THALis: tangents worth keeping as questions

tangents parking lot: Takens, PLLs, bifurcations, time scales... a loop still doesn't tell me speed or phase. keep coming back to that

• manifold question: does low-dimensional activity reflect a circuit constraint, a restricted task, or both? How does the representation change during learning or when transitioning between movements? A diagram with one long loop is not enough to answer these biological questions.

• timescale question: which states can be treated as fast and which as slow, and over what operating range? The companion material links this to singular perturbation, center-manifold reduction and normal hyperbolicity; each connection requires explicit assumptions before it can justify dropping states.

• readout question: topology can preserve a loop while changing its metric shape, timing or phase-to-output map. Generalizing from a slow movement to a faster one is therefore a hypothesis to test, not something guaranteed solely by equal Betti numbers.

• further reading categories in the tangent document: Takens embeddings, echo-state networks, SPD covariance geometry, canonical correlation analysis, random-matrix spectra, Hopf/Neimark–Sacker bifurcations and phase-locked loops. Use the companion as a map to primary references; several strong claims in its explanatory prose go beyond the evidence recorded with it.

• reference task: for each tangent, write the theorem's actual hypotheses beside the measured quantities the project could use to check them. Keep biological speculation separate from a software test or proved mathematical statement.

Neural population dynamics during reaching

Takens embedding theorem