Andreas Schadschneider

Institute for Theoretical Physics

University of Cologne

www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic

Cellular Automata Modelling of Traffic in Human and

Biological Systems

Introduction

Modelling of transport problems:

space, time, states can be discrete or continuous

various model classes

Overview

1. Highway traffic

2. Traffic on ant trails

3. Pedestrian dynamics

4. Intracellular transport

Unified description!?!

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

Asymmetric Simple

Exclusion Process

Asymmetric Simple Exclusion Process

Asymmetric Simple Exclusion Process (ASEP):

1. directed motion2. exclusion (1 particle per site)

qq

Caricature of traffic:

For applications: different modifications necessary

Highway

Traffic

Cellular Automata Models

Discrete in • Space • Time• State variables (velocity)

velocity ),...,1,0( maxvv

Update Rules

Rules (Nagel-Schreckenberg 1992)

1) Acceleration: vj ! min (vj + 1, vmax)

2) Braking: vj ! min ( vj , dj)

3) Randomization: vj ! vj – 1 (with probability p)

4) Motion: xj ! xj + vj

(dj = # empty cells in front of car j)

Example

Configuration at time t:

Acceleration (vmax = 2):

Braking:

Randomization (p = 1/3):

Motion (state at time t+1):

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

Fundamental Diagram

Relation: current (flow) $ density

Metastable States

Empirical results: Existence of

• metastable high-flow states

• hysteresis

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

NaSch model

VDR-model: phase separation

Jam stabilized by Jout < Jmax

VDR model

Simulation of VDR Model

Dynamics on

Ant Trails

Ant trails

ants build “road” networks: trail system

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.

q q Q

1. motion of ants

2. pheromone update (creation + evaporation)Dynamics:

f f f

parameters: q < Q, f

Ant trail model

q q Q

Fundamental diagram of ant trails

different from highway traffic: no egoism

velocity vs. density

Experiments:

Burd et al. (2002, 2005)

non-monotonicity at small

evaporation rates!!

Spatio-temporal organization

formation of “loose clusters”

early times steady state

coarsening dynamics

Pedestrian

Dynamics

Collective Effects

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

Pedestrian Dynamics

More complex than highway traffic

• motion is 2-dimensional• counterflow • interaction “longer-ranged” (not only nearest neighbours)

Pedestrian model

Modifications of ant trail model necessary since

motion 2-dimensional:• diffusion of pheromones• strength of trace

idea: Virtual chemotaxis

chemical trace: long-ranged interactions are translated into local interactions with ‘‘memory“

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

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

Lane Formation

velocity profile

Intracellular

Transport

Intracellular Transport

Transport in cells:

• microtubule = highway• molecular motor (proteins) = trucks• ATP = fuel

• Several motors running on same track simultaneously

• Size of the cargo >> Size of the motor

• Collective spatio-temporal organization ?

Fuel: ATP

ATP ADP + P Kinesin

Dynein

Kinesin and Dynein: Cytoskeletal motors

Practical importance in bio-medical research

Disease Motor/Track Symptom

Charcot-Marie tooth disease

KIF1B kinesin Neurological disease; sensory loss

Retinitis pigmentosa KIF3A kinesin Blindness

Usher’s syndrome Myosin VII Hearing loss

Griscelli disease Myosin V Pigmentation defect

Primary ciliary diskenesia/

Kartageners’ syndrome

Dynein Sinus and Lung disease, male infertility

Goldstein, Aridor, Hannan, Hirokawa, Takemura,…………….

ASEP-like Model of Molecular Motor-Traffic

q

D A

Parmeggiani, Franosch and Frey, Phys. Rev. Lett. 90, 086601 (2003)

ASEP + Langmuir-like adsorption-desorption

Also, Evans, Juhasz and Santen, Phys. Rev.E. 68, 026117 (2003)

position of domain wall can be measured as a function of controllable parameters.

Nishinari, Okada, Schadschneider, Chowdhury, Phys. Rev. Lett. (2005)

KIF1A (Red)

MT (Green)10 pM

100 pM

1000pM

2 mM of ATP2 m

Spatial organization of KIF1A motors: experiment

Summary

Various very different transport and traffic problems can be described by similar models

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

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