Trajectory tangling

Trajectory tangling quantifies how often a population's state trajectory nearly revisits a location while heading in a very different direction. Formally it compares, over all pairs of time points, the squared difference in the state derivatives relative to the squared distance between the states: high tangling means two nearby states have divergent futures, which cannot arise from a smooth autonomous dynamical system and would make the dynamics fragile to noise. Motor cortex during reaching and cycling exhibits conspicuously low tangling, consistent with activity generated by a noise-robust internal dynamical system.

The measure is useful comparatively. Areas that are more input-driven or representational, such as some sensory and supplementary motor signals, show higher tangling than primary motor cortex, and a network can lower its tangling by adding dimensions that keep otherwise-crossing trajectories apart — which is one reason condition-independent and other extra signals may be present. For BCI the appeal of low tangling is that untangled, smoothly flowing trajectories are more predictable and therefore more decodable; the caveat is that tangling depends on how the state space is estimated and on the set of conditions included, so it is a property of a dataset and embedding rather than an absolute constant of an area.