Biological Cybernetics (2021) 115:643-653 649 in the squid giant axon, studies of mammalian neocortical neurons have shown that many additional ion currents arepresent. In particular, excitatory neocortical neurons self-adapt as the result of a, Ca ++mediated K+current that slowly hyperpolarizes the cell (Mc Cormick and Williamson 1989 ; Sanchez-Vives et al. 2000 ). This current has an exponen-tial time constant of a second or more, almost two orders of magnitude longer than typical synaptic currents. This ﬁts well with the rate of alternations in binocular rivalry. Cru-cially, it is primarily excitatory rather than inhibitory neurons that possess this slow adapting current. The Wilson-Cowan equations have been extended to include this slow currentby introducing an equation for a hyperpolarizing variable H (Wilson 1999,2007 ; Wilson et al. 2000 ) Thus, the equation for E(x,t)i n E q. ( 2) is replaced by the two equations: τ E∂E ∂t/equal1-E+SE( βEE(x)⊗E-βIE(x)⊗I+P(x,t)-g H) τHd H dt/equal1-H+E (6) withτHapproximately 1. 0 s, almost two orders of magni-tude greater than τE. (Alternatively, Hcan be added to the semi-saturation constant σin Eq. ( 3) with no signiﬁcant qual-itative differences. ) The parameter g was assigned a valuesuch that adaptation ultimately reduced the Eﬁring rate to about 1/3 of its maximum, in agreement with electrophys-iology. Finally, the stochastic component of rivalry can besimulated, if desired, by adding a Gaussian noise term to the Hequation in Eq. ( 6), which produces a dominance time distribution that is well ﬁt by either a log-normal or gammadistribution in accord with data (Fox and Herrmann 1967 ). Given this embellishment of the Wilson-Cowan equations to include adaptation, binocular rivalry can now be explained. Separate equations describe excitatory neurons driven bythe left and right eyes, E Land ERrespectively. Competi-tion between them is driven by inhibition, IL, and IR,f r o m the opposite eyes. Thus, limit cycle competition emerges inwhich one eye ﬁrst suppresses the other eye, but it then grad-ually adapts via its H current so the second eye can escape from the suppression and itself become dominant. This isillustrated in Fig. 5. The slow oscillation on a several sec-ond time scale is a result of the very long time constant τ H in Eq. ( 6). Note that the alternation is far too slow to be explained by ordinary synaptic inhibition. Thus far, binocular rivalry had been treated as though it were a unitary phenomenon in which one monocular image uniformly replaced the other, but this is inaccurate. Rather,the suppressed image will ﬁrst begin to replace the visible image at one point, and it will then transform into a trav-elling wave moving across the image from that point. Tomeasure the travelling wave properties rivalry was restrictedto a circular, effectively one-dimensional annulus or “race track” (Wilson et al. 2001 ). A wave of the suppressed pat-tern could then be triggered at any point around the circle, with the subject indicating when it reached the ﬁnish line. Using this psychophysical technique, it was shown that thewave travelled at a roughly constant speed across the cortex,which was estimated using the cortical mapping in Eq. ( 5)t o be about 2. 24 cm/s (Wilson et al. 2001 ). In an elegant subse-quent f MRI experiment, wave speed was directly measuredon the human cortex and found to be in good agreement with the psychophysical estimate (Lee et al. 2007 ). The observed rivalry wave propagation was shown to be predictable by a model based on the Wilson-Cowan equa-tions with adaptation (Wilson et al. 2001 ). The left and right eye patterns were represented by independent groups of E L and E Rneurons that were mutually inhibitory. As the sup-pressed neurons become dominant at the moving front of the wave, they inhibit previously dominant neurons in front of the wave, thereby generating a release from inhibition. A detailed mathematical analysis of this wave propagation was later developed based on a variant of the Wilson-Cowan equations (Bressloff and Webber 2012 ). Rivalry only occurs when the two monocular images are so different that they prevent fusion and the extraction of depth. If the two images are oriented cosine gratings; for example, aninterocular orientation difference up to about ±6° will result in fusion and the depth percept of a grating tilted forward or backward in depth (Blake and Wilson 2011 ). For an interocu-lar orientation difference greater than that, however, fusion isimpossible and rivalry ensues. Importantly, when the interoc-ular orientation difference is continuously varied it has been shown that the switch from fusion to rivalry involves hystere-sis (Buckthought et al. 2008 ). Fusion, rivalry, and hysteresis have all been captured using Wilson-Cowan equations with H current adaptation as shown in Eq. ( 6) (Wilson 2017 ). To particularize the model for V1, the model comprised 12 dif-ferent excitatory populations (difference of 15° in preferred orientation between them) for each eye plus four separate inhibitory populations at each orientation (two for each eye):one for small orientation contrast normalization and one for long range orientation rivalry respectively. By incorporating additional neural populations and particularizing the connec-tivity among cell populations, hysteresis between fusion and rivalry can be explained by the Wilson-Cowan approach. 6 Epilepsy The Wilson-Cowan equations have also been applied to epilepsy. This was based upon the original observations that spatially localized limit cycles could exist and that travelling waves could occur should the inhibition be too weak (Wil-son and Cowan 1973 ). Shusterman and Troy ( 2008 ) later 123