This paper presents an in-depth theoretical analysis of the orientation selectivity properties of simple cells 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.

Additionally, we also derive closed-form expressions for the resultant and the bandwidth descriptors of the orientation selectivity curves, which have previously been used as compact descriptors of the orientation selectivity properties for biological neurons. These results together show that the properties of the affine Gaussian derivative model for visual receptive fields can be analyzed in closed form, which can be highly useful when to relate the results from biological experiments to computational models of the functional properties of simple cells and complex cells in the primary visual cortex.

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.

The affine Gabor model does, however, comprise one more degree of freedom in its parameter space, compared to the affine Gaussian derivative model, where a variability within that additional dimension of the parameter space does also strongly influence the orientation selectivity of the receptive fields. In this respect, the relationship between the orientation selectivity properties and the degree of elongation of the receptive fields is more direct for the affine Gaussian derivative model than for the affine Gabor model.