Quantitative measures

We fit data from size tuning curves with the integral of a difference of Gaussians (DeAngelis et al., 1994). We chose the size of the classical receptive field (CRF) to be the smallest diameter grating stimulus for which the fitted curve reached 95% of its maximum. Optimal sizes for Glass pattern and random dot stimuli were chosen in the same manner. We also fit descriptive functions to spatial frequency tuning curves for gratings to find the optimal spatial period, $\lambda$ (the inverse of the optimal spatial frequency) for each cell (Levitt et al., 1994).

To characterize orientation tuning curves, we determined the selectivity and preferred angle by calculating a tuning bias vector (Leventhal et al., 1995; O'Keefe and Movshon, 1998), similar to the vector strength calculation introduced by Levick and Thibos (1982). We represented an orientation tuning curve as a set of vectors, ($\theta_n$,$R_n$), where $\theta_n$ is stimulus orientation, $R_n$ is the response magnitude (with baseline subtracted), and $n$ is an index from 1 to the number of points, $N$, in the tuning curve. The preferred orientation is given by the circular mean angle:

$$\frac{1}{2}\arctan\left(\frac{\displaystyle \sum_{n=1}^{N} R... ..._n)}{\displaystyle \sum_{n=1}^{N} R_n \cos(2\theta_n)}\right).$$

To measure selectivity, we calculated the summed response vector,

$$v = \sum_{n=1}^{N} R_n e^{(i2\theta_n)}$$

and normalized its magnitude by the summed magnitude of all the response vectors:

$$\mbox{selectivity index} = \frac{\vert v\vert}{\displaystyle \sum_{n=1}^{N} R_n \vert R_n\vert}.$$

The selectivity index is 0 for a cell responding equally at all orientations and 1 for a cell that responds only to a single orientation. To estimate the significance of each selectivity estimate, we used the permutation technique described in O'Keefe and Movshon (1998). For each tuning curve, we performed the selectivity index analysis on 2000 random permutations of the data, and considered a measured selectivity index to be significant if it exceeded the 90th percentile of the permuted distribution.

To estimate analogous quantities for Glass pattern tuning curves with four lobes (rather than two), which we term ``quadropoles'', we modified the first two equations simply by substituting $4\theta_n$ for $2\theta_n$ and taking $1/4$ rather than $1/2$ of the arctangent. This results in a measure of preference and bias appropriate for functions with periodic peaks and troughs every 90$^\circ$, rather than every 180$^\circ$.