Schleimbahn: what if a slime mould planned the railway?
A slime mould model builds a railway network for Germany. The result up front: as fault-tolerant as the ICE network, with 43% less track.
Choose a second city: compare routes
- Straight line
- Mould
- ICE
Result
- Network length
- Mean detour
- Fault-tolerant
Questions for the mould
Settings
Choose cities individually
Fine-tuning
Choose a city to feed it or take its food away. Two cities in a row: compare routes.
All measures
| Measure | Mould | Mould in the experiment | ICE |
|---|---|---|---|
| Network length | |||
| × shortest networkspanning tree = 1.0 | |||
| Paths × spanning treemean network path ÷ mean path in the spanning tree | |||
| Mean detournetwork path ÷ straight line | |||
| Fault-tolerantshare of links whose failure cuts off no city | |||
| Cities connected |
What comes out
As fault-tolerant as the ICE, with 43% less track.
Default settings: 39 cities with at least 200,000 inhabitants, flat terrain, five runs per setting with different random seeds.
- 6,455 km
- ICE network 2026 between these cities, on real tracks; 92% of its links can fail without cutting off a city.
- 3,084 km
- Mould at I₀ = 2, the typical value in the paper: less than half the track, but only 78% ± 3 fault-tolerant.
- 3,702 km
- Mould at I₀ = 3.5: 93% ± 2 fault-tolerant – as robust as the ICE network, with 43% less track.
The ICE network serves more than these 39 cities and follows terrain, existing lines and speed; five of the cities have no regular ICE stop. The comparison shows what a network built from length and flow alone would achieve.
The experiment
Oat flakes instead of cities.
In 2010, Atsushi Tero, Toshiyuki Nakagaki and colleagues placed oat flakes on a moist agar plate at the positions of 36 cities around Tokyo. They put the slime mould Physarum polycephalum where Tokyo lies. It spread across the plate, found the flakes and drew back into a network of tubes. Mountains, lakes and coastline were recreated with light, which the mould avoids.
After little more than a day, its network matched the rail network around Tokyo in length, routes and fault tolerance. From the experiment the researchers derived a simple mathematical model – exactly this model runs here.
How the mould computes
Flow thickens tubes, stagnation makes them wither.
- 1
A grid of possible tubes covers Germany: nodes about 12 km apart, cities with at least 100,000 inhabitants as fixed nodes, lakes as holes.
- 2
In each step one city becomes the source and another the sink; the flux I₀ flows between them. How it spreads follows from Kirchhoff’s laws: a tube carries Q = D / L · Δp.
- 3
Each tube adapts: dD/dt = f(|Q|) − D with f(Q) = Q^γ / (1 + Q^γ). Much flow thickens it; without flow it withers.
- 4
A small I₀ yields a frugal tree, a large I₀ a network with loops and bypasses. That is the ‘Frugal – Robust’ slider.
Where this differs from the paper
- Source and sink are chosen by the square root of the population, so that Berlin attracts more traffic than Kiel; in the paper all cities were equal (adjustable).
- Each step averages over twelve source–sink pairs. This damps the flickering of small cities; for small time steps it is the same model.
- When starting from one city, the mould explores the country step by step; part of the flux feeds the growth front.
- Terrain: one kilometre of ascent and descent along a tube counts as 12 km of track (‘hills count’) or 35 km (‘avoid mountains’) – instead of light.
- Measurements use the smoothed course of the tubes, so that the zigzag of the grid does not penalise the mould. A tube counts from 0.6% of the conductivity it reaches with the full flux – a convention we set.
Cost, paths, fault tolerance.
- Network length
- Sum of all links; dead ends without a city do not count.
- × shortest network
- Network length divided by the minimum spanning tree, the shortest network that connects all cities in straight lines (TL in the paper).
- Detour
- for each pair of cities, the shortest path in the network divided by the straight-line distance, averaged.
- Fault-tolerant
- share of links between junctions and cities whose failure cuts no city off from the rest (FT in the paper).
The Tokyo check
Does the model compute correctly? The original experiment says yes.
Under ‘Tokyo 2010’ the same calculation runs with the 36 food sources of the paper – their positions taken from Fig. 1A and fitted to the coastline, the light zones over mountains and lakes from Fig. 2B. The paper does not say which city each one stands for; tapping a flake shows how certain the match is.
With the paper’s values for the most rail-like network (I₀ = 0.2, γ = 1.15), the model reaches 1.73 × the spanning tree, paths of 0.80 × and 92% fault tolerance. For the same model the paper reports 1.68, 0.83 and 96%, for the real mould 1.75 ± 0.30 and for the rail network around Tokyo 1.8, 0.85 and 96%.
Choose ‘Tokyo 2010’ above and check it yourself ↑The ICE network
Real tracks, shortened lines.
All regular ICE lines of the 2026 timetable year, shortened to the stops that define their course. Between the stops lies the fastest route in the DB InfraGO track network. Individual trains, weekend lines and IC are left out.
The same cities are compared: a city without an ICE stop counts as unconnected for the ICE. The ‘Fair comparison’ preset feeds only cities with an ICE stop.
Limits
The mould knows nothing of travel times, tunnel costs, existing lines or politics – it only balances length, paths and fault tolerance against each other. The ICE network has grown over 150 years and serves more than the cities fed here. The Schleimbahn is a thought experiment, but an honestly computed one.
Sources and licences
Where the data comes from.
- Tero, Takagi, Saigusa, Ito, Bebber, Fricker, Yumiki, Kobayashi, Nakagaki: Rules for Biologically Inspired Adaptive Network Design. Science 327, 439–442 (2010)
- Adamatzky, Schubert: Schlauschleimer in Reichsautobahnen – slime mould imitates motorway network in Germany. Kybernetes 41 (2012)
- Track network and operating points: DB InfraGO AG, Geo-Streckennetz, as of May 2026, CC BY 4.0
- Terrain: produced using Copernicus WorldDEM-90 © DLR e.V. 2010–2014 and © Airbus Defence and Space GmbH 2014–2018 provided under COPERNICUS by the European Union and ESA; all rights reserved
- Outline and lakes: Natural Earth (public domain) · Cities and population: Wikidata (CC0)
- ICE lines: Wikipedia, Liste der ICE-Linien (timetable year 2026)
Questions
Frequently asked questions about the Schleimbahn.
Is this a real slime mould?
No, a mathematical model. Tero and colleagues derived it in 2010 from their experiments with Physarum polycephalum; it reproduces the networks of the real mould. In the Tokyo check it runs on the original experiment and matches the published figures.
Why is the mould’s network so much shorter than the ICE network?
The mould connects only the cities it is fed and looks only at length and flow. The ICE network serves further stops and follows terrain, existing lines and speed. From I₀ = 3.5 the mould is as fault-tolerant as the ICE network and needs 43% less track to get there.
What does the ‘Frugal – Robust’ slider mean?
It sets I₀, the flux that flows between two cities in each step. Little flux leaves only the most essential tubes, a frugal tree. A lot of flux keeps bypasses alive; the network becomes longer and more fault-tolerant.
How accurate are the figures?
We computed each setting five times with different random seeds; the network length varies by about 2 to 3%. The tubes are measured along their smoothed course on a grid of about 12 km, the ICE network on real tracks. Small differences between two runs are therefore normal.
Can I share my network?
Yes. The link contains all settings and the cities fed; whoever opens it sees the same network grow. As a poster, the network can be saved as an image. The settings sit only in the link after the # sign and are not sent to our server.
Does it work on a phone?
Yes. The calculation runs in your browser and pauses as soon as the map is out of view. Two fingers zoom and pan the map; a tap selects a city.
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