V1. Collectively, our formulation of h(θ,t) implements the mixing rules posited by these experiments, without requiring us to arbitrarily fine-tune the relative weights of the afferentsignals. RESULTS Evolution of Network Activity We start by observing that by virtue of the invariance of w(θ-θ0) under translations of θ, the convolution operator Tww*f(θ)=Rπ-πw(θ-θ0)f(θ0)dθ0is diagonalizable by the Fourier eigenfunction basis ^eμθð Þ ¼1ﬃﬃﬃﬃﬃﬃ 2πp eiμθ(11) with μ2andêμnormalized to integrate to 1 on [-π,π]. To calculate the eigenvalues λμof the corresponding linear transformations, Zπ-πwθ-θ0ð Þ1ﬃﬃﬃﬃﬃﬃ 2πp eiμθ0dθ0¼λμ1ﬃﬃﬃﬃﬃﬃ 2πp eiμθ; (12) Figure 3. Cortical connectivity functions. (A) A difference of two Gaussians, one characterizing the excitatory interactions (here with σE= 40°) and the other the inhibitory interactions ( σI= 90°). This is the connectivity typically assumed in mean field models of cortical processing. (B)The difference of cosines formulation (Equation 9), with J 0=-2 and J1= 3, captures the local exci-tation and lateral inhibition assumed in panel A. Network Neuroscience 686Pattern 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 