Computational geometry for modeling neural populations: From visualization to simulation - Fig 2

A: As a result of the integration procedure simple quadrilaterals (left, middle) should emerge, which are usually convex (left), except near stationary points or limit cycles where concave quadrilaterals (middle) can be formed. Complex, i.e. self-intersecting, quadrilaterals can occur around strong non linearities, for example the crossing of nullclines. These definitions hold for any polygon. B: The problem of defining the Master equation: we can easily calculate how much mass per unit time leaves a given bin (i, j). This mass will reappear at a position h away from the original bin, where h is the synaptic efficacy. In the figure bin (13,7) is translated along vector (0, 0.1). This corresponds to neurons that have received an input spike, and therefore are experiencing a jump in conductance. Most neurons that are in bin (13,7), will end up in bin (13, 5) and (14,4), with some in bin (13,6) and (14,6). So C(0.,0.1)(13, 7) = {(13, 5), (14, 5), (13, 6), (14, 6)}. C: Some events will end up outside of the grid after translation. D: Fiducial quadrilaterals can be used to test where they have gone missing, and where is the best place to reassign them to the grid.