the visual cortex is an early stop in the visual pathway, we point to the need for further field theory approaches to our understanding of color perception. METHODS Linear Solution We assume in the linear case that the net input h(θ,t) is above threshold throughout the dynamics such that the activity profile is never cut off and H(h(θ,t)-T)=1∀θ2{-π,π}. Equation 18 therefore takes the linear form: τ0X∞ μ¼-∞dcμtð Þ dt^eμθð Þ ¼-X∞ μ¼-∞cμtð Þ^eμθð Þ þ βhθ;tð Þ-T ð Þ (38) Taking the inner product of Equation 38 with êνon the full domain ≡{-π,π}, we obtain the system of equations for all the coefficients cν: τ0dc0tð Þ dt¼2πβJ0-1 ð Þ c0tð Þ-ﬃﬃﬃﬃﬃﬃ 2πp βT τ0dc1tð Þ dt¼πβJ1-1 ð Þ c1tð Þ þﬃﬃﬃπ 2r βcl-isð Þ τ0dc-1tð Þ dt¼πβJ1-1 ð Þ c-1tð Þ þﬃﬃﬃπ 2r βclþis ð Þ τ0dcνtð Þ dt¼-cνtð Þ∀νjj>1(39) We may thus solve for each of the coefficients independently, yielding equations for the evolution of each. Substitution into the activity expansion of Equation 15 then gives the closed-form solution for the evolution of the activity: aθ;tð Þ ¼ K0e-1-2πβJ0 ð Þ t=τ0-ﬃﬃﬃﬃﬃﬃ 2πp βT 1-2πβJ0/C26/C271ﬃﬃﬃﬃﬃﬃ 2πp þ K-1e-1-πβJ1 ð Þ t=τ0þﬃﬃﬃπ 2rcβlþis ð Þ 1-πβJ1/C26/C271ﬃﬃﬃﬃﬃﬃ 2πp e-iθ þ K1e-1-πβJ1 ð Þ t=τ0þﬃﬃﬃπ 2rcβl-isð Þ 1-πβJ1/C26/C271ﬃﬃﬃﬃﬃﬃ 2πp eiθþ Kν1ﬃﬃﬃﬃﬃﬃ 2πp e-t/C26/C27 eiνθjνjj>1 (40) where the constants Kνare determined by the Fourier coefficients cν(0) of the initial activity a(θ, 0). For J0<1 2πβand J1<1 πβ, the solution approaches the globally asymptotic stable steady-state tuning curve a∞θð Þ ¼-βT 1-2πβJ0þcβcosθ-θ/C22 ð Þ 1-πβJ1: (41) We call the corresponding ( J0,J1) parameter space the analytical regime. Globally asymptotic stable steady state: As y s t e m 's equilibrium solution which attracts all other solutions regardless ofinitial conditions. Network Neuroscience 705Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024 