The entries of the corresponding Jacobian matrix consist of the bracketed prefactors, and may equally be obtained from the general system of equations for the global network dynamics, asfollows: J¼∂f1 ∂c0∂f1 ∂c R-1∂f1 ∂c I-1 ∂f2 ∂c0∂f2 ∂c R-1∂f2 ∂c I-1 ∂f3 ∂c0∂f3 ∂c R-1∂f3 ∂c I-12 66666666643 7777777775 c⋆ 0;c R⋆-1;c I⋆-1(33) where f1,f2,a n d f3are the right-hand sides of the equations in Equation 26 and the first-order partial derivatives are evaluated at the steady-state values of c0,c R-1, and c I-1. The stability of the tuning curve is then determined by the eigenvalues of J. We note that the existence of a steady state is a function of the cortical strengths J0and J1. By fixing the values of β,c,θ/C22, and T, we are left with a two-parameter family of differential equations, allowing us to analyze this de pendence numerically in the associated ( J0,J1) parameter space. Carrying out a parameter sweep across thi s space, we find that the system features a bifurcation curve, below which the model permits steady-state solutions and above which no equilibrium exists. To determine stability within the former region, we compute Jat the emergent steady-state tuning curves of various points in the parameter space. Solving the associated characteristic equations, we observe that the eigenvalues are always real and negative, and thus conclude that all emergent steady-state tuning curves are stable. Figure 13 shows the bifurcation diagrams for two families of equations, distinguished by their values for T. Most notable is the extended regime, which permits stable steady-state solu-tions beyond the boundary set by the linear case ( Methods :Linear Solution ). As this parameter regime is not accessible to the linear solution, the tuning curves in this regime are necessarily a product of the thresholding nonlinearity and are thus always cut off below | θ|=π. The thresh-olding nonlinearity therefore not only expands the region of stability, but also ensures that thetuning curves emerging within the extended regime are selective for hue. As we have seen, thisexpansion is pivotal when the external input is weak and the anisotropic cortical strength plays the larger role in narrowing the tuning curves. Furthermore, regardless of input strength, it allows for a larger overall network response, as the peak activity, a ∞(θ/C22), grows with increasing J1. Finally, as we will see in the following section, in the absence of any stimulus (i. e., for c= 0), the extended regime features the spontaneous generation of stable tuning curves and may thus serve as the bedrock for color hallucinations. However, looking back at Figure 13, perhaps most striking is the horizontal portion of the bifurcation curve at J0=1 2πβfor J1<1 πβ, which sets the same stability conditions on J0and J1as in the linear case. This is despite the fact that many of the points in the analytical regimes of the two featured families correspond to solutions that implement thresholding, thus signifying thatthe analytical regime is not an exclusively linear one. The key to understanding the shape of this region lies in noticing that the bifurcation dia-gram does not change for varying values of c,T, and θ/C22, as shown in Figure 13 for the two values Bifurcation curve: A curve in parameter space defininga transition in the system 's dynamics and stability. Network Neuroscience 698Pattern 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 