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 EE/H11569E/H11002b W IE/H11569I/H11001Sensory Input /H20854, /H11509I /H11509t/H11005/H11002I/H11001SI/H20853c W EI/H11569E/H11002d W II/H11569I/H11001Sensory Input /H20854. (1) Here, SEand SIare sigmoidal neural response nonlinearities, and the W/H11569Eand W/H11569Iterms are spatial convolutions of neural activity with excitatory ( E) and inhibitory ( I) spatial weighting functions; for example, WEE(x, y)/H11569E(x, y)/H11005/H20848 /H20848 WEE(/H92701,/H92702)E(x/H11002 /H92701,y/H11002/H92702)d/H92701d/H92702(/H92701and/H92702are dummy integration variables). The neural interactions are asymmetric: Neighboring excitatory cells excite each other and inhibitory cells alike, while inhibitory cellsinhibit both other inhibitory cells and excitatory cells (relatedsubtle asymmetries lie at the heart of every pattern-forming sys-tem). The Gaussian-like neural-weighting functions ( W i,j) describe the influence that neighboring neurons have on particular neurons as a function of distance; the resulting interplay between excitationand inhibition mimics the spatial derivative operator found in otherpattern-forming systems. The range of cortical excitation—about 1mm—is roughly the width of a human cortical orientation oculardominance column (a fundamental unit of cortical organization),while inhibition can span several columns. This excitatory rangematches the spatial scale seen in the migraine fortification hallu-cination—a broad flattened vertical arc made up of bright flicker-ing line segments set at roughly 60° angles to each other—whenadjusted for retinotopic location, each segment is approximately1. 2 mm long on cortex (Richards, 1971). Similarly, the hypercol-umn spacing in human cortex (about 2 mm; Horton, 1996; Horton& Hedley-White, 1984) roughly fits the maximum frequency ofhallucinatory features (for a fan geometry) with about 15 features(and blank spaces between features) per cortical hemifield (inaccord with the rough number of cortical hypercolumns that couldbe organized along the circa 60-mm extent of each hemisphere's V1; Bressloff et al., 2001). (Each 2-mm-wide hypercolumn con-tains two 1-mm orientation ocular dominance columns, one foreach eye; differential activation between the two ocular dominancecolumns could account for the sensation of depth seen in many Klu¨ver hallucinations. ) Biased flicker-induced hallucinations (see Figure 6) are a conspicuous exception to the 1-to 2-mm corticalscale, with a spatial scale similar to the scale of the biasingstimulus; this may be helpful when studying hallucinatory patternformation using functional imaging (Billock & Tsou, 2007). Relation of the Ermentrout-Cowan Model to Some Familiar Pattern-Forming Systems When first encountering self-organizing models as complex as Equation 1, there is a temptation to accept autonomous pattern for-3Because neural density is high, the neural interactions can be handled in a mathematically continuous fashion. This is sometimes called a neural field model, in analogy to the idealized fields used in physics (Amari, 1977; Coombes, 2005). Some models (Baker & Cowan, 2009; Bressloff, Cowan, Golubitsky, Thomas, &Wiener, 2001; Cowan, 1985; Henke et al., 2009; Wiener, 1994) use a singleequation that explicitly deals with excitatory neurons but builds in interactionsequivalent to the effects of inhibitory neurons. This is mathematically elegant andcomputationally advantageous, but for our neural and pedagogical purposes, it isuseful to employ a structure like Equation 1. Figure 5. A physical demonstration of cortical stripes and their perceptual correlates. The nonlinear mapping from retina to primary visual cortex (V1) introduced in Figure 2 can be seen using functional imaging of humancortex (in this case, functional magnetic resonance imaging) while viewing physical patterns. A-B: Radialpatterns on retina (s1, s2) evoke horizontal stripes on V1. C-D: Concentric patterns on retina (s3, s4) evokevertical stripes on V1. The dynamic reversing checkerboard patterns shown to retina are used to force both on-and off-cells (with both sustained and transient temporal properties) to respond nearly continuously, yielding astrong blood oxygen level-dependent response. From “Functional Analysis of Primary Visual Cortex (V1) in Humans,” by R. B. H. Tootell et al., 1998, PNAS: Proceedings of the National Academy of Sciences, USA, 95, p. 813. Copyright 1998 National Academy of Sciences, USA. Reprinted with permission. 753 HALLUCINATIONS AND CORTICAL PATTERN FORMATION