Familiar mathematical structure is found in the neural activity that governs how the body orients itself and navigates its environment. Grid cells are neurons, found in areas neighboring the hippocampus, whose individual firings line up in a coordinate-like pattern according to an animal’s movement across the full extent of its environment. Their grid pattern acts much like a coordinate system of locations and produces an abstract spatial framework or cognitive map of the environment. Partnered with place cells that code specific locations, border cells that identify the borders of an environment, and head direction cells that encode the direction we are facing, grid cells generate a series of maps of different scales that tell us where we are. John M. O’Keefe, May-Britt Moser and Edvard I. Moser received Nobel Prizes in 2014 for these groundbreaking observations. In an article for the journal Neuron, Neil Burgess makes the following observation:
Grid cell firing provides a spectacular example of internally generated structure, both individually and in the almost crystalline organization of the firing patterns of different grid cells. A similarly strong organization is seen in the relative tuning of head-direction cells. This strong internal structure is reminiscent of Kantian ideas regarding the necessity of an innate spatial structure with which to understand the spatial organization of the world.
More recently, researchers have expanded their results a bit by finding evidence that cells in this area of the brain are not entirely devoted to spatial considerations, but may also accomplish other cognitive tasks with similar action. Studies at Princeton University have investigated rats’ responses to sounds of ever-increasing frequencies on the hunch that the continuum of pitch could be compared to movement along a path. Neuroscientist Dimitriy Aronov and colleagues monitored the neuronal activity of rats, who were taught to increase the pitch of a tone being played over a loudspeaker, by pressing and then releasing a lever when the tone reached some predetermined frequency range. Cells that functioned as place cells or grid cells also fired in sequences as the rats moved through a progression of frequencies, and these were analogous to the sequences produced by their movement across the space of their environment. The significance of these findings for researchers is the light shed on what these firing patterns may have to do with memory and learning. In a Science Daily report, David Tank, one of the paper’s authors, says the following:
The findings suggest that there are common mechanisms in the hippocampal-entorhinal system that can represent diverse sorts of tasks, said Tank, who is also director of the Simons Collaboration on the Global Brain. “The implication from our work is that these brain areas don’t represent location specifically, but rather they can represent other relevant features of the animal’s experience. When those features vary in a continuous way, sequences of neural activation are produced,” Tank said.
Perhaps this is a bit of a leap but it seems to me that these findings also address the notion of a continuum, and might suggest that the generality these mathematics-like cognitive mechanisms possess resembles the kind of generality that purely mathematical ideas eventually come to exhibit. One could say that the body solved its fundamental need to navigate, to move and know where it is going, with a system that our thoughtful minds took centuries to put together. Our more willful efforts toward broadening our navigational sense perhaps begin with the ancient development of longitude and latitude. This kind of idea becomes more abstract and universal with the Cartesian coordinate system we learn in high school. And this is finally made most abstract with the vector space, which fully generalizes the notion of a coordinate. One could argue that the productive generalities of the brain’s coding mechanisms, or patterns of neuronal firing, foreshadow the generality and efficiency of the mathematical structures that they resemble.
In mathematics, the use of the notion of a coordinate was extended from being the geometric location of a point, to representing any kind of ‘variable’ or ‘parameter’ in countless physical problems. This happens when we make the transition from seeing the ordered pair of numbers (x, y) as the distance and direction from zero on the horizontal axis, and the distance and direction from zero on the vertical axis, to seeing it as an example of an n-tuple that represents any ordered list of numbers. The list can have some finite number of coordinates or it can be infinitely long. It is used to produce the tremendously useful notion of a vector space, which can be defined on that familiar cross of the x axis with the y axis, but has much broader meaning. The number of variables in the list defines the dimension of the space.
Vector spaces are used to address countless physical problems. The structure produced by our intuitive ideas of space, location and direction are effective in countless non-spatial systems. In its most general terms, a vector space is a collection of objects, that can be added together or multiplied by a number, and that satisfies a set of axioms about their arithmetic. The extent to which this generalization can be made allows the prolific application of these ideas. The collection of objects can be a collection of functions. In physics, the ‘state’ of a physical system is called a ‘state space,’ but a state space is a vector space. Vector spaces are used extensively in physics and in quantum mechanics in particular.
The broadening of a mathematical idea comes with consistently thorough and precise explorations of the meaning of concepts. And meaning comes from the relations among them, producing thought-filled weaves of ever increasing complexity. The generalizations of spatial ideas to non-spatial systems, or spatial metaphors that can clarify non-spatial notions, always captures my attention.
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