of T. The stability conditions on J0and J1are thus uniquely determined by βalone. Further-more, for the general diagram (i. e., with βfixed and c,Tunfixed), each point of the analytical regime permits linear solutions, in addition to the ones that implement thresholding. Accord-ingly, the uniqueness of the bifurcation diagram implies that at each point of the analytical ( J0, J1) subspace, the stability of the latter, nonlinear solutions is equivalent to that of the linear solutions. This means that the boundary at J0=1 2πβset by the linear case ( Methods :Linear Solution ) applies to the full, nonlinear model as well. A Turing Mechanism for Color Hallucinations Biological Turing patterns. Underpinning our hue tuning model is the mathematics of reaction-diffusion systems, for which, in particular, Alan Turing 's treatment of biological pattern forma-tion offers many valuable insights ( Turing, 1952 ). The general Turing mechanism assumes a system of two interacting chemicals, whose local reaction and long-range diffusion propertiesgovern the dynamics of their relative concentrations. In the original framework these chemi-cals are termed “morphogens ”to elicit their form-producing capabilities within a developing embryo, whose anatomical structure emerges as a result of their underlying concentrationdynamics. This, for instance, may be attributed to the morphogens 'catalysis of organ forma-tion in different parts of the developing organism. Most analogous to our model is the formulation which distributes the morphogens across a continuous ring of tissue, parameterized by the cellular position θ. Assuming that the system Reaction-diffusion system: Mathematical model of two or moreinteracting substances consisting oflocal dynamics (via the reactionterms) and spatial dynamics (via thediffusion terms). Figure 13. Bifurcation diagram for β= 1 and c= 1 for two values of threshold (shown in the leg-end). The gray and white regions correspond to the analytical and extended regimes, respectively. The black dashed line is the bifurcation curve, above which the tuning curves grow without bound. The overlaid symbols correspond to points tested in a parameter sweep over the extended regime. Notably, the parameter sweep produces the same bifurcation curve for both values of T. Here, we must note that for critical values of T, for which the input is not large enough to generate activity, the model permits the trivial a(θ) = 0 steady-state solution in both the analytical and extended regimes. This solution, however, is unstable to perturbations large enough to make the input cross the thresh-old. For more on bifurcation theory in the context of neural fields, see Gross (2021) and Hesse and Gross (2014). See also Hansel and Sompolinksy (1998) for an analogous “phase diagram ”analysis of the orientation ring model. Network Neuroscience 699Pattern 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 