Depth vs. radial distance in laminar cortex

Why a straight axis isn't enough

Mammalian cortex is strongly laminar: neurons, connectivity, and cell types are organized in layers stacked from the pial surface down to white matter. Because of that organization, two very different questions constantly come up:

• Depth: how far along the pia → white matter direction is a point? (Which layer is it in?)
• Radial distance: how far apart are two points within the laminar plane, i.e. tangential to the layers?

The oriented frame gives you a single global depth axis (y). That works where cortex is flat. But cortex curves — the true "down" direction tilts and bends as you move across the volume. If you keep using one straight axis, then "depth" and "in-plane distance" get mixed together: a point that is actually directly below another (same lamina position, deeper) can look laterally displaced, and vice versa. The error grows with curvature and with distance from wherever the straight axis happened to be correct.

The streamline

A streamline is a curvilinear depth axis: a curve that follows the local pia → white matter direction as it bends through the tissue. Think of it as the path a plumb line would trace if "down" were always the cortical down at each point.

With a streamline you can measure the two quantities separately and correctly:

• Depth along the streamline — path length down the curve from the pia, rather than a raw y difference. This is what depth_along and depth_between compute.
• Radial distance — in-plane distance measured relative to the streamline as the d = 0 reference, accounting for the fact that the curve is tilted. This is what radial_distance computes.

You can also remap points into a cylindrical-like coordinate system — radial distance as one axis, depth-along-the-streamline as the other — with radial_points. This is the basis for "straightening" a neuron along the cortical axis (see Skeletons & meshworks).

A worked intuition

Imagine two synapses. In raw oriented y they differ by 50 µm, and they also differ in x. Are they at the same lamina depth but offset sideways, or genuinely at different depths? If the cortex is curved there, the straight y answer is biased. Measuring depth along the streamline answers the anatomical question: how far down the cortical column each one really is.

Next

There are two ways a streamline can be defined: a single fixed curve applied everywhere, or a curve that varies with location. That distinction — and why the v1dd default recently changed — is covered in Streamlines vs. fields.