In contrast, our model does not apportion separate regions of the parameter space to exter-nal and recurrent mechanisms. Rather, in both the analytical and extended regimes, the rolesofcand J 1exist on a spectrum, where the effect of each parameter is suppressed by larger values of the other. Of course, this suppression is more stark in the extended regime because it covers larger values of J1. In this sense, our color model draws a similar conclusion to that of the orientation model: when the anisotropic tuning provided by the recurrent interactions islarge, the tuning from the stimulus is negligible, and vice versa. However, we emphasize that the transition is not sharp and that cdoes have an effect on the tuning curve selectivity in the extended regime (see Figure 6B ), as does J 1in the analytical regime. In this regard, the two models are more consistent in their interpretations of J0's contribution to the selectivity of the tuning curves. That is, in the two regimes of each model, the inhibitionacts cooperatively with the thresholding to sharpen the tuning curves. Here again, the orien-tation model makes a distinction between the marginal phase (i. e., /C15= 0 and J 22marginal regime), wherein the tuning curve width is completely determined by the cortical anisotropy, and all other cases, where the isotropic inhibition and stimulus come into play. For thesecases, the authors argue, J 0does not act alone to narrow the curve: though J0may sharpen the tuning curves, it is the anisotropy from the input or cortical interactions which acts as the source of the orientation selectivity. Although our color model 's tuning mechanism, too, requires a source of anisotropy, we have emphasized above that there is no single source of hue selectivity. When J1is small, in either regime, both the stimulus and the uniform inhibition are significant to the hue tuning mechanism. Ultimately, the orientation model sets up a dichotomy between two specific regions of parameter space. In the nonmarginal case, cis the primary player in the tuning mechanism, and in the marginal case, this role belongs to J2. The uniform inhibition is thus given a sec-ondary “sharpening ”role. By contrast, in choosing a fully anisotropic hlgn, our color model does not encompass an analogous marginal phase with an always dominating J1. Rather, even for large J1, the uniform inhibition is at least equally important to the modulation of the tuning width. In fact, as we have shown above, for larger values of c,J0is more effective than J1in narrowing the tuning curves, for both the analytical and extended regimes. We thus stress that the two regimes, though analogous to those of the orientation model, do not constitute a division in the hue processing mechanism. Rather, we define the boundary between the analytical and extended regimes solely by whether or not the linear case exists. It is therefore determined by the values of J0and J1for which the linear solution applies, given that the values of c,T,a n d βkeep the input above threshold throughout the dynamics (Methods :Linear Solution ). We note that for each combination of J0and J1within the analytical regime there exists also a nonlinear case, in which h(θ,t) is cut off by the thresholding non-linearity and, thereby, the linear solution does not apply. Our definition differs from that of theorientation model, which demarcates the boundary between the homogeneous and marginal phases based on the emergent steady-state tuning curves alone. For more on this approach, see the discussion of the broad and narrow profiles in Hansel and Sompolinksy (1998). A sw e will show next, the boundary is integral to the corresponding stability analysis of the steady-state tuning curves. Stability Analysis To analyze the stability of the emergent tuning curves, we turn once more to our separable activity ansatz assumed in the eigenfunction decomposition of Equation 15. This means thatwe are faced again with a nonlinearity induced coupling of the time-dependent coefficients Network Neuroscience 696Pattern 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 