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, heated frombelow and cooled from above, can form patterns in the convectingfluid. As the bottom-heated fluid rises and the top-cooled fluid falls,currents are set up, at first at random, across the container. Reinforc-ing currents combine constructively, while opposing currents cancel. If a high enough temperature difference is maintained, the system canevolve into a set of mutually reinforcing rolling concentric cylindersof convecting fluid, which (if viewed from above) resemble thecortical stripes that Ermentrout and Cowan (1979) invoked to explainhallucination patterns. Moreover, three such interacting roll patternssuperimposed at 60° angles create a hexagonal cell pattern (oftenfound in atmospheric phenomena; warm air rises in the center of eachcell, and cold air descends on the edges) like the hexagonal texturesseen in hallucinations. In simple fluid dynamics, rolls are favoredwhen surface tension is eliminated, while hexagonal cells form ifsurface tension at the top of the container is substantial. In complexitytheory, a variable whose change can force a sudden qualitative changein pattern formation is called a control parameter. Here, the temper-ature difference between top and bottom is the main control param-eter, determining if patterns can form, and surface tension is a sec-ondary control parameter, with influence over the type of patternsformed. The constant input of heat from the bottom is dissipated to thesurface by convection and requires constant renewal. Such patternsare called dissipative structures; this term has also been applied tohallucinations like those induced by flicker (Stwerka, 1993) and ratherappropriately too since the hallucinations cease promptly when theoutside forcing (the flicker) is removed. The effects of control param-eters are also analogous in the two systems: In flicker-induced hallu-cinations, hexagonal forms are more often seen for weak stimulation,and fan shapes, concentric circles, and spirals (all of which stem fromcortical stripe formation) are more often seen during stronger stimu-lation. Rule, Stoffregen, and Ermentrout's (2011) model predicts asimilar effect for flicker rate. However, although fluid-dynamic pat-tern formation resembles hallucinatory pattern formation, mathemat-ically there are closer systems. Here, we visit two other systems—reaction-diffusion and population dynamics—that mathematicallyresemble neural pattern formation to seek additional insights into thebehavior of Equation 1. Turing (1952) introduced reaction-diffusion systems. Such sys-tems consist of two diffusing agents; one agent can activate amarker of some kind (like a skin or fur pigment), while the otheragent inhibits the activation of the marker. They take the form Figure 6. Some reaction-diffusion simulations of a spotted coat growing on an animal skin of fixed shape and variable size ( /H9253). Murray (1989) found that in this case, the pattern was dependent on a particular spatial scale and suggested that this is why textured coats are common in medium-sized animals but uncommon in mice andelephants. From Mathematical Biology (p. 445), by J. D. Murray, 1989, Berlin, Germany: Springer-Verlag. Copyright 1989 by Springer-Verlag. With kind permission from Springer Science /H11001Business Media. 754 BILLOCK AND TSOU