Cellular Automata Modelling of Traffic in Human and Biological Systems

1. Cellular Automata Modelling of Traffic in Human and Biological Systems Andreas Schadschneider Institute for Theoretical Physics University of Cologne www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic

2. Introduction • Modelling of transport problems: • space, time, states can be discrete or continuous • various model classes

3. Overview • Highway traffic • Traffic on ant trails • Pedestrian dynamics • Intracellular transport • Unified description!?!

4. Cellular Automata Cellular automata (CA) are discrete in space time state variable (e.g. occupancy, velocity) Advantage: very efficient implementation for large-scale computer simulations often: stochastic dynamics

5. Asymmetric Simple Exclusion Process

6. Asymmetric Simple Exclusion Process q q • Asymmetric Simple Exclusion Process (ASEP): • directed motion • exclusion (1 particle per site) Caricature of traffic: For applications: different modifications necessary

7. Highway Traffic

8. Cellular Automata Models • Discrete in • Space • Time • State variables (velocity) velocity

9. Update Rules • Rules (Nagel-Schreckenberg 1992) • Acceleration: vj! min (vj + 1, vmax) • Braking: vj! min ( vj , dj) • Randomization: vj ! vj – 1 (with probability p) • Motion: xj! xj + vj (dj = # empty cells in front of car j)

10. Example Configuration at time t: Acceleration (vmax = 2): Braking: Randomization (p = 1/3): Motion (state at time t+1):

11. Simulation of NaSch Model • Reproduces structure of traffic on highways • - Fundamental diagram • - Spontaneous jam formation • Minimal model: all 4 rules are needed • Order of rules important • Simple as traffic model, but rather complex as stochastic model

12. Fundamental Diagram Relation: current (flow) $ density

13. Metastable States • Empirical results: Existence of • metastable high-flow states • hysteresis

14. VDR Model • Modified NaSch model: • VDR model (velocity-dependent randomization) • Step 0: determine randomization p=p(v(t)) • p0 if v = 0 • p(v) = with p0 > p • p if v > 0 • Slow-to-start rule

15. Simulation of VDR Model NaSch model VDR model VDR-model: phase separation Jam stabilized by Jout < Jmax

16. Dynamics on Ant Trails

17. Ant trails ants build “road” networks: trail system

18. Chemotaxis • Ants can communicate on a chemical basis: • chemotaxis • Ants create a chemical trace of pheromones • trace can be “smelled” by other • ants follow trace to food source etc.

19. Ant trail model qqQ • motion of ants • pheromone update (creation + evaporation) Dynamics: q q Q f f f parameters: q < Q, f

20. Fundamental diagram of ant trails velocity vs. density non-monotonicity at small evaporation rates!! Experiments: Burd et al. (2002, 2005) different from highway traffic: no egoism

21. Spatio-temporal organization • formation of “loose clusters” early times steady state coarsening dynamics

22. Pedestrian Dynamics

23. Collective Effects • jamming/clogging at exits • lane formation • flow oscillations at bottlenecks • structures in intersecting flows ( D. Helbing)

24. Pedestrian Dynamics • More complex than highway traffic • motion is 2-dimensional • counterflow • interaction “longer-ranged” (not only nearest neighbours)

25. Pedestrian model idea: Virtual chemotaxis chemical trace: long-rangedinteractions are translated intolocalinteractions with ‘‘memory“ • Modifications of ant trail model necessary since • motion 2-dimensional: • diffusion of pheromones • strength of trace

26. Floor field cellular automaton • Floor field CA: stochastic model, defined by transition probabilities, only local interactions • reproduces known collective effects (e.g. lane formation) Interaction: virtual chemotaxis (not measurable!) dynamic + static floor fields interaction with pedestrians and infrastructure

27. Transition Probabilities • Stochastic motion, defined by • transition probabilities • 3 contributions: • Desired direction of motion • Reaction to motion of other pedestrians • Reaction to geometry (walls, exits etc.) • Unified description of these 3 components

28. Lane Formation velocity profile

29. Intracellular Transport

30. Intracellular Transport • Transport in cells: • microtubule = highway • molecular motor (proteins) = trucks • ATP = fuel

31. Kinesin and Dynein: Cytoskeletal motors Fuel: ATP ATP ADP + P Kinesin Dynein • Several motors running on same track simultaneously • Size of the cargo >> Size of the motor • Collective spatio-temporal organization ?

32. Practical importance in bio-medical research Goldstein, Aridor, Hannan, Hirokawa, Takemura,…………….

33. ASEP-like Model of Molecular Motor-Traffic ASEP + Langmuir-like adsorption-desorption Parmeggiani, Franosch and Frey, Phys. Rev. Lett. 90, 086601 (2003) D A q a b Also, Evans, Juhasz and Santen, Phys. Rev.E. 68, 026117 (2003)

34. Spatial organization of KIF1A motors: experiment MT (Green) 10 pM KIF1A (Red) 100 pM 1000pM 2 mM of ATP 2 mm position of domain wall can be measured as a function of controllable parameters. Nishinari, Okada, Schadschneider, Chowdhury, Phys. Rev. Lett. (2005)

35. Summary • Various very differenttransport and traffic problems can be described by similarmodels • Variants of the Asymmetric Simple Exclusion Process • Highway traffic: larger velocities • Ant trails: state-dependent hopping rates • Pedestrian dynamics: 2d motion, virtual chemotaxis • Intracellular transport: adsorption + desorption

36. Collaborators Cologne: Ludger Santen Alireza Namazi Alexander John Philip Greulich Duisburg: Michael Schreckenberg Robert Barlovic Wolfgang Knospe Hubert Klüpfel Thanx to: Rest of the World: Debashish Chowdhury (Kanpur) Ambarish Kunwar (Kanpur) Katsuhiro Nishinari (Tokyo) T. Okada (Tokyo) + many others