170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 231 10. 2 A NEW MATHEMATICAL FORMULATION OF V1 CIRCUITRY 231 The second point to note is that the normal modes of vibration or eigenfunc-tionsof the plane are cos (x⋅k)as previously noted. What are the eigenfunctions orharmonics of the circle? The answer is that in our case they are the 𝜋-periodic functions cos2 𝜙. It follows that we expect the normal modes of vibration of our extended model of V1 to be of the form cos2𝜙⋅cos(x⋅k) that is, plane waves whose amplitude is modulated by a sinusoidal function. The third point is a little more complicated. The question we need to ask is how is the postulated Euclidean symmetry of V1 affected by the introduction of ori-entation preference? The answer to this question is as follows: we first write the Ermentrout-Cowanequationmodifiedandextendedtodealwithorientationprefer-ence in the form: 𝜕 𝜕tu(x,𝜙,t)=-𝛼u(x,𝜙,t)+𝜇∫𝜋 0∫R2𝑤(x,𝜙|x′,𝜙′)f[u(x′,𝜙′,t)]dx′d𝜙′ 𝜋(10. 7) where𝑤(x,𝜙|x′,𝜙′)is the weight of connections between neurons at xtuned to the orientation 𝜙, and neurons at x′tuned to𝜙′. We take the weighting function to decompose as: 𝑤(x,𝜙|x′,𝜙′)=𝑤loc(𝜙-𝜙′)𝛿(x-x′)+𝛽𝑤lat(x-x′,𝜙′)𝛿(𝜙-𝜙′)(10. 8) where𝑤loc(-𝜙)=𝑤loc(𝜙),and𝛽isthestrengthofthelateralcoupling. Theeffects ofthe Diracdeltafunctions 𝛿(x-x′)and𝛿(𝜙-𝜙′)actinginsidetheintegralsareto localizetheactionof 𝑤loctoxand𝑤latto𝜙. Theweightingfunctionthencontainsa localpartthatis isotropic in𝜙(andinhibitory),andalateralpartthatis anisotropic (andexcitatory). Inaddition,becauselateralconnectionslieonlyalongthedirection of the orientation preference, wecan see that 𝑤lat(x,𝜙)=𝑤(R-𝜙x) (10. 9) where R𝜃is the matrix operation that rotates the vector xby the angle 𝜃. The effect of this rotation isthat we can rewrite 𝑤lat(x-x′,𝜙)as𝑤lat(R-𝜙(x-x′)). The net effect of all this is that we can preserve Euclidean symmetry by redefin-ing the rotation operation to express the above property of the weighting func-tion, that is by defining the rotation operator to be not only R𝜃x, but also a label change of the orientation variable from 𝜙to𝜙+𝜃. This combination of a direc-tionalchangecombinedwithanorientationpreferencechangepreserves Euclidean symmetryinthelatticeversionoftheextendedspace ℝ2×𝕊,andwasreferredtoas a Shift-Twist symmetry by Zweck and Williams (2000). Thus, although I first rec-ognized the underlying shift-twist symmetry of the lateral connections a few years earlier(Cowan,1997),Ididnotthenseethatitwasamechanismforpreservingthe Euclidean symmetry ofthe neocortical plane.