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
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
Caricature of traffic:
Asymmetric Simple Exclusion
Process (ASEP):
q
q
1. directed motion
2. exclusion (1 particle per site)
For applications: different modifications necessary
Highway
Traffic
Cellular Automata Models
Discrete in
• Space
• Time
• State variables (velocity)
velocity (v  0,1,...,vmax )
Update Rules
Rules
(Nagel-Schreckenberg 1992)
1) Acceleration:
vj ! min (vj + 1, vmax)
2) Braking:
vj ! min ( vj , dj)
(dj = # empty cells
in front of car j)
3) Randomization: vj ! vj – 1 (with probability p)
4) Motion:
xj ! xj + vj
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
Simulation of VDR Model
NaSch model
VDR model
VDR-model: phase separation
Jam stabilized by Jout < Jmax
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.
Ant trail model
Dynamics:
1. motion of ants
2. pheromone update (creation + evaporation)
q
q
q
q
f
parameters: q < Q, f
Q
Q
f
f
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
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
idea:
Virtual chemotaxis
chemical trace: long-ranged interactions are translated
into local interactions with ‘‘memory“
Modifications of ant trail model necessary since
motion 2-dimensional:
• diffusion of pheromones
• strength of trace
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
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 ?
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/ Dynein
Kartageners’ syndrome
Sinus and Lung disease,
male infertility
Goldstein, Aridor, Hannan, Hirokawa, Takemura,…………….
ASEP-like Model of Molecular Motor-Traffic
ASEP + Langmuir-like adsorption-desorption
Parmeggiani, Franosch and Frey, Phys. Rev. Lett. 90, 086601 (2003)
a
q
D
A
b
Also, Evans, Juhasz and Santen, Phys. Rev.E. 68, 026117 (2003)
Spatial organization of KIF1A motors: experiment
MT (Green)
10 pM
KIF1A (Red)
100 pM
1000pM
2 mm
2 mM of ATP
position of domain wall can be measured as a function of
controllable parameters.
Nishinari, Okada, Schadschneider, Chowdhury, Phys. Rev. Lett. (2005)
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
Thanx to:
Cologne:
Duisburg:
Rest of the World:
Ludger Santen
Alireza Namazi
Alexander John
Philip Greulich
Michael Schreckenberg
Robert Barlovic
Wolfgang Knospe
Hubert Klüpfel
Debashish Chowdhury (Kanpur)
Ambarish Kunwar (Kanpur)
Katsuhiro Nishinari (Tokyo)
T. Okada (Tokyo)
+ many others