Biological Cybernetics (2021) 115:643-653 647 Fig. 3 Complex log retina to cortex mapping as deﬁned by Eq. 5. The bounding blue contour represents the verticalmeridian, while the horizontalmeridian (0°) and the twodiagonals are as indicated. Axisunits are in mm along thecortical surface. A reﬂected maprepresents the other half of thevisual ﬁeld in the opposite V1cortex. The two almost parallelhorizontal bands are theprojections of two halfconcentric circles from theretina onto the cortex 0 1 02 03 04 05 06 0 Horizontal V1 Axis (Radius)-25-20-15-10-50510152025Vertical V1 Axis (Polar Angle) ticularly near 10. 0 Hz, can induce spatiotemporal illusions, including auras in migraine sufferers (Crotogino et al. 2001 ). Ermentrout and colleagues have shown that the Wilson— Cowan equations with appropriate parameters can accuratelypredict this behaviour (Rule et al. 2011 ). A mathemati-cal analysis of the equations plus simulations demonstrated the existence of nonlinear spatiotemporal oscillations. Inresponse to a ﬂickering stimulus that was uniform across space, the initial network oscillation was unstable, but after a transient period, broke into a spatially alternating patternof synchronous oscillations. An example is illustrated in Fig. 4, where a network with appropriate parameters for an active transient mode (Wilson 1999 ) generate such a pattern in response to uniform sinusoidal stimulation. This oscil-lation consists of three spatial loci becoming active, then decaying, and the interdigitated three competing active foci beginning to ﬁre. Boundary conditions were periodic, andthe number of active populations per half cycle is deter-mined by the spatial extent of the network. Ermentrout and colleagues went on to conduct experiments using a uniformannulus within which uniform ﬁeld, counterphase ﬂicker wasemployed (Pearson et al. 2016 ). Subjects perceived a ring of equally spaced illusory grey blobs that alternated between clockwise and counterclockwise rotation. As the annuluseffectively reduced the stimulus to one-dimension (Wilson et al. 2001 ), the 1D Wilson-Cowan equations provided a neural model that effectively explained the illusion. Furtherresearch has also invoked Wilson-Cowan in explaining addi-tional spatiotemporal oscillations (Bertalmio et al. 2020 ). 4 Long-term memory The activity depicted in Fig. 2reﬂects short-range recur-rent excitation localized by longer range inhibition. As such,the activity pattern is dependent on the neural connectivity, which is the same throughout this network. This suggested, however, that network connectivity could be learned fromtraining examples and could therefore be used to encode long term memories. This was explored, ﬁrst in networks with step function neural responses and subsequently withsigmoid nonlinearities by Hopﬁeld ( 1982,1984 ). It has been pointed out that Hopﬁeld ( 1984 ) utilized a special case of the Wilson-Cowan equations in which the learned connec-tion matrix was symmetric (i. e. β ij/equal1βji), but no recurrent connections of a population to itself were permitted, so βii /equal10 (Destexhe and Sejnowski 2009 ). In addition, Hopﬁeld networks permitted neurons to have both excitatory andinhibitory connections, so no explicitly inhibitory popula-tion was incorporated. However, a more realistic model with explicit populations of excitatory and inhibitory neurons caneasily be developed (Wilson 1999 ). In Hopﬁeld networks, learning a pattern comprising N distinct neurons uses a Hebbian rule (Hebb 1949 ) to calcu-late the average cross-correlation between the responses of 123