To determine δ1and δ2, we reformulate the Heaviside as a function of θ. Given that the input h(θ,t) is a real-valued function, c02ℝand c1=c/C3-1. For mathematical convenience, we then rewrite Equation 17 in terms of c0, Re( c-1)≡c R-1, and Im( c-1)≡c I-1as hθ;tð Þ ¼λ0c0tð Þﬃﬃﬃﬃﬃﬃ 2πp þclþﬃﬃﬃ 2 πr λ-1c R-1tð Þ ! cosθð Þ þ csþﬃﬃﬃ 2 πr λ-1c I-1tð Þ ! sinθð Þ: (20) Setting q R¼clþﬃﬃﬃ 2 πr λ-1c R-1tð Þ q I¼csþﬃﬃﬃ 2 πr λ-1c I-1tð Þ q0¼λ0c0tð Þﬃﬃﬃﬃﬃﬃ 2πp(21) the input takes the form hθ;tð Þ ¼ q0tð Þ þ chtð Þcosθþγtð Þ 1/2/C138 (22) where tan( γ)=-q I q Rand ch(t)=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q2 Rþq2 Ip. The Heaviside can then be expressed as Hh-T1/2/C138 ¼ H q0þchcosθþγ ð Þ-T 1/2/C138 ¼H cosθþγ ð Þ-α 1/2/C138(23) where α≡T-q0 ch, and the time arguments are suppressed for simplicity. In this formulation, the Heaviside sets the limits of integration in Equation 19 as the angles θ=δ1,δ2where αintersects with cos( θ+γ), as shown in Figure 4. Figure 4. The limits of integration δ1and δ2in Equation 19 are the angles corresponding to the intersection of α(in gray) and cos( θ+γ) (in black). Here, c=1, β= 1, and T=-1. θ/C22=π/8. J0and J1 are as in Figure 3. Network Neuroscience 688Pattern 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 