170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 232 232 CH10 GEOMETRIC VISUAL HALLUCINATIONS 10. 2. 2 A little group theory We can now go a little more deeply into the consequences of recognizing such a symmetryin V1. Wenotethatallthesymmetry-preservingoperationswehaveintro-duced have the effect that equation 10. 7 is still left invariant under the modified action of the Euclidean group 𝔼(2). How can we use this property? The answer is that we can use the fact that every symmetry group has subgroups, each of which has some of the symmetries of the group, but not all of them. For example, the group of rotations and reflections of a circle, referred to as 𝕆(2)is a continuous group,inthatanyrotation,howeversmall,isagroupelement. Butonecaninscribe squares,rectangles,rhombuses,trianglesandhexagonsinsideacircle. Eachofthese objects has a finite set of symmetries, some rotations and some reflections which preserveitsappearance. Eachsetdefinesafinite subgroup of𝕆(2)knownasa dihe-dralgroupwithnelements,referredtoas 𝔻n. Thesearethesymmetrygroupsofthe square, rectangle or rhombus, triangle and hexagon. Interestingly, these geometric shapes are the only ones which can be used to tile a plane in repeating or periodic patterns. How can we relate these facts to neocortical neurodynamics? It turns out that there is a piece of mathematics proved by one of our collaborators on this work, Martin Golubitsky,andhiscolleagues,knownasthe Equivariant Branching Lemma (Golubitsky Stewart,and Schaeffer,1988)whichshowsthatwhenadynamicalsys-tem whose resting state has a certain symmetry becomes unstable, new stable or unstablestatesemergewithsymmetriesthataredescribedbyfinitesubgroupsofthe symmetry group of the resting state. In particular, the new states have symmetries oftheaxialsubgroupsofsuchasymmetrygroup. [Anaxialsubgroupisasubgroup that leaves only one unique planform invariant. ] We are able to use this lemma and thedemonstrationthatourmathematicalmodelof V1dynamicshas Euclideansym-metry to predict the planforms of the new stable patterns which should form when the homogeneous resting state of V1 is destabilized by the action, for example, of hallucinogens. 10. 3 Conditions for the loss of stability of the homogeneous state Before calculating such planforms we first analyze the conditions under which the stable homogeneous solution of equation 10. 7 loses its stability. We do this by lin-earizingtheequationaboutthehomogeneousstationarysolution u(x,𝜙,t)=0[The complete details can be found in Bressloff et al. (2001). ] Essentially we assume thesolutiontobeintheform u(x,𝜙,t)=e𝜆tu(x,𝜙),andsubstitutethissolutioninto equation 10. 7. The result is the eigenvalue equation 𝜆u(x,𝜙)=-𝛼u(x,𝜙)+f′𝜇[ ∫𝜋 0𝑤lat(𝜙-𝜙′)u(x,𝜙′)d𝜙′ 𝜋