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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Outline

• Motivating problem

Introduction to Math. Analysis
Mathematical Analysis
Applications
Biology
Robotics
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Motivating problem
T-Cell Receptor (TCR) triggering
T-Cell, CD3 receptor, Antigen Presenting Cell
(APC), peptide-MHC complex
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Introduction to Math. Analysis
T-Cell population
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Introduction to Math. Analysis
T-Cell population
Complex System !!!
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Introduction to Math. Analysis
How the Micro Dynamics of the Individuals
propagates to the Dynamics of Macro observations ?
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Mathematical Analysis
The Micro Agent model of the T-Cell

q3
1 never connected, 2 - connected, 3-
disconnected, a-connection, b-disconnection
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Mathematical Analysis
Micro Agent (?A)
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Mathematical Analysis
Stochastic Micro Agent (S?A)
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Mathematical Analysis
Micro and Macro Dynamics relation

• Statistical Physics reasoning (Boltzman
  distribution)

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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Mathematical Analysis
Micro Agent Stochastic Execution
A stochastic process (x(t),q(t))?X ? Q is called
a Micro Agent Stochastic Execution iff a Micro
Agent stochastic input event sequence e(?n),n?N,
?0 0 ? ?1 ? ?2 ? generates transitions such
that in each interval ?n,?n1), n?N, q(t)?
q(?n).
xn
e(?n )
e(?n1 )
e(?n2 )
V
V
V
i
1
N
q
f(x,N)
f(x,1)
f(x,i)
xn-1
X x Q
x1
Remark 1. The x(t) of a Stochastic Execution is a
continuous time function since the transition
changes only the discrete state of a Micro Agent.
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Mathematical Analysis
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.
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Mathematical Analysis
Continuous Time Markov Process Micro Agent
(CTMP?A)
A Stochastic Micro Agent is called a Continuous
Time Markov Process Micro Agent iff (x(t),q(t))?X
? Q is a Micro Agent Continuous Time Markov
Process Execution.
Stochastic system
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Mathematical Analysis
The Continuous Time Markov Chain Micro Agent with
N discrete state and state probability given by
where
is the probability of discrete
state i and
is transition rate matrix
and
is rate of transition from discrete state i to
discrete state j.
The vector of probability density functions
where
is probability density function
of state (x,i) at
time t, satisfies
is the vector of vector fields value at state
(x,i).
where
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Biological application
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 state i to state j
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Biological application
Case I solution
?12 0.9, ?23 0 , ?32 0.5, k2 0.5, k30.25
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Biological application
Case II solution
?12 0.9, ?23 0.8 , ?32 0.9, k2 0.5, k30.05
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Biological application
Case III solution
?12 0.9, ?23 0.8 , ?32 0.9, k2 0.5, k30.25
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Biological application
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Biological application
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Biological application
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Robotics application
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Robotics application
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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