The article "Vector reconstruction from Firing Rates" of Abbott and Salinas

Question

Is anyone familiar with this article?

I am reading it, and it is basically very clear. But still, some points are not clear to me.

For example, how is equation (7.1) in page 12:

$Q_{i,j}^{-1} = F(\overline C_i \overline C_j)$

It it said there that

"By rotational invariance, the correlation matrix element $Q_{i,j}$ and the corresponding inverse element $Q_{i,j}^{-1}$ can only depend on $\overline C_i \cdot \overline C_j$".

Does anyone here, by any chance, know what is "rotational invariance"?

Thank you!

Answer 1

What they mean is that as long as you rotate $C_i$ and $C_j$ by the same amount, the dot product will be the same. This is rotational invariance because the dot product is invariant to coherent rotation of the relevant vectors.

In neural terms, the correlation between the activity of two neurons (in a population representing a one dimensional circular variable) is a function of the difference in their preferred orientations and does not depend on the absolute direction of their orientations.