1. Specific Evidence of Low Dimensional Attractor Dynamics in Grid Cells
Yoon, Buice, Barry, Hayman, Burgess, Fiete JC

2. Grid Cells
Grid cell form clusters that form the vertices of an hexagonal lattice.
Similarity with place cells?
Place field
Phase precession
Place cells remaps completely under new environments

3. Attractors in networks
Given N nuerons, the phase space would be 6 N dimensional.
Brain encodes information in lower dimensional “representational space”
Regularity of grid cells response could mean LDA?

4. Spikes recorded simultaneously from the same or nearby tetrodes
4 parameters defining the lattice are almost identical (reduced states)
2 parameters defining the Spatial Phase of the firing are the only really different

5. Stability over time of the cell response (single cell and cell-cell)
Record again after 1 hour in the original environment, after confusing the rat in different environments.
Cell – Cell phase difference between experiments is more stable than single cells.
“suggesting LD internal dynamics yokes the responses of the cells within the network, rather than ... external clues ”

6. Difference of parameter values under rescaling of the environment
Rescaling gives a dramatic change in the 4 grid parameters for single cells.
Ratio of grid parameters between pair of cells is almost constant.
Relative cell-cell phases remian constant

7. Response of the grid cell to novel environments
Is this 2D manifold stability really internal or imposed by environment?
Abrupt rescaling of parameters.... BUT
Relaxation of the novel environment parameters to the original familiar environment
Response of grid cells is stabilized via internal dynamics and not by external sensory inputs!

8. Response of the grid cell to novel environments
Is this 2D manifold stability really internal or imposed by hippocampal inputs?
Ratios of the parameters between cells during trials and days are always close to 1 and
Relative phase between cells remained fixed as they shrink their grid parameters
Instead place cells (related to hippocampal representation?) suffer a complete remapping and relationship with grid- cells is uncorrelated.
“2D manifold cannot easily be ascribed to external sensory cues or hippocampal inputs because relative phases and parameter ratios are stable even when these inputs are not.”

9. Is the 2D manifold continuous?
If stability of parameters is related to cell-cell similarity, the manifold is granular!
Plotting Parameter Ratios vs Dissimilarity in Space Shift gives the answer.
Stability is similarity- independent
Spearman rank and Pearson correlation close to 0.

10. Is the 2D manifold attractive?
Stable under internal stochastic dynamics (evaluated through spiking variability).
Pertubation through velocity inputs.
SOME HELP??

11. Relationship with other (and our) works
We also relate collective motion of N neurons with 2D attractor.
Interesting models are given in Ref , 22. (spin glass model, twisted torus topology)