170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 230 230 CH10 GEOMETRIC VISUAL HALLUCINATIONS Lateral connections Hypercolu mn Local connections Figure 10. 12 Illustration of the local connections within a hypercolumn and the anisotropic lateral connections between hypercolumns. 10. 2. 1 Some more mathematical details The first point to note is that the lateral connectivity now depends on the orien-tation preference of a given patch. Let 𝜙be the variable representing orientation preference. Such a preference varies from 0∘to 18∘and is said to be 𝜋-periodic. [An example of such a 𝜋-periodic function is cos2 𝜙. ] We therefore need to repre-sentneuralactivityatanylocationin V1,notonlyintermsofthecoordinates xand ybut in terms of orientation preference 𝜙., that is, by the variables u(x,𝜙,t)and 𝑣(x,𝜙,t),respectively,forexcitationandinhibition. Thisraisesaninterestingmath-ematicalquestion:what spacedothesevariablesoccupy?The xcomponentlivesin the Euclideanplane,butwhataboutthevariable 𝜙?Theansweristhatitlivesonthe circle. Mathematically the Euclidean plane comprises all real number pairs xand y. Each by itself lives on the real line ℝ, but the(x,y)pair lives on the real plane ℝ×ℝ=ℝ2. The variable 𝜙lives on the circle 𝕊, so that the various points (x,𝜙) live onthe space ℝ2×𝕊.