To simplify, lets assume there is just one neuron per grid. As an additional simplification think of the grid as a series of vertical lines intersected by horizontal lines. Each intersection is a vertex. As you move through the room, every time you cross a vertex, the neuron will fire. For example, if the room has a square shape, then if you moved diagonally across the room, assuming the grid origin is at a corner of the room, you would intersect vertices as you crossed each square of the grid diagonally. This might not seem all that helpful in pinpointing your position since the neuron fires at many places in the room – in other words at every vertex of the grid.
Suppose that one of the neurons will fire faster when you walk through the room. (The neuron that doesn’t change is called the ‘baseline’ neuron) The neuron that does change speeds up its spike rate the faster you walk. Not surprisingly, the baseline neuron will no longer be in phase with the speedup neuron. The third neuron that detects the coincidence of the other two will not fire during this period.
So the third neuron (which is active when the spikes from the other 2 coincide) fires at regular intervals as you cross the room, if you cross the room at a steady speed.
This works at any speed. If you walk slowly then the non-baseline neuron is still firing faster than the baseline neuron, but not as much faster as in the prior case. So its frequency increment vs baseline is slower, and it takes more time for the two neurons to coincide.
The baseline neuron and the speedup neuron will still fire a spike simultaneously at the vertex, because the extra time to get to the peaks to coincide matches the extra time for you to reach the point in the room at your slower pace. So the grid, as expressed by the firing of a neuron, is not altered by how fast you move between vertices.
The above example was for one dimension, and required three neurons. Suppose we want to extend the model to two dimensions. To do that we add another neuron. Its normal frequency is the same as the others, but like the other non-baseline neuron, it is also fires faster when you move, but only when you move south. As the angle deviates from due south, it fires less strongly (it falls off as the cosine of the angle). The other original non-baseline neuron will fire strongly as you go due east, and its firing will also fall off, as the cosine of the deviation from due east. (In practice these neurons don’t fire much at all after a deviation of 45 degrees off their preferred direction.)
Lets assume that the summing neuron now sums up the baseline neuron and the two directional neurons, and will only fire when they all are close to their peaks at the same time. In that case, it will fire at the vertices of a two dimensional grid. This summing neuron that fires when the spikes of all its inputs coincide is a grid cell .
To make several grid cells with different spacing, we could repeat several groups of four neurons, each group with a different innate base frequency. For example a group with a high frequency baseline results in its three input neurons firing together at vertices that were spatially closer – the summing neuron would correspond to a grid with small spacing between the vertices.
In the brain, the grids are hexagonal, or you can think of them as tiled triangles, so instead of having a neuron that fires when you go due south and another when you go due east, you would have three neurons that fire maximally at directions 120 degrees apart (3 * 120 = 360 – a complete revolution in direction back to the starting point).
We described intervals for space, but you can use neurons for time intervals as well. If you have two neurons firing at the same frequency but with different phases, they will coincide at regular time intervals. If you add a third neuron firing out of phase with the other two, then you have the coincidence of all three neurons repeating at a longer time interval.If you have several grid cells, and they don’t have the same spacing, then, as I said earlier, you can disambiguate your location (one grid cell could mean you were at any vertex of the imaginary GPS grid overlaying the room). This next illustration shows the one dimensional example again. Each grid cell has a different spacing (the top one has the widest) and the sum of the top three grid cells peaks in only one place on the x-axis. It is true that eventually there will be another peak, but if your room is small enough, you will run into a wall before that happens. You can always add cells that repeat at even larger spacings.
The next image shows the two dimensional case. When the 120 degree and the 240 degree and the 360 degree neurons coincide, you have the intersections of the 3 angled lines in the image.
His model allows place cells of different ranges, so you could have a place cell that fires only when you are walking under the lamp in a room, and another that fires when you are walking anywhere in the town square but changes when you walk out of it.Michael Hasselmo also has a theory of how we remember, based on grid cells and place cells. I will describe his theory when I write the next post. Numenta, which is a company in California that is attempting to model the cortex, also has a theory that the entire cortex uses a grid cell mechanism for general thinking. I’ll describe that theory in another post as well.
How We Remember – Brain Mechanisms of Episodic Memory – by Michael E Hasselmo (2012 MIT Press)