never deviates far from the underlying homogeneous steady state, the two dynamical state equations for their concentrations, Xand Y, take the linear form d Xθ;tð Þ dt¼a Xθ;tð Þ þ b Yθ;tð Þ þ DX∇2Xθ;tð Þ d Yθ;tð Þ dt¼c Xθ;tð Þ þ d Yθ;tð Þ þ DY∇2Yθ;tð Þ ;(34) where a,b,c, and drepresent the chemical reaction rates, and DXand DYare the diffusion rates of Xand Y, respectively. Here, we set a,c> 0, so that increasing the concentration of X activates the production of both Xand Y, and b,d< 0 so that Yhas an inhibitory effect on the production of both chemicals ( Hoyle, 2006 ). In the absence of diffusion (i. e., with DX=DY= 0), the system has a homogeneous steady-state solution, ( X,Y) = 0, whose stability is determined by a Jacobian composed of the reaction rates,ab cd/C20/C21, and hence by the system 's local chemical properties alone ( Hoyle, 2006 ). Note that at this point the system is circularly symmetric with respect to interchanging any two cells on the ring. Assuming the existence of a stable steady-state solution, and the corresponding require-ments on the rate parameters a-d, we next set the diffusive terms DX,DY> 0, taking the sep-arable ansatz for the general solution: X¼X∞ μ¼-∞Aμeλμteiμθ Y¼X∞ μ¼-∞Bμeλμteiμθ:(35) Furthermore, we set DX<DYto generate the local excitation and lateral inhibition of the mor-phogen concentrations ( Hoyle, 2006 ;Murray, 2003 ), evoking the connectivity function ansatz Equation 9. The underlying steady state then remains stable if the real parts of the eigenvalues λμ, obtained from the modified Jacobian, are negative. With the reaction rates fixed from the stability conditions above, these eigenvalues are functions of the diffusion parameters alone. Thus, the conditions for stability may be thought of in terms of a bifurcation diagram in the ( D X, DY) phase space, comparable to Figure 13. From here, a set of additional conditions may be placed on DXand DYso that the system undergoes a Turing bifurcation, wherein at least one λμbecomes positive and the homoge-neous steady state loses its stability. With the addition of a small random perturbation, the instability results in the growth of the corresponding eigenmodes eiμθ, such that, over time, Equation 35 is dominated by the eigenmodes with largest λμ. These represent stationary waves whose wavelengths are set by the circumference of the ring (i. e., by the spatial properties of the medium) and whose growth is bounded by the higher order terms that had been initiallyignored in the near-equilibrium formulation ( Murray, 2003 ;Scholz, 2009 ;Turing, 1952 ). The underlying circular symmetry is thus broken and a spatial pattern is formed. In his seminal paper, Turing extrapolated this mechanism to explain various biological phe-nomena, such as the development of petals on a flower, spotted color patterns, and the growth Network Neuroscience 700Pattern 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 