644 Biological Cybernetics (2021) 115:643-653 dynamics were manifest in his formulation, and this led him to an analytical travelling wave solution for neural activityof the form: F(x-vt)/equal1M 2 cosh2(k(x-vt))(1) where vis the velocity of the wave (to the right here) and Fis the proportion of neurones active during passage of the wave. Mandkare constants. Beurle acknowledged that this solution was unstable, as initial excitation to an amplitude slightly greater than that for Eq. ( 1) led to a transient increase in wave amplitude to saturation, whereas initial excitation to a slightly lower amplitude led to attenuation. In fact, the solution to Beurle's equation, termed a soliton, is also a solution to the Korteweg-De Vries equation that describes propagation ofwater waves in shallow channels (Korteweg and De Vries 1895 ). It is a soliton that conserves an inﬁnite number of quantities. Although providing several major insights, Beurle ( 1956 ) made a number of errors. First, he modelled the cortex as an unstable system that must be delicately balanced to remainplausible. Under these circumstances, he produced a system that was conservative, rather than dissipative, as the brain is known to be. Finally, and perhaps, the root cause of theseproblems was the omission of inhibition as a co-equal factorin brain function. 2 Wilson-Cowan equations We designed the Wilson-Cowan equations to directly reﬂectthe nonlinear dynamics inherent in excitatory-inhibitory interactions in cortical tissue. In addition, these equations were intended to be simpler than the Hodgkin-Huxley equa-tions (1952) so that the dynamics of much larger populations of neurones could be explored. Given Beurle's ( 1956 ) insight, we chose to describe the activity of localized populations ofneurones rather than the spiking of single neurones. However, it was clear that unstable soliton solutions could not effec-tively describe neural dynamics, so we developed equationsthat could produce travelling waves only under pathologicalconditions, such as epilepsy. Crucially, we argued that studies of neural activity must focus on the balance between exci-tatory and inhibitory activity in cooperating and competingneural populations. Our ﬁrst results examined local temporal interactions of local neural populations (Wilson and Cowan 1972 ). Using phase plane techniques, it was shown that these equations could produce asymptotically stable excited states suggestive of short-term memory, limit cycle oscillations suggesting periodic motor control, and several more com-plex behaviours. Fig. 1 Sigmoid function used in the original Wilson-Cowan equations (dashed line) compared with a sigmoid (Naka-Rushton function, solidcurve) designed as a more accurate approximation of cortical dynamics Within a year, this local model was extended to include interactions among excitatory ( E) and inhibitory ( I) neural populations across space (Wilson and Cowan 1973 ), and this was published in Kybernetic, the parent of Biological Cybernetics. The original equations for the E(x,t) and I(x,t) populations are: τE∂E ∂t/equal1-E+(1-r E)SE(βEE(x)⊗E-βIE(x)⊗I+P(x,t)) τI∂I ∂t/equal1-I+(1-r I)SI(βEI(x)⊗E-βII(x)⊗I+Q(x,t)) (2) where τEandτIare the respective time constants on the order of 10-15 ms, r is the refractory period, Sis a sigmoid function increasing monotonically from its minimum at-∞ to its maximum value at + ∞, and Pand Qdescribe exter-nal inputs to the respective populations. A logistic function was originally used for S, but as the exact mathematical form of sigmoid does not change the qualitative dynamics, otherforms have more recently been used based on the experimen-tal literature. For example, responses of cortical neurones have been described by a Naka-Rushton equation (Naka and Rushton 1966 ) with an exponent of approximately 2. 4 (Sclar et al. 1990 ). A Naka-Rushton function with an exponent of 2 has also been used to facilitate mathematical solutions for equilibrium states (Wilson 1999 ). It is described by Eq. ( 3) and is plotted in Fig. 1. S(x)/equal1 M·x2 σ2+x2,x≥0 S(x)/equal10, x≤0(3) In this equation, Mis the maximum ﬁring rate, while σ deﬁnes the semi-saturation level, as when x/equal1σ,S/equal1M/2. Values of M/equal1100,σ/equal125 are shown in the ﬁgure. 123