646 Biological Cybernetics (2021) 115:643-653 Fig. 2 Dynamical response of the Wilson-Cowan equations to brief stimulation by sufﬁciently strong pulses at the pointsmarked by the vertical arrows. Due to recurrent excitation,network activity moves to twosaturated peaks of E populationactivity (red curves), but theseactivity peaks are preventedfrom spreading by the spatiallybroader inhibition (blue curve). As multiple peaks can be thusstabilized, this was interpretedto provide a basis for short termmemory 0 100 200 300 400 500 600 700 800 900 1000 Distance in V10102030405060708090100110E (red) & I (blue) Responses corresponding point of cortical projection in x,ycoordinates is: x/equal1ln(1+r) y/equal1φ (5) This mapping to one hemisphere is illustrated in Fig. 3, where the bounding blue contour represents the verticalmeridian, and the remaining three roughly horizontal lines, converging at the origin, represent orientations of 45°, 0° (horizontal), and-45° respectively. Had two concentric cir-cles with radii of 3. 6° and 7. 2° been imaged on the retina, the cortical activation would be represented by the two almost parallel red horizontal bands. Conversely, a radial patternstarting a few degrees away from the fovea would have generated almost horizontal bands of activation due to this mapping. The critical insight was that these could be approxi-mated as parallel neural activation patterns in V1 (Ermentroutand Cowan 1979 ). With the complex logarithmic mapping in 2D, plus an analysis of steady states for the Wilson-Cowan equations,Ermentrout and Cowan showed convincingly that visual hallucinations could be explained by asymptotically stable patterns of activated parallel lines of Eneurons in V1. When projected back to the retina, these would have been con-centric circles. Given drug activation of V1, higher levels of the visual system would have received the same stim-ulation from V1 as would have resulted from concentriccircles in the visual ﬁeld. In addition, drug activation of hor-izontal contours of V1 activity would have resulted in the illusory percept of radial spokes, and activation of diago-nal contours would have produced a hallucination of spirals. Complex checkerboards with checks increasing in size with distance from the fovea were also generated. Recall the min-imal properties required: 2D generalization, asymptotically stable ﬁring states of the network, and the empirically deter-mined complex logarithmic mapping in Eq. ( 5). This elegant explanation of visual hallucinations (Ermen-trout and Cowan 1979 ) was very powerful, but it simpliﬁed by ignoring a very important organizing principle of V1: ori-entation columns (Hubel and Wiesel 1977 ). A more recent study by Cowan and colleagues has reexamined this by intro-ducing multiple E populations, each tuned to a different peak orientation (Bressloff et al. 2001 ). In addition to incorporat-ing multiple arrays of orientation-tuned Eneurons, the Eto E connectivity functions were altered to incorporate collinear facilitation. Physiological data from V1 had demonstratedthe presence of long range E-Einterconnections among neurones with similar orientation preferences located at a distance from each other but aligned roughly collinearly (Ts'o and Gilbert 1988 ; Gilbert and Wiesel 1989 ). The addi-tion of multiple orientations plus collinear facilitation to the Wilson-Cowan equations permitted a much wider range of visual hallucinations to be explained (Bressloff et al. 2001 ). So far, nothing has been said about spatiotemporal hal-lucinations. It is known that uniformly ﬂickering light, par-123