hallucinations. Because each location in perceptual space is rep-resented by a cell that responds on every other cycle, this createsstanding wave patterns on cortex, with the two populations re-sponding both temporally and spatially out of phase with oneanother. The overall effect should resemble the directionally mul-tistable rotating fan shapes and pulsating bull's-eye patterns re-ported by Billock and Tsou (2007). Rule et al. found that spatialscale of flicker-induced hallucinations in their model depends onthe temporal forcing frequency. An interesting implication of themodeling is that the production of stripes rather than hexagons athigh temporal frequencies is guaranteed if the resting level ofcortical activity is sufficiently high that the average neuron will benear an inflection point in its sigmoidal input-output function. This is the hallucinatory analogue of Werner's (2007) and Plenzand Thiagarajan's (2007) contention that the nervous system iscritically poised. The result may also cast light on some otherinteractions between intrinsic neural rhythms and hallucinatoryactivity. In addition to the standing wave phenomena described by Rule et al. (2011), there are also motion instabilities in neural fieldmodels (like Ermentrout and Cowan's) that can result in travelingwaves (for reviews, see Coombes, 2005; Ermentrout & Kleinfeld,2001; Ermentrout & Terman, 2010). These waves are similar tothose generated by reaction-diffusion equations but travel synap-tically, rather than by volume transmission. Recently, it has be-come possible to study these waves in slices of living surgicallyremoved cortical tissue (e. g., Golomb & Amitai, 1997). One keytheoretical result, which fits nicely with Rule et al. 's findings, is that the speed of the traveling wave is dependent on the threshold(the lower inflection point) of the sigmoidal firing function (e. g.,S Ein Equation 1; Coombes, 2005); this has been manipulated experimentally by applying electric fields across the tissue slice (Richardson, Schiff, & Gluckman, 2005). The similarities between the optimal flicker rate for Ermentrout-Cowan simple stripe-type (see, e. g., Figure 4) hallucinations (ca. 10-20 Hz; Becker & Elliot, 2006; Billock & Tsou, 2007), thepsychophysically matched flicker rate of migraine fortificationaura (18 Hz; Crotogino et al., 2001), the roughly 10 Hz rate forkaleidoscopic changes of Klu ¨ver-like hallucinations sometimes experienced during migraine (Sacks, 1995b), and the content of thealpha and beta bands of EEG (8-12 and 13-20 Hz, respectively)are intriguing, yet poorly understood. Certainly, the frequencydependency and temporal scaling of hallucinations may be relatedto resonance between the external modulation and intrinsic neuralactivity (Ermentrout & Terman, 2010; Herrmann, 2001; Rule etal., 2011). It was recognized early on that despite reducing thebrightness of flicker, inducing hallucinations with flicker throughclosed eyes is surprisingly effective, perhaps because eye closureincreases the background alpha-band EEG signal, which the flickermay interact with (V. J. Walter & Walter, 1949). This is supportedby Shevelev et al. 's (2000) finding that when flicker is synchro-nized to the observer's alpha rhythm, pattern formation takes2-5 s, compared to 10-15 s for nonsynchronized trials. Similarly,Kanai, Chaieb, Antal, Walsh, and Paulus (2008) found that phos-phenes are best excited by 10-12 Hz current applied to cortexwhen the observer is in the dark and by 16-20 Hz current when theobserver is in the light, in accord with synergistic interactions withunderlying EEG cortical activity measured in darkness (alphaband) and light (beta band). In addition to providing evidence for interactions between in-trinsic neural activity and outside stimulation in hallucinatorypattern formation, such studies may shed light on discrepanciesbetween studies in frequency dependencies of flicker-induced hal-lucinations. Mundy-Castle (1953) pointed out that in addition tothe broad range of driving frequencies, the cortex responds with aneven wider range of harmonics and subharmonics, making itdifficult to identify the mechanism of action. Some subjects havelarger evoked responses at the first harmonic than at the funda-mental driving frequency, allowing them to respond to rather lowstimulus frequencies (V. J. Walter & Walter, 1949). Seeminglyminor stimulation differences could have a strong effect by mod-ifying the intrinsic activity. Some studies use eyelids as diffusersfor a bright flickering source, while other studies use open eyes,with rotating tops, strobe-illuminated ganzfelds (empty texturelessfields), or flickering monitors as sources. Wackerman, Pu ¨tz, and Allefeld (2008) pointed out that although alpha-band intrinsicactivity is present in both eyes-closed and eyes-open-ganzfeldviewing, the measured content of the alpha band is different for thetwo cases, with the alpha-band peak shifting to higher frequenciesfor ganzfeld viewing (similar to the viewing conditions of Kanai etal. 's, 2008, study). Becker, Gramann, Muller, and Elliott (2009)found a similar frequency shift and an additional increase ingamma-band (ca. 40 Hz) activity accompanying the developmentof the illusory percept. Figure 9. Second derivative-like spatial interaction kernels embedded in the Ermentrout-Cowan model (see Equation 1 in the text; Murray, 1989). These kernels are a systems-level mathematical analogue to the line-spreadfunctions measured in single cell electrophysiology. Here, W a(the solid line) is the interaction built into the top equation of Equation 1; for example, it represents the a WEE*E/H11002b WIE*Iinteraction in the E (excitation) growth equation. Wi(the dashed function) is the c WEI*E/H11002 d WII*Iinteraction in the I(inhibition) growth equation. Thus, inhibition is the mirror image of excitation, and every stripe of excitatory activity generated by the pattern is flanked by stripes of inhibitory activity (andvice versa). The aand iindices that Murray (1989) used mirror the activator-inhibitor terms of reaction-diffusion theory and were used toaddress anatomical stripe development on striate cortex. Each three-lobedfunction is like a scale-specific version of the Laplacian (second derivative)terms in Equations 2 and 4 in the text. Alternately, one can think of thesefunctions as Laplacians that have operated on Gaussians. Laplacian-of-Gaussian operators are ubiquitous in vision and neuroscience, especially inthe neuroscience of lateral inhibition (where they go by names like Mexican-hat functions, difference of Gaussians, and difference of offset Gaussians). From Mathematical Biology (p. 491), by J. D. Murray, 1989, Berlin, Germany: Springer-Verlag. Copyright 1989 by Springer-Verlag. With kind permission from Springer Science /H11001Business Media. 758 BILLOCK AND TSOU