Origin of the 1/n spectrum of cortical population responses
In plain words
When thousands of visual-cortex neurons respond to natural images, the variance carried by the n-th strongest activity pattern falls roughly as 1/n. This is close to the slowest decay, meaning the most high-dimensional code, that still keeps the response a smooth function of the image, and which circuit mechanism sets it is unknown.
Precise statement
In mouse primary visual cortex, principal-component variances of responses to natural images follow $\lambda_n \sim n^{-\alpha}$ with $\alpha \sim 1$ (Stringer and co-workers, 2019), near the bound $\alpha > 1 + 2/d$ required for a smooth representation of d-dimensional inputs. Identify the circuit mechanism (recurrent dynamics, plasticity, input statistics) that yields $\alpha \sim 1$, and explain why alpha stays near the bound $1 + 2/d$ for low-dimensional stimulus sets, as measured.
What would settle it
A circuit model with measured connectivity statistics that predicts $\alpha$ for several stimulus ensembles, confirmed by recordings.