648 Biological Cybernetics (2021) 115:643-653 Fig. 4 Example of a spatiotemporal illusion resulting from uniform ﬁeld ﬂicker. The plot shows one spatialdimension on the abscissa andtime increasing downward onthe ordinate. E neuron activitylevels are pseudocoloured asvery low (black), intermediate(shades of red), and high(yellow). Although the stimulusis uniform ﬂicker, the neuralactivity pattern bifurcates to aspatiotemporal alternation ofcompeting activity foci Cortical Distance Time each ijpair ( i̸/equal1j) active in the pattern. This cross-correlation (without a time lag that would encode causality) guarantees the symmetry of the connection matrix. Under these con-ditions, Hopﬁeld constructed an energy function and provedthat when stimulated with a sufﬁcient percentage of a pattern,the network would asymptotically approach activity repre-senting the full learned pattern. Neural learning models have progressed far beyond Hop-ﬁeld networks in the intervening years. Of particular impor-tance have been deep learning networks. These networks incorporate multiple hierarchical levels of model neuronsthat learn their connection weights from the previous layers. In particular, errors between the desired output and the cur-rent output are calculated, and the relative error is assignedto the weights in the various layers based on the chain rulefrom calculus (Rumelhart et al. 1986 ). The current generic deep learning networks consist of hierarchical layers in which there is ﬁrst a neighbourhood convolution with input fromthe previous layer, followed by a nonlinear transformation, such as the maximum within a neighbourhood, and then a spatial subsampling to a smaller upstream area (Reisenhuberand Poggio 1999 ; Le Cun et al. 2015 ). Recently, alternative networks using both lateral interactions and feedback from higher areas have been shown to provide greater accuracy andenhanced biologically plausiblity (Spoerer et al. 2017 ). Thesenetworks are consonant with multi-layer Wilson-Cowan net-works, as they incorporate convolution of inputs followed by a sigmoid nonlinearity, and this approach has evolved to gen-erate an enormous range of very powerful applications. 5 Binocular rivalry and travelling waves The Wilson-Cowan equations have been used to explain a range of nonlinear visual phenomena, one of the most dramatic ones being binocular rivalry. Under normal stim-ulation, the eyes have evolved to sample the visual world from two slightly different visual perspectives, which the brain then combines to generate a percept of the third dimen-sion, namely depth. However, when two radically differentimages (e. g. orthogonal gratings) are viewed independently by the two eyes, they cannot be interpreted in depth, and rivalry ensues. The brain then defaults to a stochastic oscil-lation in which ﬁrst one monocular image and then the other is perceived, with the transitions between monocular images occurring approximately once every 2 s. Before describing an explanation for binocular rivalry, it is necessary to review some more recent work on single neurons in the mammalian cortex. Since the Hodgkin-Huxley equa-tions were developed to describe action potential generation 123