I read a short article on scientificamerican.com reporting on a recent advance in the investigation of the neural systems that support navigation, or our sense of direction. When I did some follow-up on the individual who led the study, I was surprised to find another interesting collaboration between scientists and artists. While the collaboration was centered on inquiries into perception, memory, and space, it touched on things related to mathematics – at least in its discussions of space, dimension and direction. Both the study and the collaboration make some interesting points. I’ll start with the study.
It was led by Hugo Spiers of University College London. Spiers found something new in the action of head-direction cells – neural cells that fire when we face a certain direction. These cells have been known to play a role in our ability to navigate through our environment, working with place cells in the hippocampus (that establish our memory of specific locations and a kind of map of the environment) and grid cells in the adjacent entorhinal cortex (that somehow map where we are relative to where we have just been). What researchers were able to observe was that head cells also fired in response to the direction we wanted to go.
The entorhinal region displayed a distinct pattern of activity when volunteers faced each direction—consistent with how head-direction cells should behave. The researchers discovered, however, that the same pattern appeared whether the volunteers were facing a specific direction or just thinking about it. The finding suggests that the same mechanism that signals head direction also simulates goal direction.
It might help to describe the whole system as it is currently understood. Spiers and co-author Caswell Barry provide a nice description of the interaction of the cells that function in navigation in a recent paper.
Electrophysiological investigations have revealed several distinct neural representations of self-location (see Figure 1 and for review ). Briefly, place cells found in hippocampal regions CA3 and CA1 signal the animal’s presence in particular regions of space; the cells’ place fields  (Figure 1a). Place fields are broadly stable between visits to familiar locations but remap whenever a novel environment is encountered, quickly forming a new and distinct representation 17 and 18]. Grid cells, identified in entorhinal cortex, and subsequently in the pre-subiculum and para-subiculum, also signal self-location but do so with multiple receptive fields distributed in a striking hexagonal array 19 and 20] (Figure 1b). Head direction cells, found throughout the limbic system, provide a complementary representation, signalling facing direction; with each cell responding only when the animal’s head is within a narrow range of orientations in the horizontal plane (e.g. , Figure 1c). Other similar cell types are also known, for example border cells which signal proximity to environmental boundaries  and conjunctive grid cells which respond to both position and facing direction . It is likely that these spatial representations are a common feature of the mammalian brain, at the very least grid cells and place cells have been found in animals as diverse as bats, humans, and rodents .
What first struck me about the work reported in the Scientific American piece was that this navigation system, which looks fairly mechanical, has another layer at least – one that equates the direction faced with one’s intent to face it. The head cells respond to direction despite the fact that the head itself does not. From the paper on which the Scientific American piece was based:
In summary, we show that the human entorhinal/subicular region supports a neural representation of geocentric goal direction. We further show that goal direction shares a common neural representation with facing direction. This suggests that head-direction populations within the entorhinal/subicular region are recruited for the simulation of the direction to future goals. These results not only provide the first evidence for the presence of goal direction representations within the mammalian brain but also suggest a specific mechanism for the computation of this neural signal, based on simulation.
When I looked further into Spier’s research, I found links on his University College London website that provided info on work associated with art and architecture and his collaboration with artist Antoni Malinowski. In an interview that Spiers conducted with Malinowski, Malinowski talked about his own work, distinguishing it from the work of architects. Architects, he said, deal with space diagrammatically. In contrast, he explained, he dealt with space in a reduced way. His subject is the interaction of dimensions – the three and four of space and time and the two of a flat surface. He proposed that dimensions are foldable and that when he worked, he folded four dimensions into two with brushstroke and paint. These are then ‘unfolded’ in the viewing. This sounds like an inquiry, an investigation of the nature and perception of dimension.
Malinowski describes how he works:
I create a situation where you do not know where you are, and you don’t know what it is. So you have to make an effort. I want to take you to a mental area. And in order to do so I have all those tools, which are colour, rather delicious, and wonderful. So you are drawn into them. And I construct it in such a way that you want to go there.
So as viewer you notice something and you go off… But it is all done in a language of painting it is not really definable.
A review of his work by Mark Rappolt says this:
his work escapes the canvas to cover a building’s walls, Malinowski exploits architecture not as a singular fixed entity, but as a plurality of possible worlds, as an illusory reality, a space of shifting sand. Perhaps in doing this he comes closer than many architects to an understanding of what space really is. (emphasis added)
Malinowski is playing with perception and orientation, perhaps to reveal something about it. His work seems to surprise the viewer, but it’s telling us something about ourselves and how we make things sensible, something we can’t see in our day-to-day experience. Looking at the development of mathematics from it’s more familiar, more physical roots to its strange and powerful abstractions can do something similar. The investigation of what one means by ‘space’ in mathematics (Euclidean and non-Euclidean, the manifold, topological spaces, parameter spaces, etc.) has produced some of its most effective applications. Mathematics contains more than one definition of dimension, each of which produce its own results. And the vector, the mathematical description of direction, finds its way into the geometry of relativity, the phase evolution of a wave, the calculation of probabilities, the spin of fundamental particles, to name just a few. It seems clear to me that mathematics is a very thorough investigation of experience while also becoming diassociated from it. The work of building mathematics is much larger, intergenerational, shared, and more universal than Malinowski’s individual investigation of perception and orientation, but I find in his a similar inclination to pry open familiar experience to find something new.