/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 model leads to patchy and even periodic distributions of predator and prey (Segel &Levin, 1976), much like the structured coloring of the Turinganimal skins. Some similar models can lead to waves of pursuitand evasion (which is an interesting way to think about theadvancing migraine wave front, with its leading excitatory fortifi-cations and following inhibitory scotoma). By now, several of theconnections between the Ermentrout-Cowan model and thepredator-prey model should be clear, and it is not surprising thatthe Wilson-Cowan model (Cowan, 1968; Wilson & Cowan, 1972,1973) which Ermentrout and Cowan's work was based on, can beinterpreted as a sophisticated kind of population dynamicsmodel—one in which the populations become active or quietinstead of flourishing or extinct. In particular, note the terms in Equation 4 that contain the cand gcoefficients; without these terms, the model does not produce spatial patterns. The cterm says that the grasshoppers cooperate with one another; the /H11002gterm says that the predators fight with one another. Taken together with theother terms, this asymmetry mirrors the asymmetry of the Ermen-trout and Cowan equations, where excitatory cells excite bothexcitatory and inhibitory cells, while inhibitory cells inhibit theirfellow inhibitory cells as well as excitatory cells. Yet the carefulreader should wonder, Where in the Ermentrout-Cowan model arethe diffusion-like (second spatial derivative) terms that are crucialto pattern formation in both reaction-diffusion systems and popu-lation dynamics? 4The answer is that they are hidden in the very interactions of the two excitatory and two inhibitory terms in Equation 1. Because the neural excitatory and inhibitory weightingfunctions are Gaussian-like, with different variances, their effec-tive interaction kernels resemble differences of Gaussians andespecially second derivatives of Gaussians. These effective inter-action kernels dictate the periodicity and standard scale of neuralpattern formation (see Figure 9). Because a second derivative of a Gaussian is a linear operation, one can rearrange terms to isolate anaked second derivative operator, which is the Laplacian operatorused in the diffusion terms of reaction-diffusion equations andpredator-prey models (for the relevant math in a wonderfullydifferent context, see Marr & Hildreth, 1980). 5 Pattern Formation in the Ermentrout-Cowan Model If the spatial ranges of cortical excitation and inhibition differ and the level of spatially uniform excitation is raised sufficientlyhigh, the neuronal network in Equation 1 gives rise to periodicparallel stripes of neural activity and inactivity that can form at anyorientation (Ermentrout & Cowan, 1979). The patterning is due tothe local excitation-lateral inhibition connectivity built into theexcitatory cells in Equation 1. The pattern begins as a small localfiring rate increase but grows due to local excitation, resulting inan inhibited firing rate in the flanking regions. The inhibited flankscannot themselves inhibit their flanks, allowing excitation to growin the flanks of the flanks and so on. Thus, stripes of activation andinhibition form a periodic grating-like pattern whose scale is determined by the ranges of the local excitation and inhibition. There are two routes to raising the overall level of spatiallyuniform excitation. The first—used to model drug hallucinationsand epilepsy/migraine aura—involves breaking the balance be-tween the neural strengths of excitation and inhibition. Ermentroutand Cowan (1979) induced internal excitation by increasing aand cin Equation 1. However, the near symmetry of the equations suggests that a similar result could be obtained by decreasinginhibition ( bandd), as some have considered for percepts induced by migraine and epilepsy (Dahlem & Chronicle, 2004; Tass, 1995,1997). Tass (1997) studied pattern formation for every combina-tion of increasing or decreasing the four excitatory and inhibitoryparameters in Equation 1. Some parameterizations allow only onehallucinatory state (dictated by random initial conditions) toemerge; other parameterizations allow as many as four hallucina-tory states to alternate, leading to perceptual multistability. Weakercoupling between the excitatory and inhibitory elements of the network (the bandcterms of Equation 1) favors simple and stable geometric hallucinations, while stronger coupling leads to oscilla-tion between competing states (e. g., perception of fan shapesalternating with perception of spirals and concentric circles; Tass,1997). The other route is via a spatially uniform input (the Input term in Equation 1). Because visual neurons have little response to staticspatially uniform stimuli, many relatively uniform stimuli thatproduce hallucinatory pattern formation are temporally modulated(e. g., flicker, pulsed TMS, AC electric stimulation). This couldhelp account for the temporal instability of many induced hallu-cinations—each temporal modulation is both a stimulus to patternformation and a perturbation, disturbing the previously elicitedstate—but other factors likely influence the temporal scales ofhallucinations. Rule et al. (2011) experimented with variations of Equation 1 (which autonomously produces produce 13 Hz dampedoscillations). At low forcing frequencies (e. g., below 13 Hz), thesystem responds at the forcing frequency and tends to generatestable hexagonal gratings, like those that Fechner (1838) reportedon rotating black-and-white tops. At high forcing frequencies, thesystem breaks into two populations, each of which responds toevery other cycle of the stimulus, and tends to generate stripepatterns. This subsampling (called frequency demultiplication byneural theorists) was also a feature of Drover and Ermentrout's(2006) retinal-based model of electrically induced wake contour 4Here, diffusion-like indicates spread of neural activity, not movement of substances. 5The exact form of the spatial derivative term is probably not crucial; Cowan's (1985) model shows a fourth derivative-of-a-Gaussian-like inter-action kernel for the entire system (see his Figure 12. 7, p. 236), but inspatial frequency terms, this is much like a second derivative tuned to anarrower range of higher spatial frequencies. 757 HALLUCINATIONS AND CORTICAL PATTERN FORMATION