we make the change of variables θ-θ0=ϕ, so that the left-hand side of Equation 12 can be rewritten as-Zθ-π θþπwϕð Þe-iμϕ 1ﬃﬃﬃﬃﬃﬃ 2πp eiμθdϕ¼Zπ-πwϕð Þe-iμϕ 1ﬃﬃﬃﬃﬃﬃ 2πp eiμθdϕ: (13) The eigenvalues are thus: λμ¼Zπ-πwϕð Þe-iμϕdϕ: (14) Next, we assume a(θ,t) is separable in tand θand bounded on [-π,π] so that we may expand it in the eigenbasis of the convolution operator as: aθ;tð Þ ¼X μcμtð Þ^eμθð Þ: (15) Substituting the expansion into Equation 10, we have for Equation 2 hθ;tð Þ ¼X μcμtð ÞZπ-πwθ-θ0ð Þ ^eμθ0ð Þdθ0/C20/C21 þhlgnθð Þ; (16) where w(θ-θ0) is our choice for the connectivity function (Equation 9) and hlgn(θ) is defined as in Equation 6. Evaluating the integrals, we obtain hθ;tð Þ ¼ λ-1c-1tð Þ^e-1θð Þ þ λ0c0tð Þ^e0θð Þ þ λ1c1tð Þ^e1θð Þ þ ccosθ-θ/C22ð Þ ; (17) with λ0=2πJ0and λ1=λ-1=πJ1. Note here that only the zeroth and first-order complex Fourier components remain. Substituting the expansion Equation 15 and the explicit form of the activation function Equation 3 into Equation 1 yields: τ0X∞ μ¼-∞dcμtð Þ dt^eμθð Þ ¼-X∞ μ¼-∞cμtð Þ^eμθð Þ þ βhθ;tð Þ-T ð Þ H hθ;tð Þ-T ð Þ : (18) In the absence of the nonlinearity, each of the eigenmodes êμ(θ) would evolve indepen-dently of the others, and a complete analysis of the time-dependent system would seek to solve a set of equations for cμ(t) (see Methods :Linear Solution ). However, in our setup, the thresholding introduces a coupling of these coefficients, as the critical hue angles, δ1and δ2, at which the input is cut off and is determined by the combined cμ(t) at each point in time. While an analytical solution to this system is in most cases intractable, it is nonetheless infor-mative to break down the rate equation to a coupled system of equations for the evolution ofthe coefficients c μ(t). Taking the inner product of Equation 18 with êνand using < êν|êμ>=δμν, we obtain: τ0dcνtð Þ dt¼-cνtð Þ þ ^eνβh-T ð Þ H h-T ð Þ ¼-cνtð Þ þ βZδ2tð Þ δ1tð Þhϕ;tð Þ ^e/C3 νϕð Þdϕ(19) where the Heaviside restricts the domain of the inner product to [ δ1(t),δ2(t)]. The time depen-dence of the cutoff angles reflects the evolution of this curve, which requires that the thresh-olding be carried out continuously throughout the duration of the dynamics. Network Neuroscience 687Pattern 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 