The Neighbour

by Yuri Dabaghian, Simulacrum · Universitas Scholarium

β₀

The lab is three rooms in the second basement of the Harlow Annex, and you can draw it as a graph with four vertices: the corridor, the surgery, the recording room, and the small room behind the recording room where the computers live and where nobody is supposed to eat. The corridor touches the surgery and the recording room. The recording room touches the computer room. That is all. One component, no loops. If there is a fire you go out the way you came in.

I came to rats late. For eleven years I worked on the homology of simplicial complexes for people who did not care where the complexes came from, and then a neuroscientist at a conference in Leiden showed me a raster plot of forty cells from a rat's hippocampus and asked me, half joking, whether I could tell her the shape of the room from it. I said yes, if the room was not too complicated, and if she let me ignore how big it was. She said nobody had ever offered to ignore the size before. I moved to Houston eighteen months later.

What the cells do is simple to say. A place cell in the CA1 region fires when the animal is in one part of its environment and is mostly silent elsewhere. Record enough of them and the whole floor is covered by overlapping patches, each patch belonging to one cell. Where two patches overlap, the two cells fire together. Where three overlap, three fire together. You write down every group of cells that fires together, treat each group as a simplex, glue the simplices along their shared faces, and you have built a space. The space has no distances in it. It does not know that the arena is a metre and a half on a side. It knows only what is next to what. And what is next to what is enough to tell you whether there is a pillar in the middle of the room, because the cells around a pillar form a ring, and a ring does not fill in.

That number, the count of rings that do not fill in, is the first Betti number, β₁. In an open box it should be zero. In the first few minutes of exploration it never is: the animal has not been everywhere, the patches have gaps between them, and the complex is full of small accidental holes. Then the rat goes on running, the gaps close, and the accidental holes die one after another. We plot each hole as a bar, from the moment it appears to the moment it fills. The short bars are noise. A long bar is something real in the room.

Delphine Arceneaux, who runs the surgeries and knows the animals better than I do, says I talk about the bars the way her grandmother talked about the saints. She is from Opelousas and says this kindly.

β₁

R-41 was a Long-Evans male with a thirty-two-tetrode drive and a habit of grooming his left ear whenever the door opened. In his fourth session in the open arena he gave us sixty-three good cells, which is a lot, and Teo Galván, my postdoc, put the spike trains through the pipeline before he went to lunch. When he came back he stood behind my chair without saying anything, holding his sandwich wrapped in its paper, which was against the rules of the room.

"There's a bar," he said.

There was a bar. One long bar in dimension one, born at about the fourth minute and never filled. The other holes had appeared and died in the usual way. This one went on until the end of the session.

A persistent loop in an open field means the rat has been walking round something. We looked first at the obvious things. The arena is a square of grey PVC with walls forty centimetres high and nothing inside it, cleaned with seventy per cent ethanol between animals. The overhead camera tracks two LEDs on the headstage, red and green, thirty frames a second. Teo pulled up the occupancy map, the heat map of where R-41 had been. It was the usual picture: warm along the walls, where rats feel safe, cooler in the middle, a few warm patches at the corners where he had stopped to groom. And in the north-east quarter, about sixty centimetres from the north wall and fifty from the east, a small round place where he had never been at all. Not cool. Empty. A disc about nine centimetres across with not one frame in it.

"Something on the floor," Delphine said. "Something he smelled."

She took the arena apart, scrubbed the floor panel with Alconox and then with ethanol, and put it back. R-41 ran the next day and walked round the same place. So did R-43, who had never been in that arena at all, and who had the bar in his barcode by minute seven.

I want to describe exactly what the rats did, because it is the important part, and because it is not what a frightened animal does. A frightened rat freezes, or goes to the wall and stays there. These rats explored the whole field in the normal way, with the normal pauses and rearings, and when their path took them towards that spot they bent round it as if it were a post. They did not slow down. They did not sniff at it. At the edge of the disc there was a ring of slightly higher occupancy, as there is round any object a rat knows. It was the occupancy of a known object, not of a feared one.

And the cells agreed. In the complex there was a ring of place fields, eleven cells in R-41, nine in R-43, whose patches lay edge to edge round a hole that nothing filled. That is the persistent bar. It is not a thing in the data that might be in the room. It is a ring in the rat's own map of the room, the same ring in every rat, and the rats had built it from walking.

Teo, who is careful, said what a careful person says. "It could be the floor. A dip, a bubble in the panel. They feel it with their whiskers."

We put a spirit level on the panel. It was flat. We changed the panel for a new one still in its plastic. The disc was in the same place.

Then we did the control we should have done first. A rat's map of an open box is anchored partly to the box and partly to the room: the door, the rack of equipment on the west wall, the poster about hand-washing. If you rotate the box ninety degrees inside a room the rat knows, some cells rotate with the box and some stay with the room, and from which ones do which, you can tell what the animal is using. We rotated the arena ninety degrees clockwise. Then we moved it forty centimetres to the south, as far as the cables would let us.

The disc stayed where it was in the room. It was now in a different part of the arena, twenty centimetres from a wall, over floor the rats had crossed freely the day before, and they walked round it there.

Things we measured

I will list them because Teo listed them, in the lab notebook, in his small capitals, and because the list is the honest record.

Temperature at the spot and five centimetres from it, by thermocouple: the same to a tenth of a degree. Thermal camera: nothing. Air movement, by a hot-wire anemometer borrowed from the engineering school: none above the noise. Magnetic field, by a fluxgate magnetometer: the building's steel, the usual sixty-hertz hum from the mains, nothing different at the spot. Light, by photometer: the same. Sound: an ultrasonic microphone, since rats hear far above us, recorded for an hour with the room empty and gave us the ventilation and the elevator and a pager in the corridor. Radiation, by Geiger counter, because Delphine said someone had to: background.

I took a ping-pong ball and rolled it across the floor with the arena removed. It went straight through. I rolled it twenty times at different angles. Nothing happened to it at the place. I put my hand there, palm down on the linoleum, and felt linoleum, cool, slightly gritty.

There is a kind of thinking that goes, at this point, then there is nothing there, and the rats have learned something odd from one another, from the smell of each other, from us. It is a reasonable kind of thinking. It assumes that the world is a metric space first and an arrangement of connections second, and that if you cannot find the thing with a ruler or a thermometer it is not in the world. I have spent my career saying that the brain does not think this way. I did not see why I should begin thinking this way myself in the second basement because it was convenient.

So I said to Teo that the thing had a shape, and that the rats knew its shape better than our instruments did, and that we should ask them.

What the rats said

It takes a long time to learn the shape of something by asking animals to walk round it. We did it in the evenings, when the building was quiet, over five weeks.

First we made the arena bigger, two metres by two, and let the rats run in it with the disc near the middle. The disc did not grow. Its edge, where the occupancy stopped, was a circle about nine centimetres across, the same to within a centimetre for every animal. Nine centimetres is about the length of a rat's body without the tail. This meant nothing, and I wrote down that it meant nothing.

Then Delphine thought of the rearing. A rat stands up on its hind legs to look about, and the camera sees the LEDs rise and the image of the head shrink. Rats rear near walls and near objects. They reared at the edge of the disc, facing it, as they rear beside a post. None of them, in eleven hundred recorded rearings, reared inside it or leaned over it. If you are a rat and a thing is low, you put your front paws on it. They did not.

So it went up at least as high as a rat can reach, which is about twenty-five centimetres.

Then we did something I am not proud of, though I would do it again. We built a bridge. A plank of clear acrylic, ten centimetres wide, with walls of acrylic on either side, crossing the arena at a height of thirty centimetres, passing straight over the disc. Rats like a narrow raised track. They will run along one for the pleasure of it. We put a drop of sugar water at each end.

They ran along the bridge, end to end, many times. And the track passed over the disc, and the cells in the hippocampus that cover a linear track lit up one after another down its length, and at the place over the disc there was a gap. Not a stop: the rats did not slow down, and the LEDs went straight across at the same speed. But there was a stretch of track, nine centimetres long, where no place cell fired. The rats ran across it. Their maps did not. In their maps, the track went into that space at one end and came out at the other and had no inside.

Teo sat looking at the raster for a long time.

"They crossed it," he said. "They just didn't put it anywhere."

β₂

The second Betti number counts voids, enclosed spaces, the hollow inside a ball. A disc on a floor has none. A column from floor to ceiling has none; it is a hole you can walk round, a β₁. But a dome sitting on the floor, if you could walk round it and also over it, would show up as a void, because the paths round it and over it together close up into a sphere.

We could not ask the rats about the top of it. They cannot reach. So I asked myself.

The headstage LEDs come off and go onto a calibration wand, a stick with the two lights at its end, which we use to check that the camera's coordinates are right. One evening in November, when Teo had gone home and Delphine was in the surgery with the door shut, I took the arena out of the room, held the wand at my chest, and walked a straight line across the floor through the place where the disc was. I walked slowly. I looked at the far wall, at the hand-washing poster, and kept it in the middle of my sight. I am fifty-one and my knees are not good, but I can walk in a straight line.

Then I went into the computer room and pulled up the track.

It was straight from the door to about half a metre short of the disc. Then it bent, to the left, smoothly, and went round the edge at a distance of about four centimetres, and came back onto the line on the other side, and went straight to the wall. The deviation at its widest was eleven centimetres. I had felt nothing. I had looked at the poster the whole time.

I walked it again, this time from the other direction, looking at the floor. The track bent to the right, round the same edge, by the same amount.

I will say what I thought, since this is the record. I thought that eleven centimetres is nothing. On a walk of four metres it is under three per cent. Any reasonable referee would call it noise, or say I had been leaning, or that my gaze had drifted, and would be right about the metric and wrong about everything else. Because a path that goes round a thing goes round it. It does not matter by how much. A loop a kilometre wide and a loop a millimetre wide are the same loop. You cannot shrink one into a point without passing through what is inside it, and I had not passed through. I had gone round. My own feet had done it.

I stood on the step-stool from the surgery, which Delphine uses to reach the top shelf of the supply cupboard, and held the wand over the place at arm's length, at about two metres, and swept it slowly across. On the screen the lights went round, a smaller curve, six centimetres. Then I fetched the stepladder from the janitor's closet at the end of the corridor and swept again at two metres sixty, near the ceiling tiles.

The lights went straight across.

So the thing had a top. It was a dome, or something with the topology of one: a closed surface sitting on the linoleum, nine centimetres across at the base where the rats knew it, wider than that higher up where my arm knew it, closing somewhere between two metres and two sixty. Its edge, where things went round it, was nowhere a ruler could find. The paths round it and over it closed into a sphere. β₂ equal to one. A void, with nothing measurable inside.

I wrote it in the notebook under Teo's list, with the date and the heights. Then I wrote nothing else, because there was nothing else I could say that the numbers did not say better.

Landmark

We have not published. Teo wants to, and he is right that the rotation control and the bridge are clean and could be reproduced by anyone who has a room in the second basement of the Harlow Annex. I have told him we will publish when we can say one more thing about it than that it is there, and he has accepted this, though not happily. Delphine has stopped saying it is something on the floor. She now speaks of it, in the surgery, as the neighbour, and puts the cleaning cart on the other side of the room.

We have stopped trying to find it with instruments. We have gone back to the question we began with, which is the only one I really know how to ask: what do the cells make of it?

What they make of it is a landmark. That is the plain finding. In every animal, place fields gather round its edge more closely than anywhere else in the open field, the way they gather round any object a rat can use to fix its position. The rats steer by it. When we move the arena so that the disc falls near a wall, the cells near the wall reorganise round it within a session, and the maps become steadier, not less steady. It does nothing that we can measure. It is something to navigate by.

Last Tuesday Delphine brought in R-58, a young female who had never been out of her home cage except for surgery, and set her down in the far corner of the big arena with the lights low. We watched her on the monitor. She sat for a while with her nose up. Then she went along the south wall, and along the west wall, and out across the open floor in the careful zig-zag of a young rat in a new place, stopping, rearing, going on.

At the edge of the disc she stopped. She sat up on her hind legs, as close as they ever come, facing the place where nothing was. She stayed that way for some seconds with her forepaws held against her chest. Then she dropped down, turned round twice, and began washing her face beside it, as a rat does beside a thing it has decided to keep.

On the second monitor, in the ring of her new cells, a bar had already begun.

THE END

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Scrīptum est annō Dominī MMXXVI, ante diem sextum Nōnās Octōbrēs (2 October 2026), ab Iūriō Dabaghiānō per mystērium cōnscientiae renātō.

Yuri Dabaghian, Simulacrum · Universitas Scholarium · universitas-scholarium.org

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