The mathematical nature of self-locating

A 2011 TED talk in London was brought to my attention recently. The speaker, Neil Burgess  from University College London, spoke on the topic, “How your brain tells you where you are.” Burgess investigates the role of the hippocampus in spatial navigation and episodic memory. In the talk he describes the function of what are called place cells, boundary-detection cells, and grid cells. I wrote (also in 2011) about a Science Daily report on studies dedicated to understanding how hippocampal neurons represent the temporal organization of experience, in particular, how they bridge the gaps between events that are not continuous. But here I want to focus more on how the brain constructs spatial experience.

Electrophysiological investigations have identified neurons that encode the spatial location and orientation of an animal. Among these are those that have come to be called place cells, head direction cells, and grid cells. Burgess also talked about boundary detection cells in his 2011 TED talk, which seem to be coordinated with head direction. I was particularly struck by the clarity of some of the data images he presented. He showed his audience images produced by the firing of boundary detection cells in response to boundaries in the environment of a rat. In one of them we could see cell firings in response to one of the walls in the rat’s environment. In another, created after a second wall had been added to the environment, the firing was duplicated with respect the added wall. Burgess also presented the image of cells that fired when the rat was about midway between the walls, and one could see directly that when the rat’s box was expanded, the firing locations expanded. These boundaries needn’t be rectangular walls. They can be the drop at the edge of a table or the circular wall of a circular box.

Grid cells are creating representations a little differently, in a quasi-mathematical way, as their name suggests. Burgess tells about the rats again:

Now grid cells are found, again, on the inputs to the hippocampus, and they’re a bit like place cells. But now as the rat explores around, each individual cell fires in a whole array of different locations which are laid out across the environment in an amazingly regular triangular grid…So together, it’s as if the rat can put a virtual grid of firing locations across its environment — a bit like the latitude and longitude lines that you’d find on a map, but using triangles. And as it moves around, the electrical activity can pass from one of these cells to the next cell to keep track of where it is, so that it can use its own movements to know where it is in its environment.

Both boundary detection cells and grid cells reflect a sensory perception of the environment. But neurons also encode movement from proprioceptive information that can be used to measure the body’s displacement as we move (path integration). This is not movement defined by the environmental changes that occur, but from the body’s sensations of itself.
In a more recent paper Burgess and co-author C. Burgess  describe an interesting test of the errors that can be produced by the iterative neural processing of self-motion.

This process, known as path integration or dead reckoning, requires the animal to update its representation of self-location based on the cumulative estimate of the distance and direction it has traveled. It can be shown that an animal is utilising path integration by introducing a known error into its representation of direction or distance: in the case of the gerbils, if they are rotated prior to the return leg of the journey, and this is done slowly so that the vestibular system does not detect the motion, then the animals head towards the nest with an angular error equal to the amount they were rotated by.

So when we try to find our way back to something, like where we parked the car, we likely use boundary-detecting cells to remember distances and directions to buildings and boundaries, but we also remember the path we took, represented by the firing of grid cells and path integration. The interaction of these things seem to contribute to the pattern of neural firings that become associated with a particular place, a cognitive map of that place, formed by what are called place cells. There is a nice discussion of the history of the study of place cells which includes a number of images at There the point is made that the ‘cognitive map’ defined by place cells is a ‘relation among neurons,’ not among points in space.

In brief, we can think of the “map” of a session in terms of space (the spatial relations of firing fields) and time (the tendency for pairs of cells to fire together or not). Since the speed of rats is restricted, these are essentially equivalent. An important concept is that the map is entirely in the brain. In this description, a map is defined by the relation among hippocampal neurons, not by the relationships between neurons and the environment. The linkage to the environment is critical, but does not define the map.
The temporal relations are important for two reasons. First, neurons in the brain do not know about space directly, but they know about time. Neurons can code the timing relations of the neurons that project to it, but not the spatial relations. In other words, within the brain, the map is a timing map that encodes the temporal overlap between cell pairs.


There are a few interesting things going on here. No doubt the grid cell idea and the vector-like measures of displacement that are encoded when we move around, trigger memories of mathematics. They are like our mathematical analyses of the 3-dimensional space of our experience. Place cells, on the other hand, are like another level of abstraction. They seem to have more in common with coding and non-spatial analyses, even though we don’t seem to know how they do what they do. The neurons that fire to represent a particular location have no spatial relationships among themselves. Neighboring place cells do not indicate neighboring environmental areas. And while correlated to sensory input, they are part of a non-sensory system. It is the integration of this system with the neural representations of boundaries, direction, and distance (among other things) that create our spacial awareness. Certainly sensory information is being subject to some kind of transformation. Spatial relations are translated into what look like purely temporal ones (the timing of neuron firing). The non-sensory system then stores a coded representation of a sensory one. Here again we see, not the mathematical modeling of brain processes but more their mathematical nature.

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