Cellular Automata Modelling of Traffic in Human and Biological Systems

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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
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Introduction

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

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Overview

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

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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

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Asymmetric Simple Exclusion Process
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Asymmetric Simple Exclusion Process
Caricature of traffic

• Asymmetric Simple Exclusion Process (ASEP)
• directed motion
• exclusion (1 particle per site)

For applications different modifications
necessary
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Influence of Boundary Conditions

• open boundaries

Applications Protein synthesis
Surface growth
Boundary induced phase transitions
exactly solvable!
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Phase Diagram
Maximal current phase JJ(p)
Low-density phase JJ(p,?)
High-density phase JJ(p,?)
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Highway Traffic
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Cellular Automata Models

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

velocity
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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)
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Example
Configuration at time t Acceleration (vmax
2) Braking Randomization (p 1/3) Motion
(state at time t1)
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Interpretation of the Rules

• Acceleration Drivers want to move as fast as
  possible (or allowed)
• Braking no accidents
• Randomization
• a) overreactions at braking
• b) delayed acceleration
• c) psychological effects (fluctuations in
  driving)
• d) road conditions
• 4) Driving Motion of cars

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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

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Fundamental Diagram
Relation current (flow) density
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Metastable States

• Empirical results Existence of
• metastable high-flow states
• hysteresis

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VDR Model

• Modified NaSch model
• VDR model
  (velocity-dependent randomization)
• Step 0 determine randomization pp(v(t))
• p0 if v 0
• p(v) with
  p0 gt p
• p if v gt 0
• Slow-to-start rule

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Simulation of VDR Model
NaSch model
VDR model
VDR-model phase separation Jam stabilized by
Jout lt Jmax
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Dynamics on Ant Trails
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Ant trails
ants build road networks trail system
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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.

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Ant trail model

• motion of ants
• pheromone update (creation evaporation)

Dynamics
q
q
Q
f f f
parameters q lt Q, f
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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
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Spatio-temporal organization

• formation of loose clusters

early times
steady state
coarsening dynamics
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Pedestrian Dynamics
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Collective Effects

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

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Pedestrian Dynamics

• More complex than highway traffic
• motion is 2-dimensional
• counterflow
• interaction longer-ranged (not only nearest
  neighbours)

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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

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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
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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

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Transition Probabilities

• Total transition probability pij in direction
  (i,j)
• pij N Mij exp(kDDij)
  exp(kSSij)(1-nij)
• Mij matrix of preferences (preferred
  direction)
• Dij dynamic floor field
  (interaction between pedestrians)
• Sij static floor field
  (interaction with geometry)
• kD, kS coupling strength
• N normalization (? pij 1)

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Lane Formation
velocity profile
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Intracellular Transport
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Intracellular Transport

• Transport in cells
• microtubule highway
• molecular motor (proteins) trucks
• ATP fuel

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Kinesin and Dynein Cytoskeletal motors
Fuel ATP
ATP ADP P
Kinesin
Dynein

• Several motors running on same track
  simultaneously
• Size of the cargo gtgt Size of the motor
• Collective spatio-temporal organization ?

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

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Collaborators
Thanx to

• Duisburg
• Michael Schreckenberg
• Robert Barlovic
• Wolfgang Knospe
• Hubert Klüpfel

Rest of the World Debashish Chowdhury
(Kanpur) Ambarish Kunwar (Kanpur) Katsuhiro
Nishinari (Tokyo) T. Okada (Tokyo)

• Cologne
• Ludger Santen
• Alireza Namazi
• Alexander John
• Philip Greulich

many others