  This paper presents a theoretical analysis of the orientation selectivity of
simple and complex cells that can be well modelled by the generalized Gaussian
derivative model for visual receptive fields, with the purely spatial component
of the receptive fields determined by oriented affine Gaussian derivatives for
different orders of spatial differentiation.
  A detailed mathematical analysis is presented for the three different cases
of either: (i) purely spatial receptive fields, (ii) space-time separable
spatio-temporal receptive fields and (iii) velocity-adapted spatio-temporal
receptive fields. Closed-form theoretical expressions for the orientation
selectivity curves for idealized models of simple and complex cells are derived
for all these main cases, and it is shown that the orientation selectivity of
the receptive fields becomes more narrow, as a scale parameter ratio $\kappa$,
defined as the ratio between the scale parameters in the directions
perpendicular to vs. parallel with the preferred orientation of the receptive
field, increases. It is also shown that the orientation selectivity becomes
more narrow with increasing order of spatial differentiation in the underlying
affine Gaussian derivative operators over the spatial domain.
  For comparison, we also present a corresponding theoretical orientation
selectivity analysis for purely spatial receptive fields according to an affine
Gabor model. The results from that analysis are consistent with the results
obtained from the affine Gaussian derivative model,in the respect that the
orientation selectivity becomes more narrow when making the receptive fields
wider in the direction perpendicular to the preferred orientation of the
receptive field.
