Biological Cybernetics (2021) 115:643-653 https://doi. org/10. 1007/s00422-021-00912-7 60TH ANNIVERSARY RETROSPECTIVE Evolution of the Wilson-Cowan equations Hugh R. Wilson1·Jack D. Cowan2 Published online: 19 November 2021 © The Author(s), under exclusive licence to Springer-Verlag Gmb H Germany, part of Springer Nature 2021 Abstract The Wilson-Cowan equations were developed to provide a simpliﬁed yet powerful description of neural network dynamics. As such, they embraced nonlinear dynamics, but in an interpretable form. Most importantly, it was the ﬁrst mathematical formulation to emphasize the signiﬁcance of interactions between excitatory and inhibitory neural populations, thereby incor-porating both cooperation and competition. Subsequent research by many has documented the Wilson-Cowan signiﬁcancein such diverse ﬁelds as visual hallucinations, memory, binocular rivalry, and epilepsy. The fact that these equations are still being used to elucidate a wide range of phenomena attests to their validity as a dynamical approximation to more detailed descriptions of complex neural computations. 1 Introduction Astronomy and physics were the ﬁrst of the natural sciencesto be enriched by an intricate link to mathematics. Biology,although developing exceptionally powerful theories such as Darwinian evolution, lagged far behind historically in incor-porating mathematics as an integral, predictive approachin its own right. However, mathematics was beginning to encroach on biology, and particularly neurobiology, in deep ways. Hodgkin and Huxley ( 1952 ) solved the dynamics of the action potential by introducing nonlinear dynamics in four coupled equations. This was rapidly recognized to be brilliant, and they received the Nobel prize in 1962. Upon ﬁnishing my Ph D in theoretical chemistry at The University of Chicago in 1969, I (Wilson) was extremely for-tunate to be offered a postdoctoral fellowship by Jack Cowan at the university. He informed me that nonlinear dynamics Communicated by Peter J. Thomas. To highlight the scientiﬁc impact of our Journal over the last decades,we asked authors of highly inﬂuential papers to reﬂect on thehistory of their study, the long-term effect it had, and futureperspectives of their research. We trust the reader will enjoy theseﬁrst-person accounts of the history of big ideas in Biological Cybernetics. BHugh R. Wilsonhrwilson@yorku. ca 1Centre for Vision Research, York University, Toronto, Canada 2Department of Mathematics, University of Chicago, Chicago,USAwere beginning to have a major impact on neuroscience, and he encouraged me to start working on this subject. That was the beginning of the work that produced the Wilson-Cowanequations. The intellectual background must begin with the Hodgk-in-Huxley equations (Hodgkin and Huxley 1952 ). These four nonlinear differential equations are acknowledged to explainaction potentials, key to all neural computation. As empha-sized previously (Wilson 1999 ), nonlinear dynamics was the essential ingredient in providing a convincing explanation. The experiments, upon which the Hodgkin-Huxley equa-tions were based, required poisoning various ion channels selectively. Thus, Hodgkin and Huxley had at their disposal ameasured action potential plus different ionic ﬂows followingpoisoning of various channels. The glue that put the pic-ture together convincingly was nonlinear dynamics, which deﬁnitively showed that combination of the independent ionchannel results did indeed generate an exceptionally accurate squid action potential. It should be noted that the differ-ential equation computations were performed on an addingmachine and took eleven days of computation just to predict one spike. Given this background, Beurle ( 1956 ) noted the mathe-matical complexity of the Hodgkin-Huxley equations andsought a more simple approximation. Speciﬁcally, he intro-duced the concept of neural populations, which could nat-urally be described by the fraction of active neurones atany given point and time. As a physicist, he developed equations that described the propagation of neural activity waves across a one-dimensional tissue. Simpliﬁed nonlinear 123