Biological Cybernetics (2021) 115:643-653 651 Categories Fig. 6 Model for learning and recalling a series of simple patterns. Each pattern is represented by one active population from each Category (e. g. ﬁve grey circles). During Hebbian learning all connections among allﬁve units are symmetrically strengthened (only nearest neighbour con-nections are shown by double arrows to simplify diagram). Finally, allof the particular instances are mutually exclusive in any pattern and soare coupled via mutual inhibition. This is shown by the lines terminat-ing in solid circles on the far right. Other vertical population arrays ofparticulars are also mutually inhibitory, although connections are notshown for clarity. Even this small network can store up to about ﬁvepartially overlapping patterns. When a subset of these patterns are acti-vated with nearly equivalent stimulus levels, they will become dominant one after another in a generalized rivalry oscillations. This is suggested to be a model for considering several alternatives for a course of actionin deliberation. See text for other details 1982 ), the ﬁve populations in each pattern (grey for example) would have all their interconnection synapses strengthenedas shown by arrows. Within a column all the possibilities aremutually exclusive in any one thought, hence mutual inhibi-tion shown on the right by interconnections with solid circles. Under these conditions, the network will exhibit generalizedrivalry in which one thought pattern will alternate or com-pete with several others in sequence. Furthermore, if a few patterns receive fairly weak evidence or input relative to oth-ers, they are automatically excluded by the dynamics from competition within the network. For a trivially small network of only 15 neurons, about ﬁve patterns can be learned (withpartial pattern overlap), and from 2 to 5 of them will competewhen receiving roughly comparable input. If this network is extended to a more realistic 500 ×1000 neural populations or more, the link to decisions becomes quite plausible. 8 Discussion The Wilson-Cowan equations have produced useful mod-els and insightful explanations in many cortical areas andbrain functions. The original spatial model (Wilson and Cowan 1973 ) was applied to several phenomena in V1, and this has since been extended to visual hallucinations(Ermentrout and Cowan 1979 ), multiple orientations deﬁn-ing E neuron groups (Bressloff et al. 2001 ), and multiple inhibitory groups in fusion and rivalry (Wilson 2017 ). In higher cortical areas Wilson-Cowan has served as a basisfor connectionist learning and memory ﬁrst introduced in the Hopﬁeld ( 1984 ) network. In addition, Wilson-Cowan has provided interpretation for travelling waves, both in binocu-lar rivalry (Wilson et al. 2001 ; Lee et al. 2007 ) and in epilepsy (Shusterman and Troy 2008, Meijer et al. 2015 ). Finally, a generalization of rivalry has generated a possible explanationfor decisions among several plausible alternative possibilities (Wilson 2009 ; Wilson 2013 ). This range of applications of the Wilson-Cowan model depends on a number of extensions to the original model. Most obviously it is possible to generalize to multiple Eand Ipopulations reﬂective of particular cortical areas and func-tions. Among multiple populations there must be multiple population-to-population connectivity functions, which fur-ther individuate models. Furthermore, it has been shown tobe important for connectivity functions to be learned (Hop-ﬁeld 1984 ) in many cortical areas, and others have taken this much further in the development of multi-layer, deep learn-ing convolutional networks (Le Cun et al. 2015 ). To this panoply of extensions must be added the abil-ity to introduce ion currents that generate self-adaptation, particularly in E populations. Since the ﬁrst development of Wilson-Cowan, it was discovered that many cortical pyra-midal neurons, typically excitatory, incorporate slow Ca ++ mediated K+currents that serve to hyperpolarize active cells and thereby reduce their ﬁring rate (Mc Cormick and Williamson 1989 ; Mc Cormick 1998 ). The key here is that this is a slow current, with a time constant much longer than the excitatory and inhibitory postsynaptic currents onwhich the Wilson-Cowan equations were based. Other suit-ably slow potentials can obviously be introduced as they are discovered. This, however, highlights one clear limitation ofthe Wilson-Cowan formalism: it cannot deal with extremely rapid neural variation, where simulation of individual spikes would be required. To cite but one example, humans canaccurately discriminate the direction from which a soundemanates when the arrival time difference at the two ears is as small as 10 microseconds, or about 1/100 the width of an excitatory action potential. This exquisite sensitivity involvesaxonal delay lines and coincidence detection, and is clearly too fast for the Wilson-Cowan approach. As the Wilson-Cowan approach has been very success-ful at incorporating additional populations, particularizing multiple sets of interconnections, and adding slow adap-tive currents, it is appropriate to ask why this approach hassucceeded in capturing key features of neural networks innumerous areas of the cortex. When we began this work a half century ago, it was frequently claimed that nonlinear dynamics was such a vast area, effectively inﬁnite, that onlylinear approximations plus perhaps a few exact nonlinear solutions of particular equations were possible. Despite the clearly vast range of possible nonlinear systems, our argu-ment then was that the nervous system depended on strong, 123