Biological Cybernetics (2021) 115:643-653 645 The network inputs to each population are deﬁned by spa-tial convolutions, denoted by ⊗in Eq. ( 2). In the original formulation the kernels were all functions of distance, and data available then suggested that they should be decaying exponentials of distance (Sholl 1956 ). Thus, the ﬁrst convo-lution in Eq. ( 2) took the form: βEE(x)⊗E(x)/equal1∞∫-∞e-⏐⏐x-x′⏐⏐/ ωEEE( x′) dx′(4) which describes recurrent Eto Econnections. The remaining three convolutions similarly describe all possible connectionsamong the Eand Ipopulations. For different neural popula-tions, different space constants can produce different ranges of interaction, thus permitting recurrent, long-range inhibi-tion. More recent data have suggested the use of Gaussian kernels rather than decaying exponentials of distance, but this does not change the dynamics qualitatively. Finally, it rapidly became apparent that the refractory period rmainly reduced the maximum ﬁring rate but did not affect the dynamics substantially. Thus, most subsequentstudies have set r/equal10, thereby eliminating the term before the sigmoid input function. We should also emphasize here that values of parameters have largely been omitted below to emphasize major conceptual developments in the evolu-tion of these equations. Details are available in the original references. In this form the Wilson-Cowan equations exhibit a substantial range of dynamical modes, depending on the parameters chosen, that suggested explanations of a variety of cortical phenomena. The model could produce spatiallystable inhomogeneous steady states that stored informationdynamically and suggested a basis for short term memory. This is illustrated in Fig. 2, where the asymptotic E(red) and I(blue) activity is shown following brief activation at the two loci marked with arrows. Recurrent E-Econnec-tions cooperatively maintain neural activity, while longer range I-Econnections maintain the localization. Another parameter regime produced spatially localized limit cycle oscillations with likely relevance to motor control. In addi-tion, yet another set of parameters resulted in the generationof travelling waves, which were suggestive of epilepsy. Details of both the local computations, in which the con-volution in Eq. 4is replaced with simple multiplication by a weight constant, plus the one-dimensional spatial equationsincorporating convolution, can be found in the original arti-cles (Wilson and Cowan 1972,1973 ). MATLAB scripts for many simulations along with parameter values are availableelsewhere (Wilson 1999 ). Although all of the applications of Wilson-Cowan described below have resulted from simulations, it is impor-tant to note that closed-form analytic solutions have veryrecently been obtained using speciﬁc forms of the sigmoidal nonlinearity and particular parameter values (Cowan et al. 2021 ). For these particular cases, travelling soliton waves of the basic form of Eq. ( 1) have been obtained. This is true even with the extension of Beurle's formulation to includeinhibition. These, of course, are solutions under particu-lar functional conditions and do not encompass the much broader range of Wilson-Cowan dynamics as applied to par-ticular areas of cortex. In the years since the original publication of the Wilson-Cowan equations in Kybernetik, computer power has increased phenomenally. Relative to the Digital Equipment Corporation PDP8 on which the original simulations were done, a desktop i Mac Pro runs more than 10 7times faster (Wilson 2019 )! This has engendered two major developments in neural modelling. First, much larger two-dimensionaland multi-layer network simulations have become possi-ble. Furthermore, these have incorporated more than just the two original Eand Ipopulations, thereby reﬂecting greater accuracy in describing cortical networks. In parallel, vastly more detailed simulations of individual neurones have been developed, some incorporating more than 1000 differen-tial equations (Mainen et al. 1995 ). This reﬂects a trade-off between network complexity and single unit complexity con-strained by available computer power. As the Wilson-Cowanequations emphasize the network approach, we shall focuson salient applications of this approach below. Details of parameter values are seldom given, as they are available in the original references. 3 Visual hallucinations Drug-induced visual hallucinations frequently display one of a small number of geometric spatial patterns: concentric circles, radial spokes or arms spiralling outward from the cen-tre (Siegel 1977 ). The natural question this raised was: what neuronal activity in the visual cortex might generate this per-cept in the absence of appropriate stimulation? The criticalinsight of Ermentrout and Cowan ( 1979 ) was that halluci-nations could be explained by the spatially inhomogeneous steady states of the Wilson-Cowan equations if two addi-tional factors were incorporated. First, the equations must beextended to two dimensions to represent the surface of visual cortex. The second key insight for explaining hallucinations was that the gradient of ganglion cells in the retina, rang-ing from densest in the fovea to very sparse in the periphery, meant that there must be a nonlinear mapping from the retina to the cortex. This mapping was shown to be well approx-imated by a complex logarithm in polar coordinates for the left and right half visual ﬁelds (Schwartz 1977, Schwartz 1980 ). If the retinal location of a stimulus point is described in polar coordinates as radius rand orientation ϕ, then the 123