and, consequently, the analytical intractability of the associated stability analysis. We therefore set up the Jacobian matrix for a numerical analysis of the local stability. We begin by adding a small perturbation of the form δaθ;tð Þ ¼X μDμtð Þ^eμθð Þ (29) and substituting the resulting activity into Equation 1. The eigenmodes then evolve according to the following equation for the coefficients Dμ(see Methods :Linear Stability Analysis ): τ0d Dνtð Þ dt¼-Dνtð Þ þ βZδ⋆ 2 δ⋆ 1δq0tð Þ þ δq Rtð Þcosϕð Þ þ δq Itð Þsinϕð Þ 1/2/C138 ^e/C3 νϕð Þdϕ; (30) where δ⋆ 1andδ⋆ 2are the critical cutoff angles of the steady-state solution, obtained numerically. We observe that the integrand of Equation 30 is a function of D0,DR-1, and DI-1alone, and, as such, the stability of the steady-state tuning curve is completely determined by the stability of these first-order coefficients. Evaluating the integrals for ν= 0 and ν=-1, and noting from Equation 21 that δq0¼ﬃﬃﬃﬃﬃﬃ 2πp J0δc0≡ﬃﬃﬃﬃﬃﬃ 2πp J0D0 δq R¼ﬃﬃﬃﬃﬃﬃ 2πp J1δc R-1≡ﬃﬃﬃﬃﬃﬃ 2πp J1DR-1 δq I¼ﬃﬃﬃﬃﬃﬃ 2πp J1δc I-1≡ﬃﬃﬃﬃﬃﬃ 2πp J1DI-1;(31) we obtain the following system of equations for the evolution of the characteristic coefficients: τ0d D 0 dt¼βJ0δ⋆ 2-δ⋆ 1/C0/C1-1/C2/C3 D0 þβJ1sinδ⋆ 2-sinδ⋆ 1/C0/C1/C2/C3 DR-1 þβJ1cosδ⋆ 1-cosδ⋆ 2/C0/C1/C2/C3 DI-1 τ0d DR-1 dt¼βJ0sinδ⋆ 2-sinδ⋆ 1/C0/C1/C2/C3 D0 þβJ1 42δ⋆ 2-2δ⋆ 1þsin 2 δ⋆ 2/C0/C1-sin 2 δ⋆ 1/C0/C1 /C0/C1-1/C20/C21 DR-1 þβJ1 4cos 2 δ⋆ 1/C0/C1-cos 2 δ⋆ 2/C0/C1 /C0/C1/C20/C21 DI-1 τ0d DI-1 dt¼βJ0cosδ⋆ 1-cosδ⋆ 2/C0/C1/C2/C3 D0 þβJ1 4cos 2 δ⋆ 1/C0/C1-cos 2 δ⋆ 2/C0/C1 /C0/C1/C20/C21 DR-1 þβJ1 42δ⋆ 2-2δ⋆ 1þsin 2 δ⋆ 1/C0/C1-sin 2 δ⋆ 2/C0/C1 /C0/C1-1/C20/C21 DI-1(32) Network Neuroscience 697Pattern 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 