving arc with irregular jagged lines that come and go,similar to the flickering serrations of fortification illusions. Dahlem and Chronicle (2004) presented an improved reaction-diffusion model thatclosely matches patient's migraine fortification patterns. Methods devel-oped for studying traveling waves in reaction-diffusion systems haveproven useful in studying similar waves produced by Ermentrout-Cowan-like models. Perhaps closer to Ermentrout-Cowan models are population-based models that resemble the simple reaction-diffusion model. As a bridge between models, Murray (1989) gifts us with the Parable of the Sweaty Grasshoppers. Imagine a host of grasshop-pers in a field of dry grass. The grasshoppers have a mutation thatmakes them sweat excessively if they get hot. If the dry grasscatches fire, nearby grasshoppers fly a distance away and startsweating so much that their surroundings become too wet to ignitewhen the flame-front arrives. If the sparks from the initial flameignite many random areas, causing many random grasshoppermigrations-cum-sweating-attacks, the result is a scattered patternof burnt and unburnt areas, not unlike a Dalmatian's spotted coat. The key is that the grasshoppers must move faster than the fire(D I/H11022DAin Equation 2), buying time to sweat a firebreak. Now consider a slightly different system: The unfortunate grass-hoppers avoid the fire this time but attract the attention of a group ofhungry praying mantises. The interaction between the two popula-tions is asymmetrical in several ways. The mantises inhibit the grass-hoppers' numbers Gdirectly by consuming them and indirectly by reducing the number of breeders available. The grasshoppers excitethe mantis population Mby contributing nutrients for the mantis to use for reproduction. By itself, this can lead to a temporal pattern—aboom-and-bust cycle, where the mantis numbers soar until the grass-hopper numbers plummet and the mantises starve, allowing the grass-hopper numbers to recover. However, there is another possibility. Thegrasshoppers diffuse away from the mantises' territory faster than themantises can breed pursuers while continuing to cover their existingterritory. Segel and Jackson (1972) expressed a model of a similarsystem (sans insect drama) as Figure 8. Biasing a classic pattern-forming system—convection patterns formed in a pan of fluid heated from below and cooled from above. Convec-tion organizes into parallel cylinders of rising hot and falling cool fluid, whichlook like Ermentrout and Cowan's (1979) cortical stripes when viewed fromabove. The orientation of the stripes forms randomly unless biased by anoutside force. The top row of the simulations shows fluid injections (whitestripes) delivered at the beginning of the simulation (first studied by Bestehorn& Haken, 1991). In Columns a and b, by 200 time steps, the biasing geometryhas imposed its orientation on the entire emergent pattern. In Column c, twoinjections are made, one slightly stronger than the other, which compete untilthe stronger injection eventually dominates the entire pattern. From Principles of Brain Functioning (pp. 37 & 241), by H. Haken, 1996, Berlin, Germany: Springer-Verlag. Copyright 1996 by Springer-Verlag. With kind permissionfrom Springer Science /H11001Business Media. 756 BILLOCK AND TSOU