Radial sampling interval ($\Delta r$)

The radial sampling interval $\Delta r$ is the separation between consecutive radii at which a radial measurement is made. For Sholl analysis, it determines the spacing of concentric Euclidean circles used to sample intersection counts about a fixed center. The documentation distinguishes this step from the starting and ending radii.^[ref]

For constant positive spacing, the sampling grid iswhere $R$ is the chosen upper radius. For $r_{\mathrm{start}}=1$, $\Delta r=1$ and $R=4.4$, the sampled radii are 1, 2, 3 and 4. The upper bound 4.4 is not itself sampled. The software documentation explicitly notes this endpoint issue when the step does not divide the difference between ending and starting radii.^[ref]

A purely geometric example shows what coarse spacing can miss. Begin with a straight line segment from $(0,0)$ to $(4,0)$ and use the origin as center. Add a short branch from $(2.1,0)$ to $(2.4,0.7)$. Its distance from the center increases continuously from $2.1$ to $2.5$, since $2.4^2+0.7^2=2.5^2$. Counting shared endpoints only once, the added branch contributes one extra distinct intersection exactly forAt radii 1, 2, 3 and 4, the branched and unbranched traces both give $N(r)=1$. An additional sample at $r=2.25$ instead gives two intersections for the branched trace. The feature exists in the geometric input but falls between the coarse sample radii.

Smaller $\Delta r$ supplies a denser set of probes, while changing $r_{\mathrm{start}}$ shifts their positions. Neither operation restores structure already lost through image segmentation or finite pixel pitch. In particular, closer numerical spacing is distinct from improved physical spatial resolution. Report the center, calibrated distance units, starting radius, interval and sampled endpoint list with a Sholl profile.

The program's zero step setting is a special request for finite image-derived sampling, rather than the literal choice $\Delta r=0$ in the grid above. For image input, the documentation describes voxel-scale increments and a calibration-dependent rule for unequal pixel dimensions.^[ref]