Stochastic Model of a Micro Agents population

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Stochastic Model ofa Micro Agents population

• Dejan Milutinovic
• dejan_at_isr.ist.utl.pt

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Introduction

• Motivated by the work of the Immune system
  modeling as a Multi-Agent system

• Modelling framework for large Multi-Agents
  populations

• Control of large Multi-Agents population

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Immune system
Innate immune system
Adaptive immune system
- Virus, Bacteria (Antigen) - Antigen Presenting
Cell (APC) - T-Cell
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T-Cell Receptor (TCR) triggering
T-Cell

• CD3

peptide
APC
MHC
T-Cell, CD3 receptor, Antigen Presenting Cell
(APC), peptide-MHC complex
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T-Cell population
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T-Cell population
Complex System !!!
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Complex System
- 1000 Cells, 8000 variables
- Simulation analysis
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The Micro Agent model of the T-Cell

q3
1 never connected, 2 - connected, 3-
disconnected, a-connection, b-disconnection
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Micro Agent (?A)
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Stochastic Micro Agent (S?A)
A Stochastic Micro Agent is a pair S?A(?A,e(t))
where ?A is a Micro Agent and e(t) is a Micro
Agent stochastic input event sequence such that
the stochastic process (x(t),q(t))?X ? Q is a
Micro Agent Stochastic Execution.
u1
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Micro and Macro Dynamics relation
Dual Meaning of the State Probability Density
Function

• PDF function describes the state probability of
  one ?A
• Looking to the large population of ?A, PDF is a
  normalized distribution of the state occupancy by
  all ?A

• Micro dynamics of ?A and macro dynamics of ?A
  population are related through the state PDFs

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State probability dynamics
Markov chain transition over discrete space,
transition rate matrix
Vector of pdfs
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The Micro Agent model of the T-Cell
q1
q2
u(t)a
?12

?32
u(t)a
u(t)b
?23

q3
0 never connected, 1 - connected, 2-
disconnected, a-connection,
b-disconnection, ?ij event rate which leads to
transition from stat i to state j
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The Micro Agent model of the T-Cell
PDF DYNAMICS
By assumption that distribution is log-normal
STEADY STATE PDF
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Experimental distributions
( Valitutti, Nature 1995)
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Experiment-Mean Value
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Experimental distributions

• Stochastic model of flow Cytometry Measurements

Laser light
T-cell

• Data processing QQ-plot pdf estimation,
  Richardson-Lucy de-convolution

• Fitting data to the model with different
  hypothesis (only contact)

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Experimental distributions
Amount of TCRs
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Experimental distributions
Amount of TCRs
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Robotics application
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Robotics application
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Formation Control
?120.5, ?210.1, ?230.9, ?320.1
?120.1, ?210.5, ?230.5, ?320.4
t 0, 0.39, 0.79, 1.18, 1.57, 1.96
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Optimal Control
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Optimal Control
J(u)
DISCRETE APPROXIMATION
FINITE ELEMENT METHOD
Jm
ODE
Maximum principle
BOUNDARI VALUE PROBLEM
Approximate solution
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Finite Element Approximation
Test function
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Finite Element Approximation
ODE
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Criterion Approximation
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Pontryangin Minimum Principle
If u then
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Two Points Boundary Problem
Guess p(0), solve and using p(T) correct guess
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Approximate solution inequality
?
?
ODE rm
Jm(.)
J(.)
PDE
Jm(um) ? J(u)
J(u) ? J(um)
Jm(um) ? J(u) ? J(um)
J(um) ? J(u) ? J(um)
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Conclusions
- Hybrid Automata Model of individual

• Hybrid Automata Model of population based on
  Stochastic approximation

• Relation between Micro dynamics and Macro
  Dynamics based on Statistical physics reasoning

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Conclusions
- Data Processing of Flow Cytometry Data
- Model test against real data
- Formation control of large population of
individuals
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Future work

• Real experimental data analysis assuming
  different hypothesis

- Parameters uncertainty

• Control application and theory for such systems

• Other kind of applications in biology,
  nano-robotics

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Publications
Milutinovic, D., Athans, M., Lima, P., Carneiro,
J. Application of Nonlinear Estimation Theory in
T-Cell Receptor Triggering Model
Identification, Technical Report RT-401-02,
RT-701-02, 2002, ISR/IST Lisbon, Portugal
Milutinovic, D., Stochastic Model of a Micro
Agents Population, Technical Report ISR/IST
Lisbon, Portugal (working version)
Milutinovic D., Lima, P., Athans, M.
Biologically Inspired Stochastic Hybrid Control
of Multi-Robot Systems, submitted to the 11th
International Conference on Advanced Robotics
ICAR 2003,June 30 - July 3, 2003 University of
Coimbra, Portugal
Milutinovic D., Carneiro J., Athans, M., Lima, P.
A Hybrid Automata Modell of TCR Triggering
Dynamics , submitted to the 11th Mediterranian
Conference on Control and Automation MED
2003,June 18 - 20 , 2003, Rhodes, Greece
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Stochastic Model ofa Micro Agents population

• Dejan Milutinovic
• dejan_at_isr.ist.utl.pt