652 Biological Cybernetics (2021) 115:643-653 but understandable nonlinearities. Four key ideas incorpo-rated in the original formulation epitomize this. First, the simpliﬁcation to neural populations and ﬁring rates as espoused by Beurle ( 1956 ) simpliﬁed the descrip-tion of neural networks relative to a detailed descriptionusing multiple equations to describe each individual spike(Hodgkin and Huxley 1952 ). As suggested previously (Wil-son 1999 ), this is analogous to reporting data in terms of post stimulus time histograms (PSTH) rather than describ-ing individual spike trains. Bins in PSTH are typically about 10-20 ms wide, which is reﬂected in the Wilson-Cowan time constants. Regarding human understanding, it is clear that theoverwhelming wealth of our knowledge lies on the several second time scale, with memory vastly extending this back into the past. Thus, the time scale engendered by popula-tion dynamics in Wilson-Cowan ﬁts naturally with humanunderstanding as well. Implicit in the paragraph above is the notion of an exponential time scale. The Wilson-Cowan equations werederived on the assumption that changes in ﬁring rates of populations followed the time constants of typical neural post-synaptic potentials, EPSPs and IPSPs. Given multiplepopulations, multiple time constants have been incorporated. But by constraining our equations to time constants rep-resenting postsynaptic potentials, which implied ignoringindividual spikes, a major simpliﬁcation was accomplished:only population ﬁring rates mattered. Compared to simulat-ing individual spikes, this reduced computational require-ments by almost two orders of magnitude. This meant that Wilson-Cowan could simulate networks about 100 ×larger than spiking networks, given equivalent computing power. Third, Wilson-Cowan was established on the extremely important physiological observation that excitatory and inhibitory neurons formed distinct, interacting populations (Wilson and Cowan 1972 ). This is now a commonplace, but it was ignored by all earlier attempts at network modelling. One can think of neural excitation as active and cooperative (“yang”), while inhibition functions to shut down excitatory activity by introducing competition (“yin”). This generatesa cooperative-competitive dynamic that forms the basis of neural computation. Both components have been integral to Wilson-Cowan from the beginning. This cooperative-competitive theme has been indepen-dently developed into a canonical model for neocortex (Douglas et al. 1989 ; Douglas and Martin 1991 ). These authors developed a model with two excitatory neuron popu-lations, one comprising neurons in the supragranular cortical layers 2 and 3, and one comprising neurons in infragranular layers 5 and 6. Recurrent excitation both within and betweenthe excitatory populations is incorporated into the model. The ﬁnal group contains inhibitory neurons that are connected to themselves and to both excitatory groups to generate nega-tive feedback. All neural populations are described by spikerate dynamics. Thus, this candidate for a canonical circuit for neocortex may be regarded as an embellishment of Wilson—Cowan dynamics. Related E-Idynamics have also been used by Grossberg to develop theories of a large number of cortical functions (Grossberg 2021 ). Finally, the sigmoid function is a key to the power of the Wilson-Cowan equations. The sigmoid captures the impor-tance of neural thresholds, followed by the roughly linear activity increase with increasing stimulation, and ﬁnally by acompressive nonlinearity and saturation at high input levels. Unlike many nonlinear dynamical systems, his nonlinearity has facilitated state space analysis, as neural activity is con-strained by the threshold to be ≥0, and sigmoid saturation guaranteed that activity must be ≤M, the maximum value. This restriction to a hyper-cube in state space has provedvaluable in many mathematical analyses, so the importanceof the sigmoid to physiologically relevant neural modelling cannot be overstated. The Wilson-Cowan equations have clearly proved their value at explaining a wide variety of neural functions in diverse cortical areas. Keys to this success are cooperative— competitive dynamics, population dynamics on a postsynap-tic potential time scale, and sigmoid nonlinear boundedness by thresholds and saturation. Within this framework, there has been a rich evolution, and we suspect that they will con-tinue to enhance our understanding of brain function. References Bertalmio M, Calatroni L, Franceschi V, Franceschiello B, Gomez-Villa A, Prandi D (2020) Visual illusions via neural dynamics:Wilson-Cowan-type models and the efﬁcient representation prin-ciple. J Neurophys 123:1606-1618 Beurle RL (1956a) Properties of a mass of cells capable of regenerating impulses. 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