… and dilation structures might exist, physically, in some parts of the brain. (See also section 2.4. in “Computing with space …” arXiv:1103.6007 .) I will surely come back to this subject, after learning more, but here are some facts.
The primary source of this post is the article “From A to Z: a potential role for grid cells in spatial navigation” Neural Syst Circuits. 2012; 2: 6. by Caswell Barry and Daniel Bush.
From wikipedia entry for grid cell:
A grid cell is a type of neuron that has been found in the brains of rats, mice, bats, and monkeys; and it is likely to exist in other animals including humans. In a typical experimental study, an electrode capable of recording the activity of an individual neuron is implanted in the cerebral cortex of a rat, in a part called the dorsomedial entorhinal cortex, and recordings are made as the rat moves around freely in an open arena. For a grid cell, if a dot is placed at the location of the rat’s head every time the neuron emits an action potential, then as illustrated in the adjoining figure, these dots build up over time to form a set of small clusters, and the clusters form the vertices of a grid of equilateral triangles. This regular triangle-pattern is what distinguishes grid cells from other types of cells that show spatial firing correlates. By contrast, if a place cell from the rat hippocampus is examined in the same way (i.e., by placing a dot at the location of the rat’s head whenever the cell emits an action potential), then the dots build up to form small clusters, but frequently there is only one cluster (one “place field”) in a given environment, and even when multiple clusters are seen, there is no perceptible regularity in their arrangement.
So, there are place cells and grid cells. Here is what wikipedia says about place cells:
Place cells are neurons in the hippocampus that exhibit a high rate of firing whenever an animal is in a specific location in an environment corresponding to the cell’s “place field”. These neurons are distinct from other neurons with spatial firing properties, such as grid cells, border cells, barrier cells, conjunctive cells, head direction cells, and spatial view cells. In the CA1 and CA3 hippocampal subfields, place cells are believed to be pyramidal cells, while those in the dentate gyrus are believed to be granule cells.
The behaviour of these cells is explained in the Figure 1 from the mentioned article by Barry and Bush:
(Copyright ©2012 Barry and Bush; licensee BioMed Central Ltd.)
The explanation of the Figure 1 reads:
Single unit recordings made from the hippocampal formation. a) CA1 place cell recorded from a rat. The left-hand figure shows the raw data: the black line being the animal’s path as it foraged for rice in a 1m2 arena for 20minutes; superimposed green dots indicating the animal’s location each time the place cell fired an action potential. Right, the same data processed to show firing rate (number of spike divided by dwell time) per spatial bin. Red indicates bins with high firing rate and blue indicates low firing rate, white bins are unvisited, and peak firing rate is shown above the map. b) Raw data and corresponding rate map for a single mEC grid cell showing the multiple firing fields arranged in a hexagonal lattice. c) Three co-recorded grid cells, the center of each firing field indicated by a cross with different colors corresponding to each cell. The firing pattern of each cell is effectively a translation of the other co-recorded cells as shown by superposition of the crosses (right). d) Changes made to the geometry of a familiar environment cause grid cell firing to be distorted (rescale) demonstrating that grid firing is, at least, partially controlled by environmental cues, in this case the location of the arena’s walls. Raw data are shown on the left and the corresponding rate maps on the right. The rat was familiar with the 1 m2 arena (outlined in red). Changing the shape of the familiar arena by sliding the walls past each other produced a commensurate change in the scale of grid firing. For example, shortening the x-axis to 70cm from 100cm (top right) caused grid firing in the x-axis to reduce to 78% of its previous scale, while grid scale in the Y-axis was relatively unaffected. Numbers next to the rate maps indicate the proportional change in grid scale measured along that axis (figure adapted from reference ).
And now, the surprise: scale is indeed a place in the brain. Let’s see Figure 2. from the same article:
(Copyright ©2012 Barry and Bush; licensee BioMed Central Ltd.)
The caption of this figure is:
Grid scale increases along a dorso-ventral gradient in the mEC. Two grid cells recorded from the same animal but at different times are shown, both cells were recorded in a familiar 1m2 arena. Approximate recording locations in the mEC are indicated. The more ventral cell exhibits a considerably larger size of firing fields and distance between firing fields than the dorsal cell.
… and, from the article, (boldfaced by me):
The scale of the grid pattern, measured as the distance between neighboring peaks, increases along the dorso-ventral mEC gradient, mirroring a similar trend in hippocampal place fields [15,25]. The smallest, most dorsal, scale is typically 20 to 25cm in the rat, reaching in excess of several meters in the intermediate region of the gradient [15,26] (Figure (Figure2).2). This may explain how this remarkable pattern was missed by early electrophysiology studies, which targeted ventral mEC and found only broadly tuned spatial firing (for example, ). Interestingly, grid scale increases in discontinuous increments and the increment ratio, at least between the smaller scales, is constant . Grid cells recorded from the same electrode, which are, therefore, proximate in the brain, typically have a common scale and orientation but a random offset relative to each other and the environment . As such, their firing patterns are effectively identical translations of one another and a small number of cells will ‘tile’ the complete environment (Figure (Figure1c).1c). It also appears that grids of different scale recorded ipsilaterally have a common orientation, such that the hexagonal arrangement of their firing fields share the same three axes, albeit with some localized distortions [15,28,29].
That’s just amazing!
Concerning the hypothesis (Hafting, T.; Fyhn, M.; Molden, S.; Moser, M. -B.; Moser, E. I. (2005). “Microstructure of a spatial map in the entorhinal cortex”. Nature 436 (7052): 801–806. Bibcode:2005Natur.436..801H. doi:10.1038/nature03721.) that the grid cells firing fields encode the abstract structure of an euclidean space, I think this is not following from the observations. My argument is that the translation-invariance (of firing patterns in this particular case) emerge by the mechanism of dilation structures and it is, at least up to my actual understanding, an evidence for the existence of these structures in the brain. But of course, there is much to learn and think about.