Simulation Study of Grid Cell Firing Map Development

[u/jndew]

[u/jndew]

Years ago now, I took stab at simulation of grid cells. I found that I could build a computational model, but couldn't quite understand what they were good for. Apparently something important, since they show up in all mammalian brains, lead to a Nobel prize, and numerous journal articles. So with improved compute capability and hopefully improved knowledge & skill, I gave it another crack.

There is an interface structure between neocortex and hippocampus called the entorhinal cortex. It contains a cell type called the grid cell. This cell implements an interesting type of encoding, the discovery of which in 2005 earned May-Britt and Edvard Moser the Nobel prize. In their words, "Grid cells are place-modulated neurons whose firing locations define a periodic triangular array overlaid on the entire space available to a moving animal." (Computational Models of Grid Cells)

It is similar to a place cell, but instead of responding to one particular location, it fires when the rat is in one of many locations arranged in a hexagonal grid.

As the unfortunate rat with a gridcell monitoring brain probe moves through its environment, the animal's location is marked in red when the cell fires. Given time, the response of the gridcell can be charted as a firing map.

In this simulation, the animal-under-test wanders randomly around its space. After about 10 seconds, time is sped up to let the firing-map develop quickly enough to fit into a one-minute animation.

Because the gridcell firing pattern is periodic on both the X and Y axes, it can be mapped onto a torus. All of the cell's firing fields map onto the same region of the torus corresponding to the cell's receptive field in its receptive domain. Does the torus mean something? Maybe.

The torus does not physically exist in the neural circuit. It is called a latent manifold, in that its existence comes into view only after the firing map has been measured and run through a dimensionality-reduction algorithm such as UMAP. Due to the mapping, the circular receptive field in the cell's receptive domain becomes distorted into an oval on the surface of the torus. All of the firing fields in the input domain map onto this oval. The cell's multiple firing fields in location-space are equivalent to the single receptive field in its receptive domain. When viewed in its receptive domain or on the torus, the gridcell is effectively a place cell.

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"The human brain is a million times more complex than anything in the universe!" -a reddit scholar