170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 243 10. 4 EXTENSIONS OF THE MODEL 243 effects of external visual stimuli, or by chemical stimulation. Here we consider anotherpossibility:thatstabilitychangesarecausedbytheeffectsofintrinsicnoise in V1 circuits. But to consider such a possibility required extending the underlying mathematical model of large-scale neocortical activity, that is, the Wilson-Cowan equations (Wilson and Cowan, 1972, 1973) to deal with such effects. This was accomplished in a series of papers which were summarized in Buice and Cowan (2009). The resulting Stochastic Wilson-Cowan equations were then used to analyze the effects of intrinsic fluctuations on the triggering of hallucinations (Butler et al. 2012). Theresultsareveryinformative. Itturnsoutthatsuchfluctuationscanindeed trigger hallucinations. Indeed, in a randomly organized network with as many long-range inhibitory connections as long-range excitatory ones, hallucinations wouldbeubiquitous. Howeverthecircuitryof V1is,aswehaveindicated,farfrom random. In fact it is highly organized with only sparse direct long-range inhibitory connections, compared with the excitatory long-range connections (Stepanyants etal. 2009). Sucharestrictionofconnectivityincreasesthestabilityofthehomoge-neous state so that spontaneous pattern formation in the form of the hallucinations occurs, more or less, only under the conditions given above, which we refer to as mean-field pattern formation. Such conditions correspond exactlyto those reported by many observers viahallucinogens, or sensory deprivation and so on. Thus the anatomy we introduced to trigger the form constants is precisely that required to stabilize V1 against the effects of intrinsic fluctuations. Interestingly, this same anatomy was used by Kaschube etal. (2010) in a study of the development of the iso-orientation patches in V1 shown in figure 10. 10. It turns out that the horizontal connections between such patches, shown in figures 10. 11, play a key role, not only in generating the form constants, but also to stabilizethe development of the functional architecture of V1. 10. 4. 4 Flicker phosphenes Sofarwehaveconsideredthetriggeringofformconstantstobetheresultofchanges in thecontrolparameters caused by, for example, ingesting psychedelic substances or by the after-effects of anesthetics such as ketamine. However, the early work by Purkinje (1918) on flicker phosphenes indicated that flickering external stim-uli can trigger some of the form constants. Purkinje's 1819 observations were the subject of a recent paper by Rule etal. (2011), in which they added the flickering external stimulus h(t)∝Θ[ sin2𝜋t T-𝜏] (10. 15) whereΘis the Heaviside step function, and 𝜏is a constant, to a form of equations 10. 5, equivalent to a stimulus-driven version of equations 10. 6, the Ermentrout-Cowan equations.