As before, we note that the evolution of cμ(t), and therefore of Fμ(t),∀μ2ℤdepends only on the first-order coefficients c|μ|≤1(t). Therefore, the steady-state values of the higher order coefficients c R⋆-μ¼βﬃﬃﬃﬃﬃﬃ 2πp F⋆ μcosμγ⋆ð Þ c I⋆-μ¼-βﬃﬃﬃﬃﬃﬃ 2πp F⋆ μsinμγ⋆ð Þ(48) and the corresponding ϕμ, that is, tanϕ⋆ μ/C16/C17 ¼c I⋆-μ c R⋆-μ¼-tanμγ⋆ð Þ ; (49) are fully determined by the solution to Equation 26. Substitution of Equation 48 into Equation 15 then gives: a∞θð Þ ¼1ﬃﬃﬃﬃﬃﬃ 2πp c⋆ 0þβ πX μ≥1F⋆ μcosμγ⋆ð Þ cosμθð Þ-F⋆ μsinμγ⋆ð Þ sinμθð Þ/C16/C17 ¼1ﬃﬃﬃﬃﬃﬃ 2πp c⋆ 0þβ πX μ≥1F⋆ μcosμθþγ⋆ð Þð Þ ;(50) so that θ=-γ⋆represents the peak angle of the steady-state profile a∞(θ). Further, from Equation 22, we have tanγ⋆ð Þ ¼-q⋆ I q⋆ R ¼-csinθ/C22-β πλ-1F⋆sinγ⋆ ccosθ/C22þβ πλ-1F⋆cosγ⋆(51) which requires γ⋆¼-θ/C22: (52) That is, the steady-state peak-γ⋆is equivalent to the LGN hue input θ/C22. Linear Stability Analysis This section presents the mathematical details for obtaining Equation 30. Adding a small perturbation δa(θ,t) to the steady-state tuning curve and substituting the resulting network activity aθ;tð Þ ¼ a∞θð Þ þ δaθ;tð Þ (53) into Equation 1, we obtain: τ0dδaθ;tð Þ dt¼-a∞θð Þ þ δaθ;tð Þ ð Þ þ βh∞θð Þ þ δhθ;tð Þ-T ð Þ H h∞θð Þ þ δhθ;tð Þ-T ð Þ ;(54) Network Neuroscience 707Pattern 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 