Chapter 05 · Transport networks

Slime that builds networks

A yellow film creeps across a dish of oat flakes. It has no brain and no nerves. Within a day or so it has joined the flakes with a web of tubes that an engineer would not be embarrassed by.

The film is Physarum polycephalum, usually called a slime mould, though it is not a mould or any kind of fungus. In the stage people study, the plasmodium, it is a single cell: one enormous bag of cytoplasm holding many nuclei, with no walls between them. It can spread over many centimetres.

Inside, the cell is a network of tubes. Cytoplasm sloshes back and forth through them, reversing roughly every minute, so that a full back-and-forth takes a minute or two. Tubes that carry a lot of flow tend to thicken, and tubes that carry little thin out and are abandoned. Each bit of tube responds only to what is flowing through it.

In 2000 Toshiyuki Nakagaki, Hiroyasu Yamada and Ágota Tóth published a one-page paper in Nature called "Maze-solving by an amoeboid organism." They filled every corridor of a small maze with pieces of plasmodium, which fused into one cell, and put food at two exits. Over the next several hours the cell pulled its tubes out of the dead ends, leaving a thick tube joining the two food sources, usually along the shortest route.

The slime did not search the maze the way a mouse would. It already filled it, and it pruned: it started with every path at once and let the poorly used ones wither. That is a useful way to find short paths, but a different trick from the one the word "solve" calls to mind.

The rule

In 2010 Jeff Jones, a computer scientist, published a model that grows Physarum-like networks out of a crowd of particles. It does not simulate cytoplasm or flow. Its agents each leave a chemical trail and follow the trails of others, more like ants than like a cell.

The world is a grid holding a trail value in every square. Each agent has a position, a heading and three sensors a fixed distance ahead of it, one straight ahead and one angled to each side. Every step, each agent does four things:

Sense the trail under its three sensors. Turn toward the strongest: if the forward sensor is strongest, keep straight; if left or right is stronger, rotate that way by a fixed turn angle; if forward is the weakest of the three, turn left or right at random. Move one square forward. Deposit a little trail where it lands.

Then the trail map diffuses, each square becoming the average of itself and its eight neighbours, and decays, losing a fixed fraction everywhere. That is the whole model.

No agent knows where any network is. It knows three numbers. But every agent that follows a trail also adds to it, so busy trails get busier and quiet ones fade. In Figure 2, tens of thousands of agents start scattered at random and do exactly this.

What the dials do

Within a few seconds the scattered agents gather into strands, and the strands join into a lacy network of cords and cells. Press Jones 2010 to run the model with the values from Jones's paper. The lace doesn't last. Thin strands drain into thicker ones, and after a couple of thousand steps only a few fat cords are left. That echoes the organism, which also abandons most of its tubes, but it makes for a dull dish. So the other presets add a dial of our own that is not in Jones's model: wobble, a small random jitter in each agent's heading. It keeps agents spilling out of the big cords, and with about 25° of it the network stays a mesh.

Jones found that the sensor angle, the turn angle and how far ahead the sensors reach matter most. A short sensor distance gives a fine, crowded mesh (Fine mesh). A long one lets agents notice trails from far away, so the network has fewer, longer cords (Coarse cords). decay sets how long a trail is remembered, and low decay helps the network coarsen. Turn wobble to zero on any preset and watch the network prune itself.

There is no tube, no thickness, no flow and no food in the rule. A "cord" is just a place where many agents happen to be following one another.

The Tokyo experiment

A maze has one right answer. A rail system has to trade off goals that fight. It should be cheap, meaning short total track, and efficient, meaning short trips between stations. It should also be robust, so that cutting one link does not strand anyone.

In 2010 Atsushi Tero, Nakagaki and colleagues tested how Physarum handles that trade-off. They placed 36 oat flakes on a wet surface in the pattern of cities around Tokyo, with the slime starting at the flake for Tokyo itself. Because Physarum avoids light, they used lighting to mark mountains and coastline where it should not grow. As in the maze, the plasmodium spread out and then thinned its tubes down to a network.

On total length, travel efficiency and how well the network survived losing a random link, the slime's networks were comparable to the region's real railway. Some did a little better on one measure and worse on another. The paper also introduced a mathematical model in which tubes carrying more flow grow thicker. That flow model, not the agent model on this page, is the one built to explain the experiment.

The agents start everywhere at once, the way the plasmodium filled the maze. Trail is always strong where food sits, so agents gather there, and the cords between nearby sources are reinforced from both ends. Drag a source and watch new cords grow toward it as the old ones slacken. Press Remove all food and the network forgets the layout and goes back to the free-floating pattern of Figure 2.

The honest caveats

Jones's model imitates the look of a Physarum network, not its biology. Its agents are not pieces of a cell and its trail is not cytoplasm. A model that grows networks like Physarum does not prove that Physarum works like the model. This version also differs from Jones's original in two ways. It leaves out his rule that no two agents may share a grid square, and, except in the Jones 2010 preset, it adds the wobble described above.

"Solves mazes" needs the asterisk from earlier. The 2000 result is about pruning a maze the cell already filled.

The Tokyo comparison was made on specific measures. It does not mean that a slime mould could plan a railway, which answers to terrain, land prices and politics. What the experiment showed is narrower: a feedback rule that thickens busy tubes and starves idle ones finds good compromises between cost, speed and robustness, with no one weighing them up.

Five rules

Each chapter has been a version of the same small surprise. Sand grains that topple at four built a pile that tunes itself to the edge of collapse. Boids that steer by their neighbours made a flock that turns as one. Chemicals that react and spread at different speeds laid down spots and stripes. An ant that turns left or right by the colour of its square made a mess for ten thousand steps and then built a highway. And a crowd of particles that follow the strongest trail drew a network between the food. In none of them does any part see the whole, and in all of them the whole has a shape. That gap between what each piece knows and what the pieces make together is what this field guide has been about.

Try this

In Figure 3, press Remove all food, wait for the network to relax, then tap to add three sources in a triangle. Watch whether the agents connect them along the edges of the triangle, through a single junction in the middle, or both. Then slowly drag one corner away.

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