Evolution of Peak Angle We first assume that upon receiving a stimulus θ/C22at time t= 0, the network has a random spon-taneous firing rate a(θ, 0). Using Equation 15, with c02ℝandcμ=c/C3-μ, we expand the activity profile in terms of the initial values of the corresponding coefficients cμ(0): aθ;0ð Þ ¼X μcμ0ð Þ^eμθð Þ ¼1ﬃﬃﬃﬃﬃﬃ 2πp c00ð Þ þ X μ≥12c R-μ0ð Þcosμθð Þ þ 2c I-μ0ð Þsinμθð Þ/C16/C17() ¼1ﬃﬃﬃﬃﬃﬃ 2πp c00ð Þ þ X μ≥12rμ0ð Þcosμθ-ϕμ0ð Þ/C0/C1/C2/C3()(42) with tan( ϕμ)=c I-μ c R-μand r2 μ=(c I-μ)2+(c R-μ)2such that ϕμ(0) are completely determined by the initial conditions. Thus, at t= 0 the activity profile is composed of an infinite sum of cosine functions, each peaked about a corresponding disparate angle ϕμ, and therefore has no dis-cernible peak. To characterize the evolution of the network activity from these initial condi-tions to its hue tuning profile at t∞, we seek to obtain the steady-state values of ϕμand the corresponding tuning curve peak inductively as follows. Let us first take μ= 1. As seen in Figure 4, we note that δ1(t) and δ2(t) are symmetric about γ(t) such that δ2+γ=2π-(δ1+γ). Using this symmetry, we factor out cos( γ) and sin( γ), respectively, in the equations for c R-1and c I-1in Equation 26: τ0dc R-1 dt¼-c R-1þβﬃﬃﬃﬃﬃﬃ 2πp F1cosγ τ0dc I-1 dt¼-c I-1-βﬃﬃﬃﬃﬃﬃ 2πp F1sinγ(43) with F1¼ch 2δ2-δ1 ð Þ þ sinδ2-δ1 ð Þ 1/2/C138 þ 2T-q0 ð Þ sinγþδ1 ð Þ (44) and time arguments suppressed. We let F⋆and γ⋆denote the steady-state values of Fand γ, respectively, allowing for the following expressions for the steady-state values of c R-1and c I-1: c R⋆-1¼βﬃﬃﬃﬃﬃﬃ 2πp F⋆ 1cosγ⋆ c I⋆-1¼-βﬃﬃﬃﬃﬃﬃ 2πp F⋆ 1sinγ⋆:(45) Thus, we have tanϕ⋆ 1/C0/C1 ¼c I⋆-1 c R⋆-1¼-tanγ⋆ð Þ : (46) Similar calculations for the steady-state values of the higher order coefficients yield the general equations τ0dc R-μtð Þ dt¼-c R-μtð Þ þβﬃﬃﬃﬃﬃﬃ 2πp Fμtð Þcosγtð Þð Þ τ0dc I-μtð Þ dt¼-c I-μtð Þ-βﬃﬃﬃﬃﬃﬃ 2πp Fμtð Þsinγtð Þð Þ :(47) Network Neuroscience 706Pattern 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 