where δh(θ,t) is a perturbation to the input due to δa(θ,t). Taylor expanding the right-hand side of Equation 54 in h(θ,t)≡h∞(θ)+δh(θ,t) about h(θ,t)=h∞(θ) then yields τ0dδaθ;tð Þ dt¼-a∞θð Þþδaθ;tð Þ ð Þ þβh∞θð Þ-T ð Þ H h∞θð Þ-T ð Þ þ δhθ;tð Þ H h∞θð Þ-T ð Þ þ Oδh2/C0/C1 /C8/C9 : (55) For small perturbations, the higher order terms in δh(θ,t) are negligible, and, using a∞(θ)= β(h∞(θ)-T)H(h∞(θ)-T), we rewrite Equation 55 as τ0dδaθ;tð Þ dt¼-δaθ;tð Þ þ βδhθ;tð Þ H h∞θð Þ-T ð Þ : (56) Next, expanding δa(θ,t) as in Equation 29, we obtain τ0X∞ μ¼-∞d Dμtð Þ dt^eμθð Þ ¼-X∞ μ¼-∞Dμtð Þ^eμθð Þ þ βδhθ;tð Þ H h∞θð Þ-T ð Þ ; (57) wherein we express δh(θ,t) in terms of Equation 21 to yield: τ0X∞ μ¼-∞d Dμtð Þ dt^eμθð Þ ¼-X∞ μ¼-∞Dμtð Þ^eμθð Þ þβδq0tð Þ þ δq Rtð Þcosθð Þ þ δq Itð Þsinθð Þ 1/2/C138 H h∞θð Þ-T ð Þ : (58) Finally, taking the inner product of Equation 58 with êν(θ), and reformulating the thresh-olding nonlinearity in terms of the critical cutoff angles δ1and δ2as in section Evolution of Network Activity, we arrive at 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; (59) where δ⋆ 1andδ⋆2are the steady-state values of the cutoff angles. ACKNOWLEDGMENTS The authors acknowledge the fruitful and stimulating discussions with Wim van Drongelen and Graham Smith, and research support from the Oberlin College libraries. AUTHOR CONTRIBUTIONS Zily Burstein: Conceptualization; Data curation; Formal analysis; Investigation; Methodology;Project administration; Resources; Software; Supervision; Validation; Visualization; Writing-original draft; Writing-review & editing. David D. Reid: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Project administration; Software; Supervision;Validation; Visualization; Writing-review & editing. Peter J. Thomas: Conceptualization; Methodology; Resources; Writing-review & editing. Jack D. Cow an: Conceptualization; Methodology; Project administration; Resources; Writing-review & editing. FUNDING INFORMATION Peter J. Thomas, Directorate for Mathematical and Physical Sciences ( https://dx. doi. org/10. 13039/100000086 ), Award ID: DMS-2052109. Peter J. Thomas, Office of Research Infrastruc-ture Programs, National Institutes of Health ( https://dx. doi. org/10. 13039/100016958 ), Award ID: R01 NS118606. Network Neuroscience 708Pattern 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 