Let $\mathcal M$ be a manifold

(old graphics for a presentation; neural manifolds can drift in many ways. Some are disruptive, some are not. Some are easier to compensate for than others)

Things that lead to tangent variability (affect coding):

• Variability within the embedding space
• In-band variability: changes in neuronal tuning to behavior
• Coordinate distortions: Generally disruptive
• Contraction or expansion, rigid transformations: Require compensation, some component will disrupt coding (how much?)

Things that do not lead to tangent variability:

• Variations that are in the null space of the embedding space
• Some changes in manifold expansion/contraction is orthogonal to coding direction

Things that lead to correctable variability:

• Rigid transformation (+translation): generally, these changes can be corrected for by re-alignment.
• Sufficiently large-scale variability looks like rigid transformation (+translation) locally
• High-frequency / fine-scale variability can be removed by filtering