'Eigen-view' on Grid Cells - September 21, 2020 

If we had a random walk where at small times $\Delta t$ the motion was described by a multivariate normal distribution associated with the matrix $(\Delta t) \Sigma(x)$ (where x is the current location and \Sigma(x) is a positive definite matrix that is a function of position), Then, I think we could replace the Laplacian with $\tr(\Sigma(x) H)$ where H is the Hessian (or maybe that should be $\tr(\Sigma(x)^{-1} H)$? I may have gotten mixed up.) (The usual Laplacian is obtained when \Sigma(x) is the identity matrix for all x) I don’t know how the random walk should be described in the case of flight. It seems to me that with flight, momentum would be a bigger deal than it is when walking around on the ground. Maybe one would have to move to a phase space of position and momentum together? Or... idk. I don’t know what bats actually do when exploring. But, if moving vertically were just more costly, both up and down (... which doesn’t seem right...), or more generally if there’s a bias to move along some axiis over others, and where which axiis are preferred can vary in space, then I think the differential operator I described above should work to describe the diffusion of a probability distribution over position, and, seems to me like it could *maybe* result in some nice distorted versions of the usual eigenfunctions of the Laplacian? Like, if you apply some coordinate transform, and have the image of the uniform random walk under this transform, and if the resulting random walk in these new coordinates be described in terms of such a \Sigma(x) , then... uhh... I think the transformation might sorta end up relating the different eigenfunctions? ... but maybe not quite?

@@drdca8263 As I remember it, my comment was about the observation that the eigenvalues (!) are the same for all eigenfunctions (cosines; well, technically we should be talking about complex exponentials, but I'll be sloppy) with the same frequency, no matter the direction of the cosine. This requires isotropy. I believe the sinusoids will always be eigenfunctions, no matter what differential operator / diffusion / convolution you choose. At least so long as we're talking about Euclidean space and not something more general like a graph. When looking at their discretized versions, both difference operators and diffusions ARE convolutions. Since all convolutions commute, they must share eigenvectors. In continuous spaces the rigorous math might be more interesting; a quick websearch turns up the book "Convolution-like Structures, Differential Operators and Diffusion Processes" by Sousa et al. I think you are right in focusing on the _actions_ animals take. And on what the relevant state space is. At the end of the day, the actions determine the operator that would benefit from diagonalization. E.g. a human might model the effect of taking a step forward (a common action for a human) as a convolution of their current position estimate with a Gaussian centered around a point that's in the direction they are facing and a "mean step-size" away from the origin. Since it's a convolution, sinusoids are again eigenfunctions, but the set of eigenvalues representing this "step-action" is different from those that represent, say, a uniform random walk. If the idea is that the entorhinal cortex and the hippocampus implement a kind of SLAM algorithm ( en.wikipedia.org/wiki/Simultaneous_localization_and_mapping ), then by measuring the grid cells' activity we're likely taking a peek at the odometry. If the brain uses a spectral representation for the animal's location-estimate, updates due to an action are just a per-grid-cell multiplication of its activity, rather than e.g. a convolution. The interesting question then becomes: what part of this is biologically hardcoded and what part can be learned? Perhaps the cortical neurons do run something like en.wikipedia.org/wiki/Generalized_Hebbian_algorithm to do PCA and when they happen to look at covariances caused by convolutional actions, the principal components they learn are these "rotated sinusoids" we observe as grid cells.

Laplace transform is good idea too. I suggested DMD in the thread. There is a paper called DYAN on arxiv. She is a colleague of mine. She used Laplace transform. It has same benefits as DMD. They are pretty equivalent. I use them interchangeably for my projects whereever spatiotemporal coupling is involved.

Regarding warping: They are linear locally. May be that's why. I guess we have to think of it like a manifold because of local linear interactions.