With this reformulation, the system of equations for the evolution of the coupled cν (Equation 19) takes the more explicit form: τ0dcνtð Þ dt¼-cνtð Þ þ βZδ2 δ1q0tð Þ þ chtð Þcosϕþγtð Þ 1/2/C138 1/2/C138 ^e/C3 νϕð Þdϕ: (24) Note that, for all cν, the integrand of Equation 24 is a function of q0(t),ch(t), and γ(t) and therefore, implicitly, only of the coefficients c0(t),c-1(t), and c1(t). Thus, the dynamics are determined in full by the evolution of c|ν|≤1(t): τ0dc0tð Þ dt¼-c0tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp Zδ2 δ1hϕ;tð Þ1/2/C138 dϕ τ0dc1tð Þ dt¼-c1tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp Zδ2 δ1hϕ;tð Þ1/2/C138 e-iθϕð Þdϕ τ0dc-1tð Þ dt¼-c-1tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp Zδ2 δ1hϕ;tð Þ1/2/C138 eiθϕð Þdϕ;(25) with h(ϕ,t) as in Equation 22. Separating Equation 25 into its real and imaginary parts, and noting that a real-valued activity profile a(θ,t) requires c02ℝand c1=-c/C3-1, reduces the system to a set of equations forc0(t),c R-1(t), and c I-1(t). Evaluating the integrals, we obtain: τ0dc0tð Þ dt¼-c0tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp chsinδ2þγ ð Þ-sinδ1þγ ð Þ 1/2/C138 þ T-q0 ð Þ δ1-δ2 ð Þ fg τ0dc R-1tð Þ dt¼-c R-1tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp/C26ch 2cosγδ 2-δ1 ð Þ þ cosγþδ1þδ2 ð Þ sinδ2-δ1 ð Þ 1/2/C138 þ T-q0 ð Þ sinδ1-sinδ2 ð Þ/C27 τ0dc I-1tð Þ dt¼-c I-1tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp/C26ch 2sinγδ 1-δ2 ð Þ þ sinγþδ1þδ2 ð Þ sinδ2-δ1 ð Þ 1/2/C138 þ T-q0 ð Þ cosδ2-cosδ1 ð Þ/C27 ; ð26Þ where the time arguments of q0,ch,γ,δ1, and δ2are suppressed for clarity. Written in this form, the system provides a representation of the time evolution of a(θ,t)i n terms of the coupled evolution of the constants c|ν|≤1. It is important to note that these equa-tions are nonlinear due to the implicit Heaviside in our determination of δ1(t) and δ2(t). While our reformulation of the right-hand side of Equation 24 allows for the explicit representation ofthe coupling of c νvia the nonlinearity, it is also this coupling that proves the analytical solution of the trajectories intractable. Thus, to describe the behavior of the time-dependent solution, we turn next to a numerical analysis of the system 's phase portrait —that is, to an exploration of the features and stability of the system 's emergent steady states. Steady-State Solution We approach the solution to Equation 1 with a Forward Euler method, propagating the activity from a random array of spontaneous initial values between 0 and 0. 2 spikes/sec to its steady-Network Neuroscience 689Pattern 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 