170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 227 10. 1 INTRODUCTION 227 patterns of stripes and spots. Ermentrout and I were able to show 27 years later thataplanararraycomprisingexcitatoryandinhibitoryneutralscouldalsogenerate Turing patterns. The starting point of our analysis was a neural field theory introduced by Wilson and Cowan (1973), the equations of which can be written in the form: 𝜕 𝜕tu(x,t)=-bu+∫𝑤EE(x-x′)f[u(x′)]dx′-∫𝑤EI(x-x′)g[𝑣(x′)]dx′ 𝜕 𝜕t𝑣(x,t)=-𝑣+∫𝑤IE(x-x′)f[u(x′)]dx′-∫𝑤II(x-x′)g[𝑣(x′)]dx′(10. 5) where the variables uand𝑣denote, respectively, excitatory and inhibitory neural densities, and the various functions 𝑤EEand so on denote the density and strength of synaptic connections from neighbouring neurons a distance x-x′away. The effects of each pair of excitatory and inhibitory weighting functions produce an effective “Mexican Hat” coupling function which is the neural analogue of Tur-ing's diffusion-coupled autocatalytic and auto inhibitory molecular interactions, as showninfigure10. 9. [Notethataneffectiveinverted Mexican Hatwithshortrange inhibition and longer range excitation will also work. ] Wecaneliminateoneoftheseequationsandrewriteequation10. 5intheabbrevi-ated form 𝜕 𝜕tu(x,t)=-𝛼u+𝜇∫𝑤lat(|x-x′|)f[u(x′)]dx′(10. 6) where𝑤latis the effective Mexican Hat weighting function shown in figure 10. 9, and𝜇is the coupling strength parameter. [Note that because 𝑤latdepends on the magnitude of the distance between xandx′it is invariant to rotations, and is thus isotropic. ] Ermentrout and I were able to show that these equations are an exact neural ana-logue of Turing's equations, so they can also generate stripe and spot patterns in +--+w(s) s Figure 10. 9 Theeffective “Mexican Hat” weighting function.